DEVELOPMENT OF AN EXPLICIT NUMERICAL …...DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC...

QU XIAOLEI

SCHOOL OF CIVIL AND RESOURCE ENGINEERING

THE UNIVERSITY OF WESTERN AUSTRALIA

SEPTEMBER 2013

DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD

FOR DYNAMIC STABILITYANALYSIS OF ROCK

SLOPE

DEVELOPMENT OF THE NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY ANALYSIS OF ROCK SLOPE FAILURE

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DEVELOPMENT OF AN EXPLICIT

NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITYANALYSIS

OF ROCK SLOPE

QU XIAOLEI

School of Civil and Resource Engineering

A thesis submitted to

The University of Western Australia

in fulfillment of the requirement for the degree of

Doctor of Philosophy

SEPTEMBER 2013


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TABLE OF CONTENTS

Table of Contents ............................................................................................................ ii ACKNOWLEDGEMENT ............................................................................................. vi SUMMARY ................................................................................................................... vii LIST OF FIGURES ....................................................................................................... ix LIST OF TABLES ....................................................................................................... xiv LIST OF SYMBOLS .................................................................................................... xv LIST OF ABBREVIATION ...................................................................................... xviii Chapter 1. Introduction ............................................................................................... 1.1

1.1 Background ..................................................................................................... 1.1

1.2 Objectives of research ..................................................................................... 1.5

1.3 Origanization of the thesis .............................................................................. 1.6

Chapter 2. Literature review ...................................................................................... 2.9 2.1 Origin of the manifold .................................................................................... 2.9

2.2 Basic concepts of the NMM ......................................................................... 2.10

2.3 Current developments of the NMM .............................................................. 2.11

2.3.1 Improvement of the accuracy of the NMM ................................. 2.11

2.3.2 Extension of the NMM for discontinuity problems ..................... 2.12

2.3.3 Development of 3-D NMM ......................................................... 2.12

2.3.4 Other developments and applications of the NMM ..................... 2.13

2.4 Comparison with other numerical methods .................................................. 2.14

2.4.1 Comparison with FEM ................................................................. 2.14

2.4.2 Comparison with DDA ................................................................ 2.15

2.4.3 Comparison with DEM ................................................................ 2.16

2.5 Time integration algorithms for numerical methods .................................... 2.19

2.5.1 Numerical properties for time integration .................................... 2.21

2.5.2 Numerical examples for time integration .................................... 2.29

2.6 Methods for dynamic stability analysis of rock slope .................................. 2.33

2.6.1 LEM ............................................................................................. 2.36

2.6.2 Newmark method ......................................................................... 2.39

2.6.3 Numerical methods ...................................................................... 2.44

Chapter 3. Theory of the numerical manifold method and its integration schemes ........................................................................................................................ 3.46

3.1 Introduction ................................................................................................... 3.46

3.2 Fundamentals of the NMM ........................................................................... 3.46

3.2.1 Finite cover system in the NMM ................................................. 3.47

3.2.2 Contact algorithm ......................................................................... 3.57


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3.3 Integration schemes in the NMM .................................................................. 3.62

3.3.1 Simplex integration ...................................................................... 3.63

3.3.2 Time integration ........................................................................... 3.64

3.4 Summary ....................................................................................................... 3.65

Chapter 4. An explicit time integration scheme for the numerical manifold method ......................................................................................................................... 4.66

4.1 Introduction ................................................................................................... 4.66

4.2 Brief descrptions of the NMM ...................................................................... 4.67

4.3 Explicit time integration for the NMM ......................................................... 4.68

4.3.1 Mass matrix .................................................................................. 4.69

4.3.2 Internal force ................................................................................ 4.71

4.3.3 Damping algorithm ...................................................................... 4.72

4.4 Contact force in the ENMM .......................................................................... 4.74

4.4.1 Contact force calculation approach .............................................. 4.74

4.4.2 Calculation of contact force ......................................................... 4.75

4.5 Open-close algorithm in the ENMM ............................................................ 4.78

4.6 Numerical simulations .................................................................................. 4.80

4.6.1 Simply-supported beam subjected a concentrated load ............... 4.81

4.6.2 Numerical simulation of plan stress field problem ...................... 4.82

4.6.3 Single block sliding along the inclined surface ........................... 4.84

4.6.4 Highly fractured rock slope stability analysis .............................. 4.85

4.6.5 Rock tunnel stability analysis ...................................................... 4.91

4.7 Summary ....................................................................................................... 4.94

Chapter 5. Verification of computational efficiency and accuracy of the explicit numerical manifold method with wave propagation problems ............................. 5.96

5.1 Introduction ................................................................................................... 5.96

5.2 The brief overview of the NMM ................................................................... 5.98

5.2.1 The NMM and its cover system ................................................... 5.98

5.2.2 The explicit scheme of the NMM ................................................ 5.98

5.3 Stress wave propagation in a continous bar .................................................. 5.99

5.3.1 Effect of mesh size ..................................................................... 5.100

5.3.2 Effect of time step ...................................................................... 5.103

5.3.3 Computational efficiency ........................................................... 5.107

5.4 Stress wave propagation through fractured rock mass ................................ 5.108

5.4.1 P-wave propagation through homogeneous medium ................. 5.108

5.4.2 P-wave propagation through joint between different mediums . 5.109

5.4.3 Stress wave propagation through the multiple parallel joints .... 5.110

5.5 Seismic wave effect for a fractured rock slope ........................................... 5.112


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5.6 Summary ..................................................................................................... 5.115

Chapter 6. The temporal coupled explicit-implicit algorithm for dynamic problems using the numerical manifold method ................................................... 6.117

6.1 Introduction ................................................................................................. 6.117

6.2 The NMM and its cover system .................................................................. 6.119

6.2.1 Dual cover system in the NMM ................................................. 6.119

6.2.2 Contact problems in the NMM .................................................. 6.120

6.3 Temporal coupled explicit-implicit algorithm in the NMM ....................... 6.121

6.3.1 Summary of equations of motion and time integration.............. 6.121

6.3.2 The coupled explicit-implicit algorithm .................................... 6.123

6.3.3 Transfer algorithm for the E-I algorithm ................................... 6.124

6.4 Contact algorithm of the coupled algorithm ............................................... 6.126

6.4.1 Contact force calculation ........................................................... 6.126

6.4.2 Damping algorithm .................................................................... 6.127

6.5 Numerical examples .................................................................................... 6.128

6.5.1 Calibration of the temporal coupled E-I algorithm .................... 6.128

6.5.2 Open-pit mining stability analysis ............................................. 6.129

6.6 Summary ..................................................................................................... 6.141

Chapter 7. The spatial coupled explicit-implicit algorithm for dynamic problems using the numerical manifold method ................................................... 7.142

7.1 Introduction ................................................................................................. 7.142

7.2 Coupled algorithm ...................................................................................... 7.145

7.2.1 Summary of equations of motion and time integration.............. 7.145

7.2.2 Coupled explicit-implicit algorithm in the NMM ...................... 7.147

7.3 An alternative approach for the coupled E-I ALGORITHM ...................... 7.152

7.3.1 Onefold cover system ................................................................ 7.152

7.3.2 Contact algorithm on the onefold cover system ........................ 7.154

7.3.3 Contact matrices of the coupled E-I algorithm .......................... 7.157

7.3.4 Spring stiffness problems ........................................................... 7.162


7.4.1 Calibration of the spatial coupled E-I algorithm ........................ 7.164

7.4.2 Simulation of discrete blocks sliding on an inclined surface ..... 7.168

7.5 Summary ..................................................................................................... 7.169

Chapter 8. Dynamic stability analysis of rock slope failure using the explicit numerical manifold method .................................................................................... 8.171

8.1 Introduction ................................................................................................. 8.171

8.2 Numerical methods for rock slope dynamic stability ................................. 8.174

8.2.1 Continuum methods ................................................................... 8.175

8.2.2 Discontinuum methods .............................................................. 8.177


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8.2.3 Hybrid methods .......................................................................... 8.180

8.3 The new development of the NMM for dynamic stability analysis of rock slope ............................................................................................................ 8.181

8.3.1 Explicit NMM ............................................................................ 8.183

8.3.2 Coupled E-I NMM ..................................................................... 8.184

8.4 The parallel computation of the NMM ....................................................... 8.187

8.4.1 Parallelization with openMP ...................................................... 8.188

8.4.2 Speedup ...................................................................................... 8.191


8.5.1 A dynamic case study of rock slope stability analysis ............... 8.195

8.5.2 Dynamic stability analysis of Jinping I hydropower station ...... 8.198

8.6 Summary ..................................................................................................... 8.200

Chapter 9. Conclusions and recommendations ..................................................... 9.202 9.1 Summaries ................................................................................................... 9.202

9.2 Conclusions ................................................................................................. 9.203

9.3 Recommendations ....................................................................................... 9.206

References ................................................................................................................. 9.208


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ACKNOWLEDGEMENT

First and foremost, I would like to express my sincere gratitude to my supervisor,

Professor Ma Guowei for his warm encouragement, patient guidance and constant

persistence throughout this research. His unwavering enthusiasm and interest in

scientific research especially on the development of the numerical manifold method are

much appreciated and unforgettable. It is honoured and proud to work with him.

I would also like to thank Professor Shi Genhua for his advices and suggestions to

my research work. He has enhanced me a lot in my knowledge in the theoretical part.

I extend my gratitude to Professor Jing Lanru from Royal Institute of Technology

(KTH) for his suggestions, Sweden and Dr. Li Xu from Bejing Jiaotong University

(BJU), Beijing for his great helps in my research.

I am thankful to our whole NMM group, including Prof Ma Guowei, Dr. Hu

Jianhua, Dr Zhang Huihua, Mr Fu Guoyang, Mr Ren Fen, Mr Li Jinde, Mr Xu Zhenhao,

Mr Yi Xiawei, Mr Yang Shikou, Mr Wu Wei. We had a regular meeting each week. We

discussed the problems we met and tried to figure out ways to solve the problems

together. I benefited a lot from the discussions with them.

I take this opportunity to thank Professor Andrew Deeks, Professor Cheng Liang

and Dr. James Doherty for their helps and recommendations for my research work. I

would like to thank Professor Hao Hong to be as my vice advisor in my research.

The scholarship provided by China Scholarship Council (CSC) joints the

University of Western Australia (UWA) is gratefully acknowledged.

Last but not the least, I would like to dedicate this work to my beloved wife, Zhang

Li, who gives me infinite support in the shade and brings me a warm happiness family,

and to my parents for their love and support throughout the past years.


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SUMMARY

In this thesis, an explicit version of the numerical manifold method (NMM) has

been developed for dynamic stability analysis of rock slope.

Firstly, Newmark integration scheme used in the NMM is brief introduced and

derived in detail to further deepen understanding of the NMM and its implementations.

The numerical results present the explicit scheme is more efficient in solving the

nonlinear dynamic systems and such problems compared to implicit scheme. Then, an

explicit time integration scheme for the NMM is proposed to improve the computational

efficiency. The developed explicit NMM (ENMM) is validated by several examples.

The calibration study of the ENMM on P-wave propagation across a rock bar has been

conducted. Various considerations in the numerical simulations are discussed and

parametric studies have been carried out to obtain an insight into the influencing factors

in wave propagation simulation. The numerical results from the ENMM and NMM

modelling are accordant well with the theoretical solutions, but the ENMM is more

efficient than the original NMM.

The temporal and spatial coupled explicit-implicit (E-I) algorithms for the

numerical manifold method (NMM) are proposed. The time integration schemes,

transfer algorithm, contact algorithm and damping algorithm are studied in the temporal

coupled E-I algorithm to merge both merits of the explicit and implicit algorithms in

terms of efficiency and accuracy. In particular, onefold cover system is drawn into the

coupled spatial E-I algorithm, in which the contact algorithm based on the onefold cover

system is discussed and derived in detail. The simulated results are well agreement with

the implicit and explicit algorithms simulations, but the efficiency is improved

evidently.

The dynamic stability analysis of rock slope failure using the NMM is studied.

Conservational pseudo-static methods (PSMs), Newmark method and numerical

methods applying into the seismic stability analyses are investigated, the advantages and

limitations of which are studied by contrast of the NMM. An alternative ENMM and

coupled E-I algorithms are applied to study the seismic stability of rock slope.


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Furthermore, parallel computing with OpenMP is evaluated to improve efficiency of the

NMM. To reveal the validity and applicability of the developed ENMM, some

numerical examples of rock slope stability analysis are investigated, in which one

example of rock slope is taken into account to present the coupled ENMM with

discontinuous deformation analysis (DDA) in terms of efficiency. Simulated results of

the NMM will compare with the field measurements to illustrate the applicability of the

NMM.

The present study showed the developed ENMM is more efficient while without

losing the accuracy, comparing to the original implicit version of the NMM. Therefore,

it can be predicted that the proposed ENMM is promising and can be extend applied to

larger scale project of rock slope with dynamic stability analysis.


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LIST OF FIGURES

Figure 2.1 The scales of fish and fish body. ................................................................ 2.10

Figure 2.2 Size separation process with the vibrating screen modeled by the NMM (Ma et al. 2010). ...................................................................................... 2.14

Figure 2.3 Corner-to-corner contact in the NMM and DEM: (a) The shortest path method in the NMM; (b) The corner rounding technique in the DEM. .. 2.19

Figure 2.4 Efficiency versus DOFs between explicit and implicit algorithms. ........... 2.20

Figure 2.5 Spectral radius versus sampling frequency for Newmark integration. ....... 2.24

Figure 2.6 Period errors for the undamped case of Newmark integration methods. ... 2.25

Figure 2.7 Numerical damping versus sampling frequency in the NMM. ................... 2.26

Figure 2.8 Computational cost by the Newmark implicit scheme. .............................. 2.28

Figure 2.9 Computational cost by the Newmark explicit scheme. .............................. 2.29

Figure 2.10 Simplified model of n-DOF spring-mass-dashpot system........................ 2.30

Figure 2.11 Total displacement responses for 100-DOFs of system. .......................... 2.32

Figure 2.12 classification of instability and project examples for rock slopes. ........... 2.34

Figure 2.13 Equilibrium of forces on a sliding block (Chang et al. 1984). ................. 2.38

Figure 2.14 Displacement of rigid block on rigid base (Newmark 1965): (a) block on moving base; (b) acceleration plot; (c) velocity plot. ......................... 2.40

Figure 2.15 Integration of accelerograms to determine block movement (Goodman and Seed 1966). ....................................................................................... 2.42

Figure 3.1 A schematic of basic concepts in the NMM. (a) The physical domain and two MCs; (b) Overlapping of MCs and physical domain; (c) Corresponding PCs; (d) Six corresponding MEs. ................................... 3.48

Figure 3.2 The cover system in the NMM: (a) General cover system; (b) Generation of physical covers. ................................................................ 3.50

Figure 3.3 NMM model for the discontinuity problem. .............................................. 3.51

Figure 3.4 Construction of finite cover system in the NMM. ...................................... 3.53

Figure 3.5 Construction of PCs on the cover system. .................................................. 3.54

Figure 3.6 Structured mesh-based cover system in the NMM. .................................... 3.55

Figure 3.7 Construction of manifold elements on the cover system: (a) continuous elements; (b) discontinuous elements. ..................................................... 3.56

Figure 3.8 Three types of contacts: (a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge. .................................................................................................... 3.58

Figure 3.9 Entrance distance nd between a vertex and its entrance line. ..................... 3.59

Figure 3.10 Triangulate an element oij using coherent orientation. ............................ 3.64


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Figure 4.1 Two distinct contact force approaches: (a) point approach; (b) area approach. ................................................................................................. 4.74

Figure 4.2 Two schemes for contact problem: (a) Normal penetration method; (b) Direct penetration method. ...................................................................... 4.75

Figure 4.3 The proposed contact model: (a) contact elements; (b) contact points. ..... 4.78

Figure 4.4 Flowchart of the OCI in the ENMM. ......................................................... 4.80

Figure 4.5 Geometry of the simply supported beam bending problem. ...................... 4.81

Figure 4.6 Comparisons between simulated results and theoretical solution. ............. 4.82

Figure 4.7 Numerical model for an infinite plate with a traction free circular hole: (a) Geometry of model; (b) NMM meshing. ........................................... 4.83

Figure 4.8 Comparison of numerical results and analytical solution for infinite plate problem. .......................................................................................... 4.84

Figure 4.9 Comparison of simulated results and analytical solution. .......................... 4.85

Figure 4.10 Geometry of the slope modelling. ............................................................ 4.85

Figure 4.11 Simulation results by NMM and ENMM: (a) NMM, ∆t=2ms; (b) NMM, ∆t=1ms; (c) NMM, ∆t=0.1ms; (d) ENMM, ∆t=0.1ms................ 4.88

Figure 4.12 Real step-time used in NMM vs. ENMM. ................................................ 4.88

Figure 4.13 Displacements of measured point 1: (a) Horizontal; (b) Vertical. ........... 4.89



Figure 4.16 Geometry of the tunnel modelling. ........................................................... 4.92

Figure 4.17 Simulation results used by ENMM vs. NMM. ......................................... 4.92

Figure 4.18 Displacements of measured point 4. ......................................................... 4.93

Figure 4.19 Displacement of measured point 9. .......................................................... 4.94

Figure 5.1 Schematic of the rock bar model. ............................................................... 5.99

Figure 5.2 Eigenlength in the manifold mesh. ........................................................... 5.100

Figure 5.3 Percent errors at the end of first wavelength for different wave frequencies and element ratios. ............................................................. 5.101

Figure 5.4 Stress wave simulation using NMM and ENMM by two typical element ratios of 1/4 and 1/32 ( }0.1 st ). ....................................................... 5.102

Figure 5.5 Percent errors along the distance from wave source for different element ratios ( }0.1 st ). ................................................................... 5.103

Figure 5.6 Peak pressure attenuation for different time step. .................................... 5.106

Figure 5.7 Comparison between the simulated results and the Pyrak-Nolte’s analytical solution for a single joint. ..................................................... 5.109

Figure 5.8 Comparison between the simulated results of NMM and ENMM. .......... 5.110


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Figure 5.9 Simulated results at measure point by the NMM and ENMM. ................ 5.111

Figure 5.10 Schematic cross-section of the rock slope .............................................. 5.113

Figure 5.11 Record of acceleration of the seismic wave. .......................................... 5.113

Figure 5.12 Rock slope model in the NMM and ENMM. ......................................... 5.114

Figure 5.13 Simulated results of the failure of the rock slope using the NMM and ENMM (Time step in NMM =1ms, ENMM =0.2ms; total time =20s). 5.115

Figure 5.14 Measure point displacement simulated by the NMM and ENMM......... 5.115

Figure 6.1 A regularly-patterned triangular mesh in the NMM. ................................ 6.120

Figure 6.2 Transfer algorithm from the explicit to implicit integration. .................... 6.124

Figure 6.3 Geometry of the Newmark sliding modeling. .......................................... 6.129

Figure 6.4 Block displacement under horizontal ground acceleration. ...................... 6.129

Figure 6.5 Geology section of the open-pit mining. .................................................. 6.130

Figure 6.6 Study model of the layer 4#: a. Integrated Model; b. Refined Model 1; c. Refined Model 2. ............................................................................... 6.131

Figure 6.7 A stochastic horizontal seism acceleration. .............................................. 6.132

Figure 6.8 Simulation results for Refined Model 1 (Total time: 20s): (a) ϕ=100; (b) ϕ=150. .................................................................................................... 6.134

Figure 6.9 Simulation results for Refined Model 2 (Total time: 20s): (a) ϕ=100; (b) ϕ=150. .................................................................................................... 6.134

Figure 6.10 Measured point 1 with model 1: (a) ϕ=100; (b) ϕ=150. .......................... 6.135





Figure 6.15 Measured point with model 2: (a) ϕ=100; (b) ϕ=150. ............................. 6.140

Figure 7.1 An elastic body with a traction vector t. ................................................... 7.146

Figure 7.2 Sub-domain partition algorithm in the coupled E-I method: (a) Element partition method; (b) MCs partition method. ........................................ 7.150

Figure 7.3 Explicit and implicit covers for the contact problem in the NMM........... 7.152

Figure 7.4 Construction of onefold cover in the proposed NMM.............................. 7.153

Figure 7.5 Coefficient matrix of the coupled E-I algorithm: (a) Coupled E-I algorithm global coefficient matrix; (b) Implicit coefficient matrix; (c) Explicit coefficient matrix. .............................................................. 7.157

Figure 7.6 Flowchart of two alternative contact schemes: (a) shared contact algorithm; (b) separated contact algorithm. .......................................... 7.157

Figure 7.7 Contact representation in the coupled E-I algorithm. ............................... 7.158


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Figure 7.8 Domain profile suggested in stiffness estimation (Itasca 1993). .............. 7.164

Figure 7.9 Geometry of the multi-block system. ....................................................... 7.165

Figure 7.10 Displacement in the different cases: (a) case of ϕ=0; (b) case of ϕ=450. 7.166

Figure 7.11 Displacement of the top 2nd block: (a) case of ϕ=0; (b) case of ϕ=450. . 7.167

Figure 7.12 Rock slope stability analysis using the I-NMM, E-NMM and E-I NMM, respectively: (a) Initial modelling; (b) Explicit NMM; (c) Implicit NMM; (d) Explicit-implicit NMM. ......................................... 7.169

Figure 7.13 Displacement of the measured point 1 and 2. ......................................... 7.169

Figure 8.1 Location of the rock slope and aero-view from Google maps. ................ 8.172

Figure 8.2 Photograph of rock instability example in Cottesloe, Western Australia (photographed by X.L. Qu): a. right bank of the slope; b. left bank of the slope. ................................................................................................ 8.173

Figure 8.3 Continuum modeling of a rock slope by Abaqus/CAE 6.11: (a) slope meshing and 6-noded triangular element; (b) contour of the maximum strain ratio. ........................................................................... 8.176

Figure 8.4 Displacement nephogram at different times under seismic loading: (a) t=1.5s; (b) t=12.0s; (c) t=21.0s; (d) t=30.0s. ......................................... 8.177

Figure 8.5 Maximum displacement vectors and shear strain contours of the modelling in 2008 Wenchuan earthquake, China (Luo et al. 2012)...... 8.178

Figure 8.6 Simulation of a rock slope stability and failure under dynamic excavation using PFC technique (Wang et al. 2003). ........................... 8.179

Figure 8.7 Seismic simulation of Chiu-fen-erh-shan landslide by the Chi-Chi earthquake using DDA (Wu 2010). ....................................................... 8.180

Figure 8.8 Displacement distribution for each block after applying seismic loads (Miki et al. 2010). .................................................................................. 8.181

Figure 8.9 Modelling of rock fall failure under earthquake by NMM and DDA (Ning et al. 2012). ................................................................................. 8.181

Figure 8.10 Construction of onefold cover system from manifold cover system. ..... 8.185

Figure 8.11 Contact between explicit and implicit OEs based on onefold cover system. ................................................................................................... 8.186

Figure 8.12 Assembly of contact matrices in the coupled E-I algorithm. ................. 8.187

Figure 8.13 Parallel processing model: (a) UMA; (b) NUMA. ................................ 8.189

Figure 8.14 Construction of parallel computation of the NMM using OpenMP. ...... 8.190

Figure 8.15 Code segment of the parallel programming to the explicit NMM. ........ 8.190

Figure 8.16 Simulation results of the serial and parallel NMM codes. ..................... 8.192

Figure 8.17 CPU usage of the serial and multi-core NMM codes. ............................ 8.193

Figure 8.18 Computing time of the serial and parallel codes. ................................... 8.194


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Figure 8.19 Location of area of Lake Anderson slope in California USA (Keefer et al. (1980). .............................................................................................. 8.196

Figure 8.20 Numerical modelling of Lake Anderson slope. ...................................... 8.196

Figure 8.21 Record of acceleration of the earthquake. .............................................. 8.197

Figure 8.22 Simulated results of landslide under earthquake. ................................... 8.198

Figure 8.23 Scale map of geomechanical model (Zhou et al. 2008). ......................... 8.199

Figure 8.24 Modelling of the slope transfers from ENMM to DDA. ........................ 8.200


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LIST OF TABLES

Table 2.1 Values of based on different damping ratios. .......................................... 2.27

Table 2.2 CPU time for the proposed explicit and implicit scheme. ........................... 2.32

Table 4.1 Input parameters for the NMM simulation of rock slope. ........................... 4.86

Table 4.2 Maximum displacement of measure points in the NMM vs. ENMM. ........ 4.91

Table 5.1 Material properties of the rock bar. .............................................................. 5.99

Table 5.2 Comparison of CPU cost between the NMM and ENMM. ....................... 5.108

Table 5.3 CPU cost comparison between the NMM and ENMM. ............................ 5.112

Table 6.1 FoS using LEM by the integrated models. ................................................. 6.131

Table 6.2 Input parameters for the simulation of the modeling. ................................ 6.132

Table 6.3 CPU cost for the different study cases (hr.). .............................................. 6.141

Table 7.1 Contact types between two domains. ......................................................... 7.154

Table 8.1 Parameters of the used multi-core PCs. ..................................................... 8.191


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LIST OF SYMBOLS

iM = ith mathematical cover

iP = ith physical cover, reallocated with a single index i

jiP = jth physical cover generated from mathematical cover

iM

)(xi = Partition of unity function for mathematical cover iM

)(xi = Partition of unity function for mathematical cover iM

)(xui = local approximation function defined on physical cover

iP

)(xu = global approximation on the displacement field

= slope angle of an inclined plane

= a parameter of the Newmark method

= a parameter of the Newmark method

= Cauchy stress tensor

i = incident stress of P-wave

r = reflected stress of P-wave

t = transmitted stress of P-wave

= strain vector

= damping ratio

= algorithmic damping ratio

u = acceleration vector

u = velocity vector

u = displacement vector

= gradient operator

= density of material

b = body force

w = weight function

= arbitrary material domain

E = Young’s modulus

v = Poisson’s ratio


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= sampling frequency

= shear strength

c = shear strength

pc = P-wave velocity

= shear strength

T = shear strength

= natural frequency of the system

max = highest eigenfrequency of the system

t = time step size

ct = critical time step

Et = time step in the explicit algorithm

It = time step in the implicit algorithm

k = contact spring stiffness

nk = normal spring stiffness

sk = normal spring stiffness

d = penetration distance

nd = normal penetration distance

sd = shear penetration distance

u = prescribed displacement vector on u

t = traction vector on t

Nt = normal vector components

st = tangential vector components

= Lagrange multiplier vector

= shear modulus

= Kolosov constant

A = amplification matrix

B = strain displacement matrix

D = displacement matrix

D = velocity matrix


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D = acceleration matrix

K = stiffness matrix

M = mass matrix

C = damping matrix

critC = critical damping matrix

F = loading matrix

L = differential operator matrix

T = deformation matrix


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LIST OF ABBREVIATION

BEM = Boundary element method

CPU = Computer processing unit

DDA = Discontinuous deformation analysis

DEM = Distinct element method

EFGM = Element-free Galerkin method

ENMM= Explicit numerical manifold method

FCM = Finite cover method

FDM = Finite difference method

FEM = Finite element method

LEM = Limit equilibrium method

NMM = Numerical manifold method

PFC = Particle flow code

PSM = Pseudo static method

PUM = Partition of unity method

UDEC = Universal distinct element code

DEVELOPMENT OF AN EXPLICIT NUMERICAL MANIFOLD METHOD FOR DYNAMIC STABILITY

ANALYSIS OF ROCK SLOPE

1.1

CHAPTER 1. INTRODUCTION

1.1 BACKGROUND

The stability of rock slope under dynamic effect is often significantly influenced by

the structural geology of the rock such as bedding planes, joints and faults (Wyllie and

Mah 2004). These properties are generally termed discontinuities. The significance of

discontinuities is that they are planes of weakness in the much stronger intact rock, so

failure tends to occur preferentially along these surfaces. For this reason, various efforts

for rock slope stability are often rated on how effectively they incorporate

discontinuities.

In rock engineering, pseudo-static methods (PSMs) (Baker et al. 2006; Seed 1979;

Loukidis, Bandini, and Salgado 2003; Li, Lyamin, and Merifield 2009) are treated as

limit equilibrium methods (LEMs)(Bishop 1955; Morgenstern and Price 1965) to

calculate a factor of safety (FoS) for a specified discontinuous surface, and then find a

critical failure surface that associated with the minimum safety factor. In PSMs,

earthquake effects are represented by an equivalent static force, the magnitude of which

is a product of a seismic coefficient k and the weight of the potential sliding mass. This

approach, however, is incapable of quantifying the extent to which rock slope has

displaced. The sliding block theory was firstly proposed by Newmark (1965) to evaluate

the permanent displacement of slopes on dams or embankments induced by

earthquakes. The principle of this method was assumed that the potential sliding block

is a rigid body on a yielding base. Lateral displacement of block was expected to take

place when the base acceleration exceeded the critical or yield acceleration of the block.

This method is a more realistic method of analysing seismic effects on rock slopes than

the PSM of analysis. However, this method is only applicable for a single, rigid block

analysis. Geological structures in jointed rock involve complexities associated with

geometry, material anisotropy, nonlinear behavior, in situ stresses and the presence of

coupled processes (e.g. pore pressure, thermal loading, etc.). PSMs and Newmark’s



1.2

sliding block theory cannot be used for such complex problems. Numerical modeling

techniques address most of the limitations of both methods, and can model different

failure mechanism of the rock slope instability.

For the seismic stability analysis of rock slopes, the numerical methods are more

suitable because the behaviour of a rock slope is much more dependent on characteristic

and integrity of the rock mass. In general, they can be classified into three types: (i)

continuum-based numerical methods; (ii) discontinuum-based numerical methods; and

(iii) hybrid continuum-discontinuum methods.

The continuum approach introduces discontinuous interfaces in the form of “joint

elements” (Goodman, Taylor, and Brekke 1968; Ghaboussi, Wilson, and Isenberg 1973)

or “displacement discontinuities” (Katona 1983) to model discontinuities explicitly.

During the past several decades, various continuum-based numerical methods have been

developed, such as the finite element method (FEM), the finite difference method

(FDM), the boundary element method (BEM), and various meshless methods (e.g.

Element-Free Galerkin Method (EFGM)) and have been used successfully in

applications where the rock mass does not undergo large deformations. However, when

the rock mass behaviour is governed by the geometry and strength characteristics of the

discontinuities, the interactions between the individual blocks defined by the

discontinuities must be considered.

Though great efforts have been made to the continuum-based numerical methods,

block rotations, complete detachment and large-scale opening still cannot be properly

treated, the number of discontinuities which can be dealt with is also limited. To solve

such problems, the discrete element method has been developed. The discrete element

method considers interaction between rock blocks by representing the rock mass as an

assembly of rigid or deformable discrete blocks, and is capable of capturing the

kinematics of the block system. It allows for opening/closing of discontinuities,

movement of blocks relative to each other, and sliding and toppling along the

discontinuities (Jing 2003). The discrete element method is especially suitable for

simulation of large-scale displacements of individual blocks, block rotations, and

complete detachment. Typical examples of the discrete element method include the



1.3

distinct element method (DEM) (Cundall 1971a, 1971b) and the discontinuous

deformation analysis (DDA) pioneered by Shi (1988). One of the major limitations of

the discrete element approaches is their high computational requirements when the

interaction of large number of discrete blocks with contacts is involved. Thus, it is

pivotal that the selected DEM not only solve the discontinuities problem but also with

acceptable computational efficiency.

The DEM adopts an explicit time integration scheme based on finite difference

principles (Cundall 1971a). It has been developed into the commercial 2D code UDEC,

3D code 3DEC and the particle versions PFC-2D and PFC-3D by Cundall and his

colleagues (Itasca 1993, 1994, 1995), which has been enjoying a wide application range

in rock engineering. The main benefit of the DEM is that its computational efficiency is

high due to its explicit time integration nature. However, it has also been argued that the

accuracy of simulated results may be sacrificed in some particular cases (O’Sullivan and

Bray 2001; Luccioni, Pestana, and Taylor 2001). To ensure numerical stability, a DEM

simulation requires that the time step must be small enough. Furthermore, artificial

damping is required to dissipate the energy in the DEM, but the selection of an

appropriate value of damping is difficult for different cases.

On the other hand, the DDA derived based on the variational method takes the

benefit of the implicit time integration method (Shi and Goodman 1985; Shi 1988). The

formulation of DDA is similar to that of the FEM. Due to its implicit time integration,

the DDA is inclined to be unconditionally stable and it is expected to accommodate

considerably large time steps. Additional features include simplex integration method

which is a closed-form integration for the element and block stiffness matrices and the

open-close iteration (OCI) contact algorithm. The DDA method has emerged as an

attractive model because its advantage in simulating a discrete system cannot be

replaced by continuum-based methods or explicit DEM formulations. Since the

initiation of the DDA, various developments and applications have been achieved

during the last two decades. Most of the publications are included in the series of

proceedings of the International Conference on Analysis of Discontinuous Deformation

(ICADD) symposia (Li, Wang, and Sheng 1995; Salami and Banks 1996; Ohnishi 1997;



1.4

Amadei 1999; Bicanic 2001; Hatzor 2002; Lu 2003; Maclaughlin and Sitar 2005; Ju,

Fang, and Bian 2007; Ma and Zhou 2009).

The major considerable drawback of the DDA is that much larger computational

cost will be required, especially when the system contains huge number of discrete

blocks and contacts. The convergence efficiency of the OCI for a complex discrete

system is also not clear. This has been long time the challenge for the development of

the 3D DDA method. A number of articles on DEM and DDA have been published over

the past several decades, detailed mathematical formulations and discussions between

the DEM and DDA can be found in the state-of-the-art articles organized by Jing (Jing

1990, 1998) and the book written by Jing and Stephansson (2007).

The recently developed numerical manifold method (NMM) (Shi 1991, 1992,

1995, 1996a, 1996b, 1997) is such a hybrid method. The NMM is an evolvement of the

DDA and combines the merits of FEM and DDA. It inherits all the attractive features of

the DDA, such as the implicit time integration scheme, the contact algorithm and the

minimum potential energy principle (Chen, Ohnishi, and Ito 1998). It adopts a dual

cover system, i.e. a mathematical cover system overlapping the domain of interest and a

physical cover system and considers the contained discontinuities in a united manner. In

the past two decades, many developments have been carried out to improve the

performance of the NMM. Review articles on the recent development of the NMM have

been published by Ma, An, and He (2010) and An, Ma, et al. (2011a). It has been

reorganized that the NMM has great potential to be further developed in simulating a

medium with massive discontinuities.

However, enjoying the benefits of the FEM and DDA, the NMM is also suffering

with the high computational costs arising from the inherent implicit time iteration

scheme and the OCIs for contacts since the DDA had no properly developed

constitutive models of rock fractures representing the contact surfaces between blocks.

The implicit time integration algorithm involves in the solution of a system of

equations, the computational cost increases dramatically with the degrees of freedom

(DOFs) of the system is increased since the large-scale simultaneous algebraic

equations must be solved in each time step (Newmark 1959, 1965). The OCI requires



1.5

zero-tension and zero-penetration at all contacts which additionally highs up the

computation costs in order to achieve a convergence state at each time instance.

Reducing the time step to a smaller one in the OCI when the tolerance of the penetration

depth is violated losses the benefit of using a large time step in an implicit time

integration algorithm.

It has been proved that an explicit time integration scheme derives as accurate

results as an implicit one if the time step is small enough (Belytschko, Yen, and Mullen

1979; Hughes 1983). For a discrete system, the zero-tension and zero-penetration

requirement at contacts determines the accuracy of the simulation results. Considering

that the computational accuracy is equally important with the computational efficiency

for engineering problems, a proper balance of the high computational efficiency based

on an explicit time integration scheme and the high accuracy based on an appropriate

OCI process is highly demanded.

1.2 OBJECTIVES OF RESEARCH

In the traditional NMM, simulations of the rock slope stability are computationally

expensive. Thus, it is essential to make a general investigation of the traditional NMM

not only in terms of its capability to handle the discontinuous nature of the problem but

also its computational efficiency. The purpose of this research is to develop an explicit

version of the NMM for dynamic stability analysis of rock slope. The specific targets

are outlined as follows:

To develop an explicit version of the NMM:

Investigate the traditional NMM in terms of computational accuracy and

efficiency;

Propose an explicit time integration scheme for the NMM and to verify it

with respect to the computational efficiency and accuracy

To combine the explicit and implicit algorithms for the NMM:

Couple the temporal explicit and implicit NMM;

Couple the spatial explicit and implicit NMM;

To extend the explicit NMM for the rock slope stability analysis:



1.6

Implement a seismic version of the explicit NMM for the dynamic stability

analysis of rock slope;

Apply the developed program to simulate the dynamic stability of rock

slope.

1.3 ORIGANIZATION OF THE THESIS

The thesis is organized into nine chapters. The contents of each chapter are briefly

summarized as follows:

Chapter 1 introduces the background and objectives of this research and the

organization of this thesis.

Chapter 2 presents a literature review on the original numerical manifold method

(NMM), time integration algorithms and numerical methods for dynamic stability

analysis of rock slope. The basic concepts of the NMM are briefly introduced firstly.

The current development of the NMM in the improvement of the accuracy, extension of

the NMM for discontinuity problems, development of 3D NMM and other

developments and applications of the NMM are reviewed. Comparison between the

NMM and other typical numerical methods, such as finite element method (FEM),

discontinuous deformation analysis (DDA) and discrete element method (DEM), is

conducted as well. Then, time integration algorithms for numerical methods and

methods for dynamic stability analysis of rock slope are reviewed, in which numerical

properties, numerical examples for time integration and the traditional methods for

dynamic stability analysis of rock slop are investigated, respectively.

In Chapter 3, the theory of the NMM and the integration schemes are introduced,

respectively. In the former part, the fundamentals of the NMM is presented, including

the finite cover system based on the finite element mesh and contact algorithm applied

in the NMM. In the later part, the integration schemes including the spatial simplex

integration and temporal time integration are introduced, respectively, to extend the

further understanding of the NMM.



1.7

The traditional NMM is extended to an explicit version in Chapter 4. The basic

concepts consist of the frame of the NMM is briefly introduced firstly. An explicit time

integration scheme is proposed, in which the mass matrix, internal force and damping

item based on the finite cover system are derived in detail, respectively. Contact force in

the NMM is presented, including contact force approach and calculation of the contact

force based on the explicit time integration. As the contact algorithm such as OCI used

in the developed explicit version of the NMM, this chapter gives a briefly review of the

OCI in the NMM. Several numerical simulations are carried out to calibrate the

developed explicit version of the NMM, including simply-supported beam subjected a

concentrated load, single block sliding along the inclined surface and highly fractured

rock slope stability analysis.

Chapter 5 verifies the developed explicit version of the NMM (ENMM) in terms of

computational efficiency and accuracy with wave propagation problems. The NMM and

its cover system and the explicit scheme are briefly reviewed firstly. Calibration of the

stress wave propagation using the ENMM are carried out, in which the effect of mesh

size, effect of the time step and computational efficiency are discussed, respectively.

stress wave propagation through fractured rock mass, including the homogeneous

medium and joints conditions are verified with respect to the efficiency and accuracy.

Finally, seismic wave effect in the fractured rock slope is verified as well.

The temporal and spatial coupled explicit-implicit (E-I) algorithms for dynamic

problems are presented to extend the developed the explicit NMM in Chapter 6 and 7,

respectively. The temporal coupled E-I algorithm, including transfer algorithm, contact

force calculation and damping algorithm, is developed to maximize the advantages of

the explicit and implicit schemes in terms of the temporal aspect. The spatial coupled E-

I algorithm, including the coupled algorithm and contact algorithm, is proposed to

enlarge the both merits in the spatial aspect. Then, an alternative approach for the spatial

E-I algorithm is developed, in which onefold cover system is introduced to simplify the

E-I algorithm. The contact algorithm and contact matrices based on the onfold cover

system are derived, respectively. Spring stiffness problems in the coupled E-I algorithm

is discussed as well in Chapter 7.



1.8

In Chapter 8, the dynamic stability analysis of rock slope failure using the

developed explicit NMM is carried out to extend the capability of the developed

ENMM. Traditional numerical methods on the dynamic stability of rock slope are

briefly introduced firstly, including different kinds of advanced numerical methods. The

developed explicit NMM (ENMM) and the coupled explicit-implicit (E-I) algorithm are

briefly introduced as well. The parallel computation of the NMM is carried out, in

which parallelization with OpenMP and speedup of it are discussed in this chapter. Two

typical examples of rock slopes are simulated using the developed methods, in which a

dynamic case study of rock slope stability analysis is conducted using the seismic NMM

code, and a project of Jinping hydropower station is simulated using the developed

ENMM coupled discontinuous deformation analysis (DDA).

Chapter 9 draws the conclusions and gives the recommendation for future study.



2.9

CHAPTER 2. LITERATURE REVIEW

2.1 ORIGIN OF THE MANIFOLD

The numerical manifold method (NMM) is a newly developed numerical approach

appeared in a series of conference papers (Shi 1996a, 1991, 1992, 1995, 1997). It

provides a natural bridging to represent the physical change during numerical

simulations. It is also advantageous the over conventional numerical methods in

analyzing both continuous and discontinuous problems. The term of manifold comes

from the topological manifold and the differential manifold, the difference between

them lies in that the global functions of the differential manifold are highly

differentiable and defined irrelevant to the covers, while the global functions of the

numerical manifold here are defined based on covers and only piecewise differentiable.

In the traditional NMM, the manifolds connect many overlapped small patches

together to cover the entire problem domain. Each small patch is called a cover. A local

function is defined on each cover. The global behaviour is then determined by the

weighted average of local functions defined on each physical cover. Based on the finite

covers, the NMM combines the well developed analytical methods, widely used finite

element method (FEM) and the block-oriented discontinuous deformation analysis

(DDA) in a unified form.

Here, we give an example of natural world as represented in Fig. 2.1. Many

overlapped small patches scales connect together to cover the whole body of the fish.

Each piece of scale can be treated as one cover. The state of motion of fish body can be

represented as the connected scales motions. Manifold is the main subject of differential

geometry, algebraic topology, differential topology and modern algebra of mathematics,

even the natural world.



2.10

Figure 2.1 The scales of fish and fish body.

2.2 BASIC CONCEPTS OF THE NMM

The NMM approximation is based on three basic concepts, i.e. the mathematical

cover (MC), the physical cover (PC) and the manifold element (ME).

The MCs are user-defined small patches. They may or may not overlap each other,

but their union must be large enough to cover the entire problem domain. One distinct

feature of the NMM is that its mathematical covers do not need to conform to neither

the external boundaries nor the internal discontinuities, thus can always be regular. The

meshing task in the NMM is very convenient. The re-meshing is totally avoided for

discontinuity propagation. On each MC, a partition of unity function is defined, which

satisfies non-zero value only on its corresponding MC, but zero elsewhere.

The PCs are the subdivision of the mathematical covers by the physical features

such as the external boundaries and the internal discontinuities. Each physical cover

inherits the partition of unity function from its associated mathematical cover as

The ME is defined as the common region of several physical covers. On each

manifold element, we use the partition of unity functions to paste all the local functions

together to give a global approximation as

In the NMM, the global approximation is first constructed in each ME. Element

stiffness matrices are constructed and then assembled into a global stiffness matrix. In



2.11

this respect, the ME is similar to the finite element in the FEM. Just due to this

similarity, the NMM retains all the advantageous features of the system equations in the

FEM such as the symmetry and sparsity.

As aforementioned descriptions, an example shown in Fig. 2.1 can be referred to

introduce the three basic concepts of the NMM. Each scale of fish can be regarded as

one MC, all scales of fish to be formed a finite cover system to cover the fish body and

represent the activity of it.

2.3 CURRENT DEVELOPMENTS OF THE NMM

Since the initiation of the NMM in 1991 (Shi 1991), various developments and

applications have been achieved during the last two decades. Most of the publications

are included in the series of proceedings of the International Conference on Analysis of

Discontinuous Deformation (ICADD) symposia (Li, Wang, and Sheng 1995; Salami

and Banks 1996; Ohnishi 1997; Amadei 1999; Bicanic 2001; Hatzor 2002; Lu 2003;

Maclaughlin and Sitar 2005; Ju, Fang, and Bian 2007; Ma and Zhou 2009). The main

developments and applications of the NMM are categorized into several groups as

follows:

2.3.1 Improvement of the accuracy of the NMM

The traditional NMM is based on triangular finite element covers and chooses

constants as the local approximation space for each physical cover. Shyu and Salami

(1995) implemented quadrilateral isoparametric element into the NMM. However, for

some particular problems such as the bending problem, the precision and efficiency of

the quadrilateral isoparametric elements are not adequate either. And Cheng et al. (2002)

incorporated Wilson non-conforming elements into the NMM for a cantilever slab

bending problem. In terms of developing high-order NMM, Chen et al. (1998)

introduced high-order polynomials into the local approximation space for each physical

cover; Su et al. (2003) carried out a simple method to automatically produce the

expressions and writing the subroutines with the software Mathematica; and Lin et al.

(2005) developed the formulations of three-dimensional NMM with high-order local

functions.



2.12

2.3.2 Extension of the NMM for discontinuity problems

The NMM can be used to explore the strong discontinuity problems such as crack

problems and shear failures. Tsay et al. (1999a) applied the NMM together with a local

mesh refinement and auto-remeshing schemes to predict the crack growth. Zhang et al.

(1999a, 1999b) coupled the NMM with the boundary element method (BEM) to

simulate the crack propagation problems. Chiou et al. (2002) proposed the NMM

combined with the virtual crack extension method to study the mixed-mode fracture

propagation. Li et al. (2005) and Gao and Cheng (2010) developed enriched meshless

manifold method for two-dimensional crack modeling. Ma et al. (2009) exhibited the

advantageous features of the NMM for multiple branched and intersecting cracks, in which

singular physical covers are enriched with asymptotic crack tip functions, the stress

intensity factors are evaluated by virtue of domain form of interaction integral. Zhang et al.

(2010) extended it to simulate the growth of complex cracks, adopting the maximum

circumferential stress criterion to determine the crack growth.

In terms of the weak discontinuity, the NMM describes the weak discontinuities by

splitting mathematical covers into physical covers attached with independent cover

functions. Terada et al. (2003) introduced the finite cover method (FCM) as an alias of

the NMM, presented the formulation for the static equilibrium state of a structure with

arbitrary physical boundaries including material interfaces, and extended the FCM to

analyze heterogeneous solids and structures involving the discontinuities in strains and

discontinuities in displacement. The more extensions of the FCM are involved modeling

of evolving discontinuities in heterogeneous media can be referred in (Terada and

Kurumatani 2004; Kurumatani and Terada 2005, 2009).

2.3.3 Development of 3-D NMM

Lin et al. (2005) developed the formulations of 3-D NMM with high-order cover

functions, and proposed a fast simplex integration method based on special matrix

operations. Terada and Kurumatani (2005) introduced an integrated procedure for three-

dimensional structural analysis utilizing the FCM. They provide the formulations of the

FCM with interface elements for the static equilibrium of a structure. Cheng and Zhang

(2008) proposed a three-dimensional numerical manifold method with tetrahedron and



2.13

hexahedron elements. Jiang et al. (2009) proposed a three-dimensional numerical

manifold method based on tetrahedral meshes and derived the matrices of

corresponding equilibrium equations from the minimum potential energy principle. Ma

and He (2009) also proposed a three-dimensional NMM based on tetrahedron elements.

Generation of mathematical covers, formulation of discrete equations, generating

discrete blocks from a given fracture pattern were discussed in details.

2.3.4 Other developments and applications of the NMM

Li et al. (2005) derived the governing equations of the NMM from the method of

weighted residual (MWR), which enriched the mathematical foundation of the NMM

and extended it to problems such as head conduction and potential flow, where the

governing equations cannot be obtained from the minimum potential energy principle or

other variational principles. A coupled discontinuous deformation analysis and

numerical manifold method (NMM-DDA) has been developed by Miki et al. (2010) to

take both methods’ advantages while avoiding their shortcomings.

The NMM has been applied to simulate various problems as well. For example,

saturated/ unsaturated unsteady groundwater flow analysis by Ohnishi et al. (1999),

rock masses containing joints of two different scales by Lin et al., (1999), data

compression by Fang et al. (2005), dynamic non-linear analysis of saturated porous

media by Zhang and Zhou (2006), the shear response of heterogeneous rock joints by

Ma et al. (2007), dynamic friction mechanism of blocky rock system Ma et al. (2007a),

fluid-solid interaction analysis by Su and Huang (2007) and plane micropolar elasticity

by Zhao et al. (2010), etc.

The NMM is also able to deal with the well-known example of multiple discrete

blocks well. (2010) Ma et al. (2010) conducted a toppling run process in a typical

domino problem and mineral separation process using the vibrating screen with a set of

coal blocks. As shown in Fig. 2.2, a vibrating screen and container are numerically

modeled by the NMM and the results are consistent with the experimentally observed

phenomenon.



2.14

Figure 2.2 Size separation process with the vibrating screen modeled by the NMM (Ma et al. 2010).

2.4 COMPARISON WITH OTHER NUMERICAL METHODS

2.4.1 Comparison with FEM

In the FEM, the problem domain is divided into a collection of elements of smaller

sizes and standard shapes with fixed number of nodes at vertices and/or on the sides. On

the other hand, the approximation in the NMM is established based on covers. Under

such circumstances, the mathematical cover in the NMM is equivalent to the nodal

support in the FEM. The constant unknowns of each physical cover are equivalent to the

nodal unknowns in the FEM. The manifold elements in the NMM are equivalent to the

finite elements in the FEM.

The NMM is more flexible than the FEM in discontinuity modeling. The

conventional FEM requires the finite element mesh to be consistent with the internal

(a) Step 0 (b) Step 6000

(c) Step 12000 (d) Step 18000

(e) Step 24000 (f) Step 30000



2.15

discontinuities, which often complicates the meshing task. When discontinuity

evolution needs to be modeled, remeshing is inevitable. Although having achieved great

success, the meshing and the subsequent remeshing process is absolutely not an easy

task. Moreover, additional inaccuracy may arise due to the state variable mapping in

such processes. In contrast, the mathematical covers in the NMM do not need to

conform to neither the external boundaries nor the internal discontinuities, which makes

the meshing task in the NMM very convenient and discontinuity evolution be modeled

without remeshing. The meshing task is very convenient in the NMM, which also

makes the NMM more suitable than the FEM for complex domain problems with

hundreds of inclusions, voids, and/or cracks.

The FEM is a continuum-based method, thus cannot well represent the block

rotations, complete detachment and large-scale opening. In addition, the number of

discontinuities, which can be handled by the FEM is also limited. In contrast, because of

its discrete root, the NMM can easily deal with the aforementioned problems

2.4.2 Comparison with DDA

The discontinuous deformation analysis (DDA) Shi (1988) was developed

originally to solve problems in which a rock mass is delimited into blocks by joints.

Thus, in addition to the contact detection and frictional contact modeling, the DDA also

presents algorithm to form blocks from joints. Since rock blocks generally undergo

small deformation, the DDA simulates each discrete body as a simple deformable block

with only six specific degrees of freedom (DOFs).

The DDA derived based on the variational method takes the benefit of the implicit

time integration method (Shi and Goodman 1985; Shi 1988). The formulation of DDA

is similar to that of the FEM. Due to its implicit time integration, the DDA is

unconditionally stable and it is expected to accommodate considerably large time steps.

Additional features include simplex integration method which is a closed-form

integration for the element and block stiffness matrices and the open-close iteration

(OCI) contact algorithm. The DDA method has emerged as an attractive model because

its advantage in simulating a discrete system cannot be replaced by continuum-based

methods or explicit DEM formulations. The NMM inherits all the attractive features of



2.16

the DDA, such as the implicit time integration scheme, the contact algorithm and the

minimum potential energy principle Chen et al. (1998). However, the NMM employs a

number of covers to raise the DOFs to more accurately describe the displacement field

and stress field in each block. Thus, we can say the NMM is the DDA with each block

discretized into finite covers. In other words, if one discrete block is one manifold

element with linear displacement field, then the NMM will be degraded to the DDA.

2.4.3 Comparison with DEM

There is another class of numerical methods called discrete element method

(DEM). Typical examples of the DEM include the distinct element code UDEC (Itasca

1993) and the DDA pioneered by Shi (1988). The DEM is an explicit version, while the

DDA is an implicit version. The DEM adopted an explicit time integration scheme

based on finite difference principles (Cundall 1971a). The DEM has been developed

into the commercial 2D code UDEC, 3D code 3DEC and the particle versions PFC2D

and PFC3D by Cundall and his colleagues (Itasca 1993, 1994, 1995), which has been

enjoying a wide application range in rock engineering. The main benefit of the DEM is

that its computational efficiency is high due to its explicit time integration nature. The

low computational cost of DEM is mainly due to that the explicit time integration

algorithm does not involve in the solution of coupled equations, so fewer computations

are needed per time step. However, it has also been argued that the accuracy of

simulated results may be sacrificed in some particular cases (O’Sullivan and Bray

2001). To ensure numerical stability, a DEM simulation requires that the time step must

be small enough. The NMM differs from the DEM (e.g. UDEC, 2-D version of DEM)

in the following aspects:

The UDEC is an explicit method. It calculates the state of a system at current

step from the states of previous steps. It requires small time step ∆ to keep the

error in the result bounded. The NMM is an implicit method. It finds a

solution by solving an equation involving both the previous step and the

current step. Much larger time step ∆ can be used. To achieve given accuracy,

the NMM takes much less computational time than the UDEC, even taking

into account that the NMM needs to solve the equilibrium equation;



2.17

The UDEC uses numerical integration techniques (e.g. Gauss quadrature)

while the NMM adopts closed-form integrations (i.e. simplex integration

method) to evaluate the weak form;

The joints in the UDEC are always assumed as persistent, failure occurs along

pre-existing persistent joints. However, it is highly unlikely that such a

network of fully persistent discontinuities exist in nature. The NMM allows

non-persistent joints. In the NMM, cracks propagate and coalesce with each

other to form a continuous failure path, thus seems more realistic;

The UDEC uses a finite difference mesh while the NMM uses a cover system

to resolve the stress / strain vibration within each block;

The UDEC sometimes uses artificial joints to connect all the blocks together

to represent an intact material, and then fracturing in intact material is realized

by changing artificial joints to real joints. The fracturing follows pre-defined

artificial joints, thus the results are sensitive to the block configuration. The

NMM allows the cracks arbitrarily align with the elements, thus mesh

dependency is avoided to some extent;

The UDEC is a relatively mature method. Various material models (e.g.

elastic, Mohr-Coulomb plasticity, double-yield, strain-softening, etc) are

available in the code. In addition, relative motion along the discontinuities can

be linear or non-linear. The UDEC has been applied to various engineering

applications. The NMM is a relatively new method. Only linear-elastic

material model is available, and only linear motion along the discontinuities

can be accounted for. Its application is also limited. Further developments are

required.

It is noted that the contact problems is a crucial factor affecting the corresponding

time integration algorithms. Thus, it is essential to investigate the contact methods in the

NMM and DEM. The NMM is an implicit method, which uses penalty method (Shi

1988; Jing 1998) to prevent interpenetration between the contact blocks. And the

behaviour of the contact can be classified three types: corner-to-edge, edge-to-edge and

corner-to-corner in two dimension, which are similar to the DEM (Cundall 1971a;



2.18

Cundall and Hart 1992; Itasca 1993) and different to the FEM such as ANSYS software,

in which the contact types are determined as single surface, nodes-to-surface and

surface-to-surface respectively by means of searching the corresponding slave and

master surfaces (ANSYS 2009). When the corner-to-corner contact is detected, the

NMM employs a direct scheme for the corner-to-corner contact to improve the contact

accuracy, which is referred as the shortest path method. If two potential contact edges

are passed by the corresponding corners simultaneously, a penetration takes place, as

represented in Fig. 2.3(a), where the normal penetrating distances between the reference

edges and the corresponding point are 1d and

2d , respectively. When the distances

satisfy 21 dd , then a contact spring is attached between the point and the reference edge

which is penetrated by distance 1d to assemble the global stiffness matrix. Otherwise,

the other reference edge is selected. The physical meaning of applying the stiff contact

spring is ‘‘to push the invaded angle out of the block along the shortest path” (Shi

1988). Resort to the implicit time integration algorithms and OCI criteria, the contact

problems can be solved successfully. However, it is a time-consuming work to obtain

contact convergence as the repeating iterations and solving global equation. Here, it is

noted that when the corner-to-corner contact is detected, the DEM (e.g. UDEC code)

uses rounded corners to avoid stress concentration in the explicit modeling (Itasca

1993). As shown in Fig. 2.3(b), a rounded corner is represented by an arc of a circle

tangent to the two adjacent edges, in which r is radius of the arc and d is distance to the

corner. It is the rounded corner makes contact blocks can smoothly slide past one

another when two opposing corners interaction. If the two corners are in contact, the

point of contact is the intersection between the line joining the two opposing centers of

the radius r and the circular arcs. The directions of normal and shear forces acting at

the contact are defined with respect to the direction of the contact normal. Combine the

central difference algorithm, the contact problems can be simulated in the DEM

explicitly. Thus, the efficiency of the UDEC is more efficient than that of the NMM.

However, it has also been argued that the accuracy of simulated results may be

sacrificed in some particular cases (O’Sullivan and Bray 2001). To ensure numerical

stability, a DEM simulation requires that the time step must be small enough.



2.19

Figure 2.3 Corner-to-corner contact in the NMM and DEM: (a) The shortest path method in the

NMM; (b) The corner rounding technique in the DEM.

2.5 TIME INTEGRATION ALGORITHMS FOR NUMERICAL

METHODS

In general, there are two general classes of algorithms for dynamic problems:

implicit and explicit. Implicit algorithms tend to be numerical stable, permitting larger

steps, but the computational cost per step is high and storage requirements tend to

increase dramatically with the contacts between elements and these degrees of freedom

(DOFs), so they are suited to simulate the lower dynamics problems with less non-

linearities, resulting in more numerical stability and accuracy (Gelin, Boulmane, and

Boisse 1995; Yang et al. 1995; Sun, Lee, and Lee 2000). On the contrary, explicit

algorithms tend to be inexpensive per step and require less storage than implicit

algorithms, but numerical stability requires that small steps be employed, thus, they

generally used for highly non-linear problems with many DOFs (Dokainish and

Subbaraj 1989; Subbaraj and Dokainish 1989). Using the explicit algorithm, the

computational cost is proportional to the number of elements. On the other hand, using

d2 d1

d1 < d2

(a) The shortest path method in the NMM;

r

d

d = rd d >> r

r

(b) The corner rounding technique in the UDEC (Itasca 1993).



2.20

the implicit algorithm, experience shows that for many problems the computational cost

is roughly proportional to the square of the DOFs. To compare the explicit and implicit

algorithms in terms of efficiency, Fig. 2.4 illustrates the comparison of CPU time versus

DOFs using the explicit and implicit algorithms. When the DOFs increase up to some

extent, the explicit algorithm becomes more efficient than that of the implicit algorithm.

Figure 2.4 Efficiency versus DOFs between explicit and implicit algorithms.

It is noted that there are some dynamic problems, implicit algorithms are very

efficient and others explicit algorithms are very efficient. To take advantage of the

merits of implicit and explicit algorithms, many methods have been developed in

temporal and spatial discretizations, in which it is attempted to simultaneously achieve

the maximum contributions of both classes of algorithms. Belytschko and Mullen

(1976, 1978b) proposed an explicit-implicit (E-I) nodal partition and proved the

conditional stability of E-I partitions using energy methods and represented the time

step is limited strictly by the maximum frequency in the explicit partition of the mesh.

Hughes and Liu (1978) proposed an alternate element-by-element E-I partitions, in

which a similar stability condition is proven for the algorithm. Liu and Belytschko

(1982) proposed a general mixed time E-I partition procedure which permits different

time steps and different integration methods to be used in different parts of the semi-

discrete equations. Belytschko and Mullen (1978a) proposed a multi-time step

integration method involving different time steps in different zones of the model, in

which the nodal partition approach is employed for E-I systems and linearly interpolated

Explicit algorithm

Implicit algorithm

DOFs

CP

U t

ime



2.21

displacements at the interface. It differs from the referred mixed methods, which consist

of defined zones where different algorithms apply, but with a single time step defined

for the whole domain. Various other improvements in transient algorithms have also

been achieved referred in (Hughes, Pister, and Taylor 1979; Smolinski, Belytschko, and

Neal 1988; Miranda, Ferencz, and Hughes 1989; Belytschko and Lu 1992; Sotelino

1994; Smolinski, Sleith, and Belytschko 1996; Gravouil and Combescure 2001).

In many structural dynamics simulations, low mode response problems are usually

interested. For these cases, the use of implicit time integration scheme is generally

preferred due to its unconditionally stable algorithms over the explicit conditionally

stable scheme. The implicit scheme possesses the numerical property of numerical

dissipation to damp out any spurious participation of the higher modes, thus the larger

time-step size can be employed, but it is restricted in terms of considering

computational efficiency (Katona, Thompson, and Smith 1977), which is the point of

investigation of the time integration in the NMM. The purpose of investigation of

numerical dissipation is to reduce the spurious, non-physical oscillations and

computational effort to the most extent. Numerous improvements have been developed

while maintaining second-order stability and accuracy referred in (Wilson 1968; Hilber,

Hughes, and Taylor 1977; Hughes 1987; Miranda, Ferencz, and Hughes 1989; Chung

and Hulbert 1993; Zhai 1996). Moreover, Numerical dissipation is significant when

solving structural dynamic problems using the explicit integration scheme as well

(Hulbert and Chung 1996). The principal use of the explicit time integration scheme is

limited by time-step size with respect to the stability and accuracy. Newmark β method

when β=0, γ=1/2 is generally called explicit method which possessed no numerical

dissipation itself, results in the consequence is that oscillations occur in the solutions of

numerical simulations. The study of the proposed explicit scheme contrasts with the

implicit scheme in the NMM will be presented to investigate the numerical properties

for the time integration systematically. In addition, the computational efficiency of the

explicit scheme compared to the implicit scheme is explored as well.

2.5.1 Numerical properties for time integration

In order to investigate the time integration, the behaviour of an equivalent linear



2.22

system is considered. In such a system, the minimization of the system potential energy

will produce an equation of motion, which is similar to that in the FEM. Let

displacement term of and denote the approximation to the displacements

and ∆ for a time step ∆ , the discrete equation of motion can be expressed as

(2.1)

The component form of the mass , damping and stiffness terms are extensively

discussed in Shi (1991). A single step integration method can be used to solve Eq. (2.1)

for given initial conditions of and based on Newmark method, which can

be represented as

∙ ∆ ∆ ∙ ∙ ∆ ∙

1 ∆ ∙ ∆ ∙ (2.2)

where β and γ are velocity and acceleration weight parameters respectively.

In terms of numerical properties of the integration scheme, a single spring-mass

system is investigated by simplifying it into a single-degree-of-freedom (SDOF) system.

Eq. (2.1) can be degenerated in standard form with frequency and damping ratio as

follows:

fddd 22 (2.3)

where d ,d , and d are acceleration, velocity and displacement, respectively, MK/ is the

frequency of the undamped oscillator, M and K are the mass and stiffness of the

oscillator, KMCCC crit 2// is the damping ratio, KMCcrit 2 is the called critical damping,

and MFf / is the applied load.

At the n time step, extending Eq. (2.3) to be rewritten as

i

n

i

innn QAzAz

1

0 (2.4)

where A is the amplification matrix, iQ is the load vector;

0z is initial data; and

T

nnnn dtdtdz 2 . For the Newmark scheme, A is defined by (Bathe and Wilson

1972; Hilber and Hughes 1978; Hughes 1983) as:



2.23

333231

333231

333231

)1(11

)1(2/111

AAA

AAA

AAA

A

(2.5)

1

31

K

AQ

(2.6)

and

21

2/211

2

2

33

22

32

2

31

DD

A

DA

DA

(2.7)

where t , called sampling frequency. An integration method is stable if the spectral

radius of its amplification matrix remains bounded by 1, which determined by the

maximum modulus of eigenvalues denoted by i

iA max)( , 3,2,1i .

To find the spectral radius of A , the eigenvalues of A are required, which can be

determined by the characteristic equation

020)det( 322

13 AAAIA (2.8)

where1A , 2A and

3A are invariants of matrix A . tracAA2

11 , 2A =sum of principal

minors of A and AA det3 , respectively. Since 03 A , the eigenvalues of Eq. (2.8) are

given by

22

112,1 AAA (2.9)

In terms of the parameters and , the scheme is unconditional stable and

produces under-damped oscillatory response and spectral stability requirement when the

following conditions hold:

10 , 2

1 , 4/)

2

1( 2 (2.10)

At bif the complex conjugate roots bifurcate into two real distinct roots, and the

spectral radius of the amplification matrix A attains its minimum at this point, and



2.24

2)2/1(4

1

1bif

(2.11)

Bifurcation indicates more about the characteristic of the eigenvalues, which can be

evaluated by 4 bif.

The variation of spectral radius for the time integration is presented in Fig. 2.5, in

which the spectral radius 1 with time along. It satisfies the unconditional stability

conditions of Eq. (2.11). For best results from stability and numerical dissipation point

of view, a time-step size t select as(Doolin and Sitar 2004; Doolin 2005):

eet

maxmax

4,

1

(2.12)

where emax is element maximum angle frequency, which is related with minimum

value of element eigenlength.

Figure 2.5 Spectral radius versus sampling frequency for Newmark integration.

In the study, we study the accuracy property of Newmark method in terms of the

exact solution of the homogeneous SDOF model equation. As f is zero in Eq. (2.3), the

eigenvalues of A can be shown to take on a form as:

2/12 )1(

2,1 )( ieA (2.13)

where t and

2ln2

1A

Z

(2.14)

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100 1000

ρ(A

)

Z

β=0.50, γ=1.0

β=0.25, γ=0.5 0.3333



2.25

As measures of the numerical dissipation, we take the algorithmic damping ratio (in

Eq. (2.14) and the period error TT / respectively, where /2T and /2T . The

following cases can be verified:

Case 1: 0 and 2/12 , then =0, which points no numerical dissipation;

Case 2: 0 and 2/12 , then >0, which is applied into NMM value of 1 is

associated with numerical dissipation.

Figure 2.6 Period errors for the undamped case of Newmark integration methods.

In Fig. 2.6, period errors are presented for undamped Newmark methods with case

1(i.e., 2/1 ) and case 2 (i.e., 1 ). We can find that for 4/1 , the methods presented

are conditionally stable, which is made evident by the abrupt decreases in period errors

with . And the central difference method (i.e. 0 ) tends to shorten periods whereas

the trapezoidal rule increases periods (i.e. 4/1 ). About the central difference method

will be investigated in the subsequently sections.

As we noted, the NMM time integration scheme possesses numerical dissipation,

which is shown in Fig. 2.7. It is can be found that the maximum value of 0.2845 of in

the NMM beyond the other parameter values of , which suggests that a time-step size

in this range will lead to faster convergence.

0

1

2

3

4

0 1 2 3 4 5 6

T

Z

β=0, γ=1/2

β=1/4, γ=1/2

β=1/2, γ=1

T=1.0

Z=4.0



2.26

Figure 2.7 Numerical damping versus sampling frequency in the NMM.

The accuracy of the solution depends on the numerical stability and conditioning of

the global stiffness matrix ce KKK as well, where eK is element stiffness matrix,

and cK is contact stiffness matrix, respectively. For the SDOF problem above, one

lower limit of time-step size t can be estimated in terms of element mass and spring

stiffness as KMt /2 . If t is smaller than required value, the characteristics of K are

loosened and the solution becomes unstable. Whereas one larger t will lead to small

ratios of the inertia term of 2/ tM , the effect of M and numerical damping will be

insignificant.

In the Newmark integration schemes, when the parameters are 2/1,0 , recall

Eq. (2.2), we can get an explicit forward central different scheme used by velocity and

acceleration at step n to )1( n . It can be expressed as

∆∆ ∙

/ /

∆∆ ∙ / /

(2.15)

where subscripts of n+1/2 and n-1/2 denotes the central at the step n to )1( n and

)1( n to n , respectively.

Comparing with the implicit time scheme in the NMM, the explicit scheme is

conditionally stable. From the discussion above, the range of bif can be classified by

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4

ξ

Z

β=0.50

β=0.52

β=0.54

β=0.60

β=1.00

0.28450.2659

0.25190.2222

0.1457



2.27

around two general cases: 0 and 0 . In the first case, the spectral radius is equal to

unit throughout the range of 2 , when 2 the oscillation happen even approach

infinite. It is conditionally stable and time-step size t is considered as

eet

maxmax

2,

0

(2.16)

It is noted that there is no numerical dissipation when 0 . In the second case, the

spectral radius is at the range of 1)(0 A , in which the minimum appears at the point

of bif . As discussed on the implicit time integration scheme, numerical dissipation in

the proposed explicit scheme can be improved resort to the damping. In general, smaller

damping ratio of the cover system corresponds to larger time-step size t as presented

in Table. 2.1. Here, we propose )1,0(,/2 max et , in which is one coefficient to

determine the time-step size based on different damping ratios. Compared to implicit

scheme in the NMM, the proposed explicit is more suitable to solve the high frequency

problems as the numerical property of conditionally stable. When the damping is taken

into account in the explicit scheme, the time-step size is sensitive to damping and the

proposed time-step size can be selected in accordance with the coefficient of in Table

2.1. Further more, an appropriate value of should be considered against different

damping effect in terms of solution convergence.

Table 2.1 Values of based on different damping ratios.

In terms of efficiency, the solution convergence is taken into consideration between

the Newmark implicit and explicit schemes. Convergence of Newmark implicit scheme

has been proven in the above discussion. Here, it is demonstrated through solving the

SDOF problem using smaller and larger time-step sizes, respectively. To track CPU

time, a high resolution timer function which can measure up to 1/100000th of a second is

added to keep track of CPU time for each time step. All analyses are run on computer

ξ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

δ ≤ 1 0.89 0.79 0.70 0.64 0.60 0.54 0.49

ξ 0.4 0.45 0.5 0.6 0.7 0.8 0.9 1

δ ≤ 0.44 0.40 0.39 0.35 0.29 0.28 0.25 0.23



2.28

with the system configuration: processor speed = 3.0 GHz and RAM = 3.0 GB. As

shown in Fig. 2.8, it is clearly that there is a range of time-step sizes, which produces

the shortest computational cost, larger and smaller time-step sizes produce more

computational cost. Comparing the convergence with the spectral radius plotted in Fig.

2.5, the convergence trend closely accords with which of the spectral radius .

Figure 2.8 Computational cost by the Newmark implicit scheme.

Due to the proposed explicit integration algorithm is conditionally stable, the

spectral radius for the scheme is constant and equal to unit throughout the stable

range of )2,0( . Comparing to the implicit scheme, the explicit scheme uses viscous

damping to improve numerical dissipation in order to achieve faster solution

convergence. Fig. 2.9 gives one special case of 1.0 for computing the SDOF problem

using a variety of step-step sizes. When the value of t is near that of critt (critical value

of t ), the computational cost taken is less than the level of 0.1 second to achieve

convergence. If the selected value of t is smaller or larger than that of critt , more

computational cost happens even over 1.0 second or more. Furthermore, it is noted that

the convergence orientation is consistent with the spectral radius .

0

0.2

0.4

0.6

0.8

1

1.2

0.0001 0.001 0.01 0.1 1 10

Com

pu

tati

onal

cos

t (s

)

Time-step size (s)



2.29

Figure 2.9 Computational cost by the Newmark explicit scheme.

2.5.2 Numerical examples for time integration

To further investigate the time integration properties, such as numerical stability

and numerical dissipation, one numerical example of n -DOF problem constituted by

multiply mass-spring-dashpot systems is considered here. The analysis is run on the

same computer with the system configuration: processor speed = 3.17 GHz and RAM =

4.0 GB.

In the present analysis, a simplified model of n -DOF mass-spring-dashpot system,

as shown in Fig. 2.10, is studied to investigate the proposed explicit and implicit

schemes in terms of computational accuracy and efficiency. The structural properties of

the model are assumed to be im = 1000kg and

ik = 10MPa, in which im and

ik ( i =1, 2,

…, n) are ith mass and stiffness, respectively. A harmonic excitation of )(tF =1.0e4

)sin( , in which 100/ and is the number of time-step, is applied to five different

systems by taking n =10, 50, 100, 200 and 500, respectively. The initial conditions of

the systems are assumed )0(iu = )0(iu =0, where )0(iu and )0(iu are the displacement

and velocity of the ith mass, respectively. Different time-step sizes are selected for both

integrations based on stability and accuracy considerations.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02

Com

pu

tati

onal

cos

t (s

)

Time-step size (s)



2.30

Figure 2.10 Simplified model of n-DOF spring-mass-dashpot system.

The system displacements responses time histories of a typical case of n =100

under damping and no-damping are studied, respectively. The simulated results are

presented in Fig. 2.11. When no-damping case is taken into account, a proper t is

selected to investigate the explicit and implicit schemes, although a larger time-step size

can be used in the implicit scheme as it is unconditionally stable. The system obtained

displacements under no-damping reveal that the solutions of the explicit scheme are

very close to the implicit scheme in Figs. 2.11(a) and (b). On the other hand, since the

damping factor in the explicit scheme can improve the numerical dissipation and

solution convergence effectively, smaller time-step sizes are applied into the under

damping case. In Fig. 2.11(c), damping ratio of 1.0i is used in the explicit scheme,

the amplitude of the solution is stable and identical compared with the implicit scheme.

And critical damping of 0.1i is applied in the Fig. 2.11(d), the system displacements

converge to zero fleetly both in explicit and implicit schemes, which indicates that the

explicit scheme exhibits no overshooting in terms of numerical accuracy when the

appropriate t is selected. It is noted that under damping case, where accuracy

requirements restrict the t to be very small, the implicit scheme uses a dynamic

coefficient (i.e. dd in the code) of 0.999 to achieve a stable solution, which means the

velocity is reduced or damped by 0.1% before it is set as the initial velocity for the next

step. This is similar to damping item in the explicit scheme.

nu 1u2uiu

nk

nc

nm

ic

ik 2k

2c

1k

1c

im2m 1m

)(tF



2.31

(a) no-damping, t =10ms;

(b) no-damping, t =1ms;

(c) 1.0i , t =0.1ms;

-0.12

-0.06

0

0.06

0.12

0 0.5 1 1.5 2D

isp

l. (m

)

Time (s)

Newmark implicit

Newmark explicit

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

Dis

pl.

(m)

Time (s)

Newmark implicitNewmark explicit

-0.04-0.03-0.02-0.01

00.010.020.030.04

0 0.2 0.4 0.6 0.8 1

Dis

pl.

(m)

Time (s)

Newmark implicit

Newmark explicit



2.32

(d) 0.1i , t =0.01ms.

Figure 2.11 Total displacement responses for 100-DOFs of system.

In order to compare the computational efficiency between the explicit and implicit

schemes conveniently, mst 1.0 is employed in both schemes. The CPU time involved

for the simulation is recorded and assembled in Table 2.2, in which CPU time consumed

by employing the implicit scheme is denoted by ICPU1 and ICPU 2

, while that involved by

the proposed explicit scheme is represented by ECPU1 and ECPU 2

based on two cases of

i =0 and 0.1 respectively. From sixth to eighth columns of Table 2.2, where the values

of I

E

CPU

CPU

1

1 , I

E

CPU

CPU

2

2 and E

E

CPU

CPU

2

1 reveals the CPU time consumed by the explicit scheme is less

than one fifth of that in the implicit scheme, and which is decreased sharply as the

number of degree of freedom (i.e. n) growing in both damping and no-damping

conditions. It is clear that the explicit scheme is efficient to solve multiply DOF

problem as it doesn’t involve iterations of equations such as in the implicit scheme.

Table 2.2 CPU time for the proposed explicit and implicit scheme.

n-DOF 0i 1.0i

I

E

CPU

CPU

1

1 I

E

CPU

CPU

2

2 E

E

CPU

CPU

2

1

ICPU1 ECPU1

ICPU 2 ECPU2

10 0.266 0.063 0.266 0.063 0.236842 0.236842 1

50 0.969 0.094 0.969 0.094 0.097007 0.097007 1

100 1.766 0.110 1.813 0.110 0.062288 0.060673 1

200 3.516 0.156 3.485 0.141 0.044369 0.040459 1.106383

500 8.672 0.266 8.485 0.282 0.030673 0.033235 0.943262

-0.0004

0

0.0004

0.0008

0.0012

0 0.2 0.4 0.6 0.8 1

Dis

pl.

(m)

Time (s)

Newmark implicit

Newmark explicit



2.33

The numerical results present good accuracy and stability of the Newmark explicit

scheme compared to the implicit scheme. The Newmark explicit scheme is more

efficient in solving the nonlinear systems and such problems compared to implicit

scheme with respect to computational efficiency.

2.6 METHODS FOR DYNAMIC STABILITY ANALYSIS OF

ROCK SLOPE

Rock slope stability analyses are routinely performed and directed towards

assessing the safe and equilibrium conditions of slopes. The analysis technique chosen

depends on both in-site conditions and potential type of rock failure, with careful

consideration being given to the varying strength, weaknesses and limitations inherent

in each methodology. In general, the primary objectives of rock slope stability analyses

can be summarized as:

Determination of the factor of safety (FoS) of the rock slope to investigate potential

failure mechanisms;

Determination of sensitivity to different triggering mechanism (i.e. seismic loading,

blasting effect and pore pressures, etc.);

Test and comparison of different support and stabilization schemes; and

Optimization design for the slope to cover safety, reliability and economics.

To properly conduct such investigations, and analyse and evaluate the potential

hazard relating to an unstable rock slope, it is essential to understand the processes and

mechanisms pushing the failure of slopes. Basically, the instability of rock slope can be

categorized into planar failure, wedge failure, toppling failure, circular failure and rock

fall failure, and they are summarized and presented in Fig. 2.12. To study these five

kinds of typical failure modes conveniently, the corresponding simplified models are

plotted, respectively. In practices, many performances have been carried out to study the

stability of rock slope. There are the analytical methods: i.e. Limit Equilibrium Methods

(LEMs) (Bishop 1955; Morgenstern and Price 1965) and Newmark methods (Newmark

1959, 1965), and numerical methods: i.e. Shear Strength Reduction Method of Finite

Element Method (SSRM-FEM) (Zienkiewicz, Humpheson, and Lewis 1975; Griffiths



2.34

and Lane 1999), Distinct Element Method (DEM-UDEC/3DEC) (Itasca 1999b, 1999a),

Discontinuous Deformation Analysis (DDA) (Shi 1988, 1993) and Numerical Manifold

Method (NMM) (Shi 1991, 1992), etc.).

Figure 2.12 classification of instability and project examples for rock slopes.

Further more, some real examples photographs are attached to represent these

typical modes of rock slope failure in order to illustrate these rock slope failure visually.

1. ref. technology.infomine.com; 2. ref. www.rocscience.com; 3. ref. www.rocscience.com;

4. ref. elkorose.schopine.com; 5. ref. www.capetownskies.com.

Type Simplified mode Project example

Planar failure containing persistent joints striking parallel to the rock face for jointed rock slopes.

Wedge failure on two intersecting discontinuities for jointed rock slopes.

Toppling failure of columns separated from the rock mass by steeply dipping parallels or near parallels to the slope.

Circular failure in weak rock mass or heavily jointed rock slope with a spoon-shaped surface.

Rock fall failure contains sliding, rotating, falling and bouncing of loose rocks and boulders on the slope.

2.

1.

3.

4.

5.



2.35

For the category of rock slope instability, critical parameters, methods of analysis and

acceptability criteria, more details can be referred as (Hoek and Bray 1981; Hoek 1991;

Cruden and Varnes 1996; Wyllie and Mah 2004).

In rock slope engineering, the analysis of seismic stability of rock slope is one of

important issues attracted the attention by geotechnical engineering and earthquake

engineering. Many researchers have attempted to develop and elaborate on the methods

for slope stability analysis. Exiting methods for evaluating the stability of rock slope

subjected to seismic effect can be classified into three categories: (1) force-based

pseudo-static methods; (2) displacement-based sliding block methods; (3) numerical

methods.

Conventional pseudo-static approach is a widely used based on the LEM (Bishop

1955; Seed 1979; Chang, Chen, and Yao 1984) to evaluate slope stability where the

dynamic effects are major from earthquake can be simplified as horizontal and/or

vertical dynamic coefficients ( and ), in which the magnitude of the coefficients is

expressed in terms of a percentage of gravity acceleration. Due to the simplicity of the

pseudo-static approach, it has drawn the attention of a number of investigators (Seed

1979; Chang, Chen, and Yao 1984; Ling, Leshchinsky, and Mohri 1997; Baker et al.

2006; Li, Merifield, and Lyamin 2008). Li et al. (2008) investigated the seismic effects

on rock slope stability coupled with Hoek-Brown failure criterion using the pseudo-

static method (PSM), in which the advantage of limit theorems were exploited to

bracket the true solutions for rock slope stability numbers and to provide a range of

seismic stability charts for rock slopes (Lyamin and Sloan 2002a, 2002b). However, the

pseudo-static approach has certain limitations, since it cannot simulate the transient

dynamic effects of earthquake shaking, because it assumes a constant unidirectional

pseudo-static acceleration (Cotecchia 1987; Kramer 1996).

Dynamic analysis has also been employed using a simplified manner. Huang et al.

(2001) conducted Newmark’s sliding block analysis considering the whole failure rock

mass as one single block, where they demonstrated that the surface-normal acceleration

played a vital factor in the initiation of the Chiu-fen-erh-shan landslide. Jibson (1993)

and Jibson et al. (1998) have developed procedures for estimating the probability of



2.36

landslide occurrence as a function of Newmark displacement based on observations of

landslides caused by the 1994 Northridge earthquake in California. In this section, we

give a brief overview of the developed methods in the past several decades.

2.6.1 LEM

The LEM of determining the FoS of a sliding block can be modified to incorporate

the effect on stability of seismic ground motions. The analysis procedure, known as the

pseudo-static method, involves simulating the ground motions as static horizontal forces

acting in a direction out of the face. The magnitude of this force is the product of a

seismic coefficient Hk and the weight of the sliding block w . The value of Hk may be

taken as equal to the design peak ground acceleration (PGA), which is expressed as a

fraction of the gravity acceleration (i.e. Hk = 0.1 if the PGA is 10% of gravity).

However, this is a conservative assumption, the actual transient ground motion is

controlled by constant forces acting on the entire design of the slope.

In the design of soil slopes, it is common that Hk is fraction of the PGA, provided

that there is no loss of shear strength during cyclic loading (Seed 1979; Pyke 1999).

Study of slopes using Newmark analysis with a yield acceleration yk equal to 50% of

the PGA (i.e. gaky /5.0 max ) showed that permanent seismic displacement would be less

than 1m (Hynes-Griffin and Franklin 1984). Based on these studies, the California

Department of Mines and Geology (CDMG 1997) suggests that it is reasonable to use a

value of Hk equal to 50% of the design PGA, in combination with a pseudo-static FoS of

1.0 -1.2. With respect to rock slopes where the rock mass contains no distinct sliding

surface and some movement can be tolerated, it may be reasonable to use the CDMG

procedure to determine a value for Hk . However, for rock slopes there are two

conditions for which it may be advisable to use Hk values somewhat greater than 0.5

times the PGA. First, where the slope contains a distinct sliding surface for which there

is likely to be a significant decrease in shear strength with limited displacement; sliding

planes on which the strength would be sensitive to movement include smooth, planar

joints or bedding planes with no infilling. Second, where the slope is a topographic high

point and some amplification of the ground motions may be expected. In critical



2.37

situations, it may also be advisable to check the sensitivity of the slope to seismic

deformations using Newmark analysis. The FoS of a plane failure using the pseudo-

static method is given by modifying as follows (assuming the slope is drained):

)cos(sin

tan))sin(cos(

pHp

pHp

kW

kWcAFoS

(2.17)

The equation demonstrates that the effect of the horizontal force is to diminish the FoS

because the shear resistance is reduced and the displacing force is increased.

Under circumstances where it is considered that the vertical component of the

ground motion will be in phase with, and have the same frequency, as the horizontal

component, it may be appropriate to use both horizontal and vertical seismic

coefficients in stability analysis. If the vertical coefficient is Vk and the ratio of the

vertical to the horizontal components is Kr (i.e. HVK kkr / ), then the resultant seismic

coefficient Tr is

2/12 )1( kHT rkr (2.18)

Acting at an angle )/( HVk kkatn above the horizontal, and FoS is given by

))cos((sin

tan)))sin((cos(

kpTp

kpTp

kW

kWcAFoS

(2.19)

Study of the effect of the vertical component on the FoS has shown that incorporating

the vertical component will not change the FoS by more than about 10%, provided that

HV kk ((NHI) 1998). Furthermore, Eq. (2.19) will only apply when the vertical and

horizontal components are exactly in phase. Based on these results, it may be acceptable

to ignore the vertical component of the ground motion.

A model presented by Chang et al. (1984) developed for the evaluation of the

critical condition and the subsequent response to earthquakes, is here represented. The

model applies the pseudo-static limit equilibrium analysis for the determination of the

critical condition of the slope and the Newmark analytical procedure to assess the

displacement of the rigid block.

The computation procedure of the model can be subdivided into the following

steps:



2.38

1). A designed earthquake is selected and the seismic accelerations,gk are determined

for all the time steps considered during the earthquake shaking. The choice of the

constant time interval of 0.001s was suggested by the model authors to designate the

time and to subsequently estimate all the corresponding accelerations. The acceleration

within a time interval is assumed to be linear, but not necessarily constant.

2). The motion acceleration 1x acting on the sliding block can be calculated for the

case of a block on a plane (where Ckk and 1x ) as described in Fig.2.13.

Figure 2.13 Equilibrium of forces on a sliding block (Chang et al. 1984).

According to equilibrium equations of forces on the block, Newton second law

can be employed to express the seismic force as

xg

WF 1

1 (2.20)

where

2cos

)cos()(

gkkx ci .

3). By using the results obtained in step 2 and starting from the beginning of the

seismic event, the first positive motion accelerationix , which corresponds to the

starting of the sliding motion at the timeit , is determined. If

ix is the first positive

motion acceleration, then 1ix at time

1it must be negative, except that particular case in

which 01 ix . Time t , at which x =0 must then be computed. The motion velocity x

will start to increase from zero from this time. By linear interpolation one obtains:

11

11 )(

i

ii

iii txx

ttxt

(2.21)

(a) (b)

1W



2.39

At the time in which the acceleration induced by the seismic event exceeds the yield

acceleration, the sliding block velocity increases from zero and the motion displacement

occurs.

4). The motion velocity ix , at the time t , can be computed by assuming a linear

variation of the acceleration as )(2/ ttxx i . By knowing ix , the motion velocity

1ix

can similarly be calculated as:

]2/))([( 111 iiiiii ttxxxx (2.22)

The value of 1ix , in this equation, is obtained from Eq. (2.21). All the velocities, in the

selected time, can be calculated by using the same procedure. The resistance to the

uphill movement can be assumed as being indefinitely large without causing serious

errors, as Newmark pointed out. The time 2nt can be expressed as:

112

1212

)(

iii

iiin t

xx

ttxt

(2.23)

Two non-consecutive displacements can be computed during the time which passes

from 1it to 2it . However, the calculations for these two separated displacements

are required only in the case in which the motion acceleration is negative at the time

1it and positive at the time 2it . Otherwise, only the time 1nt (i.e.

2211 inni tttt )

will be required and the movement will cease at time 2it .

5). The displacement 1ix , between time

it and time 1it can be calculated as:

6/]))(2[()( 211111 iiiiiiiii ttxxttxxx (2.24)

Thus the block overall displacement can be determined to all the times of the seismic

effect.

2.6.2 Newmark method

When a rock slope is subject to seismic shaking, failure does not necessarily occur

when the dynamic transient stress reaches the shear strength of the rock. Furthermore, if

the FoS on a potential sliding surface drops below one at some time during the ground

motion it does not necessarily imply a serious problem. What really matters is that



2.40

magnitude of permanent displacement caused at the times that the FoS is less than 1.0

(Lin and Whitman 1986). The permanent displacement of rock and soil slopes as the

result of earthquake motions can be calculated using a method developed by Newmark

(1965). This is a more realistic method of analysing seismic effects on rock slopes than

the pseudo-static method of analysis.

Figure 2.14 Displacement of rigid block on rigid base (Newmark 1965): (a) block on moving base;

(b) acceleration plot; (c) velocity plot.

The principle of Newmark’s method is illustrated in Fig. 2.14. It is assumed that

the potential sliding block is a rigid body on a yielding base. Displacement of a block

occurs when the base is subjected to an uniform horizontal acceleration pulse of

magnitude ag with duration 0t . The velocity of block is a function of the time t and is

designated )(ty , and its velocity at time t is y .Assuming a frictional contact between

the block and the base, the velocity of the block will be x , and the relative velocity

between the block and the base will be u where yxu .

The resistance to motion is accounted for by the inertia of the block. The maximum

force that can be used to accelerate the block is the shearing resistance on the base of

the block, which has a friction angle . This limiting force is proportional to the weight

of the block W and magnitude of tanW , corresponding to a yield acceleration ya of

tang , as shown in Fig. 2.13(b) by the dashed line on the acceleration plot. The shaded

area shows that the ground acceleration pulse exceeds the acceleration of the block,

resulting in slippage. Fig. 2.13(c) shows the velocity as a function of time for both the

ground and the block accelerating forces. The maximum velocity for the ground

accelerating force has a magnitude v , which remains constant after an elapsed time of

Time

Acceleration Velocity

Time (b) (c)(a)



2.41

0t . The magnitude of the ground velocity gv is given by

0agtvg , while the velocity of

the block bv is tangtvb . After time mt , the two velocities are equal and the block

comes to rest with respect to the base, the relative velocity, 0u . The value of mt is

calculated by equating the ground velocity to the velocity of the block to give the

following expression for the time mt :

tang

vt b

m (2.25)

The displacement m of the block relative to the ground at time

mt is obtained by

computing the area of the shaded region as follows:

)tan

1(tan22

1

2

1 2

0

ag

vvtvtmm

(2.26)

Eq. (2.26) gives the displacement of the block in response to a single acceleration pulse

0t and ag that exceeds the yield acceleration tang , assuming infinite ground

displacement. The equation shows that the displacement is proportional to the square of

the ground velocity. While Eq. (2.26) applies to a block on a horizontal plane, a block

on a sloping plane will slip at a lower yield acceleration and show greater displacement,

depending on the direction of the acceleration pulse. For a cohesionless surface where

the FoS of the block is equal to (p tan/tan ) and the applied acceleration is horizontal,

Newmark shows that the yield acceleration ya can be given as

py gFoSa sin)1( (2.27)

where is the friction angle of sliding surface, and p is the dip angle of this surface.

Note that for 0p , tangay . Also Eq. (2.27) shows that for a block on a sloping

surface, the yield acceleration is higher when the acceleration pulse is in the down-dip

direction compared to the pulse in the up-dip direction.

The displacement of a block on an inclined plane can be calculated by combining

Eqs. (2.26) and (2.27) as follows:

)1(2

)( 2

a

a

ga

agt y

ym (2.28)



2.42

In an actual earthquake, the pulse would be followed by a number of pulses of

carrying magnitude, some positive and some negative, which will produce a series of

displacement pulses. This method of displacement analysis can be applied to the case of

a transient sinusoidal acceleration gta )( illustrated in Fig. 8.6(Goodman and Seed 1966)

. If during some period of the acceleration pulse the shear stress on the sliding surface

exceeds the shear strength, displacement will take place. Displacement will take place

more readily in a downslope direction; this is illustrated in Fig. 2.14 where the shaded

areas are the portion of each pulse in which movement takes place. For the conditions

illustrated in Fig. 2.15, it is assumed that the yield acceleration diminishes with

displacement, that is, 321 yyy aaa due to shearing of the asperities in the manner.

Figure 2.15 Integration of accelerograms to determine block movement (Goodman and Seed 1966).

Integration of the yield portions of the acceleration pulses gives the velocity of the

block. It will start to move at time 1t when the yield acceleration is exceeded, and the

velocity will increase up to time 2t when the acceleration drops to zero at time 3t as the

acceleration direction begins to change from down slope to up slope. Integration of the

velocity pulses gives the displacement of the block, with the duration of each

displacement pulse being (13 tt ). The simple displacement models shown in Fig. 2.14

has since been developed to more accurate model displacement due to actual earthquake



2.43

motions, with much of this work being related to rock slopes (Jibson 1993; Jibson et al.

1998).

The Newmark method transcends the LEM to provide an estimate the displacement

of a landslide block subjected to seismic motion. On this method, that portion of the

design accelerogram above a critical acceleration ca , is intergrated twice to obtain the

displacement, the critical acceleration is defined using Eq. (2.29)

sin)1( gFoSac (2.29)

where ca is critical acceleration, in meters per-second squared, FoS is static factor for

safety and is slope angle, respectively. A rigorous Newmark analysis involves

subtracting ca , from the accelerogram and integrating the difference twice to compute

the total displacement. When the earthquake acceleration does not overcome the initial

limit equilibrium of the slope, and the rigorous Newmark analysis predicts no slope

displacement. In Newmark’s approximation, the total slope displacementnD , is a

function of the peak particle velocity (ppv) and the critical acceleration ca , as shown in

Eq. (2.30)

),6(2

2

Maxa

ppvD

cn (2.30)

where ppv is peak partical velocity and repsents a dimensionless constant equal to the

duration of strong motion, respectively.

Another approximation to the Newmark method by mapping landslide hazard in

southern California is presented by Jibson et al. (1998). The development of this

approximation was motivated by the difficulty of using a rigorous Newmark approach

within the geographic information system (GIS) framework, commonly adopted for

regional hazard mapping. The Arias intensity has recently found use in representing

earthquake shaking likely to cause landslides. It is given by

dttag

I a

0

2)]([2

(2.31)



2.44

where aI is the Arias intensity and )(ta is the ground acceleration as a function of time,

respectively.

2.6.3 Numerical methods

With respect to the numerical methods, Zhang et al. (1997) carried out studies on

the dynamic behaviour of a 120-m high rock slope and a blocky arch structure of the

Three Gorges Shiplock using DEM, which demonstrate that the effects of different

seismic input and parameters on the dynamic response behaviours and failure mode of

the slope are significant. Hatzor et al. (2004) carried out dynamic two-dimensional

stability analysis of a highly discontinuous rock slope, which is upper terrace of King

Herod’s Palace in Masada using a fully dynamic version of DDA where a 2% kinetic

damping is introduced to predict the realistic damage of the slope. Bhasin and Kaynia

(2004) performed static and dynamic rock slope stability analyses for a 700-m high rock

slope in western Norway using a numerical discontinuum modelling technique, in which

three cases have been simulated for predicting the behaviour of the rock slope under

existing environmental and earthquake conditions. Liu et al. (2004) studied the dynamic

response of Huangmailin Phosphorite rock slope in China under explosion using

DEM/UDEC and compared with field measurements, the numerical results show that

UDEC is efficient to simulate the dynamic response of jointed rock slope. Crosta et al.

(2007) performed 3D dynamic analysis of the thurwieser rock Avalanche, Italian Alps,

where the propagation of rock avalanche and runout were studied. Latha and Garaga

(2010) carried out a comprehensive study on seismic slope stability of a natural slope in

jointed rock mass using FLAC (Itasca 2002) through a case study in the Himalayan

region of India. Wu (2010) and Wu et al. (2009) carried out a seismic landslide

simulation in DDA and dynamic discrete analysis of an earthquake-induced large-scale

landslide, respectively. In the simulations, three available algorithms incorporate

seismic impacts into DDA simulations for earthquake-induced slope failure are

investigated. Chiu-fen-erh-shan landslide, triggered by 6.7wM Taiwan Chi-Chi

earthquake is studied using the DDA, in which the main objectives of the study were to

investigate if it was possible to numerically model the landslide progression, including

slope disintegration, and to reproduce the post-failure configuration. Miki et al.(2010)



2.45

studied earthquake response analysis of rock slope using the coupled DDA-NMM, the

simulated results of the earthquake response analysis indicated that the failure modes

and travelling distance of the collapsed rock blocks are affected by the joint strength

between blocks significantly. An et al. (2012) investigated the seismic stability of rock

slope using the NMM, in which the validity of the NMM in predicting the ground

acceleration induced permanent displacement is verified by comparing its results with

the analytical solutions and the Newmark- numerical integration solutions. Ning et al.

(2012) studied rock fall of earthquake-induced failure with pre-existing non-persistent

joints located on the crest of a rock slope, in which the failure of rock mass is modelled

by the NMM coupled with a fracturing algorithm based on the Mohr-Coulomb criterion,

and DDA model based on a strain and kinematic energy conservation transition

technique, respectively.



3.46

CHAPTER 3. THEORY OF THE NUMERICAL MANIFOLD

METHOD AND ITS INTEGRATION SCHEMES

3.1 INTRODUCTION

Since the birth of the NMM in 1991 (Shi 1991), its proposed concepts and

perspectives are distinct from the definitions of traditional numerical method. The

NMM employs a dual cover system to represent physical domain, which increases its

difficulty in terms of understanding and further development to some extent. To

recognize these distinct characteristics of the NMM clearly, the finite cover system, the

cover-based contact algorithm and integration scheme are introduced in this chapter.

The basic concept and constructed finite cover system are firstly illustrated in Section

3.2. Then, the integration schemes based on the cover system are explored with respect

to the spatial and temporal domain in Section 3.3 as well.

3.2 FUNDAMENTALS OF THE NMM

NMM is one of the newest developed numerical methods in recent decades. It

merges the FEM and DDA nicely and thus it reconciles the Continuum-based methods

with Discontinuum-based methods. The discretization involves two domains: Physical

domain is used to describe geometry property and Mathematical domain is used to build

global displacement function. The mathematical treatment to these two domains is

relatively independent, but these two domains interact with each other according to

special integration and interpolation scheme. Because its geometrical model can be

prepared independently without considering the meshing issue, NMM becomes very

suitable for the numerical simulations of crushing-like material (rock mass) or relevant

complex engineering applications.

As present in Section 2.2, mathematical cover (MC) is the essential component in

the mathematical domain. Each MC possesses the separate cover function and been

represented on its nodal star. The nodal stars are connected by weight function in a



3.47

mathematical sense to describe the mathematical domain. As a result, the global

displacement function is generated by the weighted average of local independent cover

functions on the common part of corresponding MCs. Physical cover (PC) is generated

as an inter-media between the physical domain and the corresponding MC. The PCs are

overlapping to each other by the nodal stars. Manifold element (ME) is the final product

following the PCs of the Physical domain. It includes the geometrical information

mapping back to the corresponding PCs. In a sense, it is equivalent to the integration

field of element in FEM.

3.2.1 Finite cover system in the NMM

The finite cover system is defined as a combination of the mathematical domain

and physical domain. The mathematical mesh defines the fine or rough approximation

of unknown functions, which is used to build MCs that present small regions of the

whole field and can be any of shape and size. They can overlap each other and do not

need to coincide with the PC as long as they are large enough to cover the physical

domain.

To visualize these concepts, an example illustrated in Fig. 3.1 is used. There are

two MCs in total, a regular hexagon and a circle . The thick lines define the

physical domain ]2,1[ . Intersected with the physical domain, and . are divided,

respectively, into two PCs, i.e. , and , , as shown in Fig. 3.1(b). Here, notation

represents the jth PC generated from the Ith MC.

On each MC , a weight function is defined, which satisfies

Ii

Ii

Mxx

MxCCx

,0)(

),10()( 00

(3.1)

With

J

JMifx

J x 1)( (3.2)

Eq. (3.1) indicates that the weight function has non-zero value only on its

corresponding MC, but zero otherwise, whereas Eq. (3.2) is just the partition of unity

property to assure a conforming approximation. The weight function )(xi associated



3.48

with IM will be accordingly transferred to

iP , any of the PCs ][ jiP in

IM , which is

expressed as )(xi on iP hereafter.

Figure 3.1 A schematic of basic concepts in the NMM. (a) The physical domain and two MCs; (b) Overlapping of MCs and physical domain; (c) Corresponding PCs; (d) Six corresponding MEs.

So far, each MC is associated with several PCs, and each PC has two indices, i.e. I

and j. Considering the simple example, for instance, the four PCs can be renumbered by

1 11 , , and in light of different MCs. These four PCs

finally form six MEs as shown in Fig. 3.1(d), i.e. )( ]1[11 PEE , ),( ]2[

1]1[

12 PPEE ,

)( ]2[13 PEE , )( ]1[

24 PEE , ),( ]2[2

]1[25 PPEE and )( ]2[

26 PEE .

(c)

Overlapping of MCs and

physical domain

Physical domainMCs

PC PC

PC

(a)

(b)

(d)

PC



3.49

With these concepts, the interpolation approximation can be constructed. First, a

cover function )(xui is defined individually as a local approximation on the PC

iP for

the displacement field, which can be constant, linear, high order polynomials or other

functions with unknowns (also termed DOFs) to be determined. Then, the global

displacement )(xu on a certain CE e is approximated to be

i

iPe

ii xuxxu )()()( (3.3)

Here, we give another example as shown in Fig. 3.2(a), the mathematical cover

system, which is united by six rectangle patches denoted by , , , , and

respectively. The overlapping patches cover the whole material domain without

considering any physical properties, so any arbitrary shape of mathematical cover can

be chosen. And then, physical covers can be obtained from these mathematical covers

intersect with the physical domain , a manifold element can be produced as the

common area of physical covers. Each small rectangle patch is termed as a

mathematical cover (MC), denoted by iM (i= 1, 2, 3, …, 6). External boundary and

internal joints or cracks may intersect one MC into several separate sub-patches, then

each one within the material domain is termed as a physical cover (PC), denoted by jiP (

Nj ). As can be seen in Fig. 3.2(b), material domain is intersected by patch to

generate one PC within its material domain, denoted by 11P . When the internal

discontinuities (i.e. cracks or joints) are taken into accounted in the NMM, each

discontinuous boundary is considered as one special material domain to form a new PC.

If the crack passes through the whole patch within the material domain, two isolated

PCs form by the crack surface just as 4M and

6M , two separated PCs, denoted by 14P ,

24P based on

4M and 16P , 2

6P based on 6M , respectively. On the other case, when the

crack cuts MC partially, only one PC forms within the material domain, which can be

seen by 2M ,

3M and 5M , only one PC generates denoted by 1

2P , 13P and 1

5P respectively.

Furthermore, the common area of several overlapping PCs is termed as a manifold

element (ME).



3.50

(a) General cover system in the NMM;

(b) Generation of physical covers for the NMM.

Figure 3.2 The cover system in the NMM: (a) General cover system; (b) Generation of physical

covers.

For a two-dimensional problem, the MCs can be generated from a regularly-

patterned triangular finite element mesh or a regularly-patterned rectangular finite

element mesh. Fig. 3.3 shows a NMM model for discontinuity problem, we can use a

finite element mesh composed of equilateral triangles to construct the MCs. Each node

in the triangular finite element mesh is termed as a nodal star. The union of six triangles

sharing a common star forms a hexagonal MC. When one MC is interacted by two

physical domains, to simplify procedure for construction of PCs and MEs, the MC will

24P

15P



3.51

be divided into a pair MCs automatically with neighboured indices, i.e. star 17 and 18,

25 and 26, 27 and 28, 29 and 30, 33 and 34, 35 and 36, 40 and 41, respectively.

Figure 3.3 NMM model for the discontinuity problem.

Here, each physical cover has two indices, i.e. I and j, in which I and j are the index

number of the MCs and physical domains (2 domains in the example), respectively. To

simplify an implementation, a reallocated single index to each PC can be referred,

which can be expressed as ≜ with i calculated by

jmjIiI

ll

1

1

),( (3.4)

Continuing the above discussions, each PC is assigned a local function. In the

NMM, the local functions are usually taken as the polynomials as

iT

i xpxu )()( (3.5)

where i is an array of constant coefficients, and )(xpT is the matrix of polynomial bases

as

pppp

ppppT

yxyyxxyx

yxyyxxyxxp

00000010

00000001)(

11

11

(3.6)

Though the polynomials can approximate smooth functions well, for strong

singularity problems, the smooth basis polynomial local approximations cannot

Nodal star

1 PC

2 PCs

MC



3.52

properly capture the high gradient solutions. Therefore, various singular functions may

need to be used to enrich the approximation space for capturing the singularities without

refining the meshes.

Consider the NMM model in Fig. 3.3 for an example. The common area of MCs

and the two physical domains forms total 42 PCs, denoted as )42,2,1( iPi, where the

subscript represents the number of the MCs. Then, the overlapping of the PCs and finite

element mesh (represents the MCs) generates MEs, denoted as )42,2,1( iE ji

, where the

subscript represents the number of the associated PCs and superscript represents the cut

element number by the associated finite element mesh. Here, we choose three typical

nodal stars, marked as 9, 25 and 26, 35 and 36, respectively, to illustrate the

construction of the MEs based on the finite cover system. Taking stars 25 and 26 for

instance, two complete overlapped MCs, 25M and

26M form two associated PCs, 25P and

26P , respectively. The intersection of the PCs and finite element mesh generates 9 MEs,

denoted as 125E , 2

25E , 325E , 4

25E , 525E , 6

25E , 725E and 1

26E , 226E , respectively. The other two

constructions of MEs and the associated PCs can be obtained in a similar way, as

illustrated in Fig. 3.4. Then, the NMM adopts the partition of unity functions to paste all

the local functions together to give the global approximation.



3.53

Figure 3.4 Construction of finite cover system in the NMM.

As shown in Fig.3.5, the structured mesh-based cover system is built on the

triangular finite element mesh, in which each node is termed as a star. The union of six

triangles sharing a common star forms a hexagonal MC. Each PC coincides with the

corresponding MC and physical domain of the block, thus each MC generates one or

more PCs, in which a local function is assigned to each PC, such as that is allocated an

index of ① generates a PC denoted by 11P , ② created two PCs for the joint in the block

denoted by 12P and 2

2P , respectively. Similarly, following the criterion we have indices of

④, ⑤ and ⑥ generate one PC denoted by 14P , 1

5P and 16P ; ③ generates two PCs denoted

by 13P and 2

3P , respectively. Each triangular element e is constructed by the associated

three PCs starred at its three nodes.



3.54

its three nodes.

Figure 3.5 Construction of PCs on the cover system.

When internal discontinuities such as joints and cracks present in the physical

domain as shown in Fig. 3.6, the structured mesh-based cover system is built on the

triangular finite element mesh, in which each node is termed as a star. The union of six

triangles sharing a common star forms a hexagonal MC. To the continuous media, each

PC coincides with the corresponding MC, thus each MC generates a PC, in which a

local function is assigned. Each triangular element e is constructed by the associated

three PCs starred at its three nodes. When the discontinuities (i.e. ① and ②) are taken

into account in the problem domain, a MC can be sub-divided two and more PCs shared

the original star (i.e. 2 PCs and 4 PCs). If one MC is not or partly cut by the

discontinuities, only one PC is constructed. In this case, we usually apply refining mesh

technique and cutting off the discontinuity tips by the element edges to avoid singular

matrices occurrence at utmost extend. Since each cover has two degrees of freedom,

thus each element formed by the overlapping of the three PCs has six degrees of

freedom. For the discontinuities and physical boundaries are considered in the cover

system, many new generated PCs will be reallocated updated indices to give the global

approximations.

MC

①

②

②

①

③

④

⑤

⑥

block

MC Overlap PCs Joint

1 PC

2 PCs



3.55

Figure 3.6 Structured mesh-based cover system in the NMM.

Here, we give a simple example to illustrate the constructions of the PCs and MEs

on the cover system as plotted in Fig. 3.7. To the continuum as shown in left of the

figure, two interconnected MEs ie and je sharing two PCs indexed )2( and )3( , in

which ie is constructed by the associated three PCs indexed by )1( , )2( and )3( , and je

is constructed by the PCs indexed by )2( , )4( and )3( , accordingly. Since the global

approximation involves the assembling of a global stiffness matrix, the interactions

between ie and je can be offset in the implicit time integration of the NMM. On the

other hand, when a discontinuity passes through the PCs coving the MEs as presented in

right of the figure, each MC is divided into two PCs and each ME is separated into two

MEs subsequently. A formula of single index is used to reallocate the new generated PC

and the new generated MEs are rerecorded as follows: 1ie is built by )1( , )3( and )5( ; 2

ie

is built by )2( , )4( and )6( ; 1je is built by )3( , )7( and )5( ; 2

je is built by )4( , )8( and )6( ;

respectively. The contacts between MEs 1re and 2

re ( jir , ) occur and the interactions

between the MEs are calculated by adding and removing the linear stiff springs on

contact points in the normal and shear directions.

① ②

①, ②: discontinuities

1 PC

2 PCs 4 PCs

1 PC

①

①

②

①



3.56

(a) continuous elements;

(b) discontinuous elements.

Figure 3.7 Construction of manifold elements on the cover system: (a) continuous elements; (b)

discontinuous elements.

On each ME, the NMM adopts the partition of unity functions to paste the three

associated local functions together to give the global approximation as

),(

),(),(

).(

).( 3

1 yxv

yxuyxw

yxv

yxu

i

i

ii

e

e (3.7)

Where ),( yxwi is the weight function defined on the associated PC iP with the

expression as

ii

ii

Pyxyxw

Pyxyxw

),(,0),(

),(,0),( (3.8)

The weight function is a partition of unity and satisfies

discontinuity



3.57

ePyx

i yxyxwi

),(,1),(),(

(3.9)

where e is the problem domain. When the constant local function are considered, the

global displacement on each manifold element can be rewritten as

)3(

)2(

)(

)3()2()1( ),(),(),(),(),(

),(

e

e

ie

eeeeee

e

D

D

D

yxTyxTyxTDyxTyxv

yxu (3.10)

in which

),(0

0),(),(

)(

)(

)( yxw

yxwyxT

ie

ie

ie and 3,2,1,

),(

),(

)(

)(

)(

iyxv

yxuD

ie

ie

ie.

3.2.2 Contact algorithm

Contact algorithm is a requisite tool for further developing the numerical

simulation technique. The NMM aims at the discontinuous problems, even with

movements. When intersected with physical features like cracks and material interfaces,

each MC forms several independent PCs associated with different local functions. Thus,

the adjacent MEs formed by these PCs are independent on each other in the framework

of NMM. An example is the problem with discrete bodies, in which the displacement

across each body boundary is discontinuous; however, one body cannot penetrate into

another body. Such constraints are normally termed as non-penetration or unilateral

condition, and attributed to a contact problem in physics.

Since the frictional contact problems are inherently nonlinear and irreversible, for

the sake of generality, an incremental approach is adopted in the NMM. It is assumed

that:

The contact state at the initiation of the time step, ntt , is known and the contact

state at 1 ntt , the end of the step, after time interval

nn ttt 1 is to be solved;

The time incremental for each time step is chosen small enough so that the

displacements of all the contact points within the problem domain are less than a

predefined maximum displacement limit .

In the NMM, contact detection at the beginning of each time step can be seperated

into two steps:



3.58

A global search is first done to find the bodies which might possible come into

contact by representing the bodies by bounding boxes, i.e. if the distance between

the bounding boxes of any two bodies is less than 2, they might contact within

this time step;

A local search is then conducted to figure out all the vertex-edge contact pairs.

There are generally three types of contacts: angle-to-edge, angle-to-angle and edge-

to-edge, as shown in Fig. 3.8. Among them, edge-to-edge contact can be treated as two

angle-to-edge contacts. The contact will be of angle-to-angle if both of the following

conditions are satisfied: (1) minimum distance of the two angle vertices is less than 2;

(2) the maximum overlapping angle of the two angles is less than 2 when one angle

vertex translates to the vertex of the other angle without rotation. The contact will be of

angle-to-edge if both of the following conditions are satisfied: (1) minimum distance

between the vertex and the edge is less than 2; (2) maximum overlapping angle of the

angle and the edge is less than 2 when the angle vertex translates to the edge without

rotation.

(a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge.

Figure 3.8 Three types of contacts: (a) angle-to-angle; (b) angle-to-edge; (c) edge-to-edge.

For each contact pair, there are three possible contact modes: open, sliding and

sticking. At the beginning of each time step, the contact modes for all contact pairs are

assumed as sticking except those contact pairs inherited from the last time step. For a

sticking contact pair, a normal spring is applied to push the vertex away from the

entrance line in the normal direction and another shear spring is applied to avoid the

tangential displacement between the vertex and the entrance line.

1P 2P 3P 2P

3P 1P

4P



3.59

To calculate the penetration, an entrance distance nd , as shown in Fig. 3.9, is

introduced, which is defined as the distance from the vertex 1P of the angle to the

entrance line 32 PP at the end of the step. Contacts are pre-determined according to the

entrance distance. On this basis, the stiff springs are applied to push the vertex 1P away

from32 PP .

Assume 1P is the vertex, 32 PP is the entrance line and ),( qq yx and ),( qq vu are the

coordinates and displacement of )3,2,1( qPq, respectively; point 1P is in element i, and

points 2P , 3P are in element j, stiffness of the normal and shear springs are

nk and sk . If

three points 1P , 2P , 3P rotate anticlockwise, then the normal distance

nd from vertex 1P

to 32 PP is

3333

2222

1111

1

1

11

vyux

vyux

vyux

ldn

(3.11)

where l is the distance between points 2P and 3P .

Figure 3.9 Entrance distance nd between a vertex and its entrance line.

Since the contact distance nd is such small, the second-order infinite small terms in

Eq. (3.7) can be omitted. The equation can be re-expressed as

l

SDGDHd j

Ti

Tn

0)()( (3.12)

where

1P

2P 3P 0P nd

i

j



3.60

33

22

11

0

1

1

1

yx

yx

yx

S (3.13)

23

3211)( ),(

1

xx

yyyxT

lH T

i (3.14)

12

2133)(

31

1322)( ),(

1),(

1

xx

yyyxT

lxx

yyyxT

lG T

jT

j (3.15)

The potential energy due to the normal spring stiffness nk can be expressed as

2

0)(

0)(

0

)()()()()()(

2

0)()(

2

22

22

2

2

l

SGD

l

SHD

l

S

DGHDDGGDDHHDk

l

SDGDH

k

dk

Tj

Ti

jTT

ijTT

jiTT

in

jT

iTn

nn

n

(3.16)

Thus, the sub-matrices due to the normal spring can be obtained and assembled

into global stiffness matrix and global loading vector as

3,2,1,

3,2,1,

3,2,1,

3,2,1,

)()()()(

)()()()(

)()()()(

)()()()(

srKGGk

srKHGk

srKGHk

srKHHk

sjrjT

sjrjn

sirjT

sirjn

sjriT

sjrin

siriT

sirin

(3.17)

3,2,1

3,2,1

)()(0

)()(0

rFGl

Sk

rFHl

Sk

rjrjn

ririn (3.18)

The point ),( 000 yxP is the projection point of vertex 1P on entrance line 32 PP , which

is also the hypothetical contact points, with its coordinates given as

30200

30200

)1(

)1(

ytyty

xtxtx

(3.19)

The shear displacement of point 1P within the current step can be expressed as

2233

2233001100113210 )(

11

vyvy

uxuxvyvyuxux

lPPPP

lds (3.20)



3.61

Omitting the second-order infinite small terms, Eq. (3.20) can be rewritten as

l

SDGDHd j

T

i

T

s0

)()(

~~~

(3.21)

where

23

2301010

~yy

xxyyxxS (3.22)

23

2311)( ),(

1~yy

xxyxT

lH T

i

30201

3020133)(

30201

3020122)(

32

3200)(

2)12(

2)12(),(

1

)21()1(2

)21()1(2),(

1

),(1~

ytyty

xtxtxyxT

lytyty

xtxtxyxT

l

yy

xxyxT

lG

Tj

Tj

Tj (3.23)

The potential energy induced by the shear spring sk can be obtained as

2

0)(

0)(

0

)()()()()()(

2

0)()(

2

~~

~2

~~

2

~~2

~~~~2

~~~

2

2

l

SGD

l

SHD

l

S

DGHDDGGDDHHDk

l

SDGDH

k

dk

Tj

Ti

j

TTij

TTji

TTi

s

j

T

i

Ts

ss

n

(3.24)

Then, the sub-matrices due to the shear spring are gained and assembled to the

global stiffness matrix and global loading vector as

3,2,1,

~~

3,2,1,~~

3,2,1,~~

3,2,1,~~

)()()()(

)()()()(

)()()()(

)()()()(

srKGGk

srKHGk

srKGHk

srKHHk

sjrj

T

sjrjs

sirj

T

sirjs

sjri

T

sjris

siri

T

siris

(3.25)

3,2,1~

~

3,2,1~

~

)()(0

)()(0

rFGl

Sk

rFHl

Sk

rjrjs

riris (3.26)

For the sliding case, besides the normal spring, a pair of frictional forces instead of

a shear spring will be implemented. Based on the Coulomb’s friction law, the frictional

force can be calculated as



3.62

tan nn dksignF (3.27)

where ‘sign’ is assigned as ‘+’ or ‘-’ according to the direction of relative sliding, nndk

is the normal force and is the friction angle, respectively.

Then, the potential energy due to frictional force F on the element i is

HDFyy

xxvu

l

F Tif

ˆ)(

23

2311

(3.28)

where

23

2311)( ),(

1ˆyy

xxyxT

lH T

i (3.29)

The potential energy due to frictional force F on the element j is

GDFyy

xxvu

l

F Tjf

ˆ)(

23

2300

(3.30)

where

23

2300)( ),(

1ˆyy

xxyxT

lG T

j (3.31)

Then, the sub-matrix due to the frictional force are obtained and assembled to the

global loading vector as

3,2,1ˆ

3,2,1ˆ

)()(

)()(

sFGF

rFHF

sjsj

riri (3.32)

After the load increment of the current time step is applied, the equilibrium

equation is solved and the state quantities such as displacements, stresses, etc, are

obtained.

3.3 INTEGRATION SCHEMES IN THE NMM

In terms of spatial aspect in the NMM, MCs are usually formed by a regularly-

patterned triangular or rectangular finite element mesh, the MEs may have arbitrary

shapes because of the intersection with the external boundaries and/or the internal

discontinuities. Thus, direction integration scheme over the whole element like that in

the conventional FEM is not applicable. Simplex integration method can be adopted to

evaluate the integrations over an arbitrary ME. On the other hand, in the temporal aspect,



3.63

the discrete equations are based on a dynamic analysis. For a static analysis, discrete

equations can be given by removing the inertia item from the governing equation. In the

NMM, the Newmark time integration scheme is used to solve the governing equation by

single time step.

3.3.1 Simplex integration

In both the ph cloud method and the meshless method, the integration is carried

out using Gauss quadrature. In the NMM, integrations are evaluated analytically. The

domain of integration is represented as a simplicial complex via a simplicial chain

regardless of the convexity of the domain. Analytical integration maybe carried out

using the generalized Stoke’s equation (Flanders 2012). Shi (1996b), on the other hand,

suggests direct integration over each simplex in the chain, because many functions, in

particular polynomials, can be integrated analytically on a simplex. With a coordinate

transformation, analytical results can be obtained for integration over an arbitrary n-

dimensional Euclidean space.

This scheme is explained with an 2R example. A physical domain ),,,,( mlkji and a

triangulation are shown in Fig. 3.10. For both, the oriented boundaries are considered of

the same sequence ordered edges: ),(),(),(),(),( immllkkjji . A boundary preserved

triangulation can easily be achieved by connecting each pair to one single point. The

coordinate origin, )0,0(o is a desirable choice of the three ordered vertices is a two-

simplex by Li et al. (2005) .

With a simplex chain representation of a ME e , the integration becomes a sum of

simplex integration as follows

iS ie

e

Eie

dKdKdKA (3.33)

Each of the integration on a simplex can be evaluated analytically. This is carried

out in two steps. First, an integration in terms of area coordinates, 1L , 2L and 3L , over a

coordinate simplex, i.e., simplex with vertices )0,0(0U , )0,1(1U , )1,0(2U , can be evaluated

analytically. Namely,



3.64

)!2(

!!!

210

21021210

21

210

0

nnn

nnnLdLLLL nn

UUU

n (3.34)

Second, a general integration in terms of x and y over a simples with vertices

),( 00 yxo , ),( 11 yxi and ),( 22 yxj is evaluated by the coordinate transformation

221100

221100

2101

LyLyLyy

LxLxLxx

LLL (3.35)

Such that

21221100221100 )()()( LdLLyLyLyLxLxLxJsigndxdyyx ba

oij

ba (3.36)

where, sign(J) is a signal Jacobian. The more details on the simplex integration method

can be found in Shi (1996b).

Figure 3.10 Triangulate an element oij using coherent orientation.

3.3.2 Time integration

In order to investigate the time integration in the NMM, the behaviour of an

equivalent linear system is considered. In such a system, the minimization of the system

potential energy will produce an equation of motion, which is similar to that in the

FEM.

When a single step integration method is used to solve Eq. (2.1) for given initial

conditions of 00 dD and

00 VD based on Newmark method as expressed in sub-section

2.5.1 , taking the parameters of 2/1 and 1 , Eq. (2.1) can be simplified as

j

i

k

m

l

o

i

j

o

o - Coordinate origin



3.65

FDK

(3.37)

where nn DDD 1,

Kt

MK 2

2 and

nn D

t

MFF

21

, respectively.

3.4 SUMMARY

This chapter presents a brief review on the fundamentals of the NMM, and

emphatically introduces the finite cover system and cover-based contact algorithm,

including the constructions of the MC, PC and ME using the finite element mesh, the

builds of local and global approximation for the NMM, and the derivations of the

contact sub-matrices based on the finite cover system.

Then, the integration schemes are reviewed with respect to spatial and temporal

aspects in this chapter to build up the foundation of the thesis work. The simplex

integration scheme in terms of the spatial aspect is illustrated firstly, and the Newmark

integration scheme is brief introduced and derived in detail to further deepen the

understanding the NMM and its implementations.



4.66

CHAPTER 4. AN EXPLICIT TIME INTEGRATION

SCHEME FOR THE NUMERICAL MANIFOLD METHOD

4.1 INTRODUCTION

Discrete element methods have the advantage in simulating discrete block systems

under different actions, which has been widely used applied in rock engineering. There

are two major representatives in the discrete element method family. The distinct

element method (DEM hereafter) adopted an explicit time integration scheme based on

finite difference principles (Cundall 1971a, 1971b). The discontinuous deformation

analysis (DDA) derived based on the variational method takes the benefit of the implicit

time integration method (Shi and Goodman 1985; Shi 1988). Since the initiation of the

DDA, many developments and applications have been implemented by Ohnishi et al.

(1995), Hatzor et al. (2004) and Zhao et al. (2011), etc.

The numerical manifold method (NMM) involved in this chapter is an evolvement

of the DDA, which combines the merits of FEM and DDA. The NMM inherits all the

attractive features of the DDA, such as the implicit time integration scheme, the contact

algorithm and the minimum potential energy principle (Chen et al. 1998). It adopts a

dual cover system, i.e. a mathematical cover system overlapping the domain of interest

and a physical cover system, which considers the contained discontinuities, such as

material joints, voids, interfaces and aggregates, in a united manner. In the past two

decades, many efforts have been carried out to improve the performance of the NMM,

which are stress intensity factors (SIFs) problems (Ma et al. 2010; Zhang et al. 2010),

crack propagation problems (Tsay et al. 1999; Zhang et al. 2010), high order NMM

theory (Chen et al. 1998; Lin et al. 2005), and extensions of the NMM (Terada et al.

2003; Miki et al. 2010), etc. A review article on the recent development of the NMM

has been published by Ma et al. (2010) and An et al. (2011). It has been reorganized that

the NMM has great potential to be further developed in simulating a medium with

massive discontinuities.



4.67

In this chapter, a modified version of the NMM based on an explicit time

integration algorithm is derived. The original NMM based on displacement method is

revised into an explicit formulation of a force method. The governing equations are built

up on the dual cover system and the global stiffness matrices used in the traditional

NMM are no longer necessary. A diagonal mass matrix is derived for the dual cover

system which makes the solution highly efficient at each time step. The OCI is still

employed, however, the relative cost is much lower because of the explicit time

integration scheme without solving simultaneous algebraic equations in each step and

the smaller penetration incurred due to a smaller time step used. The developed method

is validated by two examples, one static problem of a continuous simply supported

beam, and one dynamic problem of a single block sliding down on a slope. Results

showed that the accuracy of the explicit numerical manifold method (ENMM) can be

ensured when the time step is small for both the continuous and the contact problems. A

highly fractured rock slope is subsequently simulated. It is shown that the computational

efficiency of the proposed ENMM can be significantly improved, while without losing

the accuracy, comparing to the implicit version of the NMM. The ENMM is more

suitable for large-scale rock mass stability analysis and it deserves to be further

developed for engineering computations of practical rock engineering problems.

4.2 BRIEF DESCRPTIONS OF THE NMM

The traditional NMM is based on the dual cover system, which consists of

mathematical covers (MCs), physical covers (PCs) and manifold elements (MEs). The

MCs are user-defined small patches, and their union covers the entire problem domain.

The PCs are the subdivision of the MCs by the physical features such as the external

boundaries and the internal discontinuities, and each PC inherits the partition of unity

function from its associated MC. The ME is defined as the common region of several

PCs. On each ME, partition of unity functions is used as well to assemble all the local

functions associated the PCs to offer a global approximation.



4.68

For a two-dimensional problem, a regularly structured mesh is employed in the

NMM to form the cover system, which is similar as that in the FEM. More details of the

descriptions of the NMM can be referred in Section 3.2 of the previous chapter.

4.3 EXPLICIT TIME INTEGRATION FOR THE NMM

In order to investigate the time integration in the NMM, the behaviour of an

equivalent linear system is considered. In such a system, the minimization of the system

potential energy will produce an equation of motion, which is similar to that in the

FEM. Let displacement term of nD and 1nD denote the approximation to the

displacements )(tD and )( ttD for a time step t , the discrete equation of motion can

be expressed as

1111 nnnn FDKDCDM (4.1)

The component form of the mass M , damping C and stiffness K terms are

extensively discussed in (Shi 1991). A single step integration method can be used to

solve Eq. (4.1) for given initial conditions of 00 dD and 00 VD based on

Newmark method (described in Section 3.3). And a variety of well-known members of

the Newmark family methods are developed with reference in (Hughes 1983). In the

NMM, the implicit scheme is carried out by minimizing the potential energy associated

with an increment of time t , which can be represented as:

FDK ˆˆ (4.2)

where nn DDD 1,

Kt

MK 2

2ˆ and

nn D

t

MFF 2ˆ

1.

Since the NMM uses an implicit time integration scheme which provides numerical

damping, the explicit damping term C is assumed to be zero in Eq. (4.1). Eq. (4.2)

requires assembling the global stiffness matrix and solving the coupled system of

equations using successive over relaxation iteration method.

The DEM uses an explicit scheme for the discontinuous problem, this motivates us

to develop an explicit version for the NMM. When the parameters are 2/1,0 , we



4.69

can get an explicit forward central different scheme, which can be modified using the

Verlet algorithm, where the velocity is calculated at each half time step (Langston

1995), it can be expressed as:

2/11

2/11

2/12/1

2

1

nnn

nnn

nnn

DtDD

DtDD

DtDD

(4.3)

Eq. (4.3) is required as the calculation of the damping force depends on the velocity at

the next step )1( n . Then, substituting Eq. (4.3) into Eq. (4.1), we can get

FDM n

~1

(4.4)

where }~

{}~

{}{}~

{ 1 IFFF dn is the force item, in which }]{[}

~{ 1 nd DCF is the damping

item and the direction is such that energy is always dissipated, and }~

{I is the internal

force item assembled in the PC, respectively. When the explicit scheme is used, the

mass matrix on the PC can be diagonalizable as an equivalent lumped matrix ]~

[M , which

is different from that in the implicit scheme of the traditional NMM. Eq. (4.4) is

essentially the Newton’s second law of motion, and it is used as the main equation of

motion in the proposed explicit scheme. In contrast to the implicit scheme in the NMM,

the proposed explicit scheme eliminates the assembly of global stiffness matrix and

inversion of the global matrix, which uncoupling of the equation of motion is one of

major advantage than the implicit scheme. Since ]~

[M is diagonal, the }{ 1nD on the PCs

can be solved explicitly without assembling the global stiffness matrix. Thus, it is more

efficient than the implicit scheme.

4.3.1 Mass matrix

The mass matrix in Eq. (4.4) is traditionally called consistent mass matrix, which

uses the same weight functions as that used for displacement. Given an element e, we

can get a mass consistent matrix as

3,2,1,)()( srdxdyTTM se

T

A ree (4.5)

in which is the element mass density. Since the element generated by the three

associated PCs, eM is allocated to the PCs to form M~ on each PC. A common

procedure to obtain the lumped mass matrix M~ uses the row-sum lumping technique.



4.70

And many other strategies of the mass matrix diagonalization can be referred in (Wu

and Qiu 2009).

In this chapter, we can gain the M~ based on the displacement matrix ),( yxTe

.

Considering the matrix ),(),( yxTyxT eT

e for the contributions of element e, we have

L

yxwyxwyxwyxwyxw

yxwyxwyxwyxwyxw

yxwyxwyxwyxwyxw

yxwyxwyxwyxwyxw

yxwyxwyxwyxwyxw

yxwyxwyxwyxwyxw

yxTyxTyxT

yxT

yxT

yxT

TT

eeeee

eeeee

eeeee

eeeee

eeeee

eeeee

eeeT

e

Te

Te

eT

e

2)3()3()2()3()1(

2)3()3()2()3()1(

)3()2(2

)2()2()1(

)3()2(2

)2()2()1(

)3()1()2()1(2

)1(

)3()1()2()1(2

)1(

)3()2()1(

)3(

)2(

)1(

),(0),(),(0),(),(0

0),(0),(),(0),(),(

),(),(0),(0),(),(0

0),(),(0),(0),(),(

),(),(0),(),(0),(0

0),(),(0),(),(0),(

)],([)],([)],([

)],([

)],([

)],([

(4.6)

Then, the lumped mass matrix M~ can be obtained using the row-sum lumping

technique. Decompose matrix L , we can get

2,1),,()1(

6

1

iyxwL ej

ij (4.7)

4,3),,()2(

6

1

iyxwL ej

ij (4.8)

6,5,),(6

1)3(

iyxwLj

eij (4.9)

Eq. (4.7) maps the first PC associated with the element, the subsequent Eqs. (4.8) and

(4.9) map the second and third PCs, respectively. Thus, the allocated mass at the each

PC can be expressed as

3,2,1),(~

)()( rdxdyyxTMA rere (4.10)

Since the element shape in the NMM is generally polygon, )(

~reM can be calculated

using the analytical method such as simplex integration method. Then, the lumped

matrix M~ at each PC can be assembled by the associated )(

~reM on the cover system. We

can find that when all the PCs use the lumped mass matrices, the globally assembly

mass matrix is diagonal as well.



4.71

4.3.2 Internal force

In this chapter, internal force item I~ in Eq. (4.4) can be formulated by element

stiffness eK and contact element stiffness cK . To the continuous problem, each element

internal force eI~ is generated by the eK , which satisfies the equation of motion as

}{}~

{}]{[}]{[}]{[ 11111e

nnne

nne ffDKDCDM (4.11)

in which }~

{ 1nf is a unknown force vector from the neighbour elements and }{ 1e

nf is the

external force vector but the neighbour elements, respectively. When the whole system

is assembled into a global lumped mass matrix on the cover system, the item of }~

{ 1nf

can be offset, which satisfies

e

enn

en

fF

f

}{}{

0}~

{

11

1 (4.12)

Thus, only ][ eK is considered. Since the mass matrix is lumped, ][ eM of each element

can be uncoupled to calculate }{ 1nD explicitly. The internal force }~

{ eI in the element can

be expressed as

}]{[}~

{ 1 ne

e DKI (4.13)

On the other hand, to discontinuous problems, as the PCs associated with the contact

elements have no overlap with each other, the contact elements are taken into account to

form contact matrix ][ cK . In each contact pair, a linear stiff spring is added to generate

force item and cause the stiffness change of the contact elements. To the contact

element, }~

{ 1nf can not be neglected and is determined by the contact matrix, which

satisfies

en

e

enn

en

ffF

f

}~

{}{}{

0}~

{

111

1 (4.14)

Then, the internal force }~

{ eI in the element can be rewritten as

}]{[}]{[}~

{ 11 nc

ne

e DKDKI (4.15)

For a discrete block system involving m elements, there are N contact pairs have



4.72

been detected to an element i ( mi ,2,1 ). Here, we assume one element denoted by j (

Nj ,2,1 ) and i are detected in contact state, ][ cK between i and j can be expressed as

][][

][][cjj

cji

cij

ciic

KK

KKK (4.16)

in which ][ cijK , ji and mji ,2,1, , is defined by the contact spring between the

contact elements i and j, and the value is zero if the elements i and j have no contact.

Since each element is consisted by three associated PCs, thus the matrix ][ cijK is a 6×6

sub-matrix. It is noted that the displacement 3,2,1,)( rD ri on the PCs can be predicted by

the previous step n. Then contact forces associated with ][ cijK on the contact element i

are assembled as

NjDKIN

jn

cij

ci ,,2,1,][

11

(4.17)

The total internal forces on the element i can be represented as

NjDKKIDKI n

N

j

cij

eii

cin

eiii ,,2,1},{][][}{}{][}

~{ 1

11

(4.18)

in which iI~ is the element internal force vectors and e

iiK is the stiffness matrix of

element. Since each element is formed by the three associated PCs, thus iI~ can be

rewritten as

)3(

)2(

)1(

~

~

~

~

i

i

i

i

I

I

I

I (4.19)

in which )1(

~iI maps the first PC associated the element, the subsequent )2(

~iI and )3(

~iI

map the second and third PCs, respectively. Then, I~ at each PC can be assembled by

the associated iI~ on the cover system.

4.3.3 Damping algorithm

Damping algorithm is used to dissipate the excessive energy in the contact

problems due to the use of linear springs between contact elements. As referred in

(Cundall 1982), we suggest an alternative scheme to simulate the damping, in which the



4.73

damping force with the inertial force is in direction proportion. The total potential

energy from the damping item is summered in an element e, which can be written as

3,2,1,}]{~

[}{ )()()( rDMDd

erere

Tre

(4.20)

where is damping ratio. Substituting Eq. (4.16) using the variational principle, the

equivalent damping force assembled on each PC can be expressed as

}]{~

[~

1 nd DMF (4.21)

which is then added to Eq. (4.4) to form the matrix item F~ . To the discrete systems,

there are usually some domains in stable conditions and the others in motion statuses, in

which the different damping force can be considered in the different domains as an

appropriate method. The damping force described in Eq. (4.17) is proportional to the

inertial force and varied in the whole system, thus it is more adaptive even the system

approached the stable conditions.

In terms of the numerical stability, since the applied explicit time integration as

same as the DEM (Cundall 1971a) is conditionally stability, the size of the selected

time-step t is usually smaller than that of the implicit scheme. When the damping item

is taken into account in the system motions, smaller damping ratio of the PCs system

corresponds to larger time-step size t . Here, we propose

max

2

t (4.22)

where λ ∈ 0,1 is a coefficient associated with damping ratio ; is the highest

eigenfrequency of the system. For no damping case, the value of λ can be considered as

1. More details of discussions about the selection of t can be referred in (Cundall 1982;

Bath 1982; Hughes 1983). It is noted that the explicit scheme employs dynamics

method to solve the uncoupled equations, in which the generated kinetic energy can not

be neglected. It is noted that the time-step t for dynamic problems depends not only on

damping ratio and , but contact stiffness, mesh resolution, deformation stiffness

of rock mass.



4.74

4.4 CONTACT FORCE IN THE ENMM

4.4.1 Contact force calculation approach

In general, there are two different types of techniques to treat the contact forces

acting on the contact elements in the numerical methods, as shown in Fig. 4.1: (i) point

contact forces or (ii) area contact forces. The point approach usually assumes vertex

contact force is a function of penetration of an individual contact element vertex into

another contact element, while area method is evaluated from the shape and area of the

overlap between two contact elements. A thorough discussion and formulations of these

approaches can be found in (Munjiza 2004).

(a) Point contact force approach; (b) Area contact force approach.

Figure 4.1 Two distinct contact force approaches: (a) point approach; (b) area approach.

In the present paper, we apply the point contact force approach combining the

contact point and entry area to simulate the contact interaction problems and calculate

the contact forces. As shown in Fig. 4.2, two different strategies of normal penetration

method and direct penetration method can be employed to determine the penetration

distance d. Firstly, the entry area n can be determine by the penetration vertex

),( iii yxP and entry element vertices ),( jjj yxP and ),( kkk yxP , which is represented as

kk

jj

ii

n

yx

yx

yx

1

1

1

2

1 (4.23)

The penetration distance d in the normal penetration method can be determined as

ld n

2 (4.24)

j

i

ifjf

i jj

i

);,( jiNji

i1i

f

3if

1jf

3jf

2jf

ajf

j

;,( Nba );, jiNji

2if

bif



4.75

in which 22 )()( kjkj yyxxl is the entry element boundary length. On the other

hand, the direct penetration method calculates d directly as

22 )()( nini yyxxd (4.25)

in which ),( nnn yxP is assumed as contact point between the contact elements. In the

present study, the direct penetration method is employed into the calculation of contact

force in the ENMM.

(a) Normal penetration method; (b) Direct penetration method.

Figure 4.2 Two schemes for contact problem: (a) Normal penetration method; (b) Direct

penetration method.

4.4.2 Calculation of contact force

As the explicit equations are not coupled, contact force is calculated explicitly. Fig.

4.3 describes the proposed contact model for the ENMM. The proposed contact force

approach treats contact problem explicitly by adding and subtracting the normal and

shear spring when contact is detected. Here we choose the contact block pair of block I

and J with the contact vertex of P1 to study the proposed contact algorithm in the present

paper. The contact elements i and j , penetration vertex of P1 i can be found by the

associated with partition of domain qU (i.e. q is the index number in the programming

code) as can be seen in Fig. 4.3(a). Then the contact pair is constructed between contact

element i and j , in which the contact point P1 i , the contact position is represented by

P0, and entry line P2 P3 (i.e. (P0, P2, P3) j ) are presented in Fig. 4.3(b). The penetration

distance of element i can be presented as d, which can be expressed as two components

of nd andsd satisfying equation of 22

sn ddd . The penetration angle can be expressed

d

Pn

Pi

Pk Pj

Entry area ∆n

Pn

Pi

Pk Pj

Entry area ∆n

d



4.76

as ddn /sin or dd s /cos .The penetration distance d between contact elements can be

written as

20011

20011 )()( vyvyuxuxd (4.26)

where ),( 11 yx , ),( 11 vu and ),( 00 yx , ),( 00 vu are the coordinates and displacement increments

of contact point P1 and penetration point P0 in one step time. Once the contact elements

in the loops are detected, the normal and shear stiffness springs and damper are applied

at each penetration point. The stiffness of contact springs and damper can be expressed

as nk ,

sk and nc , sc respectively. The strain energy stored in the contact springs c is

minimized with respect to the displacement variable vector ][ D and added to the contact

element stiffness sub-matrix to produce internal force vector. Then, the contact block

displacement is expressed as

DyxTyxv

yxu),(

),(

),(

(4.27)

where ),( yxT is displacement matrix and D is displacement vectors on the PC system

respectively. The strain energy of stiffness springs can be written as

222 )cossin(2

1dkk snC (4.28)

Substituting d in Eq. (4.26), Eq. (4.28) can be rewritten as

))(

][}{2][}{2}]{[][}{

}]{[][}{2}]{[][})({cossin(2

1

01

010101

01

01

01

01

22

yy

xxyyxx

yy

xxTD

yy

xxTDDTTD

DTTDDTTDkk

Tj

Tj

Ti

Tijj

Tj

Tj

jjT

iT

iiiT

iT

isnC (4.29)

The contribution of the contact normal and shear stiffness to the equilibrium

coefficient matrix ck can be given by

)][}{2}]{[][}{2

}]{[][}({)cossin(2

1

01

01

222

2

yy

xxTDDTTD

DTTDdd

kkdd

k

Ti

Tijj

Ti

Ti

iiT

iT

iirit

snirit

ccii , 6,2,1, rt (4.30)

forms a 66 sub-matrix

][])[])([cossin( 22 ciii

Tisn KTTkk (4.31)



4.77

which is then added to the sub-matrix ][ ciiK in Eq. (4.16).

)][}{2}]{[][}{2

}]{[][}({)cossin(2

1

01

01

222

2

yy

xxTDDTTD

DTTDdd

kkdd

k

Tj

Tjjj

Ti

Ti

jjT

jT

jirit

snjrjt

ccjj

, 6,2,1, rt (4.32)


][])[])([cossin( 22 cjjj

Tjsn KTTkk (4.33)

which is then added to the sub-matrix ][ cjjK in Eq. (4.16).

})]{[][}{2()cossin(2

1 222

2

jjT

iT

ijrit

snjrit

ccij DTTD

ddkk

ddk

, 6,2,1, rt (4.34)


][])[])([cossin( 22 cijj

Tisn KTTkk (4.35)

which is then added to the sub-matrix ][ cijK in Eq. (4.16).

})]{[][}{2()cossin(2

1 222

2

jjT

iT

iirjt

snirjt

ccji DTTD

ddkk

ddk

, 6,2,1, rt (4.36)


][])[])([cossin( 22 cjii

Tjsn KTTkk (4.37)

which is then added to the sub-matrix ][ cjiK in Eq. (4.16). And the equivalent force

matrix can be described as

)][}{2()cossin(2

1)0(

01

0122

yy

xxTDd

kkd

F Ti

Ti

irsn

ir

ci

, 6,2,1, rt (4.38)


iTisn Fyy

xxTkk

)])([cossin(

01

0122 (4.39)

which is then added to the force item F in Eq. (4.4).

)][}{2()cossin(2

1)0(

01

0122

yy

xxTDd

kkd

F Tj

Tj

jtsn

jt

cj , 6,2,1, rt (4.40)




4.78

jT

jsn Fyy

xxTkk

)])([cossin(01

0122 (4.41)

which is then added to the force item F in Eq. (4.4).

Figure 4.3 The proposed contact model: (a) contact elements; (b) contact points.

4.5 OPEN-CLOSE ALGORITHM IN THE ENMM

The NMM uses a penalty-constrain approach to treat the contact problems in which

the contact is assumed to be rigid. When two elements boundaries overlap, an

impenetrability constraint is employed by applying a numerical penalty function

analogous to stiff springs at the contact points with the direction of the penetrating

vertex to arrest interpenetration. Numerically, it can be carried out by means of adding

or subtracting penalty springs values to the contact elements to produce contact stiffness

and contact forces, then assembling them into the coupled global equation (i.e. equation

in the dotted box in Fig. 4.4) to get the global displacements of the whole cover system.

Assuming n contact pairs have been detected, the terms are selected by checking the

previous and current statuses of the contact, each controlled by a vector of values i =

-1, 0 and 1, ni ,,2,1 . Accordingly, within each time step, the assembled global

equations are solved iteratively by repeatedly adding and removing contact springs until

each of the contacts converge to a constant state, which is known as OCIs proposed by

Shi (1991).

Here, we use the well-known open-close algorithm as well to obtain contact

convergences in the ENMM. As we can see in Fig. 4.4, from the term of initial contact

condition to the end of contact computation in each step, the vital process concentrates

a.

Block J

Block I qU

i

j

b.Contact element i

dn

ds

d

P0

P1

P2 P3

Contact element j



4.79

on achieving the contact statuses identically as the merged in the dotted box. In the

implement, the contact statues don’t stop change until that is constant in one step as

presented with blue diamond shadow: open-open, lock-lock and slide-slide, then the

open-close processor completes in this time step. In the OCI, the no-penetration and no-

tension contact constraint is imposed at each detected contact point. To satisfy this

constraint, the open-close iterative solver proceeds until this is no penetration at any

contact point. Number of OCIs (i.e. OCIN ) is required for contact convergence in each

time step. When it is over the set value of limN (i.e. value of 6 in the code) , the time-step

will decrease by 02 t even

03 t , in which 2 is prescribed as one third and 3 is

the reciprocal of the maximum displacement ratio (which represents the tolerance

criteria of maximum penetration distance nd is less than the prescribed allowable value

0d ) in the OCIs. The value OCIN dramatically increases for problems involving more

contact elements, which increases the computational cost as well. Since the small time-

step is used in the ENMM, the large penetration between the contact elements can be

avoided to achieve contact convergence. Thus, the unnecessary OCIs procedures can be

removed to improve efficiency. Further more, the explicit equations are uncoupled

without assembling the global stiffness as in the NMM, the ENMM is more efficient in

terms of solving equations even the OCIs. An alternative scheme for simplifying open-

close algorithm in the ENMM can be considered, in which only the maximum

displacement ratio 3 (i.e. 23 in the code) is regarded as the judgement criteria to

obtain contact convergences. Since the size of t used in the ENMM is very small,

contact convergence can be obtained within one time of iteration. Then the penetrations

are beyond the prescribed displacement ratio, t will decrease by 03 t directly. The

OCI runs until the system penetration is less than the prescribed ratio. As the small

time-step is employed, the accuracy of contact normally can be satisfied in the OCIs as

well. On the other hand, the efficiency is improved dramatically.



4.80

Figure 4.4 Flowchart of the OCI in the ENMM.

4.6 NUMERICAL SIMULATIONS

In order to further investigate the proposed explicit scheme for the NMM, two

calibration examples are simulated firstly, in which one static problem of a continuous

simple-supported beam and dynamic one of a single block sliding along on a slope are

validated using the ENMM, respectively. Then a discrete rock slope modelling is

studied, in which the proposed ENMM using the open-close algorithm is investigated

and compared with the NMM in terms of computational accuracy and efficiency. In

Previous status Open Lock Slide

Open-open Lock-lock Slide-slide

Current status Open Lock Slide

Contact check

Tolerance criteria

0ddn

no

yes

Initial contact condition Contact detection Contact transfer

Set ntt ;

0tt

START

Step(n)

Calculate contact stiffnesses and contact forces

Solve explicit equation

Solve implicit equation

0OCIN

yes 1 OCIOCI NN

limNNOCI

no

Next Step

limNNOCI

Storage and update geometry and physical data

ntt no

yes

ntt ;03 tt

ntt ;02 tt

Resume contact status and other variables

0tt



4.81

order to track CPU time for each time step, a high resolution timer function which can

measure up to 1/100000th of a second is added to keep track of CPU time for each time

step. All analyses are run on the same computer with the system configuration:

processor speed = 3.17 GHz and RAM = 4.0 GB.

4.6.1 Simply-supported beam subjected a concentrated load

This simulation studies a simply supported beam subjected to a concentrated load

at its center. As shown in Fig. 4.5, the dimension of the beam is 10 m long, 1 m deep

and 1m wide. The point loading is P = 200 N. The material of the beam is assumed

elastic without damage with properties of E = 1×105 Pa and ν= 0.24. There are totally 11

measure points shown in Fig.4.5.

Figure 4.5 Geometry of the simply supported beam bending problem.

The theoretical solutions for simply supported beam under central point loading

can be expressed as (Chen et al. 1998):

)2/0(1612

)(2

3 lxxEI

Plx

EI

Pxv (4.42)

where E is the Young’s modulus, I is the moment of inertia and l is the length of the

beam. The simulated results of the proposed ENMM, NMM and the theoretical

solutions are compared in Fig. 4.6. It can be found that the numerical result converges

to the theoretical solutions using the proposed ENMM.

5 m 5 mP=200 N

x

y

1 m



4.82

Figure 4.6 Comparisons between simulated results and theoretical solution.

4.6.2 Numerical simulation of plane stress field problem

To verify the accuracy of the proposed explicit NMM code in accurately describing

the displacement field and stress field with the refinement of the mathematical covers.

An infinite plate with a traction free circular hole subjected to a unidirectional tension is

considered here. The geometry of model is presented in Fig. 4.7(a), in which the radius

of the hole is prescribed by 0.1m, 8 prescribed measuring points distant the top of

circular hole in vertical direction 0.05m, 0.1m, 0.2m, 0.3m, 0.4m, 0.5m, 0.6m, 0.7m and

0.8m respectively. The material properties of the model, i.e., Mass density, Young’s

modulus and Possion’s ratio, are assumed to be =2000 kg/m3, E =25 GPa and

=0.25 accordingly. The far-field uniaxial tension is prescribed as 4KPa. The manifold

element mesh topology representing the problem is shown in Fig. 4.7(b). All elements

consist of divided regular-patterned triangular mathematical mesh and physical

boundary. A refinement scheme of the mesh is employed by gradually refining the mesh

from level of coarseness to fineness in this study.

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 1 2 3 4 5 6 7 8 9 10x (m)

Dis

plem

ent (

m) Analytical

NMMENMM



4.83

Figure 4.7 Numerical model for an infinite plate with a traction free circular hole: (a) Geometry of

model; (b) NMM meshing.

The theorical solution of stress in x direction for the model can be expressed as:

4

2

342

2

31

4

4

2

2

cos)coscos(r

d

r

dTx

(4.43)

4

2

342

2

14

4

2

2

cos)coscos(r

d

r

dTy

(4.44)

where T is the uniaxial tensile stress applied to the plate, d is the radius of the circular

hole, r and are polar coordinates associated with its origin located at the center of

the hole.

The refined element mesh gives satisfactory results for the present analysis. Fig.4.8

gives a plot of computed x versus the analytical solution and the NMM. A good

agreement has been observed.

Measuring points (a) Geometry of model;

y

xTT

(b) NMM meshing.



4.84

Figure 4.8 Comparison of numerical results and analytical solution for infinite plate problem.

4.6.3 Single block sliding along the inclined surface

In this simulation, one single block sliding along the inclined surface is studied

here using the implicit and explicit NMM respectively. In the numerical model, one

square block with the size of 1 m×1 m locates on the top of the inclined surface, the

bottom triangle block with the size of 5 m×8.66 m is fixed by the fixed point. The angle

of the inclined surface is 300.

In theory, one block slides along the inclined surface by itself gravity. The

displacement along the inclined surface by time history can be derived based on

Newton’s second law as

2)tancos(sin

2

10

tgd g

(4.45)

where dg is the displacement of the block, g is the acceleration of gravity, is the angle

of inclined surface, is the friction angle, t is the time. The case with friction angle of

5° is simulated using the NMM and proposed ENMM. The displacement time history is

compared with the analytical solution as shown in Fig. 4.9, and good agreements have

been found. In the simulation, the two blocks are divided into 292 MEs to calibrate the

proposed ENMM. To obtain the computational accuracy of the proposed ENMM, a

damping ratio value of 0.001 is used to simulate the damping item. Comparing to the

residual error of -1.2883% using the NMM, the explicit NMM has approximate

1.6391% to the analytical solution. This calibration example shows that the proposed

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

5 10 15 20 25 30 35 40 45 50

σ x/ K

Pa

r /m

Analytical

NMM

ENMM



4.85

ENMM can be accepted with respect to the computational accuracy. Furthermore, the

CPU time by the ENMM and NMM methods taken are 192.05s and 293.46s as the

larger step-time of 2ms is employed and the scale of the model is not large.

Figure 4.9 Comparison of simulated results and analytical solution.

4.6.4 Highly fractured rock slope stability analysis

In this simulation, one highly fractured rock slope modeling is taken into account

as shown in Fig. 4.10. 6 measure points are prescribed to investigate the computational

accuracy and efficiency of the proposed ENMM compared with that of the NMM. The

modeling is divided into 390 discontinuous blocks including 627 elements in total. In

the present study, selections of t are taken as 2ms, 1ms and 0.1ms in the NMM, 0.1ms

in the proposed ENMM, respectively. The input parameters for the simulation of the

rock slope can be seen in Table 4.1.

Figure 4.10 Geometry of the slope modelling.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5Time (s)

Dis

plac

emen

t (m

) Analytical NMM

ENMM

110m

20m

80m Measured point 5

Measured point 1, 2, 3

Measured point 4

Measured point 6



4.86

Table 4.1 Input parameters for the NMM simulation of rock slope.

Physical property Parameter

Unit weight (kN/ m3) 26.0

Young’s modulus (GPa) 1.0

Possion’s ratio 0.2

Internal friction angle(deg.) 20.0

Cohesion (MPa) 5.0

Tensile strength (MPa) 0.0

NMM

Joint normal stiffness (GPa) 1.0

ENMM


Joint shear stiffness (GPa) 0.25

In order to represent the discontinuity of the rock slope, the cut blocks are

displayed explicitly, and the corresponding subdivided elements in the NMM and

ENMM are concealed, which can be seen in Fig. 4.11. The total time of one second is

prescribed to simulate the rock slope stability, the different selections of t result in

different computational steps and CPU time, which are shown that when t =2ms is

applied in the NMM, the code runs 650 steps to reach 1 second with 1.28 hours. As the

decline of the step-time applied, both computational steps and CPU time increase from

1090 steps with 1.83 hours to 10000 steps with 5.41 hours, respectively (see Fig. 4.11

(b) and (c)). Alternatively, the proposed ENMM gains a well agreement with the

simulation results of the NMM in terms of computational accuracy, and the most

importance is the CPU time of the ENMM code falls to 0.28 hour dramatically, which is

shown in Fig. 4.11(d).



4.87

(a) NMM, ∆t=2ms (1.28 hrs);

(b) NMM, ∆t=1ms (1.83 hrs);

(c) NMM, ∆t=0.1ms (5.41 hrs);



4.88

(d) ENMM, ∆t=0.1ms (0.28hrs).

Figure 4.11 Simulation results by NMM and ENMM: (a) NMM, ∆t=2ms; (b) NMM, ∆t=1ms; (c)

NMM, ∆t=0.1ms; (d) ENMM, ∆t=0.1ms.

It is noted that the NMM employs implicit scheme and OCI to treat the contact

problem, which cut down the computational velocity and reduce the computational

efficiency. Fig. 4.12 shows that the real used step-time decreases dramatically as the

increase of t . When large t can not achieve contact convergence, t will be cut down

until which satisfies the OCI convergence. This increases the CPU cost and declines the

computational efficiency. On the other hand, t used in the proposed EMM is stable

and efficient.

Figure 4.12 Real step-time used in NMM vs. ENMM.

In terms of computational accuracy, the horizontal and vertical displacements of

the measure points 1, 3 and 6 are plotted in Figs. 4.13, 4.14 and 4.15, respectively. It is

0

0.001

0.002

0.003

0.004

0 0.2 0.4 0.6 0.8 1Total time (s)

Rea

l ∆t (

s)

NMM, ∆t=2msNMM, ∆t=1msNMM, ∆t=0.1msENMM, ∆t=0.1ms



4.89

noted that a damping ratio value of 0.001 is used to simulate the damping item in this

simulation. The simulated results using the ENMM are well agreement with that of the

NMM. The maximum relative error is under 0.27%, which is shown in Table 4.2. It can

be concluded that the ENMM can be taken into account in the application to rock

engineering, especially involving the computational efficiency of the modelling for

large scale engineering project.

(a) horizontal displacement;

(b) vertical displacement.

Figure 4.13 Displacements of measured point 1: (a) Horizontal; (b) Vertical.

-0.012

-0.01

-0.008

-0.006

-0.004

-0.0020

0.002

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1ms

NMM, Δt=0.1ms

ENMM, Δt=0.1ms

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1msNMM, Δt=0.1ms

ENMM, Δt=0.1ms



4.90





0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1ms

NMM, Δt=0.1ms

ENMM, Δt=0.1ms

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1ms

NMM, Δt=0.1ms

ENMM, Δt=0.1ms

-0.25

-0.2

-0.15

-0.1

-0.05

0

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1ms

NMM, Δt=0.1ms

ENMM, Δt=0.1ms



4.91



Table 4.2 Maximum displacement of measure points in the NMM vs. ENMM.

Measure point 1 3 6

Max. Displ. (m) Vertical Horizontal Vertical Horizontal Vertical Horizontal

NMM, ∆t=2ms -0.01179 -1.61859 1.3911 -2.73225 -0.21954 -0.62965NMM, ∆t=1ms -0.00966 -1.49954 1.36399 -2.71774 -0.23232 -0.63504

NMM, ∆t=0.1ms -0.00613 -1.58604 1.40826 -2.76374 -0.23629 -0.66333

ENMM, ∆t=0.1ms -0.01053 -1.58071 1.73911 -2.60442 -0.21174 -0.55114

Max. rel. error (%) 0.718 0.054 0.275 -0.057 -0.104 -0.169

4.6.5 Rock tunnel stability analysis

In this study, a tunnel modelling is taken consideration to extend the capability of

the proposed explicit version of the NMM. The tunnel consists of two sets of

intersecting joints by the orientation of 600 and 1300 as shown in Fig. 4.16. Two

neighbouring tunnels with same sectional dimension are surrounded by the fractured

rock blocks. The width of the tunnel a is 7.8 m, the interval distance between the two

tunnels b is 4.2 m, the high of the arch h is 2.7 m, and the high of the tunnel H is 3.0 m.

The whole cross section is with the length of 66.0 m and high of 33.0 m, and it is

symmetrical. 8 measure points locate on the top of the tunnels, one measure point is at

the centre of the modelling. The input physical parameters are identical with the last

rock slope modelling, which can be referred as Table 4.1.

-0.8

-0.6

-0.4

-0.2

0

0 0.2 0.4 0.6 0.8 1Time (s)

Dis

pl. (

m)

NMM, Δt=2ms

NMM, Δt=1ms

NMM, Δt=0.1ms

ENMM, Δt=0.1ms



4.92

Figure 4.16 Geometry of the tunnel modelling.

To compare the NMM and ENMM conveniently, the same time step is chosen to

simulate the stability of the tunnel modelling. The simulated results are presented in Fig.

4.17, in which the total time is set as 0.5 second. It is noted that CPU time taken by the

ENMM (i.e. 0.381 hour) is less than that of the NMM (i.e. 1.174 hours). Thus, the

ENMM is obviously more efficient than the NMM in terms of efficiency.

Figure 4.17 Simulation results used by ENMM vs. NMM.

66.0m

h

H

a ab33.0m

h

H

15.0

m

18.0

m

Measured point: 1, 2, 3, 5, 6, 7, 8.9,

(a) ENMM (CPU time 0.381hr);

(b) NMM (CPU time 1.174hr).



4.93

Meanwhile, the horizontal and vertical displacements of the measure point at the

top of the tunnels are plotted in Figs. 4.18, respectively. It is noted that a damping ratio

value of 0.002 is used to simulate the damping item in this simulation. The simulated

results using the ENMM are well agreement with that of the NMM. The simulation of

the tunnel stability further reveals the capability of the developed ENMM in terms of

efficiency, and accepted level of accuracy. Especially, the ENMM is promising in

simulations of the large scale under-ground engineering such as tunnel stability analysis.



Figure 4.18 Displacements of measured point 4.

‐0.2‐0.18‐0.16‐0.14‐0.12‐0.1

‐0.08‐0.06‐0.04‐0.02

00.02

0 0.1 0.2 0.3 0.4 0.5

Displement (m

)

Time (s)

NMM

ENMM

‐0.4

‐0.35

‐0.3

‐0.25

‐0.2

‐0.15

‐0.1

‐0.05

0

0 0.1 0.2 0.3 0.4 0.5

Displacement (m

)

Time (s)

NMM

ENMM



4.94



Figure 4.19 Displacement of measured point 9.

4.7 SUMMARY

In this chapter, an explicit time integration scheme for the NMM is proposed to

improve the computational efficiency, in which a modified version of the NMM based

on an explicit time integration algorithm is derived on the dual cover system. The

original NMM based on displacement method is revised into an explicit formulation of

a force method. Although the ENMM requires small time-step due to numerical stability

of the scheme, it is efficient without assembling the stiffness equations. Compared to

the OCI used in the NMM, the open-close algorithm is more efficient in the ENMM

because of the explicit time integration scheme without solving simultaneous algebraic

0

0.002

0.004

0.006

0.008

0 0.1 0.2 0.3 0.4 0.5

Displacement (m

)

Time (s)

NMM

ENMM

‐0.7

‐0.6

‐0.5

‐0.4

‐0.3

‐0.2

‐0.1

0

0 0.1 0.2 0.3 0.4 0.5

Displacement (m

)

Time (s)

NMM

ENMM



4.95

equations in each step and the smaller penetration incurred due to a smaller time step

used. The developed method is validated by three examples, two static problems of a

continuous simple-supported beam and plan stress field problem, the other dynamic one

of a single block sliding down on a slope. Results showed that the accuracy of the

ENMM can be ensured when the time step is small for both the continuous and the

contact problems. A highly fractured rock slope and tunnel modelling are subsequently

simulated. It is shown that the computational efficiency of the proposed ENMM can be

significantly improved, while without losing the accuracy, comparing to the original

implicit version of the NMM. The ENMM is more suitable for large-scale rock mass

stability analysis and it deserves to be further developed for engineering computations

of practical rock engineering problems.



5.96

CHAPTER 5. VERIFICATION OF COMPUTATIONAL

EFFICIENCY AND ACCURACY OF THE EXPLICIT

NUMERICAL MANIFOLD METHOD WITH WAVE

PROPAGATION PROBLEMS

5.1 INTRODUCTION

Wave propagation problems are always hot issues in the engineering analysis

drawing researchers’ attention. For rock engineering, the damage criteria of rock mass

under dynamic loads are generally governed by the threshold values of wave amplitudes

(Zhao et al. 2006). Therefore, the prediction of wave attenuation across the fractured

rock mass is important on assessing the stability and damage of rock mass under

dynamic loads. In the past several decades, many analytical approaches and numerical

methods are developed for the solution of wave propagation problems in both the

continuous and discontinuous media.

The numerical manifold method (NMM), originally proposed by Shi (1991, 1992),

is based on topological manifold and differential manifold, which combines both the

continuum-based finite element method (FEM) and the discontinuum-based

discontinuous deformation analysis (DDA) (Shi and Goodman 1985; Shi 1988) in a

unified form. Since the NMM uses a mesh-based partition of unity method (PUM)

(Babuska and Melenk 1995; Melenk and Babuška 1996; Babuška and Melenk 1997) to

combine all the local approximations together to give a global approximation, thus

which can also be viewed as a PU-based extension to the conventional FEM. One of the

most innovative features of the method is that it employs a dual cover system, i.e.

mathematical covers (MCs) and physical covers (PCs) to formulate the physical

problem. These two covers are interrelated through the application of the PUM. More

importantly, due to the cover-based property, the NMM is in essence different from the

classical FEM, and particularly suitable for modelling arbitrary discontinuities. With

respect to the discontinuum, the NMM inherits all the attractive features of the DDA,



5.97

such as the implicit time integration scheme, the contact algorithm and the minimum

potential energy principle (Chen et al. 1998; Jing 1998).

On the other hand, when the implicit time integration algorithm involves in the

solution of a system of equations, the computational cost increases dramatically with the

DOFs of the system is increased since the large-scale simultaneous algebraic equations

must be solved in each time step (Newmark 1959; Wilson, Farhoomand, and Bathe

1972). The OCI requires non-tension and non-penetration at all contacts which

additionally highs up the computation costs in order to obtain a convergence state at

each time instance (Doolin and Sitar 2004). On the contrary, the finite difference

method (FDM) (Kaczkowski and Tribillo 1975; Mitchell and Griffiths 1980) and

distinct element method (DEM) (Cundall 1971a) employ the explicit time integration

schemes based on finite difference principles. To the DEM, the main benefit of the

DEM is that its computational efficiency is high due to its explicit time integration

nature. However, it has also been argued that the accuracy of simulated results may be

sacrificed in some particular cases (O’Sullivan and Bray 2001). To ensure numerical

stability, a DEM simulation requires that the time step must be small enough. In the past

several decades, a great mount of discussions have been carried out between the implicit

and explicit time integrations (Bathe and Wilson 1972; Belytschko and Hughes 1983;

Dokainish and Subbaraj 1989a, 1989b).

In chapter 4, a modified version of the NMM based on an explicit time integration

algorithm is derived. The original NMM based on displacement method is revised into

an explicit formulation of a force method. The governing equations are built up on the

dual cover system and the global consistent mass, damping and stiffness matrices are no

longer necessary. A diagonal mass matrix is derived for the dual cover system which

makes the solution is highly efficient at each time step. OCI is still employed, however,

the relative cost is much lower because of the explicit time integration scheme without

solving simultaneous algebraic equations in each step and the smaller penetration

incurred due to a smaller time step used. To validate the proposed scheme, the stress

wave propagation problems through rock mass are simulated with continuous and

discontinuous considerations. The wave propagation problems depend on many factors,



5.98

such as the computational model and the algorithm used, the model size, the mesh size,

the initial condition, the boundary condition, the material model and the damping and

time step size, etc. (Lysmer and Kuemyer 1969; Chen 1999; Zhao et al. 2008; Gu and

Zhao 2009; Ma, Fan, and Li 2011).

In this chapter, the calibration study of the explicit numerical manifold method

(ENMM) on the P-wave propagation along a one-dimensional elastic rock bar is

conducted to investigate the accuracy of the ENMM on wave propagation problems.

Parametric studies are carried out to obtain an insight into the influencing factors of the

ENMM model on wave propagation problems. The reflected and transmitted waves

through the fractured rock mass are also numerically simulated. Furthermore, to verify

the capability of the proposed ENMM in modelling of seismic wave effect in fractured

rock mass, a dynamic stability assessment for fractured rock slope under seismic effect

is analysed as well. It is shown that the computational efficiency of the proposed

ENMM can be significantly improved, while without losing the accuracy, comparing to

the original implicit version of the NMM.

5.2 THE BRIEF OVERVIEW OF THE NMM

5.2.1 The NMM and its cover system

In the NMM, the manifolds connect many overlapped small patches together to

cover the entire problem domain. Each small patch is called a cover. A local function is

defined on each cover. One manifold element is generated through a set of overlapping

covers, and the behaviour is then determined by the weighted average of local functions

defined on the associated physical covers. It is the cover system, distinguishes the

NMM from other numerical methods and it equips the NMM as a robust tool for both

continuous and discontinuous problems. More details can be referred in Section 3.2 of

Chapter 3.

5.2.2 The explicit scheme of the NMM

More details of derivation of the explicit scheme of the NMM can be referred in the

Section 4.3 of Chapter 4.



5.99

5.3 STRESS WAVE PROPAGATION IN A CONTINOUS BAR

To further investigate the proposed ENMM, stress wave propagation problems

through the rock mass are studied using the proposed explicit scheme to compare with

the implicit version with respect to computational efficiency and accuracy. In order to

track CPU time for each time step, a high resolution timer function which can measure

up to 1/100000th of a second is added to keep track of CPU time during the analysis.

Both analyses are run on the same computer with the system configuration: processor

speed = 3.17 GHz and RAM = 4.0 GB.

In the present simulation, a slender rock bar model subjected to impact loading is

studied as shown in Fig. 5.1. For the sake of decreasing transverse effect caused by

wave propagation, the one-dimensional elastic rock bar with width of 0.05 m, the length

of 2.00 m. The rock bar model consists of divided regular-patterned triangular manifold

element. A half sinusoidal impacting loading with the amplitude of 1MPa and different

frequencies are employed at the left edge of the model. The physical material properties

of the model can be seen in Table 5.1. The manifold element mesh topology

representing the model is shown in Fig. 5.1.

Figure 5.1 Schematic of the rock bar model.

Table 5.1 Material properties of the rock bar.

Property Value

Density(kg/m3) 2650

Young's Modulus(GPa) 66.25

Poisson's Ratio 0.25

P-wave velocity (m/s) 5000

0.05m

2.0m

P-wave Measuring point y

xo



5.100

5.3.1 Effect of mesh size

It is commonly recognized that the mesh size of the numerical model significantly

influences the accuracy of numerical results for wave propagation problems whether the

model is based on the continuum or the discontinuum approach. This is because the

dynamic problems may lead to a computational instability if the following condition is

violated: xvt max, where maxv is the maximum particle velocity and x is the length

interval. One key parameter in the design of the mesh size is the mesh ratio ( rl ), defined

as the ratio between the eigenlength ( eigenl ) (see Fig. 5.2) of the largest element along the

wave propagation direction and the smallest wavelength ( min ).

mineigen

r

ll (5.1)

Based on the study on the mesh size limitation in the FEM, Kuhlemeyer and

Lysmer (1973) proposed that the mesh ratio must be smaller than 1/8~1/12 for the

accurate modeling of one-dimensional wave propagation through a semi-infinite

continuous medium. Theoretically, a numerical model with a refined mesh can produce

more accurate results, because the model is hardened by predefining a function to

represent the displacement or stress field in the elements. An element size larger than a

certain critical value may result in the numerical oscillation of wave presentation.

Figure 5.2 Eigenlength in the manifold mesh.

In the numerical model, the mesh ratio varies as 4

1 , 8

1 , 12

1 , 16

1 , 32

1 and 64

1 in the

longitudinal direction. Three cases corresponding to the same one half sinusoidal P-

wave with pressure amplitude of 1MPa and frequencies of 2500, 5000 and 10000 Hz,

respectively, are studied in this modeling. The corresponding wavelengths are 2.0, 1.0,



5.101

and 0.5m, respectively. The relative error, defined as the ratio of peak pressure

difference between the ENMM modeling results and the theoretical solution is used to

assess the accuracy of the numerical results, which can be repressed as

%

T

TE

P

PP (5.2)

As the rock material in the computational model is treated as a linear elastic medium,

there is no geometrical damping occurring in the one-dimensional wave propagation

problem. Thus, an elastic wave propagates through the rock with virtually no

attenuation and the theoretical solution is the same as the incident wave.

Figure 5.3 Percent errors at the end of first wavelength for different wave frequencies and element

ratios.

The relative error at the end of the first wavelength for different wave frequencies

and mesh ratio in the middle of the bar is shown in Fig. 5.3. It can be seen that with the

decrease of the mesh ratio, the percent error will decline evidently. When the mesh ratio

is equal to or less than one-eighth of the element dimension along the wave propagation

direction, the errors are less than 5%. Furthermore, the higher incident wave frequencies

will generate less accurate simulation result. The result is consistent with the previous

results obtained by the other researcher using the DEM and DDA (Chen 1999; Gu and

Zhao 2009). To further describe the differences of accuracy between the explicit and

implicit time integrations, lumped and consistent mass matrices used in the ENMM and

NMM, respectively. Two typical stress waves by the mesh ratios 1/4 and 1/32 in the

0

5

10

15

20

1/4 1/8 1/12 1/16 1/32 1/64

Rel

ativ

eer

ror

(%)

Mesh ratio

2500Hz

5000Hz

10000Hz



5.102

case of wave frequency 5000 Hz are presented in Fig. 5.4. It is found that the refined

mesh brings more accurate result to the theory solutions, but on the other hand, the

computational cost is more expensive whether using the NMM or ENMM.

Figure 5.4 Stress wave simulation using NMM and ENMM by two typical element ratios of 1/4 and 1/32 ( }0.1 st ).

Fig. 5.5 shows the error along the distance from wave source for different mesh

ratios with the wave frequency of 5000 Hz. It can be seen that the error increases with

the increasing distance whether in the NMM or ENMM, and the mesh ratio equal to or

less than one-sixteenth provides stable results with errors less than 2.5% even at the

furthest distance. It can also be seen that the fluctuation of percent errors occurs along

the distance from the wave source for the smaller mesh ratio in which the error is less

than 5%. This indicates that too finer a mesh may not necessarily produce more accurate

results in which the error may be dominated by other factors such as time step,

boundary condition and convergence error. Since the original NMM using the implicit

algorithm, the percent errors are less than the ENMM when the same time step is used.

On the other hand, the solution of the equations in the NMM is uneconomical as the

assembly of the coupled global stiffness matrix and the repeated over-relaxation

iterations, even as the increase of the DOFs in the large scale simulations, the efficiency

of the NMM is declined dramatically.

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 0.0002 0.0004 0.0006 0.0008 0.001

Str

ess

(MP

a)

Time (s)

NMM, lr=1/4ENMM,lr=1/4NMM, lr=1/32ENMM, lr=1/32



5.103

(a) NMM;

(b) ENMM.

Figure 5.5 Percent errors along the distance from wave source for different element ratios ( }0.1 st ).

5.3.2 Effect of time step

The time step selection in a dynamic analysis is crucial to ensure the accuracy and

stability of the numerical results. In the absence of damping, both the ENMM and the

NMM employ the same discrete element framework to solve the equilibrium equations

with inter-block constraints, which are similar as the DEM and DDA. One of the main

differences between them is the time integration algorithm used to implement the time-

marching equations. O’Sullivan and Bray (2001) discussed the relative merits of

implicit and explicit schemes for discrete element modeling. Cundall and Strack (1979)

0

5

10

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Per

cen

t er

ror

(%)

Distance from wave source (m)

1/4

1/8

1/12

1/16

1/32

0

5

10

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Per

cen

t er

ror

(%)

Distance from wave source (m)

1/41/81/121/161/32

·



5.104

used the explicit, central difference time integration scheme in the DEM. A limitation of

this approach is that it is only conditionally stable and the critical time increment needs

to be calculated additionally. The implicit time integration scheme avoids the stability

issues arising from the explicit time integrator. However, the implicit scheme is

computationally more expensive because it needs a significant number of iterations to

form the stiffness matrix that is compatible with the contact state at the end of each time

step. The modified explicit time integration algorithm applied in the present ENMM

program constructs the lumped mass matrices and force vectors based on the state of

contact between the blocks at the end of the previous time step. Resorting to the solved

acceleration in each cover, the displacement increments are determined and the related

stiffness matrix and force item are updated, depending on whether contacts are open or

close over the course of the time step. For a discontinuous problem, the time step

required for the initial assumption is determined by the minimum of two time steps: one

is for the internal element system calculation and the other one is for the calculation of

contacts (de Lemos 1997).

The internal mesh calculations of the cover system assume that no information is

transmitted from a cover to another cover in the same zone in less than one time step.

The time step required for the stability of the zone of computations is proportional to the

mesh size and is estimated as

}/min{ min pa chat (5.3)

where pc is the P-wave velocity and

minh is the minimum height of the element, a is a

user-supplied factor intended to account for the increase of apparent stiffness due to the

contact springs attached to the boundary zones.

For the calculation of contacts of a rigid block system, the time step is calculated

by analogy to a single DOF system as (Last and Harper 1990)

maxmin /2 KMbtb (5.4)

where minM is the mass of the smallest block in the system;

maxK is the maximum contact

stiffness; b is user-supplied factor in order to account for that a block may be in contact

with several blocks.



5.105

When damping item is used, the following formula is used to adjust the time step to

maintain the calculation stability (Belytschko and Hughes 1983):

}/))1((2min{ 2iiict (5.5)

where i is the critical damping ratio for mode with angular frequency

i . The

representation of the critical damping ratio can be found from Bathe and Wilson (1972):

)/(2

1iii (5.6)

where and are the mass-proportional and stiffness-proportional damping constant,

respectively.

The time step for a discontinuous problem is thus chosen as:

},,min{ cba tttt (5.7)

The maximum time increment will be set based on the value smaller than this

selected t that can be used in a time step. The initial calculation will be processed to

check whether this value satisfies the assumption of infinitesimal displacements and

equilibrium state for all the blocks with appropriate contact conditions within a time

step. Within a specified number of iterations, the procedure of contact lock selection so

called OCI will be repeated for solving the motion equations until there is no

penetration and no tension occurrence at each contact point. The program will

automatically adjust a smaller time step if iteration number becomes too large.

The simulation model is taken account stress wave with frequency of 5000 Hz and

the mesh ratio 1/32, thus the minimum size of the element is 0.03125 m. All the other

parameters used are the same as in the previous study. From Eq. (5.7), we can obtain the

time step required for both mesh ratio and contacts as: at =3.61 μs (here, a is selected

as 3/1 ), bt =54.27 μs (assumed one single joint in the middle of the bar, b is selected as

0.5, maxK is 45 GPa) and

ct =22.48 μs (here, critical damping ratio i selected as 1.0).

Therefore, t is selected to be 3.61μs. In this simulation, five different time increments

have been set to study the influence of the time step, the time increments are 0.1, 0.5,

1.0, 2.0 and 5.0μs, respectively.



5.106

As shown in Fig. 5.6, it can be found that all the results are stable by both the

NMM and ENMM simulations when the maximum time increment is lower than the

selected t . With the decrease of the time step, the wave attenuation along the bar

becomes smaller and the results are more accurate for the peak pressure. But on the

other hand, the computational cost will be lengthened. In addition, the fluctuation

phenomena before the arriving wave become larger when larger time increment is used,

especially at the case of the ENMM modelling. Considering both accuracy and

efficiency, the time increment 0.5μs is a good choice for the ENMM model.

(a) NMM;

(b) ENMM.

Figure 5.6 Peak pressure attenuation for different time step.

0.4

0.6

0.8

1

0 0.4 0.8 1.2 1.6Distance from wave source (m)

Pea

k pr

essu

re (M

Pa)

0.1μs

0.5μs

1.0μs2.0μs

5.0μs

0.4

0.6

0.8

1

0 0.4 0.8 1.2 1.6Distance from wave source (m)

Pea

k pr

essu

re (M

Pa)

0.1μs

0.5μs1.0μs2.0μs5.0μs



5.107

5.3.3 Computational efficiency

Since the ENMM applies the explicit time integration scheme, the motion

equations can be solved explicitly without assembling the coupled global stiffness

matrix and repeating iterative solvers. In the ENMM, the lumped mass matrix is

employed instead of the consistent mass matrix used in the NMM. Thus, the

computational efficiency of the ENMM is significantly higher than that of the NMM

when the selected time step satisfies the numerical stability of the explicit integration

algorithm. To further investigate the efficiency of the ENMM comparing with the

NMM, five different time increments of 0.1, 0.5, 1.0 and 2.0 are used to simulate the

stress wave propagation in the bar by the different mesh ratios of 1/4, 1/8, 1/12, 1/16

and 1/32, respectively. In order to track CPU time for each time step, a high resolution

timer function which can measure up to 1/100000th of a second is added to keep track

of CPU time for each time step. All analyses are run on the same computer with the

system configuration: processor speed = 3.17 GHz and RAM = 4.0 GB.

The CPU time used by the ENMM and NMM are listed in Table 5.2, in which the

total time is set 1.2 ms and the stress wave with frequency of 5000 Hz is employed to

study the wave propagation in the modelling. We can find that the efficiency of the

ENMM is over four times of the NMM when the same time step size is used. Even

though the different time steps are applied in the NMM and ENMM, such as 0.5μs used

in the ENMM and 1.0μs in the NMM with the mesh ratio 1/32, the ENMM is more

efficient than that of the NMM as well.



5.108

Table 5.2 Comparison of CPU cost between the NMM and ENMM.

Mesh ratio 1/4 1/8 1/12 1/16 1/32

Δt=0.1μs NMM(hr) 0.2731 0.2971 0.3236 0.3406 0.4295

ENMM(hr) 0.0436 0.0434 0.0547 0.0572 0.1047

Efficiency 6.264 6.846 5.916 5.955 4.103

Δt=0.5μs NMM(hr) 0.0603 0.0565 0.0576 0.0606 0.0817

ENMM(hr) 0.0079 0.0098 0.0116 0.0156 0.0218

Efficiency 7.6317 5.766 4.961 3.890 3.752

Δt=1.0μs NMM(hr) 0.0274 0.03 0.028 0.0294 0.0393

ENMM(hr) 0.0041 0.0045 0.005 0.0056 0.0099

Efficiency 6.683 6.667 5.600 5.250 3.970

Δt=2.0μs NMM(hr) 0.014 0.0144 0.0148 0.0156 0.0179

ENMM(hr) 0.0022 0.0023 0.0025 0.0028 0.0051

Efficiency 6.364 6.261 5.920 5.572 3.510

5.4 STRESS WAVE PROPAGATION THROUGH FRACTURED

ROCK MASS

When the wave propagates through fractured rock mass, both the reflection and the

transmission waves will be generated. In this section, two situations are considered, one

is to verify the transmission/reflection coefficients for the wave propagation through a

single joint in a homogeneous rock bar and the other one is the wave propagating

through the joint between different materials. Furthermore, the stress wave propagation

through the multiple parallel joints is investigated using the NMM and ENMM,

respectively.

5.4.1 P-wave propagation through homogeneous medium

When a wave transmits through a joint, the wave transmission, wave reflection and

energy dissipation occur at the joint. Pyrak-Nolte (1988) proposed a complete solution

for the effect of a joint on seismic waves based on a linear joint stress-deformation

relationship. In the case of a wave normal to the joint, the absolute values of the

reflection and transmission coefficients are written as

2/1

2 1)//2(

1

ZkR (5.8)



5.109

2/1

2

2

1)//2(

)//2(

Zk

ZkT (5.9)

where k is the joint stiffness; is the wave angular frequency; Z is the wave

impedance ( c );c is wave propagation velocity; is the density.

Fig. 5.7 shows the NMM and ENMM modelling results comparing with the Pyrak-

Nolte’s analytical solution in Eqs. (5.8) and (5.9). A normalized joint stiffness, )//( Zkn,

is used to represent the x -value where nk is joint normal stiffness. It indicates that both

the NMM and ENMM modelling results are consistent with the analytical solution.

Figure 5.7 Comparison between the simulated results and the Pyrak-Nolte’s analytical solution for

a single joint.

5.4.2 P-wave propagation through joint between different mediums

When the wave propagates through the joint between two mediums, conditions

should be met at the interface for both equilibrium and compatibility. Those two

conditions can be further expressed as (Bedford and Drumheller 1994)

irEE

EE

1122

1122 (5.10)

itEE

E

1122

222 (5.11)

where i ,

r and t are the incident, reflected, and transmitted stress form, respectively.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12Normalised joint stiffness, kn/Z/ω

Coe

ffic

ient

s, R

, T

Analytical solution, RAnalytical solution, TNMM simulation, RNMM simulation, TENMM simulation, RENMM simulation, T



5.110

In the numerical simulation, the rock bar with 2.0m is divided into two half with

different materials. All the related parameters are same with the mesh ratio 1/32 and

only Young’s modulus of the second 1.0m part is modified to be 4 times that of the first

part. The theoretical solution for the reflected r is 0.33

i with a negative sign while the

transmitted t is 1.33

i .

In Fig. 5.8, the wave forms at two measured points (0.2m and 1.01m away from the

wave source) are compared with the theoretical solutions. It can be seen that the

numerical results on both the reflected and transmitted waves agree well with the

analytical solutions.

Figure 5.8 Comparison between the simulated results of NMM and ENMM.

5.4.3 Stress wave propagation through the multiple parallel joints

The wave propagation through multiple joints is more complicated than that at a

single joint. When a stress wave propagates through multiple joints, each joint transmits

and reflects the wave, and causes time delay, thus multiple transmissions and reflections

occur. Based on the previous studies, the wave propagation through 5 parallel joints

with the same spacing is investigated for the half sine P-wave with frequency of 5000

Hz using the NMM and ENMM, respectively. The joint normal and shear stiffness are

assumed as 45 and 10 GPa, respectively. Joint properties such as friction angle,

-1.6

-1.2

-0.8

-0.4

0

0.4

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012

Pre

ssu

re (

MP

a)

Time (s)

NMM, D=0.2m

NMM, D=1.01m

ENMM, D=0.2m

ENMM, D=1.01m

Analytic solution



5.111

cohesion and tensile strength are taken account as 15 degree, 0 GPa and 0 GPa,

respectively. Five joints uniformly locate at 1.0, 1.2, 1.4, 1.6 and 1.8 m of the bar. A

measure point is set behind the fourth joint (at 1.8 m) to record the transmitted wave.

Fig. 5.9 shows the simulated results at the measure points by the NMM and ENMM. It

can be seen that based on both the NMM and ENMM, the stress wave is superposed

from the transmitted and reflected waves with different time delay through the 5 joints.

Figure 5.9 Simulated results at measure point by the NMM and ENMM.

To investigate the efficiency of the ENMM comparing with the NMM, the time

steps of 0.1μs and 0.5μs with mesh ratios of 1/32 and 1/64 are employed in the

simulations, respectively. When the joints in the modelling are taken into account, the

contact computation is a time-consuming job as the repeated OCI solvers to achieve the

no-penetration and no-tension requirement. The CPU time of the simulations by the

NMM and ENMM is presented in Table 5.3. It is shown that the ENMM is more

efficient than that of the NMM to solve the discontinuous problems comparing with the

continuous problems. In particular, the more joints locate in the modelling, the higher

efficiency to the ENMM comparing with the NMM.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.0002 0.0004 0.0006 0.0008 0.001 0.0012

Pre

ssu

re (

MP

a)

Time (s)

NMM ENMM



5.112

Table 5.3 CPU cost comparison between the NMM and ENMM.

Mesh ratio 1/32 1/64

Δt=0.1μs NMM (hr) 0.4382 0.9623

ENMM (hr) 0.0841 0.1755

Efficiency 5.211 5.484

Δt=0.5μs NMM (hr) 0.0889 0.1748

ENMM (hr) 0.0167 0.0376

Efficiency 5.324 4.649

5.5 SEISMIC WAVE EFFECT FOR A FRACTURED ROCK

SLOPE

To further verify the capability of the proposed ENMM in modelling of wave

propagation in fractured rock mass, a dynamic stability assessment for a fractured rock

slope under seismic effect is analysed in this section.

Fig. 5.10 shows the schematic cross-section of the rock slope. The asperity of the

slope is denoted by abcdefghi , and the fractured zone locates at the crest of the slope.

The span of the slope is 100 m, the height of the left and right edges are 30 m and 9 m,

respectively. A horizontal seismic acceleration history as shown in Fig. 5.11 is applied

on the bottom of the slope to simulate a seismic effect. The physical material properties

of the rock, such as Young’s modulus, Poisson’s ratio and mass density are assumed as

10 GPa, 0.2 and 2200 kg/m3, respectively. To get access to computational accuracy by

the seismic wave effect, the fractured joints are simulated with the normal contact

stiffness of 20 GPa and 12 GPa in the NMM and ENMM, respectively. The friction

angle, cohesion and tensile strength of the joint are assumed as 20 degree, 0 GPa and 0

GPa, respectively.



5.113

Figure 5.10 Schematic cross-section of the rock slope

Figure 5.11 Record of acceleration of the seismic wave.

To obtain the effective accuracy of the computations, the simulation for the

fractured slope with the mesh, as shown in Fig. 5.12, is performed. The discrete blocks

in the dotted box are represented by two sets joints by the dips of 90 and 32 degrees as

the fractured zone. The input seismic wave acts on the fixed points at the bottom of the

slope. Fig. 5.13 shows the simulated results of final state of the discrete blocks by the

NMM and ENMM. The measure point displacement simulated by the ENMM and

NMM are plotted in Fig. 5.14. The final displacement of the measure point under the

seismic effect is 6.437 m in the ENMM, which is close to that of 6.212 m in the NMM.

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25

Acc

eler

atio

n (

g)

Time (s)

100m

9m

30m

a b c d

e f

g h i Fractured Zone



5.114

Figure 5.12 Rock slope model in the NMM and ENMM.

In terms of the efficiency, the ENMM is obvious efficient compared with the

NMM since the explicit time integration is used. In the NMM, the open-close algorithm

is employed to obtain contact convergence, in which the no-penetration and no-tension

contact constraint is imposed at each detected contact point. To satisfy this constraint,

the open-close iterative solver proceeds until zero penetration at any contact point. This

procedure is time-consuming and cockamamie in the computations. As the increase of

scale of the discrete blocks, the efficiency is declined dramatically, even the size of time

step decreases less than the explicit one to obtain the effective contact accuracy. In the

present simulation, although time step in the NMM (=1.0 ms) is five times of the

ENMM used (=0.2 ms), the CPU cost of the ENMM (=0.729 hour) is more efficient

than that of the NMM (=1.387 hours).

Measured point



5.115

Figure 5.13 Simulated results of the failure of the rock slope using the NMM and ENMM (Time

step in NMM =1ms, ENMM =0.2ms; total time =20s).

Figure 5.14 Measure point displacement simulated by the NMM and ENMM.

5.6 SUMMARY

In this chapter, a modified version of the NMM based on an explicit time

integration algorithm is proposed. The calibration study of the ENMM on P-wave

propagation across a rock bar has been conducted. Various considerations in the

numerical simulations are discussed and parametric studies have been carried out to

obtain an insight into the influencing factors in wave propagation simulation. The

0

2

4

6

8

0 4 8 12 16 20

Dis

pla

cem

ent

(m)

Time (s)

NMM ENMM

NMM, CPU time= 1.387 hrs

ENMM, CPU time= 0.729 hrs



5.116

numerical results from the ENMM and NMM modelling are accordant well with the

theoretical solutions. The mesh ratio is regarded as one of the major factors influencing

the simulation accuracy. With the consideration of both calculation accuracy and

efficiency, a mesh ratio of 1/16 is recommended for one dimension ENMM analysis.

Furthermore, the selection of a suitable time step depends on the internal element

system and the contact transfer between the interfaces. With the decrease of the time

step increment, the results become more accurate for the incident wave. In terms of

efficiency, the ENMM is more efficient than that of the NMM, even though the

different time steps are used. To further verify the capability of the proposed ENMM in

modelling of seismic wave effect in fractured rock mass, a dynamic stability assessment

for fractured rock slope under seismic effect is analysed as well. The simulated results

show that the computational efficiency of the proposed ENMM can be significantly

improved, while without losing the accuracy, comparing to the original implicit version

of the NMM. The various studies presented in this paper demonstrate a promising future

for the ENMM method in modelling stress wave propagation and other dynamic

problems for rock engineering.



6.117

CHAPTER 6. THE TEMPORAL COUPLED EXPLICIT-

IMPLICIT ALGORITHM FOR DYNAMIC PROBLEMS

USING THE NUMERICAL MANIFOLD METHOD

6.1 INTRODUCTION

The efficiency and accuracy are usually regarded as two indices to check the

capability of a numerical method in terms of time integration for dynamic problems. In

general, there are two classes of time integration algorithms for dynamic problems:

implicit and explicit (Gelin, Boulmane, and Boisse 1995). Implicit algorithms methods,

such as the continuum-based finite element method (FEM) and discontinuum-based

discontinuous deformation analysis (DDA), tend to be numerical stable permitting

larger steps, but the computational cost is high and storage requirements incline to

increase dramatically with the contacts between elements and these degrees of freedom

(DOFs). Thus, the algorithms are suited to simulate the lower dynamics problems with

less non-linearities, resulting in more numerical stability and accuracy (Gelin et al. 1995;

Yang et al. 1995; Sun et al. 2000). On the contrary, explicit algorithms such as finite

difference method (FDM) and discrete element method (DEM) attempt to be

inexpensive per step and require less storage than implicit algorithms, but numerical

stability requires that small steps be employed, thus, the algorithms generally are used

for highly non-linear problems with many DOFs (Subbaraj and Dokainish 1989a, 1989b;

Yang et al. 1995). To take advantage of the merits of implicit and explicit algorithms,

many methods have been developed in temporal and spatial discretizations, in which it

is attempted to simultaneously achieve the maximum contributions of both classes of

algorithms. These studies normally concentrated on the continuum-based methods with

prescribed contact states.

When more contact problems are involved in the discontinuum-based methods

such as NMM and DEM, the efficiency is significantly declined. Thus, how to treat the

contact problems balancing the efficiency and accuracy, an appropriate time integration



6.118

algorithm is required. In general, low-frequency motion states, less-nonlinear contact

states and contact without external forces can be treated as quasistatic problems to apply

the implicit time integration in the NMM. On the other hand, when high non-linear

contact problems such as high velocity impact problems, blasting problems and

earthquake problems take places, the implicit schemes can not calculate and simulate

the phases change during the contact efficiently and accurately as the large time steps

are used. The explicit algorithms can be employed to solve the above non-linear contact

cases efficiently. On account of the different phases change in the conatct, the

corresponding algorithms can be used to maximize the merits of both algorithms. Thus,

the corresponding temporal coupled algorithm is required to combine the explicit and

implicit algorithms effectively.

In this chapter, the numerical manifold method (NMM) is taken into account to

combine the two time integration algorithms. During the previous two decades, the

NMM has been widely carried out for solving various structural dynamic problems. The

traditional NMM is originally proposed by Shi (1991, 1992) based on topological

manifold and unifies both FEM and DDA. It employs the implicit time integration and

open-close contact iteration for the simulations of complicated dynamic problems in

rock engineering and rock mechanics. Since the implicit scheme requires the assembling

of the coupled global stiffness matrix for the governing equations, which may involve

many thousand DOFs, especially when such more contact problems and nonlinear

problems are encountered, the computational cost can be increased dramatically. This

has motivated us toward developing more efficient computational algorithm for the

NMM. Thus, the choice of an appropriate algorithm is essential to ensure efficiency and

robustness of the numerical simulations, but the difficulty resides in being able to

combine robustness, accuracy, stability and efficiency of the algorithms. Furthermore,

Newmark-β family methods (Newmark 1959) are employed. When the different

parameters of β and γ are taken into account, the different algorithms can be selected to

simulate the corresponding dynamic problems. The distinction between explicit and

implicit where we have considered is that the explicit uses a diagonal mass matrix and

the implicit applies a consistent inertia matrix (Liu and Belytschko 1982). Then, a



6.119

temporal coupled E-I algorithm for the NMM based on the dual cover system is

proposed, in which different time steps, time integration schemes and contact

algorithms are applied in temporal discretization. Then, some calibration examples and

numerical simulations are studied to validate the coupled E-I algorithms.

6.2 THE NMM AND ITS COVER SYSTEM

In the traditional NMM, one manifold element is generated through a set of

overlapping covers, which is the distinct characteristic differs from other numerical

methods. The cover system in the NMM provides a robust tool for both continuous and

discontinuous problems. In this section, the fundamentals of the NMM and its cover

system are described respectively.

6.2.1 Dual cover system in the NMM

Since the NMM doesn’t require MC sides to coincide with the material boundaries

and internal cracks, arbitrary shapes of MCs can be employed in the NMM simulation.

For convenience, a regularly structured mesh is employed in the NMM which is similar

as that in the FEM. As is shown in Fig. 6.1, in which a regularly-patterned triangular

mesh is presented, in which each MC is defined through six triangular elements sharing

a common node (i.e. nodal star). Each cover has two degree of freedom is similar as

node property in the FEM, each element formed by the overlapping of three

neighbouring hexagonal covers has six degree of freedom for the second order time

integration. The mathematical mesh covers the whole physical domains to form PC

system, the common areas denoted by ie , je and

ke are formed by the neighbouring

three hexagonal MCs combined with the material domains. When the linear triangular

element weight function is applied based on cover system, the global displacement

function over a ME can be expressed as

ei

iie yx，yxUyxyxu

),(),(),(),(3

1

(6.1)

where ),( yxi is the weight function over the three associated MCs, ),( yxUi is the

displacement function on the three associated PCs. Here, it is the cover system makes

the solution for both continuous and discontinuous problems practicability without any



6.120

re-meshing technique used in the FEM. Moreover, three separate material domains

constituted by smooth curved boundaries actually is approximated as polygons

composed of straight edges in order to apply the simply integration over the ME.

Figure 6.1 A regularly-patterned triangular mesh in the NMM.

6.2.2 Contact problems in the NMM

The traditional NMM uses a penalty-constrain approach in which the contact is

assumed to be rigid. Numerically, it can be carried out by means of adding or

subtracting penalty value to the contact terms into the global equation to produce

contact stiffness matrix. The terms are selected by checking the previous and current

states of the contact, each controlled by a vector of values i = -1, 0 and 1, i =1, 2, ...

n , respectively. Accordingly, within each time step, the assembled global equations are

solved iteratively by repeatedly adding and removing contact springs until each of the

contacts converges to a constant state, which is known as OCIs proposed by Shi (1988).

The purpose of the OCI is to achieve the contact modes identically. In the NMM,

the no-penetration and no-tension contact constraint is imposed at each detected contact

point. To satisfy this constraint, the open-close iterative solver proceeds until this is no

penetration at any contact point, and number of OCIs required for contact convergence

in each time step dramatically increases for problems involving more contact elements,

which increases the computational cost as well. To reveal the efficiency of the OCI, a

numerical example of rock slope referred in (Khan, Riahi, and Curran 2010) is

investigated, in which the OCI used by the DDA compares with contact algorithm

MC



6.121

applied in the DEM. The simulated results show that the OCI is one of the most

important factors affecting its computational efficiency. Since several times of iterations

are required in each time step, the computational cost will become more expensive.

There are two major issues of this approach; (i) coming up with an appropriate

penalty value for an arbitrary problem is difficult, and (ii) the open-close iterative

algorithm used to enforce penalty constraint is a time consuming process. These two

issues make NMM penalty method computationally expensive. Moreover, the

assumption that contacts are infinitely rigid is less realistic because blocks undergo

some local deformation at contact points that must be accounted for in computing

contact forces. By definition, this approach assumes zero thickness for the filling

material between the contact interfaces.

6.3 TEMPORAL COUPLED EXPLICIT-IMPLICIT ALGORITHM

IN THE NMM

6.3.1 Summary of equations of motion and time integration

Generally, the equation of motion can be derived in the strong form as

vb (6.2)

in which is the symmetric Cauchy stress tensor, is the gradient operator, is the

density of material, b is the body forces and v is the velocity, respectively. Constitutive

equations are required to couple the Cauchy stress tensor and the density to the

kinematics of the deformation, Eq. (6.2) can be written as the well-known discrete

equation

}{}]{[}]{[ extFuKuM (6.3)

where ][M is mass matrix, ][ K is stiffness matrix and }{ extF is the external force vector,

respectively; }{u and }{u are acceleration and displacement vectors, respectively. It is

noted that ][M and ][ K are symmetric, whereas they are also banded and sparse as the

local property of deformation matrix ][T of the MCs in the NMM, and it requires more



6.122

computer memory with the assembly of global stiffness matrices for solving the

governing equations.

In general, two classified methods for direct integration of the equations of motion

can be considered: explicit methods, in which the accelerations are taken into account

and then integrated to obtain the displacements; and implicit methods, in which the

equations of the motion combine with the time integration operator in order to gain

displacements directly. Both methods are developed from different formulas based on

the Newmark- methods (Newmark 1959) using an increment of step time t . In the

explicit algorithm of the integration, Eq. (6.3) can be written as

}){}({][}{ int1 FFMu ext (6.4)

where }{ intF is the internal force vector. We apply the explicit central difference

scheme, i.e. 0 and 2/1 , the velocity and displacement can be calculated as

2/11

2/11

2/12/1

2

1

nnn

nnn

nnn

utuu

utuu

utuu

(6.5)

where 2/1nu is the central velocity by half a time step. A big advantage of it is the use of

a diagonal mass matrix (see Section 4.3) to improve the computational efficiency,

whereas it is conditionally stable. Thus, the increment of step time t must be satisfied

by less than a critical step time crt , which can be expressed as

ecrt

max

2

(6.6)

where emax is element maximum angle frequency, which is related with minimum value

of element eigenlength, )1,0( is the coefficient to determine t based on different

damping ratios.

When the implicit time integration is carried out by minimizing the potential

energy associated with an increment of time t , which is similar to the Newmark-

methods with parameters 2/1 and 1 correspond to the constant acceleration

scheme. Then, substituting the parameters into Eq. (6.6), we can have



6.123

}{][2

}{}{)(

][2][

2 next ut

MFu

t

MK

(6.7)

where nn uuu 1 is the displacement increment from the current time step t to

)( tt corresponds to the step increment of n to )1( n . Eq. (6.7) requires assembling

the global stiffness matrix and solving the coupled system of equations using successive

over relaxation iteration, which increases the computational cost even the OCI is

applied into the contact problems. Since the implicit scheme which provides numerical

damping, the explicit damping term C is assumed to be zero. It is noted that the implicit

scheme is stable for larger time step, while the appropriate selection of t is made in

accordance with the required level of numerical damping. For best results from stability

and numerical dissipation point of view, a time-step t is selected as (Doolin and Sitar

2004; Doolin 2005):

et

max

4

(6.8)

6.3.2 The coupled explicit-implicit algorithm

In the computations, time integration discretized in temporal algorithm for the

dynamic problems combines the explicit and implicit algorithms as a coupled method to

expose both advantages at utmost. For the different problems, there are two types of

coupled approaches can be considered: implicit-explicit (E-I) algorithm and explicit-

implicit (I-E) algorithm. When different approaches are employed, the different step

time scale can be applied into the corresponding time integration scheme. In order to

investigate the temporal couples algorithm, the E-I algorithm is taken into account in the

present study. Furthermore, the Newmark- methods with two characteristic

parameters and for all sub-domains are assumed here.

As is shown in Fig. 6.2, the initial diagonal mass matrix and force vector are

constructed for the explicit algorithm, then the explicit central difference method, i.e.

the Newmark method with the parameters 01 and 2/11 , is employed from the

initial step time 0t to

nt at the step number n to simulate the high frequency part of the

dynamic problems. And then, the explicit integration algorithm switches to the implicit,

in which the transfer algorithm is proposed in order to achieve the conservation of the



6.124

kinematic energy from the explicit to implicit integration, and IE DD ,

IE vv and

IE are satisfied for the coupled E-I integration without the element and node

partition. Thus, it is convenient to achieve in the programming code. In the part of the

implicit integration from step time 1nt to rnt , the constant acceleration method with the

parameters 2/12 and 12 is used in the implicit integration by the end of step

)( rn for the low frequency and quasi-static problems. Continuing the explicit

procedure, the initial inertial, stiffness matrix and force vector for the implicit

integration are require to construct again as the difference items in the equations of

motion. It is noted that different step time sizes are adopted before and after the

transition in the couples E-I algorithm in terms of the numerical stability and accuracy,

respectively. Normally, the step time size It in the implicit algorithm is larger than

Et

in the explicit algorithm, which denotes EI tt , 1 is the coefficient to describe

the scale between the implicit and explicit integrations. Sequentially, the transfer

algorithm of the coupled E-I integration is exposed and discussed in the following

section.

Figure 6.2 Transfer algorithm from the explicit to implicit integration.

6.3.3 Transfer algorithm for the E-I algorithm

In the coupled E-I method, we employ the explicit algorithm to model motion of

the system at the early stage, followed by the implicit algorithm to simulate the

subsequent motions of the system. Thus, an explicit physical model in the NMM will be

transferred to the implicit one at a certain time so that the coupled method is more

Initial diagonal mass matrix and force matrix for explicit

Transfer explicit to implicit

Initial inertia, stiffness matrix and force matrix for implicit

Newmark method

Newmark method



6.125

efficient. In the transition, the geometric configurations, physical and mechanical

parameters, and status, including stress state and velocities, are consistent and continue.

Therefore, the transfer algorithm is required to satisfy the kinetic energy and potential

energy conservation from the explicit integration to the implicit one, which can be

represented as

n

i

Iy

Ix

Ii

n

i

Ey

Ex

Ei vvMvvM

1

22

1

22)(

2

1)(

2

1 (6.9)

),,(),,( I

e

Ixy

Iy

Ix

E

e

Exy

Ey

Ex (6.10)

where EiM and I

iM are the i-th explicit and implicit element mass, respectively; Exv ,

Eyv and I

xv , Iyv are the velocity components of an explicit element and implicit element

in the x and y directions, respectively; E

e

Exy

Ey

Ex ),,( and I

e

Ixy

Iy

Ix ),,( are the

explicit and implicit element potential energy, in which Exy

Ey

Ex ,, and I

xyIy

Ix ,, are

stress components of the explicit and implicit elements respectively. Furthermore,

equations of Ii

Ei MM , I

xEx vv , I

yEy vv , I

xEx , I

yEy , I

xyExy are satisfied in the

transfer algorithm to ensure the parameters of the terms are consistent and the

computation is continuous.

When the explicit integration transfers to the implicit one, the time step size control

is required as the difference between them with respect to the numerical stability. In

general case, we choose the coefficient more than ten times is efficient for the

coupled E-I method. In theory, the implicit step time size It is near the certain value

as described in Eq. (6.8) leads to the maximum numerical dissipation and more efficient

computation cost, and if the explicit step time Et is beyond the limited by Eq.(6.6) will

result in the numerical oscillations as the numerical dissipation property of the central

difference method. Even if larger magnitude of is given, the iterative solution and

open-close criteria in the NMM make the used It is close to a certain critical value.

Thus, the large value of is not always a good thing in the coupled E-I method,

especially more contacts occur in the simulations. In the present work, we suggest the

value of is taken the range from 10 to 20 according to the large or small scale of the

studied problems. Since the different styles of equations of the motion between the



6.126

explicit and implicit algorithms, the initial conditions are constructed before and after

the transfer from the explicit to implicit integrations, respectively. It is noted that the

contact detection and contact condition judgment are shared between the explicit and

the implicit algorithms in the overall simulations.

6.4 CONTACT ALGORITHM OF THE COUPLED ALGORITHM

6.4.1 Contact force calculation

As previously mentioned, contact detection and contact force calculations are done

by the NMM. Once contacts have been detected, a contact interaction algorithm is

employed to evaluate contact forces between the contact elements. Contact interaction

between two neighbouring contact elements occurs along the discontinuity loops. At

low contact condition, the discontinuity boundaries may touch only at a few points.

With increasing of penetration distance and elastic deformation of individual interface

asperities occurs, resulting in an increase in the real contact area and hence an increase

in number of contact points. A thorough discussion and formulations of these

approaches can be found in (Munjiza 2004).

For a discrete block system involving m elements, when element i and j have

contacts, cK between i and j can be expressed as

][][

][][cjj

cji

cij

ciic

KK

KKK (6.11)

in which ][ cijK , ji and mji ,2,1, , is defined by the contact spring between the

contact elements i and j, and the value is zero if the elements i and j have no contact.

Since each element is consisted by three associated PCs, thus the matrix ][ cijK is a 6×6

sub-matrix. It is noted that the displacement 3,2,1,)( rD ri on the PCs can be predicted by

the previous step n. Then contact forces associated with ][ cijK on the contact element i

are assembled as

3,2,1;,,2,1,][1

)(

rmjDKIm

jri

cij

ci

(6.12)

The total internal forces on the element i can be represented as



6.127

3,2,1;,,2,1,][~

)(1

rmjDKKI ri

m

j

cij

eiii

(6.13)

in which iI~ is the element internal force vectors and e

iiK is the stiffness matrix of

element. Since each element is formed by the three associated PCs, thus iI~ can be

rewritten as

)3(

)2(

)1(

~

~

~

~

i

i

i

i

I

I

I

I (6.14)

in which )1(

~iI maps the first PC associated the element, the subsequent )2(

~iI and )3(

~iI

map the second and third PCs, respectively. Then, intF at each PC can be assembled by

the associated iI~ on the cover system.

6.4.2 Damping algorithm

Damping algorithm is used to dissipate the excessive energy in the contact

problems due to the use of linear springs between contact elements. Two types of

damping are used: mass-proportional damping and stiffness-proportional damping. The

combined use of two damping forms is usually termed Rayleigh damping, which can be

proved as effective way of considering damping for analysis of structures. Rayleigh

damping can be given by (Bath 1982)

][][][ KbMaC (6.15)

where ][C is the damping matrix; a and b are the given Rayleigh constants, which are

also called as mass proportional and stiffness proportional respectively.

Since the explicit algorithm is conditional stability, it can be shown that the

stability limit for damping item is given in terms of the highest eigenvalue in the

system:

)1(2 2

max

t (6.16)

It is noted that the explicit scheme employs dynamics method to solve the

uncoupled equations, in which the generated kinetic energy can not be neglected. To the

static or quasi-static problems, it requires the physical damping to adsorb the kinetic



6.128

energy of the systems so that the systems achieve stable condition. More details of the

derivation of the damping force can be seen in Section 4.3.

6.5 NUMERICAL EXAMPLES

In order to investigate the validity of the proposed algorithms, some numerical

examples are simulated in terms of temporal respect. In the simulations, the proposed

coupled temporal algorithms for the NMM are calibrated between the simulation results

and analytical solutions, firstly. Then, an examples of open-pit mining slope stability

analysis are taken into account using the proposed E-I algorithms to further extend the

current NMM, which represent the potential of the proposed E-I algorithms for

simulating the larger scale projects in the further research.

6.5.1 Calibration of the temporal coupled E-I algorithm

In order to calibrate the proposed coupled E-I algorithm for the temporal problems,

one Newmark sliding modelling of block sliding under input horizontal acceleration Ha

is studied here. The geometry of the modelling is shown in Fig. 6.3, in which a block

rests on an inclined plane is taken into account as a first approximation of the Newmark

sliding model. The angle of the plane is 31.470. A sinusoidal seismic acceleration Ha is

employed to impose to excitation point as expressed in Eq. (6.17), where g is the gravity

acceleration, t is the simulation duration for the simulation. In this study, we assume the

frictional angle 030 , the total displacement of analytical solutions can be referred in

(Newmark 1965; An, Ning, et al. 2011), then the simulated results of the proposed E-I

NMM can be obtained as shown in Fig. 6.4. It is noted that when the E-I algorithm is

considered, the proposed transfer algorithm is employed from the explicit to implicit

algorithm at the time of st 1 , and the final results of the simulations are well

agreement with the analytical solution.

st

sttga H 10

14sin1.0 (6.17)



6.129

Figure 6.3 Geometry of the Newmark sliding modeling.

Figure 6.4 Block displacement under horizontal ground acceleration.

6.5.2 Open-pit mining stability analysis

An open-pit mine often encounters a fault or fracture zone or weak discontinuous

planes consisted by jointed rock masses. Joints usually occur in sets which are more or

less parallel and regularly spaced. There are usually several sets in very different

directions so that the rock mass is broken up into a blocky system (Jaeger and Cook

1979). Such geology to an open-pit mining slope have a great influence on the slope

stability. The general approach, when investigating the deformation and failure

characteristics of a slope, is to carry out numerical analysis, varying the input

parameters. The results of the numerical analysis can be verified by comparing them

with the measured displacement data at the mine site or by carrying out a test on a

laboratory-scale model. When every parameter is properly scaled down in accordance

with the scaled-down geometry, it is possible to represent the geological conditions of

0

0.02

0.04

0.06

0.08

0.1

0 0.5 1 1.5 2

Displ. (m)

Time (s)

Analytical

E-INMM

Transfer algorithm

Explicit algorithm Implicit algorithm

Sliding block

Excitation points



6.130

the operation site with the scaled model (Jeon et al. 2004). To investigate the stability of

open-pit mining, the proposed E-I algorithm is employed in the present study.

In this simulation, one open-pit mine slope modeling is assumed to study the

stability using the proposed E-I algorithm. As shown in Fig. 6.5, there are 9 layers

denoted by 1# to 9# separate the whole modeling, in which we assume the layer 4# is

the fracture zone constituted by many discontinuous joints. The inclined angle of the

slope is 42° and drop is 120 m. In order to investigate the effect of the fracture zone to

stability of the slope, two models of layer 4# are represented in Fig. 6.6. Integrated

Model considers the whole layer as one domain, on the other hand, Refined Model adds

more joints into the layer to approach the realistic condition, in which two sets of joints

with nearly perpendicular orientation (i.e. 32.470 and 139.090) are constructed as seen

Figs. 6.6(b) and (c) to simulate the fractured zone of the open-pit slope.

Figure 6.5 Geology section of the open-pit mining.

Traditional methods apply to the slope stability analysis is to determine the factor

of stability (FoS) of the slope using the limit equilibrium method (LEM) without

considering the effect of the dynamic loading with time history (i.e. seismic loading,

blasting loading, etc). Here, the FoS is computed using the LEM to the integated model,

the results can be seen in Table 6.1. We can find that the determined values of FoS of

the Integrated Model (IM) is over one and the slope approaches to the stability. It is

noted that FoS may be less than 1 when the RM is employed, since the IM is simplified

into a block without considering the fracture joints. When the model with joints are

taken into consideration, the slope tends to be more instable, such as the case of RM2 is

failure when the joint frictional angle is up to ϕ=150 .Thus the RM is closer to the

Layer 2#

Layer 8# Layer 7# Layer 6#

Layer 5#

Layer 3# Layer 4#

Layer 9#

Layer 1# Altitude (m)

160

80

10 10

40 40

120



6.131

realistic property of the fracture zone 4#. The above discussion presents that the RM

determines the low limit of the FoS and should be take into account to investigate and

design the slope of open-pit mining

Figure 6.6 Study model of the layer 4#: a. Integrated Model; b. Refined Model 1; c. Refined Model

2.

Table 6.1 FoS using LEM by the integrated models.

Study Model Integrated model

ϕ=10° ϕ=15°

FoS 1.436 2.182

Continue the above study, the RM is used into the simulation of slope stability

analysis for the seismic loading using the proposed E-I algorithm. In order to further

investigate the stability of the slope under the earthquake, a stochastic horizontal

seismic acceleration (Ml = 6.4) with maximum value of 0.2g is applied into this

simulation by the cases of ϕ=100 and ϕ=150 as shown in Fig. 6.7.The detailed of

physical parameters in this simulation are list in Table 6.2.

In this study, total time assumed as 20s. The proposed E-algorithm is used into the

simulation at the first 10s of seismic loading, then, the following I-algorithm is applied

Layer 4#

a.

Integrated Model

Refined Model 2

Measured point 3

c.

Refined Model 1

Measured point 3

b.

Measured point 1

Measured point 1

Measured point 2

Measured point 2



6.132

into the simulation in order to simulate the slope efficiently. The simulation results for

RM1 and RM2 using the E-I algorithm with the case of ϕ=100 and ϕ=150 can be seen in

Figs. 6.8 and 6.9. The fracture zone (i.e. layer 4#) in the RM2 is unstable and slides

along the interface between fracture zone and layer 5# for the both cases of ϕ=100 and

ϕ=150 when the seismic loading is employed, but the RM1 is nearly stable at the case of

ϕ=150. It is shown that the simulated results are different with the traditional LEM and

easily to approach the realistic conditions of the slopes. The instability focuses on the

fracture zone and grows at the beginning of the dynamic loading, then the other blocks

slide following the movement of the fracture zone. We can find that the fracture zone is

the key effect on the slope stability in the open-pit mining engineering and should be

paid attention more seriously. Furthermore, field investigation is important to find the

location of the fracture zone and the properties of the joints of the fracture zone in order

to study open-pit mining slope truly and correctly.

Figure 6.7 A stochastic horizontal seism acceleration.

Table 6.2 Input parameters for the simulation of the modeling.

Physical property Parameter

Unit weight (kN/ m3) 26.0

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 5 10 15 20

Acc

eler

atio

n (

g)

Time (s)



6.133

Young’s modulus (GPa) 1.0

Possion’s ratio 0.2

Internal friction angle(deg.) 20.0

Cohesion (MPa) 5.0

Tensile strength (MPa) 0.0


Joint shear stiffness (GPa) 0.5

To investigate the instability of the fracture zone under earthquake loading, the

measure points selected in the fracture zone and the displacements of them at both cases

of ϕ=100 and ϕ=150 are presented in Figs 6.10 to 6.15, respectively, in which the

original NMM, proposed ENMM and E-I NMM are used to simulate the open-pit

modeling. It is noted that the different time steps are chosen to the explicit and implicit

algorithms with respect to the numerical properties of the time integrations. We select

0.5 ms to simulate the slope in the ENMM, 5.0 ms in the NMM and 0.5 ms in the first

explicit part of the E-INMM, 5.0 ms in the second implicit part of the E-INMM. The

simulated results of the measure point are nearly identical in the three approaches with

both case of ϕ=100 and ϕ=150. It is verified that the proposed E-INMM satisfies the

computational accuracy comparing with the original NMM. We can find that the slope

is instable at the case of ϕ=100 whether static or dynamic states, but the slope

approaches to be stable after the seismic loading at the case of ϕ=150. Thus, the

fractured zone should be taken into account to the design of the open-pit slope to

improve the stability of slope.



6.134

Figure 6.8 Simulation results for Refined Model 1 (Total time: 20s): (a) ϕ=100; (b) ϕ=150.

Figure 6.9 Simulation results for Refined Model 2 (Total time: 20s): (a) ϕ=100; (b) ϕ=150.

(a) ϕ=100

(a) ϕ=100

(b) ϕ=150

(b) ϕ=150



6.135

(a) ϕ=100;

(b) ϕ=150.

Figure 6.10 Measured point 1 with model 1: (a) ϕ=100; (b) ϕ=150.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM



6.136

(a) ϕ=100;

(b) ϕ=150.


0

2

4

6

8

10

12

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

0.2

0.4

0.6

0.8

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM



6.137

(a) ϕ=100;

(b) ϕ=150.


0

50

100

150

200

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

1

2

3

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM



6.138

(a) ϕ=100;

(b) ϕ=150.


0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

0.2

0.4

0.6

0.8

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM



6.139

(a) ϕ=100;

(b) ϕ=150.


0

5

10

15

20

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM



6.140

(a) ϕ=100;

(b) ϕ=150.

Figure 6.15 Measured point with model 2: (a) ϕ=100; (b) ϕ=150.

With respect to the efficiency of the proposed algorithms, CPU time is taken into

consideration to check the computational cost of the algorithms. In order to track CPU

time for each time step, a high resolution timer function which can measure up to

1/100000th of a second is added to keep track of CPU time for each time step during the

analysis. All three algorithms are run on the same computer with the system

configuration: processor speed = 4.0 GHz and RAM = 4.0 GB. As represented in Table

6.3, the proposed E-I algorithm is more efficient comparing the explicit and implicit

algorithms in the model of RM1 and RM2 with both cases of ϕ=100 and ϕ=150. In

particular, E-I algorithm can be considered as one computational criteria for the large

0

100

200

300

0 5 10 15 20

Displacement (m

)

Time (s)

NMM

ENMM

E‐INMM

0

40

80

120

160

0 5 10 15 20

Displacement (m

)

Time (s)

ENMM

ENMM

E‐INMM



6.141

scale engineering as it combines the merits of both the explicit and implicit algorithms

in terms of accuracy and efficiency of the computations dramatically.

Table 6.3 CPU cost for the different study cases (hr.).

6.6 SUMMARY

The temporal coupled explicit and implicit algorithm for the numerical manifold

method (NMM) is proposed in this chapter. The time integration schemes, transfer

algorithm, contact algorithm and damping algorithm are studied in the temporal coupled

E-I algorithm to merge both merits of the explicit and implicit algorithms in terms of

efficiency and accuracy. Then, some numerical examples are simulated using the

proposed coupled algorithms, in which one calibration example is studied with respect

to the coupled temporal based on the dual cover system. One numerical example of

open-pit slope seismic stability analysis using the coupled E-I algorithm is investigated

as well. The simulated results are well agreement with the implicit and explicit

algorithms simulations, but the efficiency is improved evidently. It is predicted that the

couple E-I algorithm proposed in the present paper can be applied into larger scales

engineering systems to combine the merits of both the implicit and explicit algorithms

in the NMM.

Study

Case

10° 15° ICPU ECPU IECPU ICPU ECPU

IECPU

Model 1 1.467 1.376 0.882 1.816 1.515 0.911

Model 2 1.485 1.501 0.875 1.271 1.326 0.809



7.142

CHAPTER 7. THE SPATIAL COUPLED EXPLICIT-

IMPLICIT ALGORITHM FOR DYNAMIC PROBLEMS

USING THE NUMERICAL MANIFOLD METHOD

7.1 INTRODUCTION

In dynamic analysis, there are two general classes of algorithms can be referred:

explicit and implicit. The explicit algorithm in spatial discretization using finite or

discrete elements is very widespread, especially for contact or impact problems. Since

an explicit algorithm allows the implementation of large-scale models with a relatively

affordable computational cost, and the conditional stability property is not a matter if

the time step satisfies the equation of the critical time step. However, the smallest

element size in the meshes determines the time step for the whole system, thus it is

necessary to develop an approach, in which the explicit algorithm can be employed

involving different time steps in different sub-domains for the system. On the other

hand, the implicit algorithm can use larger time steps in terms of numerical

unconditionally stable, but for the complex large-scale problems, the solution of system

equations involving too many thousand degrees of freedom (DOFs) can be costly using

the implicit algorithm. Therefore, more efficient computational algorithm is required to

develop for the dynamic analysis, in which some problems applying implicit algorithms

are very efficient and others employing explicit algorithms are very efficient.

In the computations, the explicit algorithms tend to be inexpensive per step and

require less storage than implicit algorithms, but numerical stability requires that small

steps be employed, thus, they generally used for highly non-linear problems with many

DOFs (Subbaraj and Dokainish 1989a, 1989b; Yang et al. 1995). On the contrary, the

implicit algorithms tend to be numerical stable, but the computational cost per step is

high, and storage requirements tend to increase dramatically with the contacts between

elements and these DOFs. Thus, they are suited to simulate the lower dynamics

problems with less non-linearities, resulting in more numerical stability and accuracy



7.143

(Yang et al. 1995; Gelin et al. 1995; Sun et al. 2000). To take advantage of the merits of

implicit and explicit algorithms, many methods have been developed in temporal and

spatial discretization, in which it is attempted to simultaneously achieve the maximum

contributions of both classes of algorithms. Belytschko and Mullen (Belytschko and

Mullen 1976, 1978b) proposed an explicit-implicit (E-I) nodal partition and proved the

conditional stability of E-I partitions using energy methods and represented the time

step is limited strictly by the maximum frequency in the explicit partition of the mesh.

Hughes and Liu (1978) proposed an alternate element-by-element E-I partitions, in

which a similar stability condition is proven for the algorithm. Liu and Belytschko

(1982) proposed a general mixed time E-I partition procedure which permits different

time steps and different integration methods to be used in different parts of the semi-

discrete equations. Belytschko and Mullen (1978a) proposed a multi-time step

integration method involving different time steps in different zones of the model, in

which the nodal partition approach is employed for E-I systems and linearly interpolated

displacements at the interface. It differs from the referred mixed methods, which consist

of defined zones where different algorithms apply, but with a single time step defined

for the whole domain. Various other improvements in transient algorithms have also

been achieved referred in (Hughes, Pister, and Taylor 1979; Muller and Hughes 1984;

Smolinski, Belytschko, and Neal 1988; Miranda, Ferencz, and Hughes 1989)

(Belytschko and Lu 1992; Sotelino 1994; Smolinski, Sleith, and Belytschko 1996;

Gravouil and Combescure 2001).

To the time integration schemes, Newmark-β family methods (Newmark 1959,

1965) are adopted. When different parameters of β and γ are taken into account, the

different algorithms between the explicit and implicit schemes can be selected to

simulate the corresponding dynamic problems. When the two parameters of β and γ are

identical for all sub-domains, the distinction between explicit and implicit where we

have considered is that the explicit uses a diagonal mass matrix and the implicit applies

a consistent inertia matrix (Liu and Belytschko 1982), respectively. These algorithms

are used for first-order systems (Belytschko, Smolinski, and Liu 1984) and second-order

systems (Belytschko et al. 1979). In the former case, a stability study of the algorithm



7.144

with mixed method is proposed for the E-I algorithm, but no extension to this case with

element partition is proposed; for the latter case, there is no general stability analysis in

the E-I algorithm with the mixed method. Moreover, other techniques can be found to

couple arbitrary numerical schemes of the Newmark family in each sub-domain with

different time-steps (Combescure et al. 1998; Combescure and Gravouil 1999).

In the present study, the numerical manifold method (NMM) is taken into account

to couple the both algorithms. The traditional NMM is original proposed by Shi (1991,

1992, 1996, 1997) based on topological manifold and unifies both finite element method

(FEM) and discontinuous deformation analysis (DDA) (Shi 1988), which employs the

implicit time integration and open-close contact iteration for the modelling of

complicated dynamic problems in rock engineering and rock mechanics. However, the

coupled global equations may involve many thousand DOFs, especially such more

discontinuous contact problems that nonlinear problems encountered, the computational

cost can be increased dramatically. This has motivated research toward developing more

efficient computer algorithms for the NMM analysis. Thus, the choice of an appropriate

algorithm is an essential criterion to ensure efficiency and robustness of the numerical

simulations, in which the difficulty resides in being able to combine robustness,

accuracy, stability and efficiency of the algorithms. In this study, we present a coupled

E-I algorithm for the NMM based on the onefold cover system, in which different time

steps, time integration schemes and contact algorithms are applied in spatial

discretization. To illustrate the proposed algorithm systematically, this chapter is

basically organized by 5 sections. In section 7.2, the explicit and implicit algorithms are

investigated based on the dual cover system of the NMM. In section 7.3, an alternative

coupled E-I algorithms is proposed with respect to the spatial discretization, in which

onefold cover system is proposed and derived in detail, respectively. Then, in section

7.4, some calibration examples and numerical simulations are studied to validate the

coupled E-I algorithms in terms of the accuracy and efficiency. Section 7.5 concludes

and summaries the spatial coupled E-I algorithms.



7.145

7.2 COUPLED ALGORITHM

7.2.1 Summary of equations of motion and time integration

Generally, the equation of motion can be derived in the strong form as

vb (7.1)

in which is the symmetric Cauchy stress tensor, is the gradient operator, is the

density of material, b is the body forces and v is the velocity, respectively. Constitutive

equations are required to couple the Cauchy stress tensor and the density to the

kinematics of the deformation. Eq. (7.1) can be multiplied by weight function w and be

integrated over an arbitrary material domain . We employ the principle of virtual

work and rewrite Eq. (7.1), the corresponding weak form can be obtained as

t

tdwbdwdwdvw : (7.2)

where nt is the traction vector, n is the outward normal vector of the boundary, t

is the traction boundary. As is shown in Fig. 7.1, an elastic body with a traction vector

t is taken into account. The boundary ctu is constituted by the prescribed

displacement boundary u , the traction boundary

t and discontinuous surface c ,

respectively. And the boundary needs to satisfy

TN ttt on t (7.3)

uu on u (7.4)

where Nt and

Tt are the normal and tangential vector components of t along the

boundary t when the contact problem is considered, u and u are the displacement

vector and the prescribed displacement vector on u , respectively.



7.146

Figure 7.1 An elastic body with a traction vector t.

Thus, using the variational principle (Washizu 1982), Eq. (7.2) can be rewritten as

0)( ut

duuutdubdVudVdVuu TTTTT (7.5)

where is the Green strain tensor of the second order, is the first order variation, and

is the penalty value of the penalty meth (Zienkiewicz, Taylor, and Zhu 2005) to treat

the contact problem, respectively. As the MCs are not necessarily consistent with the

boundaries and the DOFs are defined on the PCs rather than nodes as in the FEM, thus

the penalty value can not be treated by prescribing DOFs of the FEM.

As similar as the principle in FEM, the whole material domain can be discretized

into a number of elements in the NMM. Following the approach by Galerkin, the shape

function matrix ][T is applied to substitute Eq. (7.5), the governing equations for a

discrete model can be repressed in the form

}{}{}]{[ intFFuM ext (7.6)

where

dxdyTTdxdyTTMM T

eV e

Te

ee

e

][][)][][(][][ (7.7)

tdTbdxdyTfF TT

eextext ][][}{}{ (7.8)

dxdyBfF T

e

][}{}{ intint (7.9)



7.147

in which ][M is the mass matrix, }{ extF the external force vector, }{ intF the internal force

vector, ][B the strain displacement matrix and }{u is the acceleration vector,

respectively. In the dynamic analysis, an increment of step-time is applied into the

continuous and discontinuous deformation analyses, linear elastic problems can be

considered as

}]{[}{ int uKF (7.10)

where ][K is the stiffness matrix, in which

dxdyBEBKK T

ee ]][[][][][ (7.11)

where ][E is the elastic matrix, thus, we can obtain the well-known equation

}{}]{[}]{[ extFuKuM (7.12)

From Eqs. (7.7) and (7.11), it is noted that ][M and ][K are symmetric, whereas they are

also banded and sparse as the local property of matrix ][T of the MCs in the NMM, and

it requires more computer memory with the assembly of global stiffness matrices for

solving the governing equations. These properties inherit that of the FEM as the

implement of the element in the computation.

7.2.2 Coupled explicit-implicit algorithm in the NMM

In general, there are two classes of algorithms for dynamic problems: implicit and

explicit. Discussions of the advantages and disadvantages of the implicit and explicit

algorithms can be referred (Belytschko and Mullen 1977, 1978; Hughes et al. 1979;

Muller and Hughes 1984; Miranda et al. 1989), in which the elements are partitioned

into three types, explicit-explicit (E-E), implicit-implicit (I-I) and explicit-implicit (E-I)

algorithms; the nodes are partitioned into two types, explicit and implicit. We can

consider two sub-domains partitioned by the elements or the covers (i.e. nodal stars),

which distinction is vital for the interface since each of the methods is specific in the

way the interface is treated. Besides, we assume Newmark- methods with two

characteristic parameters and differently for the different sub-domains between the

explicit and implicit algorithms.



7.148

Sub-domain partition algorithm in the coupled E-I method

In the present study, we assume the element partition method and MCs partition

method to couple the E-I algorithm for the dynamic problems. As shown in Fig. 7.2,

when the element partition method is taken into account for the modeling of the

systems, the whole domain is divided into two classes elements: explicit elements,

denoted by 1Ee , 2

Ee , 3Ee , 4

Ee , …, 21Ee , 23

Ee ; and implicit elements, denoted by 1Ie , 2

Ie , 3Ie ,

4Ie , …, 19

Ie , 20Ie , respectively. In Fig. 7.2(a), we can find interface between the Explicit

and the implicit elements, e.g. 4Ie and 1

Ee , 5Ie and 2

Ee , 11Ie and 6

Ee , 10Ie and 5

Ee , 16Ie and 11

Ee , 15Ie

and 17Ee , 19

Ie and 10Ee , 20

Ie and 18Ee , and ach interface is in conjunction with a pair of explicit

and implicit elements. As the ME is based on the MCs system, we suggest each MC is

split into explicit MC and implicit MC accordingly. Taking an example of interface 16Ie

and 11Ee , we assume the original indexes of the MCs are

iC , jC and

kC to form a ME, then

the coupled E-I algorithm based on element partition method is used to generate a new

pair MEs: explicit element 11Ee and implicit element 16

Ie . Furthermore, the new updated

indexes of the MCs accompany the generation of the new MEs, iC separates into E

iC and

IiC ,

jC becomes EjC and I

jC , kC develops into E

kC and IkC , respectively. Following the

above procedure, the explicit and implicit MCs are formed to apply the corresponding

algorithms, respectively. When the MCs partition method is considered, one domain is

covered by the separated explicit MCs and implicit MCs, which becomes more efficient

to solve the equations of systems explicitly.

Let notation of be the set of all MEs, 46,...,3,2,1e . The assembly of the mass

matrix ][M can be represented as

e

eMM ][][ (7.13)

where ][ eM is the element contribution matrix. Then, let I and

E be the subsets of

corresponding to the implicit and explicit elements. We can get

EI (7.14)



7.149

EI (7.15)

Consequently, substituting into Eq. (7.13) can be rewritten as

EI e

ee

e MMM

][][][ (7.16)

Accordingly, the other items of the equations can be expressed as

EI e

ee

e CCC

][][][ (7.17)

EI e

ee

e KKK

][][][ (7.18)

EI e

ee

e FFF

(7.19)

then, the explicit and implicit terms can be separated and solved based on different time

integration algorithms.

When the MCs partition method is taken into account, as can be seen in Fig. 7.2(b),

the whole domain is covered by the explicit MCs, denoted by 1En , 2

En , 3En , 4

En , …, 11En ,

12En , and the implicit MCs, denoted by 1

In , 2In , 3

In , 4In , …, 13

In , 14In , respectively. The

MCs partition method is more convenient to rearrange the index number of the original

MCs in contrast to that of the element partition method, and is efficient to solve the

coupled and uncoupled equations of motions in the system. Thus, the coupled E-I

algorithm is inclined to the MCs partition to simulate the continuous problems in the

NMM, so that the computational cost can be efficient saved. However, to the

discontinuous problems, the contact positions change along with the different contact

conditions. Neither the element partition method nor the MCs partition method can treat

the contact problems efficiently. Thus, the more efficient method is required to develop

to solve the discontinuous problems using the coupled E-I algorithm in the NMM.



7.150

(a) Element partition method

(b) MCs partition method

Figure 7.2 Sub-domain partition algorithm in the coupled E-I method: (a) Element partition

method; (b) MCs partition method.

Contact algorithm in the coupled E-I algorithm

The NMM treating the contact problems inherits the DDA with the same

methodology, details of the discussion can be referred in (Shi 1988, 1991). In general,

the contract problems are inherently nonlinear and irreversible, and then a maximum

Implicit MCs

Explicit MCs



7.151

displacement ratio is prescribed in the NMM so that the displacements of all the points

within the problem domain are less than the prescribed limit value. Meanwhile, the

contact zones may change considerably within a time step, in which a surface point of a

domain may contact any other section of the surface of another domain, and such a

point can even contact itself surface of the domain. Thus, the algorithm for detecting

contact is of the utmost importance.

In the traditional NMM, there are three possible contact conditions are assumed:

open, slide and lock. At the beginning of each time step, the contact conditions for all

the contact pairs are supposed as lock except which carries forward the previous time

step. When the contact condition of lock is taken into account using the implicit

algorithm, the normal and shear springs are applied to resist the normal penetration of

the vertex and tangential displacement between the vertex and the surface, respectively.

Consequently, all the possible contact pairs satisfy the contact conditions convergence

by means of the OCI technique, the computation then proceeds to the next time step. On

the other hand, the explicit algorithm employs the normal and shear springs explicitly to

compute the contact forces when the contact pairs are detected within each time step.

This approach saves the computational cost to some extent, whereas the explicit

algorithm is conditionally stable, the step time t is required to satisfy the

corresponding numerical property. As shown in Fig. 7.3, we assume two domains:

explicit domain and implicit domain are in contact condition, the explicit and implicit

MCs overlap the explicit and implicit domains, respectively. When the contact is

detected, the contact zone is formed in junction with both the explicit MCs and implicit

MCs. There are five common MCs in the box of the contact zone, which brings the

trouble to determine which algorithm applies to the common MCs. One alternative

scheme can be considered to rearrange the index number of the five common MCs to

generate the new explicit and implicit MCs as the above discussions.



7.152

Figure 7.3 Explicit and implicit covers for the contact problem in the NMM.

7.3 AN ALTERNATIVE APPROACH FOR THE COUPLED E-I

ALGORITHM

Since the coupled E-I algorithm involves the selection and optimization of the

partition methods as presented in the previous section, neither the element nor MCs

partition methods can be considered as one efficient method to deal with the complex

and nonlinear contact problems due to the distinguish characteristic of the MC cover

system in the NMM. Each MC (denoted by “star”) is not always corresponding to the

node as in the FEM when the interface occurs between the explicit and implicit

algorithms. Thus, it is time-consuming to rearrange the index number of the MCs no

matter the element or MCs partition method is employed in the coupled E-I algorithm.

In the study, we give one alternative approach to simulate the contact problems of the

coupled E-I algorithm.

7.3.1 Onefold cover system

In order to solve contact problems efficiently, we propose onefold cover system to

describe the contact between the explicit and implicit domains. As shown in Fig. 7.4, a

Implicit cover Explicit cover

Implicit & explicit cover

Contact domain



7.153

onefold cover system is constructed to couple the explicit and implicit algorithms, in

which one upper triangle cover system is built to combine the explicit and implicit

covers, and the constructed upper triangle is required to cover the whole domains. In

order to represent the whole domains, one box approach is assumed by searching the

maximum and minimum coordinates in x and y directions, and then the four vertices

coordinates can be determined as ),( minmin yx , ),( minmax yx , ),( maxmax yx and ),( minmin yx . Thus,

each ME represents the domain itself, and which defined by a group of index numbers

of MCs. In Fig. 7.4, there are r MEs are taken into account in the system based on

onefold cover system, in which n implicit MEs and )( nr explicit MEs are formed

based on the area algorithm as discussed in the previous section. The implicit elements

constituted by the MCs )1(Ii , )2(

Ii , )3(Ii , …, )(n

Ii ; )1(Ij , )2(

Ij , )3(Ij , …, )(n

Ij ; and )1(Ik , )2(

Ik ,

)3(Ik ,…, )(n

Ik , respectively. Accordingly, the explicit elements are built by MCs )1(Ei , )2(

Ei ,

)3(Ei , …, )( nr

Ei ; )1(

Ej , )2(Ej , )3(

Ej , …, )( nrEj ; and )1(

Ek , )2(Ek , )3(

Ek ,…, )( nrEk , respectively.

Based on the constructed onefold cover system, the displacements and deformations of

the explicit and implicit elements can be obtained using the corresponding algorithms

respectively. Furthermore, the contact problems between the explicit and implicit

elements can be solved efficiently.

Figure 7.4 Construction of onefold cover in the proposed NMM.

, , ...,

,

, , ...,

,

, , ..., ,



7.154

7.3.2 Contact algorithm on the onefold cover system

Basically, there are three kinds of contact types can be detected using the area

algorithm in the onefold system: EE , II and IE . As can be seen in Table 7.1,

when the different contact types are detected, the different algorithms with the

corresponding step time t will be applied to solve the equations of the system. The

explicit and implicit algorithms can be employed to deal with EE and II ,

respectively. EE is proposed to extend the contact type of EE with different step

time in the different domains using the explicit algorithms only, in which the whole

domains are divide into two sub-domains using the element partition method. One

partition is integrated with a time step /t , where is an augmenter greater or equal

to 1, the other with a time step t . This mixed method of the explicit algorithm is more

efficient for the linear system equations, but the interface interpolation for the

integration does not guarantee that all of the difference equations are satisfied at all

nodes. Thus, the numerical stability is required to investigate once more, which increase

the computational cost on the other hand, and the details of EE scheme can be seen

in reference (Belytschko et al. 1979). Liu and Belytschko (1982) have proposed one

mixed algorithm of IE using finite elements for transient analysis, in which an

effective implicit integration procedure with t is used in the implicit domains and an

explicit algorithm with t is applied in the explicit partitions to simulate fluid-structure

system, respectively.

Table 7.1 Contact types between two domains.

Denotation Contact type Explicit Implicit

EE Explicit-Explicit with t --

II Implicit-Implicit -- with t

IE Explicit-Implicit with t with t

EE Explicit-Explicit with /t , t --

IE Explicit-Implicit with t with t Notes: referred in (Belytschko et al. 1979); referred in (Liu and Belytschko 1982).

After selection of the coupled E-I algorithms, the corresponding integration scheme



7.155

will be adopted to form the global coefficient matrix, in which the stiffness sub-matrix,

damping item and mass item are assembled to satisfy the corresponding algorithm for

the equations of system. To illustrate these properties of the coupled E-I algorithm

based on onefold cover system, the coupled E-I coefficient matrix with r elements as

shown Fig. 7.5 is considered here. Since the assembled stiffness matrix and damping

matrix both have a sparse band-profile framework in the implicit algorithm, the

coefficient sub-matrix ][K is formed to the implicit algorithm. Accordingly, the diagonal

coefficient sub-matrix ][M is generated to the explicit algorithm, and the whole matrix

structure is symmetric and positive definite when the coupled E-I algorithm is used

based on onefold cover system. We give an example of global coefficient matrix of

)( rr , in which each element is overlapped by three MCs of sub-matrix )66( . From the

index number of elements 1 to 1s and )1( 2 s to 3s are composed of implicit elements, the

I-algorithm is used to form sub-matrix ][K for the equations of system; on the other

hand, from )1( 1 s to 2s and )1( 3 s to r are constituted by explicit elements, the E-

algorithm is applied to generate sub-matrix ][M into the global coupled coefficient

matrix. In order to improve the computational efficiency, one alternative technique is

applied to divide the global coupled coefficient matrix into implicit coefficient matrix

and explicit one, which can be re-assembled in Figs. 7.5(b) and (c). For each case, the

corresponding algorithm with time step t is applied to solve the equations directly, in

which the formation of the implicit coefficient matrix is minimized by optimizing the

MCs index. In addition, non-zero storage and SOR iteration technique are applied to

save the computational cost; the explicit algorithm employs the central different method

to solve the equations explicitly.

For the contact problems in the onefold cover system, there are two alternative

contact schemes can be considered as shown in Fig. 7.6 to the contact type of IE :

uniform contact algorithm and separated contact algorithm. In the initiation, each

domain area is calculated using the simplex integration, and the judge criteria of area

algorithm is applied to partition the whole domains to explicit domains and implicit

domains. Then, the corresponding contact algorithms and integration schemes are used



7.156

to form the matrix items of the equations of systems. The contact algorithm in the

original NMM uses the penalty technique to restrict the elements penetration at the

normal and shear directions, and the normal and shear springs are added to form tangent

stiffness matrices for the implicit Newmark- method time integration. Resort to the

OCI, the contact convergence is achieved finally, which is a time-consuming work.

Here, we define/assume the explicit contact algorithm, in which the contact springs are

added explicitly when the contact is detected. The contact forces are formed for the

explicit Newmark- method time integration, which is more efficient to solve the

contact problems without considering the OCI problems. When the uniform contact

algorithm is used (see Fig. 7.6(a)), implicit or explicit contact algorithm is chosen as the

uniform algorithm to fo6m the contact matrix and contact force, then solve the

equations of system until the contact convergence. On the other hand, the separated

contact algorithm (see Fig. 7.6(b)) is seemed to be more efficient to simulate the

coupled E-I contact problems under te demand of computational accuracy. Therefore,

the separated contact algorithm is assumed as the contact algorithm to apply into the

contact computations.

(a) Global coefficient matrix for the coupled E-I algorithm;

12 3 .

Symmetric

I-algorithm

E-algorithm

I-algorithm

E-algorithm



7.157

(b) Implicit coefficient matrix; (c) Explicit coefficient matrix.

Figure 7.5 Coefficient matrix of the coupled E-I algorithm: (a) Coupled E-I algorithm global

coefficient matrix; (b) Implicit coefficient matrix; (c) Explicit coefficient matrix.

(a) Shared contact algorithm; (b) Separated contact algorithm.

Figure 7.6 Flowchart of two alternative contact schemes: (a) shared contact algorithm; (b)

separated contact algorithm.

7.3.3 Contact matrices of the coupled E-I algorithm

In the study, the separated contact algorithm is employed as discussed in the

previous section. Once the penetration is detected, the contact matrices are added to the

12 3.

Symmetric

I-algorithm

E-algorithm

yes no

Solve equations

Calculate domain area

Judge criteria

Explicit domain Implicit domain

Uniform algorithm

yes no

Solve equations

Calculate domain area

Judge criteria

Explicit domain Implicit domain

Explicit algorithm Implicit algorithm



7.158

corresponding contact elements. Here, we focus on the contact matrices of the coupled

E-I algorithm. As illustrated in Fig. 7.7, one penetration is searched between the explicit

domain E and the implicit domain I from the time step

0t to )( 0 tt , the

corresponding contact pairs of Ee and Ie are recognized as well. The vertex of Ee is

assumed as 1P ),( 11 yx ; and the penetration edge

32 PP in the implicit element Ie is defined

as well, in which the coordinates of the vertices 2P and 3P are assumed as ),( 22 yx and

),( 33 yx . Consequently, the normal contact matrix, shear contact matrix and friction

matrix can be derived as follows

Figure 7.7 Contact representation in the coupled E-I algorithm.

Normal Contact Matrix

It is assumed that vertices 1P , 2P and 3P rotate in the same sense as the rotation of

ox to oycoordinate axis, the normal penetration distance is assumed as nd , which can be

written as

3333

2222

1111

1

1

11

vyux

vyux

vyux

lldn

(7.20)

where

223

223 )()( yyxxl (7.21)

at at



7.159

3

31221

2

23113

1

123320 )()()(

v

uxxyy

v

uxxyy

v

uxxyyS (7.22)

33

22

11

0

1

1

1

yx

yx

yx

S (7.23)

in which the step displacements ),( ii vu , 3,2,1i is small as the results of small time

step, thus contact distance nd is small from the definition of the contacts. And,

})}{,({ 111

1 EE DyxTv

u

(7.24)

})}{,({ 222

2 II DyxTv

u

(7.25)

})}{,({ 333

3 II DyxTv

u

(7.26)

Then, the normal contact distance nd is rewritten as

}{}{}{}{0 ITIETEn DGDH

l

Sd (7.27)

where

23

3211

)3(

)2(

)1(

)},({1

}{xx

yyyxT

lH

H

H

H TEi

E

E

E

E (7.28)

12

2133

31

1322

)3(

)2(

)1(

)},({1

)},({1

}{xx

yyyxT

lxx

yyyxT

lG

G

G

G TITI

I

I

I

I (7.29)

Then, the normal contact spring with the stiffness of nk is introduced between the

contact point and the implicit element. The potential energy of the normal spring is

2

02 }{}{}{}{22

I

jTIE

iTEn

nn

n DGDHl

Skd

k (7.30)

Minimising n by the derivatives in terms of }{D , the four 66 sub-matrices of ][K

and two 16 sub-matrices of load matrices can be obtained, which are

][}}{{ ))(()()(E

srTE

sEr KHHp (7.31)

][}}{{ ))(()()(IEsr

TIs

Er KGHp (7.32)



7.160

][}}{{ ))(()()(EIsr

TEs

Ir KHGp (7.33)

][}}{{ ))(()()(I

srTI

sIr KGGp (7.34)

}{}{ )()(0 E

rErn FH

l

Sk

(7.35)

}{}{ )()(0 I

sIsn FG

l

Sk

(7.36)

where 3,2,1, sr are the order number of MCs of the corresponding explicit and

implicit elements, respectively.

Shear contact matrix

As shown in Fig. 7.7, the assumed contact point 0P ),( 00 yx is on the edge

32 PP in the

explicit element. The shear spring is introduced along the edge on the direction of32 PP ,

and connects vertices 1P and

0P . Then, the shear displacement sd of 0P and

1P along the

edge32 PP can be repressed using inner product form as

0

03232

1

12323

03210 )(

1)(

1ˆ1

v

uyyxx

lv

uyyxx

ll

SPPPP

ld s

(7.37)

where

23

2301010 )(ˆ

yy

xxyyxxS (7.38)

As assumed that point 1P belongs to the explicit element and point 0P belongs to the

implicit element, so Eq. (7.37) can be rewritten as

}{}ˆ{}{}ˆ{ˆ

0 ITIETEs DGDH

l

Sd (7.39)

where

23

2311

)3(

)2(

)1(

)},({1

ˆ

ˆ

ˆ

}ˆ{yy

xxyxT

lH

H

H

H TE

E

E

E

E (7.40)



7.161

32

3200

)3(

)2(

)1(

)},({1

ˆ

ˆ

ˆ

}ˆ{yy

xxyxT

lG

G

G

G TI

I

I

I

I (7.41)

Then, the shear contact spring with the stiffness of sk is introduced on the direction

32 PP connects vertices 1P and

0P . The potential energy of the shear spring is

2

02 }{}ˆ{}{}ˆ{ˆ

22

ITIETEs

ss

s DGDEl

Skd

k (7.42)

Minimising s by the derivatives in terms of }{D , the four 66 sub-matrices of ][ K

and two 16 sub-matrices of load matrices can be obtained, which are

][}ˆ}{ˆ{ ))(()()(E

srTE

sEr KHHp (7.43)

][}ˆ}{ˆ{ ))(()()(IEsr

TIs

Er KGHp (7.44)

][}ˆ}{ˆ{ ))(()()(EIsr

TEs

Ir KHGp (7.45)

][}ˆ}{ˆ{ ))(()()(I

srTI

sIr KGGp (7.46)

}{}ˆ{ˆ

)()(0 E

rErs FH

l

Sk

(7.47)

}{}ˆ{ˆ

)()(0 I

sIss FG

l

Sk

(7.48)


implicit elements, respectively.

Friction Matrix

When the vertex 1P relative to

0P slides along the edge32 PP , the friction force can

be calculated based on the Coulomb’s law in the case of friction angle is not zero.

Then, we can get

)tan( gnnn sdkF (7.49)

where gns is a Sign function to represent the movement of 1P relative to

0P along the

vector of 32 PP .Then, the potential energy E

f and If of friction force F on the explicit

and implicit elements can be obtained, respectively, which are



7.162

23

2311

23

2311 )},({}{)(

yy

xxyxTD

l

Fyy

xxvu

l

F TETEEf

(7.50)

23

2300

23

2300 )},({}{)(

yy

xxyxTD

l

Fyy

xxvu

l

F TITIIf

(7.51)

Minimising Ef and I

f by the derivatives in terms of }{D , two 16 sub-matrices of

friction load matrices can be obtained, respectively, which can be repressed as

}{}~

{ )()(Er

Er FHF (7.52)

}{}~

{ )()(Is

Is FGF (7.53)

in which

23

23

11)(

11)()( ),(0

0),(1}

~{

yy

xx

yxw

yxw

lH E

r

ErE

r (7.54)

23

23

00)(

00)()( ),(0

0),(1}

~{

yy

xx

yxw

yxw

lG I

s

IsE

s (7.55)


implicit elements, respectively; ),( 11)( yxw Er

and ),( 00)( yxw Is

are the weight functions in

the explicit and implicit contact elements, respectively.

7.3.4 Spring stiffness problems

The selection of spring stiffness k is important to the accuracy of the solutions in

the contact problems. Cheng (1998) observed that the choice of contact spring stiffness

nk and/or sk or time-step size t has a significant effect on convergence. Thus, the

stiffness of the spring plays an important role in the solution and the quality of solution

depends on the range of stiffness values chosen. If the value of k is too small, the

penetration distance becomes large so that the constraints are poorly satisfied, which

may result in the following conditions (Shi 1992):

The closed contacts can not be detected and transferred to the next step;

The stresses in the material may be reduced; and

The deformation along the contact edges can be incorrect.



7.163

For large values of k , the diagonal items of the global stiffness matrix making it

numerically ill-conditioned. Furthermore, the OCI may not converge and the calculated

contact forces and displacements may not be realistic. Lin et al. (1996) has proposed the

augmented Lagrangian method, in which the value of k can be variable and does not

have to be a very large number, but the solution requires more iterations to reach contact

convergence increasing the computational cost.

For the explicit algorithm, it is noted that the spring stiffness of nk and/or sk are

difficult to obtain. Cundall emphasized the importance of using realistic values for the

sk by demonstrating how the ratio of sk to nk dramatically affects the Poisson

response of a rock mass (Itasca 1993). Moreover, some guidelines are suggested

estimate the contact spring stiffness for a reasonable analysis, in which nk and sk

should be kept smaller than ten times the equivalent stiffness of the stiffest

neighbouring zone of elements adjoining the contact interface, and it can be written as

min

3/4max0.10

dk

(7.56)

where and are the bulk and shear modulus, respectively; and mind is the smallest

width of the domain adjoining the contact interface in the normal direction as shown in

Fig. 7.8. If the stiffness value is greater than 10 times the equivalent stiffness, the

solution time of the model will be significantly longer, which increases the

computational cost on the side.

Contact interface



7.164

Figure 7.8 Domain profile suggested in stiffness estimation (Itasca 1993).


In order to investigate the validity of the proposed algorithms, some numerical

examples are simulated in terms of temporal and spatial respects, respectively. In the

simulations, the proposed coupled temporal and spatial algorithms for the NMM are

calibrated between the simulation results and analytical solutions, firstly. Then, a

examples of open-pit slope stability analysis are taken into account using the proposed

E-I algorithms to further extend the current NMM, which represent the potential of the

proposed E-I algorithms for simulating the larger scale projects in the further research.

7.4.1 Calibration of the spatial coupled E-I algorithm

To calibrate the proposed coupled spatial E-I algorithm in the NMM, one

numerical simulation of dynamic friction mechanism of multi-block system as shown in

Fig. 7.9 is considered. In the present study, the multi-block system with four identical

properties blocks sequentially connect with each other, which system is subjected to a

dynamic pulse loading )(tFv on the top of the block in the vertical direction and a

constant loading )(tFh on the next block in the horizontal direction. In the simulation,

the multi-block system is classified into two kinds of blocks: the top two blocks are

taken into account as explicit blocks, which are denoted by 1E and 2E , simulated using

the proposed explicit algorithm, including the explicit time integration and explicit

contact algorithm. The bottom two blocks are considered as implicit blocks, which are

denoted by 1I and 2I , simulated using the original implicit algorithm, including the

implicit time integration and open-close contact iteration, respectively.



7.165

Figure 7.9 Geometry of the multi-block system.

In the current study, a half-sine pulse loading with amplitude of 1MPa and duration

of 20 ms is considered. The analytical solution of the total vertical displacement can be

referred in (Chopra 2001; Shu et al. 2007), which can be expressed as

TtTtTv

TtTu

TttDtCtBtA

tu

DD

D

DD

)(sin)(

)(cos)(

)sinsin()sincos(

)(

(7.57)

where mkD is the natural frequency of the multi-block system; is the loading

frequency; T =20ms is the duration of pulse loading; A and B are coefficient items

related to the initial condition, C and D are coefficient items related the frequency

aspects, which are can be referred in (Ma, An, and Wang 2009). Once the vertical

displacement is obtained, the friction between the blocks can be expressed as

tan)( tukF nH (7.58)

E1

E2

I1

I2

2m

2m

2m

2m

4m 1m 1m

Fv(t)

Fh(t)



7.166

in which HF is the dynamic friction force in the horizontal direction, nk is the normal

contact spring stiffness of the blocks, and .is the friction angle.

In the simulations, two cases of friction angles are studied: 0 and 045

respectively. The density, Young’s modulus and Possion’s ratio are 2500 Kg/m3, 10

GPa and 0.25, respectively. The simulation results of horizontal displacements between

the two cases can be seen in Fig. 7.10. It can be found that the simulated results as

plotted in Fig. 7.11 of the proposed coupled spatial E-I algorithm of the NMM are well

agreement with the analytical solution. In terms of efficiency, the coupled E-I algorithm

CPU cost is 141s, t is 0.01 ms) is obviously efficient than that of the original NMM

version (CPU cost is 317s, t is 0.01 ms).

(a) case of ϕ=0;

(b) case of ϕ=450.

Figure 7.10 Displacement in the different cases: (a) case of ϕ=0; (b) case of ϕ=450.



7.167

(a) case of ϕ=0;

(b) case of ϕ=450.

Figure 7.11 Displacement of the top 2nd block: (a) case of ϕ=0; (b) case of ϕ=450.



7.168

7.4.2 Simulation of discrete blocks sliding on an inclined surface

In the present simulation, a rock slope stability analysis using the coupled E-I

algorithm is taken into account to investigate validity and applicability of the algorithm.

In the study, three models of the rock slope are considered as shown in Fig. 7.12, in

which explicit algorithm, implicit algorithm and coupled explicit-implicit algorithm are

employed to simulate the rock slope failure process, respectively. It is noted that area

algorithm is employed in the coupled explicit-implicit algorithm, base of the slope is

regarded as implicit block and blocks on the slope are assumed as explicit blocks as can

be seen in Fig. 7.12(d). The displacements of the measured point 1 and 2 are plotted in

Fig. 7.13, respectively. We can find that the coupled explicit-implicit algorithm based

on onefold cover system is workable and satisfies the numerical accuracy comparing the

implicit and explicit algorithms. The efficiency of the coupled algorithm (CPU cost is

336s, t is 0.05 ms) can be improved contrast to the original implicit version of the

NMM (CPU cost is 1161s, t is 0.05 ms) significantly under the requirement of the

computational accuracy.

In addition, selection of the contact spring stiffness is sensitive to the

computational accuracy, how to determine an appreciate value of the spring stiffness is

still current hot spot in the simulations using the NMM, which will be paid more

attention in the further research.

(a) Initial modelling;

ρ = 25 KN/m3

E = 1.0 GPa ν = 0.2 ϕ = 50, c = 0, T = 0

Measured point 1, 2

410

(b) Explicit NMM;

Kn = 0.25 GPa Ks = 0.1 GPa Δt = 0.02ms umax = 0.02mm

Explicit blocks

Explicit block


Implicit blocks

Implicit block

(c) Implicit NMM;


Explicit blocks

Implicit block

(d) Explicit-implicit NMM.



7.169

Figure 7.12 Rock slope stability analysis using the I-NMM, E-NMM and E-I NMM, respectively: (a)

Initial modelling; (b) Explicit NMM; (c) Implicit NMM; (d) Explicit-implicit NMM.

(a) Displacement of measure point 1; (b) Displacement of measure point 2.

Figure 7.13 Displacement of the measured point 1 and 2.

7.5 SUMMARY

The coupled E-I algorithm for the numerical manifold method (NMM) is proposed

in this chapter. The time integration schemes in the E-I algorithm, transfer algorithm of

the coupled E-I algorithm, the implicit contact algorithms based on the implicit

integration scheme and explicit contact algorithm based on the explicit integration are

studied in terms of accuracy and efficiency, respectively. In particular, onefold cover

system to the coupled E-I algorithm is proposed and drawn into the coupled spatial E-I

algorithm, in which the contact algorithm based on the onefold cover system is

discussed and derived in detail. Finally, some numerical examples are simulated using

the proposed coupled E-I algorithms, in which one calibration example is studied using

the proposed E-I algorithms relied on the onefold cover system; One numerical example

of rock slope seismic stability analysis using the coupled E-I algorithm is studied as

well. The simulated results are well agreement with the implicit and explicit algorithms,

0

1

2

3

0 0.2 0.4 0.6 0.8 1

Dis

pl.

(m)

Time (s)

I-NMM

E-NMM

E-I NMM

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1D

isp

l. (m

)

Time (s)

I-NMM

E-NMM

E-I NMM



7.170

but the efficiency of the coupled algorithm is obviously higher than that of the original

version of the NMM. It is predicted that the couple E-I algorithm proposed in the

present paper can be applied into larger scales engineering systems to combine the

merits of both the implicit and explicit algorithms in the NMM.



8.171

CHAPTER 8. DYNAMIC STABILITY ANALYSIS OF

ROCK SLOPE FAILURE USING THE EXPLICIT

NUMERICAL MANIFOLD METHOD

8.1 INTRODUCTION

Comparing with the static stability analysis of rock slope, the dynamic stability is

more difficult as the complexity of the triggered mechanism of dynamics, such as

earthquake load and blast effect (Hoek and Bray 1981). During the previous a few

decades, the dynamic stability analysis of rock slope is still hot issue to motivate

researchers conduct different techniques.

It is noted that when the complex dynamics is taken into account, the traditional

implicit version of the NMM will become inefficient especially in the simulations of

large scale modelling. Thus, how to simulate the dynamic stability of rock slope

efficiently draws our attentions; how to improve the efficiency of the current version of

the NMM within the appropriate accuracy motivates us to develop the new version of

the code. In this paper, the traditional NMM is further extended for earthquake-induced

rock slope stability analysis using the proposed explicit scheme. The validity of the

explicit NMM (ENMM) has been investigated by comparing its results with available

reference solutions in the previous sections. Here, we put the proposed ENMM and

other techniques into the dynamic stability analysis of rock slope failure.

To illustrate the dynamic stability of rock slope, we give one example of rock slope

failure in Perth of Western Australia as shown in Fig. 8.1. A slim north-south orientated

rock slope (described by shallow red shadow zone from Google map ref.

http://maps.google.com.au/ ) sites close to beach of Cottesloe WA, this hazard zone is

instable and danger to passengers, particularly the effect of impact from ocean tide,

typhoon and tsunamis. Furthermore, this district belongs to a classic Mediterranean

climate (Sturman et al. 1996; Linacre and Geerts 1997). Annual rainfall falling between

May and September is heavy, which is one of factors affect the slope stability. It is



8.172

noted that this is a very normal phenomenon exists around coast in Western Australia as

the broad coastline. Here, the slope example is taken into account as shown in Fig. 8.2

to investigate the stability and potential failure types. In Fig. 8.2a, right bank of the

slope is a slim slope bordered on the India Ocean. Area B is smooth and consists by

intact rock mass, this area is regarded stable. On the other hand, area A is consisted by

loose and discrete rock masses, especially rock blocks denoted by and can be

considered as potential instability such as rock sliding even rock fall failure under

dynamic conditions, thus this slope is required to investigate the stability rely on in-situ

measurement technique and other methods such as LEM and numerical methods. At the

left bank of the slope as shown Fig. 8.2b, the major type of rock failure focuses on the

rock fall from the shadow denoted by , and , respectively. It is considered

potential instable and hazard area, and a yellow warning board saying “ROCKFALL

RISK AREA NO ENTRY” and wooden fence locate in the lower left of the photo to

remind the passengers keep personal safety.

Figure 8.1 Location of the rock slope and aero-view from Google maps.

N

Location

WA



8.173

Figure 8.2 Photograph of rock instability example in Cottesloe, Western Australia (photographed

by X.L. Qu): a. right bank of the slope; b. left bank of the slope.

From the above referred examples, we can find that rock slope stability analysis

should be taken into account at the both cases of static and dynamic conditions. These

motivate researcher to study these failure mechanisms by generation to generation in the

past several decades. In general, dynamic stability analysis of rock slope concentrates

on the triggering factor such as earthquake, blasting even tsunamis respects.

The dynamic stability analysis of rock slope is studied using different techniques

by several earlier researchers. Here, a comprehensive study is carried out with an

emphasis on dynamic stability analysis of rock slope in jointed rock mass using the

proposed ENMM codes, coupled algorithms and parallel computation.

Area A

Area B

a

b



8.174

8.2 NUMERICAL METHODS FOR ROCK SLOPE DYNAMIC

STABILITY

For the seismic stability analysis of rock slopes, the numerical methods are more

suitable because the behaviour of a rock slope is much more dependent on characteristic

and integrity of the rock mass. At present, dynamic FEM techniques (Zienkiewicz et al.

1975; Griffiths and Lane 1999) have became one of the important tools in seismic

stability analysis for rock slopes, yet it still has difficulties in the simulation of

numerous rock discontinuities (such as faults, joints, etc.) which are discreted by special

elements. To overcome the disadvantage of FEM in seismic stability analysis of rock

slope, new numerical methods have been developed for the deformation based on

multiple blocks system containing a large quantity of discontinuities. Among these

methods, distinct element method (DEM) (Cundall 1971a, 1971b; Itasca 1993, 1994,

1995), discontinuous deformation analysis (DDA) (Shi 1988, 1993), and the numerical

manifold method (NMM) (Shi 1991, 1992) are typical. The NMM was developed to

integrate the DDA and the FEM. The distinct feature of NMM is which employs dual

cover system to describe a problem domain. The advantages of the NMM are releasing

the task of meshing and combining continuum and discontinuum problems into one

framework. The NMM has been successfully applied to strong discontinuity problems,

weak discontinuity problems and rock failure, etc (Tsay, Chiou, and Chuang 1999b; Ma

et al. 2008; Ma and He 2009; Zhang et al. 2010; Ning, An, and Ma 2011; An, Ma, et al.

2011b; Wu and Wong 2012; An et al. 2013).

Traditional methods of rock slope stability analysis are limited to simplified

problems. They employ simple slope geometries and basic loading conditions, which

provide little insight into slope failure mechanisms without considering complexities

relating to geometry, material anisotropy, non-linear behaviour, in situ stresses and the

realistic loading conditions, such as pore pressures, seismic and blasting loading, etc. To

address these limitations, numerical modeling techniques have been forwarded to

provide approximate solutions to problems over the traditional techniques. In general,

numerical methods for rock slope stability analysis can be classified into three

approaches: (1) continuum modeling; (2) discontinuum modeling; (3) hybrid modeling.



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Continuum method is best suited for the analysis of rock slopes that are comprised of

massive intact rock, weak rocks or heavily fractured rock masses; discontinuum

modelling is appropriate for slopes governed by discontinuity behaviour; hybrid

approach involves the coupling of above both techniques to maximize their critical

advantages.

8.2.1 Continuum methods

Continuum approaches applied into the dynamic stability analysis of rock slope

contain the finite difference methods (FDM) and FEM. Both methods divide the

problems domain into a set of sub-domains or elements. The difference is the former

relies on numerical approximations of the governing equations, on the other hand, the

latter resorts to the continuity of displacements and stresses of the conjoint elements.

Both advantages and limitations of these two methods are discussed in (Hoek,

Grabinsky, and Diederichs 1991). To slope stability analysis, earlier studies are often

limited to static and elastic analysis, which is restrained in the application. Nowadays,

most continuum based codes incorporate a facility for discrete fractures such as faults

and bedding planes and dynamic input parameters analysis such as FLAC 2D/3D and

Abaqus/CAE. Fig. 8.3 illustrates an application of an elasto-plastic constitutive model

based on a mohr-Coulomb yield criterion using the general commercial software

Abaqus/CAE 6.11 (Abaqus 2011). The input geometry parameters: section size is

100*40 m, angle of slope is 450; material parameters: ρ= 20 kN/m3, c = 38.2 kPa, φ =

17o, E = 100 MPa, ν = 0.3), which results can be simulated by FALC 2D and the same

criterion is employed to model translational slide movements of Frank Slide, Canada by

(Stead et al. 2000).



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(a) slope meshing and 6-noded triangular element;

(b) contour nephogram of the maximum strain ratio.

Figure 8.3 Continuum modeling of a rock slope by Abaqus/CAE 6.11: (a) slope meshing and 6-

noded triangular element; (b) contour of the maximum strain ratio.

In terms of seismic stability analysis of the slope modelling, the El-Centro

earthquake ( 0.7agM and 4.6lM ) acceleration in 1940, U.S.A. is taken into account to

simulate the slope displacement using software of Abaqus/CAE. In the simulation, the

horizontal seismic acceleration with time depended 25s is exerted in the bottom of the

slope base, the displacements depend on time history of seismic acceleration can be

obtained as shown Fig. 8.4, in which the horizontal displacements nephogram at

different times are plotted respectively. It is noted that the displacement is changed

smoothly at the initial stage of the earthquake, and then the maximum displacement is

achieved following the peak seismic acceleration. Thus, the peak seismic acceleration

plays virtual role in the rock slope failure under seismic effects.

1

2

3

4 5

6



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Figure 8.4 Displacement nephogram at different times under seismic loading: (a) t=1.5s; (b)

t=12.0s; (c) t=21.0s; (d) t=30.0s.

8.2.2 Discontinuum methods

When a rock slope comprises multiple joint sets, which control the mechanism of

failure, then a discontinuum modelling technique may be considered more appropriate.

In the discontinuum approaches, problems domain is treated as an assemblage of

distinct, interacting bodies or blocks that subjected to external dynamic loads such as

earthquake, blasting even tsunamis, and is expected to undergo significant motion with

time-history. The underlying basis of the discontinuum method is that the dynamic

equation of equilibrium for each block in the motion system is repeatedly solved until

the boundary conditions and laws of contact and motion are required. Nowadays, the

discontinuum technique constitutes the most commonly applied numerical approach to

rock slope analysis and generally three alternative variations of the methodology exist:

(1) DEM; (2) DDA; (3) particle flow codes (PFC).

The DEM is originally developed by Cundall (1971a, 1971b) and further described

in detail by Hart (1991), in which the algorithm is based on a force-displacement law

and a law of motion. In the respect of the dynamic stability analysis of rock slope, the

explicit solution of the DEM in the time domain used by the method is ideal for

following the time propagation of a stress wave. Fig. 8.5 provides an example of a

typical high-speed landslide hosted on consequent bedding rock induced by Wenchuan

(a) t = 1.5s; (b) t = 12.0s;

(c) t = 21.0s; (d) t = 30.0s.



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earthquake at a medium-steep hill slope (Luo et al. 2012), in which the maximum

displacement vectors and shear strain contours at time of t=10s and t=25s are studied

using UDEC, respectively. The model show that it is helpful for understanding seismic

dynamic responses of consequent bedding rock slopes, where the slope stability could

be governed by earthquakes. In addition, Eberhardt and Stead (1998) carried out

dynamic stability analysis of a natural rock slope using the UDEC, in which an initially

stable slope subjected to an earthquake, resulting in yielding and tensile failure of intact

rock at the slope toe and rotational type movements.

(a) maximum displacement vectors at time of t=10s and t=25s;

(b) shear strain contours at time of t=10s and t=25s.

Figure 8.5 Maximum displacement vectors and shear strain contours of the modelling in 2008

Wenchuan earthquake, China (Luo et al. 2012).

One more recent development in discontinuum modelling techniques is the

application of distinct-element methodologies by a new pattern of Particle Flow Code,

PFC2D/3D (Itasca 1995), in which clusters of particles can be bonded together to form

joint-bounded blocks. This code is capable of simulating fracture of the intact rock

blocks through the stress-induced breaking of bonds between the particles. This is a

significant development as it allows the influence of internal slope deformation to be

investigated both due to yield and intact rock fracture of jointed rock. Wang et al.

(2003) demonstrate the application of PFC in the analysis of heavily jointed rock slopes

unit: mm

t =10s

unit: mm

t =25s

t =10s t =25s



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under dynamic excavation as shown Fig. 8.6, which simulation results reveal an intact

rock with certain mechanical properties, such as joints (discontinuities) play a

determinant role in slope stability.

Figure 8.6 Simulation of a rock slope stability and failure under dynamic excavation using PFC

technique (Wang et al. 2003).

The DDA (Shi 1988, 1993) has also been applied in the modelling of dynamic

stability analysis with some success, both in terms of landslides (Wu et al. 2009; Wu

2010) and jointed rock slope (Hatzor et al. 2004). This approach differs from the

distinct-element method in that the unknowns in the equilibrium equations are

displacements as opposed to forces, by which the equilibrium equations can be solved in

the same manner as the matrix analysis used in FEM based on minimum potential

energy principle. With respect to slope dynamic stability analysis, the method has the

advantage of being able to model large deformations and rigid body movements, and

can simulate coupling or failure state between contacted blocks. In addition, SASAKI et

al. (2006) carried out the contact damper to controlling the surplus penetration in high

speed velocity to simulate the seismic response analysis of Myo-ken slope in Niigata,

Japan. Fig. 8.7 demonstrates a dynamic DDA version to simulate landslide by the Chi-

Chi earthquake in Taiwan, in which time-dependent accelerations and constraining

seismic displacements of the base rock are studied, respectively; and a novel algorithm

is carried out to diminish the velocity of the base rock in the seismic analysis. Further

more, seismic DDA analysis results coincide well with the topography of the Chiu-fen-

erh-shan landslide slope. Thus, DDA is a useful tool for simulating the response of a

block assembly under the impact of an earthquake as we expected.



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Figure 8.7 Seismic simulation of Chiu-fen-erh-shan landslide by the Chi-Chi earthquake using

DDA (Wu 2010).

8.2.3 Hybrid methods

Hybrid methods are increasingly being adopted in rock slope stability analysis. An

alternative coupled method using LEM and finite-element and stress analysis such as

adopted in the GEO-SLOPE suite of software (Geo-Slope 2000). In particular, a recent

new developed method of NMM is performed by Shi (1991, 1992) based on topological

manifold, unifies both the FEM (Zienkiewicz et al. 1975) and the DDA (Shi 1998,

1993), and applied into the dynamic stability analysis of rock slope. For the dynamic

response analysis of discontinuous rock slopes, seismic forces are commonly applied to

the basement block modelled using a single DDA block. However, it is necessary to

consider the local variation of seismic forces and stress conditions, especially when the

size of slopes is large and/or the slope geometry becomes complicated. There is

difficulty in DDA to consider the local displacements of the single block for the

basement due to the fact that the strain in the single block is uniform and displacement

function is defined at the gravity center. On the other hand, the NMM can simulate both

continuous and discontinuous deformation of blocks with contact and separation.

However, the rigid body rotation of blocks, which is one of the typical behaviors for

rock slope failure, cannot be treated properly because NMM does not deal with the rigid

body rotation in explicit form. Fig. 8.8 presents an application for the discontinuous

rock slope behaviour during earthquake successfully, in which the mechanical

behaviour of falling rock blocks is simulated by DDA with the basement block covered

Time =0 sec

653.1m

Time =30 sec

Time =60 sec

874.8m

Time =300 sec

1013.1m



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by the NMM mesh, where seismic forces are given. Further more, the formulation for

the coupled NMM and DDA (NMM-DDA) is presented with the programming code

developments. Ning et al (2012) carried out a numerical modelling of earthquake-

induced failure of a rock slab with pre-existing non-persistent joints using the couple

NMM and DDA, as shown in Fig. 8.9, in which the complete rock failure process

including the fracturing of the intact rock and the motions of the generated rock blocks

is wholly reproduced.

Figure 8.8 Displacement distribution for each block after applying seismic loads (Miki et al. 2010).

Figure 8.9 Modelling of rock fall failure under earthquake by NMM and DDA (Ning et al. 2012).

8.3 THE NEW DEVELOPMENT OF THE NMM FOR DYNAMIC

STABILITY ANALYSIS OF ROCK SLOPE

In the present study, a new development of the NMM is introduced to simulate

dynamic stability analysis of rock slope. In order to investigation of the performance of

geo-structures such as rock slopes and/or tunnels subjected to seismic loading, the



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NMM is further extended for earthquake-induced rock slope stability analysis. Ning et

al. (2012) proposed the seismic NMM code combining DDA to simulate the seismic

stability of rock slope, in which the earthquake loading is applied in the form of

constraining seismic displacement time history. Seismic acceleration time histories

recorded in earthquakes is transferred into displacement time histories to be loaded by

the following formulas:

2/21

1

tatvdd

tavv

nnnn

nnn (8.1)

where 1nv and

nv are the seismic velocity at step n+1 and n, respectively, and 00 v ;

1nd and nd are the constraining seismic displacement at step n+1 and n, respectively,

and 00 d ; na is the seismic acceleration at step n; t is the time interval between two

adjacent steps. To improve the transformation accuracy, especially when the sampling

frequency of the data is low, several steps can also be interpolated between two adjacent

records, where the acceleration at each step is obtained by linear interpolation between

the two adjacent acceleration records. With such seismic NMM code, the earthquake

acceleration amplitude that can lead to the initiation of the failure could be derived. The

complete rock failure process including the fracturing of the intact rock and the motions

of the generated rock blocks are wholly reproduced. Continuing the research, An et al.

(2012) investigated the seismic stability of rock slopes using the developed seismic

NMM code, in which the instability mechanisms of rock slopes under horizontal and

vertical ground accelerations are revealed, respectively. And the validity of the NMM in

predicting the ground acceleration induced permanent displacement has been verified by

comparing its results with the analytical solutions and the Newmark- numerical

integration solutions. Furthermore, the proper values of control parameters for NMM

calculations are suggested in the study. Thus, the seismic NMM adopted is promising

for the modelling of earthquake-induced rock failures and deserves to be further studied.

Following the previous study on the dynamic stability analysis of rock slope using

the NMM, here we present an alternative algorithm of time integration for the NMM.

As the explicit central different technique applied in the DEM, the proposed explicit

algorithm in the NMM can be considered for modelling rock slope to seismic events



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relating to earthquake and/or blasting as well. In addition, the explicit algorithm in the

time domain is particularly well suited for the time propagation of a stress wave and is

efficient in terms of computational efficiency. Comparing to the implicit algorithm used

in the current NMM code, the proposed explicit algorithm owns the following

advantages except the term of conditionally stability:

☑ Few computations are required per time step, low computational cost;

☑ It requires little computer memory without the assembly of stiffness matrices;

☑ Size of the explicit NMM code is shortened, especially cancelling iterative solvers;

☑ It is reliable with regards to accuracy and completion of the computation.

8.3.1 Explicit NMM

The explicit algorithm of the NMM owns distinct advantage in contrast to the

implicit algorithm with respect to the computation efficiency. To further improve the

efficiency of simulation under the accuracy, the coupled explicit NMM and DDA can be

taken into account to simulate dynamic stability of large scale rock slope, in which a

transfer algorithm is required to carry out the conversion from the NMM to DDA. Since

both NMM and DDA employ the same methodologies of global formula and contact

treatment technique, Ning et al. (2012) proposed one possible approach to transfer

NMM to DDA, in which the geometric configurations, physical and mechanical

parameters, and status, including stress state and velocities, are inherited from an NMM

model. It is noted that material area change problem and the unnatural deformation

problem can not be solved easily using the NMM as the rigid body rotation is not

represented in an explicit form (Miki et al. 2009). Thus, an alternative approach to

transfer the rotational velocity based on the principle of conservation of the kinematic

energy is applied and derived in the form as

DDANMM

n

i

ei

ei JJ 2

1

2

2

1

2

1

(8.2)

where eiJ and e

i are the moment of inertia and rotational velocity of the i-th element

with respect to its centre of the NMM, respectively; J and are the moment of inertia



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and rotational velocity of the block with respect to the block centre of the DDA,

respectively. Continuing the discussion of the explicit NMM and DDA, a numerical

example will be investigated in the following section to verify the proposed numerical

algorithm in detail.

8.3.2 Coupled E-I NMM

The explicit and implicit algorithms have their own limitations and advantages in

terms of numerical properties and computational efficiency, respectively. The implicit

algorithm can use the large time step whereas more computational cost produces

especially when the contact problems occurrence; on the other hand, the explicit

algorithm has reverse points in contrast to the implicit algorithm. The more details of

the studies between explicit and implicit algorithms can be referred in Chapter 6 and 7.

In addition, the NMM is hybrid numerical method combining the continuum and

discontinuum approaches relied on the mathematical and physical covers to simulate the

dynamic problems. In the present study, the failure process of crack propagation is not

considered in the coupled explicit-implicit NMM. Therefore, the study focuses on the

sub-domain partition of the coupled explicit-implicit NMM.

Construction of onefold cover system

In the NMM, the coupled E-I algorithm is inclined to the MCs partition to simulate

the continuous problems, so that the computational cost can be efficient saved.

However, to the discontinuous problems, the contact positions change along with the

different contact conditions, neither the element partition method nor the MCs partition

method can treat the contact problems efficiently. Further more, each MC (denoted by

“star”) is not always corresponding to the node as in the FEM when the interface occurs

between the explicit and implicit algorithms. Thus, it is time-consuming to rearrange the

index number of the MCs no matter the element or MCs partition method is employed.

Therefore, the more efficient method is required to develop to solve the contact

problems. Here, we present an alternative approach based on onefold cover system to

couple the explicit and implicit algorithms, in which a onefold cover system is built and

the contact problems is solved efficiently. To illustrate the proposed onefold cover

system, one example is given as shown Fig. 8.10, in which there are five partly



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overlapped mathematical covers denoted by 1MC , 2MC , 3MC , 4MC and 5MC to cover

the whole domain based on the manifold cover system in the current NMM, and five

overall overlapped mathematical covers denoted by 1OC , 2OC , 3OC , 4OC and 5OC based on

the onefold cover system, respectively. Comparing manifold with onefold system, the

distinct difference is 1MC to 5MC use different nodal coordinates while 1OC to 5OC share

the common nodal coordinates, which improve the computational efficiency and save

computation cost in simulations.

Figure 8.10 Construction of onefold cover system from manifold cover system.

When the onefold cover system is considered to simulate the contact problems in

the coupled E-I algorithm, the partition of explicit and implicit element (denoted by OE)

become clear based on the onefold system. Here, we give one simple example to expose

the partition technique based on onefold system as shown Fig. 8.11. When two OEs

contact is searched using the contact detect criteria, the explicit and implicit OEs can be

determined relying on certain judgement algorithm (i.e. area algorithm, stress and strain

algorithms and/or displacement algorithm etc.), then the corresponding onefold covers

coordinates can be found. Here, we present one upper triangular cover to cover the

whole contact domain, the stars of 1Ii , 1

Ij and 1Ik describe the implicit OE, the stars of

1Ei , 1

Ej and 1Ek describe the explicit OE, respectively.

5MC



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Figure 8.11 Contact between explicit and implicit OEs based on onefold cover system.

Contact matrices between the explicit and implicit elements

In the coupled explicit-implicit NMM, once the contact is detected using the search

algorithm, the corresponding contact types can be determined based on certain

judgement criteria and the contact matrices are added to the contact elements. Basically,

there are three kinds of contact types can be detected using the area algorithm in the

onefold cover system: explicit to explicit (E-E), explicit to implicit (E-I) and implicit-

implicit (I-I). After this, the corresponding integration scheme will be adopted to form

the global coefficient matrix, in which the stiffness sub-matrix, damping item and mass

item are assembled to satisfy the corresponding algorithm for the equations of

equations. Since the assembled stiffness matrix and damping matrix both have a sparse

band-profile framework in the implicit algorithm, the coefficient sub-matrix 66][ K is

formed to the implicit algorithm. Accordingly, the diagonal coefficient sub-matrix

66][ M is generated to the explicit algorithm, and the whole matrix structure is

symmetric and positive definite when the coupled E-I algorithm is used based on

onefold cover system. To illustrate these properties of the coupled E-I algorithm, one

example of assembly of contact matrices involving r contact elements as shown in Fig.

8.12 is considered here. When two implicit contact elements (i.e. i and j ) participate in

, ,

,

Boundary point

Onefold cover star Contact domain

Explicit OE Implicit OE



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contact, contact matrix can be expressed as language C++ format of IIjj

IIji

IIij

IIii kkkk ,

and two explicit (i.e. k and l ) elements contact matrix is written as EEll

EElk

EEkl

EEkk kkkk

, respectively. Contact between one explicit element (i.e. k ) and one implicit element

(i.e. l ) will produce contact matrix 22

IEjkk to the equilibrium equation of the dynamic

system.

Figure 8.12 Assembly of contact matrices in the coupled E-I algorithm.

8.4 THE PARALLEL COMPUTATION OF THE NMM

Nowadays, parallelization has become the most important way to accelerate

engineering computations and simulations, in which several processors are distributed

to execute the computations simultaneously. With development of High Performance

Computing (HPC) (Chang 2006), a variety of parallel processors have been used in

different situations. These processors can be classified into three types: CMP (Chip

Multi-Processors) (2010), GPGPU (General Purpose GPU) (Owens et al. 2008), and

Heterogeneous Multiprocessor (Baker 2000). These processors are further organized as

SMP (Shared Memory Processors), MPP (Massively Parallel Processors) or DSM

(Distributed Shared Memory) style to form cluster systems. Each type of processor has

its own features. CMP, such as Intel Core2 Duo, Xeon and AMD Phenom, has the most

rr

Symmetric

I-I contact matrixII

iik IIijk

IIjik

EEkkk

EEllk

EEklk

EElkk

IEjkk

E-E contact matrix

E-I contact



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market share. In fact, CMP with 2 or 4 cores has been used for nearly all laptops,

desktops and workstations. GPGPU based computing, proposed by NVidia and AMD,

is now a hot topic in fast developing. It requires cooperation between GPU and CPU

hardware. Heterogeneous Multiprocessor, such as IBM Cell used in the Top One HPC

cluster Roadrunner, is powerful but not easily available for common customers.

To personal computer, more multi-core processors such as quad-core CPU, 8×core

CPU, even 80-core CPU (Chang 2006) are developed to obtain the best parallel

efficiency. Parallel programming environments such as OpenMP (OpenMP 2010),

pThreads (Nichols, Buttlar, and Farrell 1996) and TBB (TBB 2010) can be used to

implement the multi-core version of an existing code. Normally, the parallelization of a

code on multi-core PC is relatively simple as it only needs to deal with the shared

memory environment. It does not need to consider the task distribution and

communication between different processors. However, there also exist some

disadvantages of multi-core processor (Merritt 2008; ). Miao et al. (2009) carried out a

parallel implementation of NMM based on multiprocessor platforms, in which parallel

Jacobi's iterative method with OpenMP is implemented to improve computing

performance for such class of engineering problems. In the present chapter, we focus on

the parallel implementation of the NMM with Open MP.

8.4.1 Parallelization with openMP

There are general two models to do parallel processing: OpenMP and Message

Passing Interface (MPI) (MPI 2010), which is presented in Fig. 8.13. OpenMP is based

on Uniform Memory Access (UMA), is suited for shared memory systems like we have

on our desktop computers. Shared memory systems are systems with multiple

processors but each are sharing a single memory subsystem. Using OpenMP is just like

writing your own smaller threads but letting the compiler do it. On the other hand, MPI

is based on Non Uniform Memory Access (NUMA), is most suited for a system with

multiple processors and multiple memory such as a cluster of computers with their own

local memory, relying on MPI to divide workload across this cluster, and merge the

result when it is finished. Both support C/C++ and Fortran languages packed in

commercial software platform of Microsoft visual studio 2010. As the NMM employs



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implicit solution, which involves more computations of matrices to assemble global

matrix of system and contact treatments to the discontinuous problems, the

computational cost always draws our attentions when the large scale engineering is

considered. These motivate us to develop more efficient programming codes to reduce

the computational cost.

(a) UMA; (b) NUMA.

Figure 8.13 Parallel processing model: (a) UMA; (b) NUMA.

One alternative method to improve the efficiency is the explicit scheme of the

NMM, in which the explicit time integration is used and the code is shortened without

the assembly of global matrix of current NMM code. Since OpenMP is an explicit

programming based on fork-join model, parallelization can be as simple as taking a

serial program and inserting compiler directives. To further improve the computational

efficiency, a hypothesis construction scheme of parallel computation of the explicit

NMM using OpenMP is considered based on Duo-core CPU (Intel E8500 @ 3.16GHz,

3.17GHz) as shown in Fig. 8.14. We can find that the current NMM code is serial and

only one master thread through out the whole computation. The parallel code uses the

fork-joint model to let one parallel region being calculated by more than one thread such

as four threads in region Ⅰand two threads in region Ⅱ. In each parallel region, the

forked NMM thread is allocated to different threads to carry out parallel computing

based on shared memory (i.e. 4GB) and the value of environment variables,

OMP_NUM_THREADS is set 4 to obtain the most computational efficiency.

System Interconnect

P1 P2 Pn

SM1 Shared Memory

I/O SMn

Processors

Message-passsing Interconnection network

(Mesh, ring, torus, hypercube, cube-connected cycle, etc.)

PM

P

M

P

M

PM

P

M

P

M

PM

PM

P M

P M



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Figure 8.14 Construction of parallel computation of the NMM using OpenMP.

It is noted that, in the case of the explicit NMM, force the item and displacement

item can be paralleled directly by dividing into two parallel regions using the OpenMP,

which solves the motion equations conveniently and efficiently. A code segment of the

parallel programming to the explicit NMM is attached as shown in Fig. 8.15, in which

the computational performance of the code can be fully increased under the original

code structure. On the other hand, as the current NMM code is an implicit solution of

the equilibrium equation by principle of minimum potential energy, the assembly of the

global matrix in the computations reduces the efficiency when the OpenMP is taken into

account in the simulations.

Figure 8.15 Code segment of the parallel programming to the explicit NMM.

T1

T2

T3

T1

T4

T2

Parallel region Ⅰ Parallel region Ⅱ

NMM thread

T1 T2 T1 T4 T2 T3

NMM thread

int i=0; #pragma omp parallel for private (j) for (i=1; i<= m1; i++) { for (j=1; j<= 2; j++) { r[i][j] = f[i][j] + c0[i][j]; } /* j */ } /* i */



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8.4.2 Speedup

Speedup is the most important factor to evaluate the performance of parallel

algorithms, which is defined as the computing time ratio between the parallel runtime

for a given number of CPUs and the serial runtime (Kumar et al. 1994). The formula

can be expressed as

s

p

t

tS (8.3)

in which pt is the runtime of the serial code using the best optimization and st is the

runtime of the parallel code for the same problem, respectively.

To illustrate the speedup of parallel computing, a simply rock slope is considered

to compare the serial and parallel computational efficiency. The parameters of the two

used multi-core PCs in the parallel computations are presented in Table 8.1. The

simulated results using the serial and parallel NMM codes (total time 20s), as shown in

Fig. 8.16, indicates they are identical between two codes, which reveals that the parallel

NMM code is succeed to implement into the simulations. Fig. 8.17 shows the CPU

usage of the serial and multi-core NMM codes. It can be seen that the serial NMM has

not taken full advantage of the Multi-core CPU, and only 54% computing resource is

used for the serial NMM, but which increases up to almost 100% for the multi-core

NMM (both in the case of 2 CPUs and 4 CPUs). It means the OpenMP implementation

is effective and the computing resources can be fully used.

Table 8.1 Parameters of the used multi-core PCs.

CPU Core Hyper Threading Speed Memory

Intel E 8500 2 No 3.16 GHz 4 GB

Intel i5 760 4 No 2.80 GHz 4 GB



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Figure 8.16 Simulation results of the serial and parallel NMM codes.

In the present study, the rock slope modelling with 357 elements and 54 blocks is

simulated by the current NMM code and the explicit NMM code, in which both serial

and parallel codes are employed, respectively. The simulated results obtained by the two

types of codes are just the same. The computing time of the parallel codes with

OpenMP comparing the serial codes of the NMM and ENMM are presented in Fig.

8.18. It can be seen that speed of the parallel code with OpenMP is obvious faster than

the serial one both the cases of NMM and explicit NMM. The speedup of the 2-core and

4-core NMM and ENMM codes have been tested as shown in Fig. 8.18 (b), in which the

maximum value is up to 1.533 and 1.414 by the case of NMM and ENMM,

respectively. Furthermore, explicit NMM is more efficient than that of the current

NMM as the explicit code removes the assembly of the global matrix of the implicit

solutions.

(a) Serial NMM;

(b) Parallel NMM (2 CPUs).



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Figure 8.17 CPU usage of the serial and multi-core NMM codes.

(a) Serial NMM; (b) 2-core NMM;

(c) 4-core NMM.



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(a) CPU time;

(b) Speedup.

Figure 8.18 Computing time of the serial and parallel codes.


In recent years, numerical methods have been widely implemented to the rock

engineering, which motivates researchers seek varied numerical technologies for the

analysis of the rock engineering. In this section, to reveal the validity and applicability

of the proposed NMM, some project examples of rock slope stability are investigated.

The first one is a dynamic case of Lake Anderson slope failure from the earthquake of 6

August 1979 Coyote Lake, California USA. Simulated results of the seismic NMM

code will compare with the field measurements to illustrate the applicability of the

0

1000

2000

3000

4000

5000

6000

7000

NMM ENMM

CP

U t

ime

(s)

Serial 2 CPUs 4 CPUs

0

0.5

1

1.5

2

1 2 3 4

Sp

eed

up

CPUs

NMM ENMM



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seismic NMM code. The other example of JinpingⅠHydropower Station dam left

abutment slope is taken into account to present the coupled ENMM with DDA in terms

of efficiency. Furthermore, the coupled explicit and implicit NMM are presented to

simulate the stability of rock slope as well.

8.5.1 A dynamic case study of rock slope stability analysis

The Coyote Lake, California, earthquake of 6 August 1979 (ML 5.7) provided a

rare opportunity to perform a dynamic numerical analysis of a seismically induced slope

failure using the NMM. This earthquake was recorded by several strong-motion

instruments located in the epicentral region (Porcella et al. 1979). The earthquake

caused a slope failure near the east shore of Lake Anderson, 9 km northwest of the

epicentre. Location of the slope can be seen in Fig. 8.19, in which denotation of dot

presents the earthquake epicentre, pentacle indicates area of Lake Anderson slope

failure and triangle expresses the location of strong-motion instruments. Wilson et al

(Wilson and Keefer 1983) carried out a extend Newmark analysis to calculate

displacements of the landslide located strong-motion area under the action of seismic

ground motion, in which an expression for critical acceleration in terms of static FoS is

determined as

sin)1( gFoSac (8.4)

where FoS is the static FoS and is the angle of slope, respectively. The Newmark

analysis satisfactorily predicts the occurrence of the slope failure and the amount of

displacement of the landslide as actually measured.

In the simulation, the Lake Anderson slope is considered to investigate the stability

under the Coyote Lake earthquake, which of modelling can be seen in Fig. 8.20, in

which the landslide block of AFGHIJ is investigated using the proposed seismic NMM.

The results of conversional Pseudo-static analysis for the slope are 492.1FoS and

gac 22.0 as referred in (Wilson and Keefer 1983), and dynamic analysis parameters of

the NMM can be seen in Fig. 8.20. Fig. 8.21 shows a record of acceleration of the

earthquake by this region, in which the peak acceleration value of ap = 0.345g is

presented as well.



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Figure 8.19 Location of area of Lake Anderson slope in California USA (Keefer et al. (1980).

Figure 8.20 Numerical modelling of Lake Anderson slope.

N

B G A F

H I

J K

+

Pseudo-static analysis (Wilson and Keefer 1983),: Strength values c=300psf ϕ=25° Static FoS = 1.492 β=26°

Critical acceleration ac = 0.22g Dynamic analysis: Mass density ρ = 22 KN/m3 Youngs modulus E = 1.0 GPa Passion ration ν = 0.2 Frictional angle ϕ = 25° Cohesion c = 0.143 MPa

×

β



8.197

Figure 8.21 Record of acceleration of the earthquake.

The simulated results can be seen in Fig. 8.22, in which the displacement of the

landslide block is simulated using proposed seismic NMM and explicit NMM,

respectively. Since the acceleration is low at the beginning stage of the earthquake, the

displacement is small, and then it increases up to 23.50mm (seismic-ENMM) and

20.71mm (seismic-NMM) around 4 seconds, respectively. The results are in excellent

agreement with the estimated displacement of 21mm from field measurements of the

slope and the Newmark analysis solution of 27mm (Wilson and Keefer 1983).

Therefore, the proposed seismic NMM and explicit NMM are applicable to analysis the

stability of rock slope, which can be considered as a convenient numerical tool to apply

into other rock engineering. It is noted that the simulated results are closely related with

the physical parameters of the modelling, so more in situ measurements are necessary

and required to build realistic modelling of the slope.

-0.4

-0.2

0

0.2

0.4

0 5 10 15 20 25

Acc

eler

atio

n (

g)

Time (s)



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Figure 8.22 Simulated results of landslide under earthquake.

8.5.2 Dynamic stability analysis of Jinping I hydropower station

Jinping I hydropower station is located at the border of Yanyuan and Muli counties

in Sichuan Province, China. It is built on the Yalong River as a controlling cascade

hydropower station in the middle and downstream of the main stem. In this simulation,

the proposed explicit NMM-DDA method described in this paper is used to analyse the

left abutment slope stability of JingpingⅠ hydropower station. The scale map of

geomechanical model taken for analysis in 2D is shown in Fig. 8.23. We can find that

the slope is cut by bedding planes that have a stride towards the hillside, with

inclinations of 55°-70°. Since rocks in the left bank mainly consist of metasand stone

and slate of group 332 zT , plus complex geological structure, together with variablestrata

and stress-relief disturbance has affected the stability of rock masses on both sides of

the river (Song et al. 2011).

0

5

10

15

20

25

30

0 2 4 6 8 10

Dis

pl.

(mm

)

Time (s)

Seismic-NMM

Seismic-ENMM

Displ. =27



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Figure 8.23 Scale map of geomechanical model (Zhou et al. 2008).

When excavation is involved in the slope, such as blasting technique is employed

in the excavation, dynamic stability analysis of the slope is required to keep slope still

stable under dynamic effect. In this case, there is one alternative scheme of coupled

NMM and DDA can be considered in terms of computational efficiency. In order to

save computational time, we can use the proposed explicit NMM to simulate the

transient analysis of the basting effect and the DDA to study the slope stability after

blasting. To illustrate the efficiency of the proposed ENMM and DDA, the modelling of

the slope is constructed using the ENMM and DDA as shown in Fig. 8.24. The first

stage of simulation uses the proposed explicit NMM as the blasting duration is transient,

it is very suited for the explicit than the implicit algorithm, and then following the DDA

simulation. Here, we choice first one tenth of the total time to use explicit NMM code

and the other left time to simplify the modeling using the DDA code. There are total

1018 NMM elements and 32 blocks both in NMM and DDA. To compare the

efficiency, the implicit NMM ( st 001.0 ) is employed to simulate the stability of this

slope, the cost of which is 763.84s, is more than twice time of the proposed explicit

NMM and DDA ( st 0005.0 ). It is clear that the proposed method is more efficient to

simulate the stability of the slope when more element and blocks involving in the study.

It can be predicated that the proposed method can be extend applied to larger scale



8.200

project by simplifying the computation model using the proposed explicit NMM-DAA

method.

Figure 8.24 Modelling of the slope transfers from ENMM to DDA.

8.6 SUMMARY

The dynamic stability analysis of rock slope failure using the numerical manifold

method (NMM) is studied in the present chapter. Firstly, conservational pseudo-static

methods, Newmark Method and numerical methods applying into the seismic stability

analyses are investigated, the advantages and limitations of which are studied by

contrast of the NMM. Then, an alternative explicit algorithm of the NMM and coupled

explicit-implicit NMM are proposed to study the seismic stability of rock slope.

Furthermore, parallel computing with openMP is evaluated to improve efficiency of the

NMM. To reveal the validity and applicability of the proposed NMM, some numerical

examples of rock slope stability analysis are investigated. The first one is a dynamic

case of Lake Anderson slope failure from the earthquake of 6 August 1979 Coyote

Lake, California USA. Simulated results of the NMM will compare with the field

measurements to illustrate the applicability of the NMM. The other example of Jinping

ⅠHydropower Station dam left abutment slope is taken into account to present the

coupled explicit NMM with DDA in terms of efficiency. Furthermore, the coupled



8.201

explicit and implicit NMM are presented to simulate the stability of rock slope as well.

Therefore, it can be predicted that the proposed method is promising and can be extend

applied to larger scale project of rock slope with respect to dynamic stability analysis.



9.202

CHAPTER 9. CONCLUSIONS AND

RECOMMENDATIONS

9.1 SUMMARIES

The stability of rock slope under dynamic effect is often significantly influenced by

the discontinuities of the rock masses. The traditional NMM employs implicit time

integration and OCI algorithm to solve the discontinuous problems in rock slope. Based

on the finite covers system, the NMM combines the well developed analytical methods,

widely used FEM and the block-oriented DDA in a unified form. However, it is less

efficient to simulate the rock slope stability under dynamic effect when many contacts

involved.

This thesis focuses on the development of the explicit version of the NMM for

dynamic stability analysis of rock masses. A thorough investigation of the traditional

NMM is made in terms of the time integration and contact mechanics. The specific

works are outlined as follows:

Developed an explicit version of the NMM:

Investigated the traditional NMM in terms of computational accuracy and

efficiency;

Proposed an explicit time integration scheme for the NMM and to verify it

with respect to the computational efficiency and accuracy;

Combined the explicit and implicit algorithms for the NMM:

Couple the temporal explicit and implicit NMM;

Coupled the spatial explicit and implicit NMM;

Extended the explicit NMM for the rock slope stability analysis:

Implemented a seismic version of the explicit NMM for the dynamic

stability analysis of rock slope;

Applied the developed programming code to simulate the dynamic

stability of rock slope.



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9.2 CONCLUSIONS

The major contribution of this research is concentrated on the development of the

explicit numerical manifold method (ENMM) and its implementation on the stability

analysis of rock slope under dynamic effect. Based on the work in this thesis, the

following conclusions can be drawn:

1). Newmark integration scheme used in the NMM is investigated. The numerical

results present good accuracy and stability of the Newmark explicit scheme

compared to the implicit scheme. The Newmark explicit scheme is more

efficient in solving the nonlinear dynamic systems and such problems

compared to implicit scheme with respect to computational efficiency.

2). An explicit time integration scheme for the NMM is proposed to improve the

computational efficiency, in which a modified version of the NMM based on

an explicit time integration algorithm is derived on the dual cover system. The

original NMM based on displacement method is revised into an explicit

formulation of a force method. Although the ENMM requires small time-step

due to numerical stability of the scheme, it is efficient without assembling the

stiffness equations. Compared to the OCI used in the NMM, the open-close

algorithm is more efficient in the ENMM because of the explicit time

integration scheme without solving simultaneous algebraic equations in each

step and the smaller penetration incurred due to a smaller time step used. The

developed method is validated by three examples, two static problems of a

continuous simple-supported beam and plane stress field problem, the other

dynamic one of a single block sliding down on a slope. Results showed that

the accuracy of the ENMM can be ensured when the time step is small for

both the continuous and the contact problems. A highly fractured rock slope

and tunnel modelling are subsequently simulated. It is shown that the

computational efficiency of the proposed ENMM can be significantly

improved, while without losing the accuracy, comparing to the original

implicit version of the NMM. The ENMM is more suitable for large-scale



9.204

rock mass stability analysis and it deserves to be further developed for

engineering computations of practical rock engineering problems.

3). A modified version of the NMM based on an explicit time integration

algorithm is proposed. The calibration study of the ENMM on P-wave

propagation across a rock bar has been conducted. Various considerations in

the numerical simulations are discussed and parametric studies have been

carried out to obtain an insight into the influencing factors in wave

propagation simulation. The numerical results from the ENMM and NMM

modelling are accordant well with the theoretical solutions. The mesh ratio is

regarded as one of the major factors influencing the simulation accuracy. With

the consideration of both calculation accuracy and efficiency, a mesh ratio of

1/16 is recommended for one dimensional ENMM analysis. Furthermore, the

selection of a suitable time step depends on the internal element system and

the contact transfer between the interfaces. With the decrease of the time step

increment, the results become more accurate for the incident wave. In terms of

efficiency, the ENMM is more efficient than that of the NMM, even though

the different time steps are used. To further verify the capability of the

proposed ENMM in modelling of seismic wave effect in fractured rock mass,

a dynamic stability assessment for fractured rock slope under seismic effect is

analysed as well. The simulated results show that the computational efficiency

of the proposed ENMM can be significantly improved for the simulation of

stress wave propagation problems.

4). A temporal coupled explicit and implicit algorithm for the numerical manifold

method (NMM) is proposed. The time integration schemes, transfer algorithm,

contact algorithm and damping algorithm are studied in the temporal coupled

E-I algorithm to combine both merits of the explicit and implicit algorithms in

terms of efficiency and accuracy. Then, some numerical examples are

simulated using the proposed coupled algorithms, in which one calibration

example is studied with respect to the coupled temporal based on the dual



9.205

cover system. One numerical example of open-pit slope seismic stability

analysis using the coupled E-I algorithm is investigated as well. The simulated

results are well agreement with the implicit and explicit algorithms

simulations, but the efficiency is improved evidently. It is predicted that the

couple E-I algorithm proposed in the present paper can be applied into larger

scales engineering systems to combine the merits of both the implicit and

explicit algorithms in the NMM.

5). A spatial coupled E-I algorithm for the NMM is proposed. The time

integration schemes in the E-I algorithm, transfer algorithm of the coupled E-I

algorithm, the implicit contact algorithms based on the implicit integration

scheme and explicit contact algorithm based on the explicit integration are

studied in terms of accuracy and efficiency, respectively. In particular, onefold

cover system to the coupled E-I algorithm is proposed and drawn into the

coupled spatial E-I algorithm, in which the contact algorithm based on the

onefold cover system is discussed and derived in detail. Finally, some

numerical examples are simulated using the proposed coupled E-I algorithms,

in which one calibration example is studied using the proposed E-I algorithms

relied on the onefold cover system; One numerical example of rock slope

seismic stability analysis using the coupled E-I algorithm is studied as well.

The simulated results are well agreement with the implicit and explicit

algorithms, but the efficiency of the coupled algorithm is obviously higher

than that of original version of the NMM. It is predicted that the couple E-I

algorithm proposed in the present paper can be applied into larger scales

engineering systems to combine the merits of both the implicit and explicit

algorithms in the NMM.

6). The dynamic stability analysis of rock slope failure using the numerical

manifold method (NMM) is studied. Firstly, conservational pseudo-static

methods, Newmark Method and numerical methods applying into the seismic

stability analyses are investigated, the advantages and limitations of which are



9.206

studied by contrast of the NMM. Then, an alternative explicit algorithm of the

NMM and coupled explicit-implicit NMM are proposed to study the seismic

stability of rock slope. Furthermore, parallel computing with OpenMP is

evaluated to improve efficiency of the NMM. To reveal the validity and

applicability of the proposed NMM, some numerical examples of rock slope

stability analysis are investigated. The first one is a dynamic case of Lake

Anderson slope failure from the earthquake of 6 August 1979 Coyote Lake,

California USA. Simulated results of the NMM will compare with the field

measurements to illustrate the applicability of the NMM. The other example of

JinpingⅠHydropower Station dam left abutment slope is taken into account to

present the coupled ENMM with DDA in terms of efficiency. Furthermore, the

coupled explicit and implicit NMM are presented to simulate the stability of

rock slope as well. Therefore, it can be predicted that the proposed method is

promising and can be extend applied to larger scale project of rock slope with

respect to dynamic stability analysis.

9.3 RECOMMENDATIONS

The present study has validated the efficiency and robustness of the developed

ENMM to account for the dynamic problems such as stress wave propagation and

seismic stability analysis of rock slope. It is demanded a long way to further improve its

capability and applications in the project. Thus, there are still lots of work to be done in

the future research.

As demonstrated in Chapter 4 and 5, the accuracy and efficiency the ENMM is

more easily apt to be controlled by the finite element mesh size than that of the NMM.

The research can be carried out as the following:

Development of the adaptive finite mesh technique to avoid the tiny and

singular elements to the most degree;

Modification of the contact mechanics of OCI to further improve the contact

accuracy of the ENMM, including the selection of the appropriate contact

spring stiffness and the optimization of the current contact algorithm;



9.207

Development of the more efficient explicit scheme to further improve the

efficiency the current ENMM.

The coupled E-I algorithms in Chapter 6 to 8 have demonstrated the viability of the

developed ENMM combining the NMM and other numerical methods. Thus, the

following work can be done:

Development of other coupled algorithms such as E-I and E-E with different

time step sizes in spatial and temporal aspects;

Development of the coupled method of ENMM and DDA;

Development of the explicit scheme of the 3D NMM to further improve the

capability of the current 3D code;

Development of the parallelization computation such as MPI parallel algorithm

and so on.

The preliminary studies have demonstrated the great potential of the ENMM

enabling simulating for stability analysis of rock slope under seismic effect with

different conditions. Thus, the more influent factors can be taken into account such as

seepage, liquefaction and mechanical effects in the future research.



9.208

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