New point-to-face contact algorithm for 3-D contact ... point-to... · New point-to-face contact...

New point-to-face contact algorithm for 3-D contact problems using the augmentedLagrangian method in 3-D DDA

S. Amir Reza Beyabanakia,*, Roozbeh Grayeli Mikolab, S. Omid Reza Biabanakic and Soheil Mohammadid

aPooyesh Rah Mandegar Consulting Engineers, No. 48, Shahr-Tash Alley, North Sohrevardi Avenue, Tehran 1559615311, Iran; bDepartment ofCivil and Environmental Engineering, University of California, Berkeley, CA, USA; cDepartment of Civil Engineering, Sharif University of

Technology, Tehran, Iran; dSchool of Civil Engineering, University College of Engineering, University of Tehran, Tehran, Iran

(Received 13 September 2008; final version received 25 March 2009)

This paper presents a new point-to-face contact algorithm for contacts between two polyhedrons with planar boundaries. A new discrete numericalmethod called three-dimensional discontinuous deformation analysis (3-D DDA) is used and formulations of normal contact submatrices based on theproposed algorithm are derived. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. Thisapproach does not need to use an iterative algorithm in each time step to obtain the contact plane, unlike the ‘Common-Plane’ method applied in theexisting 3-D DDA. In the present 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, butmay lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violatesthe physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in thisresearch. This point-to-face contact model has been programmed and some illustrative examples are provided to demonstrate the new contact rulebetween two blocks. A comparison between results obtained by using the augmented Lagrangian method and the penalty method is presented as well.

Keywords: numerical method; three-dimensional; discontinuous deformation analysis; augmented Lagrangian method; rock mechanics

1. Introduction

Many engineering materials and structures are composed of blocks

in different shapes and sizes. For example, rock masses are divided

into discrete units by joints and faults. Soils are composed of small

particles. Stones and bricks form the fabric of masonry structures.

Contact between blocks may consist of a material, such as mortar

in masonry, or it may be plain interactions of solid objects, such as

joints in rock. The explicit modelling of these contacts, represented

as structural discontinuities, is outside the capability of continuum

idealisations, which generally underlie standard finite element

models. Discrete element models are very appropriate tools to

represent blocky structures. An early numerical approach capable

of modelling the movement and interaction between distinct blocks

was introduced by Cundall under the term ‘distinct element

method’ (DEM) (Cundall 1971). More recently, the ‘discontinuous

deformation analysis’ (DDA) method was developed by Shi (1988,

1993) to model the behaviour of discontinuous media. Similar

methods for simulating blocky rock mass behaviour include the

‘block spring method’ (BSM) (Wang and Garga 1993) and ‘com-

bined DEM/FEM’ formulation (Munjiza 2004), which involves

discretising each distinct element into finite elements (e.g., the

‘discrete finite elements’). Two-dimensional discontinuous defor-

mation analysis (2-D DDA) introduced by Shi (1988, 1993) in the

late 1980s has become a rapidly developing modern numerical

simulation technique and has found wide acceptance by research-

ers and engineers in a variety of mechanical analysis applications.

In the DDA approach, equations governing the equilibrium of

discrete blocks are derived by minimising their total potential

energy. It has been shown that the kinematic interactions between

the blocks can be modelled with great accuracy (Shi 1988, 1993).

MacLaughlin and Doolin (2006) provided a review of more than

100 published and unpublished validation studies on the DDA

approach. Previous DDA studies focused on solving problems in

two dimensions, but in many engineering problems three-dimen-

sional effects have to be considered (Abe et al. 1999, Jones and

Papadopoulos 2001, Duan and Ye 2002) . Up to now, relatively

little work on DDA development in 3-D has been published. Shi

(2001a,b,c) and Wu et al. (2005) provided basic formulations for

matrices for different potential terms. Liu et al. (2004) and Yeung

et al. (2003, 2004) highlighted the application of 3-D DDA. Jiang

and Yeung (2004) developed a point-to-face model for 3-D DDA.

In these researches, contacts between the blocks are detected by

using the ‘‘Common-Plane’’ approach (1988). In this approach, a

plane is located between the blocks, so that the overlap of vertices

or edges on this plane indicates the physical interaction and defines

the contact area. The location and the orientation of the contact

plane are obtained by an iterative algorithm, which may require

many computations. This plane is updated as the blocks move.

In this paper, a new model of 3-D point-to-face contact detec-

tion and mechanics is developed and formulations of normal and

Geomechanics and Geoengineering: An International Journal

Vol. 4, No. 3, September 2009, 221--236

*Corresponding author. Email: [email protected]

ISSN 1748-6025 print=ISSN 1748-6033 online� 2009 Taylor & FrancisDOI: 10.1080=17486020903045370http:==www.informaworld.com

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shear contacts submatrices based on this new model of contact

are presented for 3-D DDA. In the proposed method, a direct

algorithm for contact resolution is provided. The sequence of

steps allows decisions regarding whether the blocks are actually

in contact and the identification of the contact type and geometry

parameters are obtained, depending on the possible block posi-

tions. In this way, unlike the ‘‘Common-Plane’’ approach, an

iterative algorithm in each time step, which may require many

computations, is not required. Also in this new method, the

contact mechanics computation does not need to project vertices

and simply uses only coordinates of block vertices, in opposition

to Jiang and Yeung’s approach(2004).

The penalty method was originally used by the above-men-

tioned 3-D DDA researchers to enforce contact constraints at the

block interface. The accuracy of the contact solution depends

highly on the choice of the penalty number and the optimal

number cannot be explicitly found beforehand. Obviously, the

penalty number should be very large to achieve zero interpene-

tration distance. However, a very high penalty number leads to

progressive ill-conditioning of the resulting system and thus one

cannot hope to achieve high-accuracy solutions with this

approach. A well-known method to overcome these problems

for equality constrained problems is the augmented Lagrangian

method (Landers and Taylor 1985). The augmented Lagrangian

method has been advocated by Lin et al. (1996) in two-

dimensional discontinuous deformation analysis. In this research,

the same method has been implemented in three-dimensional

discontinuous deformation analysis and some illustrative exam-

ples are presented for demonstrating this new approach.

2. Three-dimensional discontinuous deformation analysis

formulation

DDA calculates the equilibrium equations by minimisation of

the potential energies of single blocks and the contacts between

two blocks. To calculate the simultaneous equilibrium equa-

tions, deformation functions must be defined. The deformation

function calculates the deformation of all the blocks using the

displacement of each block centroid.

Assuming all displacements are small and each block has

constant stress and constant strain throughout, the displacement

(u,v,w) of any point (x,y, z) of a block can be represented by 12

displacement variables. In the 12 variables, (u0,v0, w0) is the rigid

body translation of a specific point (x0,y0,z0) within the block,

(rx,ry, rz) are the rotation angle of the block with a rotation centre

(x0,y0,z0), and "x, "y, "z, �xy, �yz, �zx are the normal and shear

strains in the block. The displacement of any point (x, y, z) in the

block i can be represented by Equation (1).

uvw

� �¼ Ti½ � Di½ � ¼

¼1 0 0 0 ðz� z0Þ �ðy� y0Þ ðx� x0Þ 0 0

ðy�y0Þ2

0ðz�z0Þ

2

0 1 0 �ðz� z0Þ 0 ðx� x0Þ 0 ðy� y0Þ 0ðx�x0Þ

2

ðz�z0Þ2

0

0 0 1 ðy� y0Þ �ðx� x0Þ 0 0 0 ðz� z0Þ 0ðy�y0Þ

2

ðx�x0Þ2

264

375 �

u0v0w0rxryrz"x"y"z�xy�yz�zx

0BBBBBBBBBB@

1CCCCCCCCCCA

ð1Þ

Since 3-D DDA conforms to the minimum total potential

energy principle, the total potential energy is the summation of

all potential energy sources for each block, i.e., the potential

energy contributed by the elastic deformation of the blocks, the

initial stresses, the point load on a block, the volume forces, the

inertia forces, the potential energy when the blocks contact each

other; and the potential energy contributed by the constraint

displacement points. The fixed point is the point where the

prescribed constraint displacement equals zero in DDA.

For a system with N blocks, the total potential energy can be

expressed in matrix form as follows:

� ¼ 1

2fD1gT fD2gT fD3gT : : : fDNgT� �

½K11� ½K12� ½K13� : : : ½K1N �½K21� ½K22� ½K23� : : : ½K2N �½K31� ½K32� ½K33� : : : ½K3N �

..

. ... ..

. . .. ..

.

½KN1� ½KN2� ½KN3� : : : ½KNN �

266666664

377777775

fD1gfD2gfD3g

..

.

fDNg

266666664

377777775

þ fD1gT fD2gT fD3gT : : : fDNgT� �

·

fF1gfF2gfF3g

..

.

fFNg

266666664

377777775þ C

ð2Þ

222 S.A.R. Beyabenaki et al.


where Dif g represents displacement variables and Fif g indi-

cates loading and moments caused by the external forces and

stress acting on block i. The stiffness submatrices ½Kii�depend

on the material properties of block i, with ½Kij� i�j being defined

by the contacts between blocks i and j; and C is the energy

produced by friction force.

If a first-order displacement function is chosen, there are 12

displacement variables for each block. As a result, Dif g and

Fif g are 12·1 matrices and ½Kij� is a 12·12 matrix.

By minimising the total energy, the simultaneous equations

can be expressed in matrix form as follows:

½K11� ½K12� ½K13� : : : ½K1N �½K21� ½K22� ½K23� : : : ½K2N �½K31� ½K32� ½K33� : : : ½K3N �

..

. ... ..

. . .. ..

.

½KN1� ½KN2� ½KN3�: : :½KNN �

2666664

3777775

fD1gfD2gfD3g

..

.

fDNg

2666664

3777775¼

fF1gfF2gfF3g

..

.

fFNg

2666664

3777775ð3Þ

For only one block, the equilibrium equations for each time

step are derived by minimising the total potential energy, �, in

each variable. For block i, equations:

@�

@u¼ 0;

@�

@v¼ 0;

@�

@w¼ 0 ð4Þ

represent the equilibrium of all loads and contact forces acting

on block i along X; Y and Z directions respectively. The

equations:

@�

@rx

¼ 0;@�

@ry

¼ 0;@�

@rz

¼ 0 ð5Þ

represent the moment equilibrium of all loads and contact

forces acting on block i. The equations:

@�@"x¼ 0; @�

@"y¼ 0; @�

@"z¼ 0

@�@�yz¼ 0; @�

@�zx¼ 0; @�

@�xy¼ 0

8<: ð6Þ

represent the equilibrium of all external forces and stresses on

block i.

The differentiations:

@2�

@dri@dsj

; r; s ¼ 1; 2; ::: ; 12 ð7Þ

form a 12·12 submatrix, which is the submatrix ½Kij� in the

global Equation (3). The differentiations:

� @�ð0Þ@dri

; r; s ¼ 1; 2; ::: ; 12 ð8Þ

are the free terms of the equilibrium equations derived by

minimising the total energy, �. Therefore, all terms of

Equation (8) form a 12·1 submatrix, which is the submatrix

Fif g in Equation (3).

An essential part of any 3-D discrete element method is a

rigorous contact model governing the interaction of many 3-D

discontinuous blocks. The contact model includes two main

steps, named ‘contact detection’ and ‘contact mechanics’. In

the next sections, these two parts of the proposed contact model

are explained.

3. Contact detection scheme

Contact detection is usually performed in two independent

stages. The first stage, referred to as neighbour search, is

merely a rough search that aims to provide a list of all possible

particles in contact. Among available algorithms for neigh-

bour searching, the most recent ones include the sweep and

prone algorithm (Cohen et al. 1995) and the spatial partition-

ing algorithm (Munjiza 2004). A review of neighbour search

methods is available in Bergen (2003). In the second stage,

which is studied in this paper, called geometric resolution,

pairs of contacting particles obtained from the first stage are

examined in more detail to find the contact points and calcu-

late the contact forces.

In the contact theory, it is necessary to determine the type of

contact between any two arbitrary convex-shaped polyhedral

blocks. The type of contact is important because it determines

the mechanical response of the contact. There are six types of

contact for 3-D blocks, i.e., vertex-to-vertex, vertex-to-edge,

vertex-to-face, edge-to-edge, edge-to-face and face-to-face.

Yeung et al. (2003) and Jiang and Yeung (2004) pointed out

that vertex-to-face, edge-to-face and face-to-face contact types

can be converted to the contact of a point to a face. In this paper,

this type of contact will be studied. In the next sections, a new 3-

D point-to-face contact detection algorithm for geometric reso-

lution with new contact force calculation formulas is presented.

This algorithm can be divided into two phases: finding contact

points between two possible blocks in contact, and identifica-

tion of contact type.

3.1 Finding contact points between probable blocksin contact

When determining the distance between two polyhedrons, the

immediate and most natural approach consists in computing

and comparing the distances between their boundary features

(vertices, edges and faces). If the number of vertices and edges

as well as faces is high, searching between all boundary features

of two objects would be costly. To overcome this problem in the

first step, the nearest vertices for two particles are computed

and the search is continued between all faces that share those

vertices in the two polyhedrons. As shown in Figure 1, if R and

Q are the nearest vertex of two polyhedrons A and B respec-

tively, a search for computing the closest point should be

carried out only on the neighbouring faces of R and Q (neigh-

bouring faces of a vertex are those that share that vertex).

Geomechanics and Geoengineering: An International Journal 223


In order to compute contact points between blocks A and B,

the following steps should be performed:

1.0. Find vertices on the neighbouring faces of R and Q

2.0. Let T denote the plane passing through the face (polygon) f

of block B

3.0. Find vertices of step 1.0 whose distance to plane T falls

within tolerance (in this study tolerance is 2·maximum

displacement in each time step). The following steps

should be done for each of them:

3.1. Calculate projection of the vertex obtained in step 3.0

(e.g., P) on plane T (e.g., point P¢ in Figure 2)

3.2. If P¢ is inside a given polygon f (Figure 2a), this point

and P will be contact points, then Goto 4.0

3.3. Else (If P¢ is outside the polygon f (as shown in Figure

2b)), the following steps should be performed:

3.3.1. Find the nearest point of polygon f to P’ (e.g.,

point P1 in Fig 2b)

3.3.2. Find the nearest point of neighbouring faces of

R to P1 (e.g., P2)

3.3.3. If P2 is exactly P, this point and P1 will be

contact points, then Goto 4.0

3.3.4. Else find the nearest point of polygon f to P2

(e.g., P3)

3.3.4.1. If P3 is exactly P1, these two points

are contact points

3.3.4.2. Else P P2 and P1 P3 then

Goto 3.3.2

3.3.4.3. End If

3.3.5. End If

3.4. End If

4.0. End

A similar approach is proposed by Nezami et al. (2006).

3.2 Identification of contact type

If the distance from contact points to another block falls within

the tolerance, one of the following types of contact is

probable:

� If three or more points of same face fall within the tolerance,

the probability of face-to-face contact type is applicable

(Figure 3).

� If two points of same edge fall in the tolerance, the prob-

ability of edge-to-face contact type is applicable (Figure 4).

� If one vertex falls in the tolerance, the vertex-to-face type is

applicable (Figure 5).

4. Three-dimensional point-to-face contact mechanics

When a point-to-face contact candidate is found in the compu-

tation, as shown in Figure 6, the effects of the contact can be

represented by applying two stiff contact springs in the normal

and tangential directions (Wu et al. 2005). To prevent the

blocks from penetrating each other, 3-D DDA considers the

normal contact forces shown in Figure 6(a) when the blocks

Figure 1. Best candidate for finding the closest points between twopolyhedrons A and B (the neighbouring faces are shaded).

Figure 2. Projection of vertex P of block A on face f of block B (a) projectivepoint P¢ located inside the boundary of f (b) projective point P¢ located outsidethe boundary of f.

Figure 3. Face-to-face contact type.



come into contact. However, the forces are disregarded when

the blocks are separated. The usage of the normal contact spring

satisfies the no-penetration and no-tension criteria developed in

the original DDA and the open--close iteration is used to obtain

the converged results at each time step (Shi 2001). In addition, a

shear spring as shown in Figure 6(b) is activated when the shear

force is smaller than the shear resistance of a discontinuity to

diminish the relative displacement of the two blocks in the

form:

Fs < Fn tgð’Þ þ C ð9Þ

where Fs is the shear contact force, ’ is friction angle of the

discontinuity, and Fn is the normal contact force.

The penalty method is used to calculate the potential energy

caused by the contact spring in the existing 3-D DDA. The main

features of this method are (Mohammadi 2003):

� Enforcement of constraints requires no extra equations.

� The solution is easily obtained by simply adding contact

components to the stiffness matrix.

� The constraints are only satisfied in an approximate manner

and the contact solution depends highly on the choice of the

penalty number and the optimal number cannot be explicitly

found beforehand.

� If the penalty number is too low, the constraints are poorly

satisfied, while if it is too large the simultaneous equili-

brium matrix becomes difficult to solve.

In this paper, a more efficient method, named the augmented

Lagrangian method, is used and the results are compared with

the penalty method results. Since (1) shear contact spring is

applied only when the shear force is smaller than the shear

resistance of a discontinuity; (2) its calculation using the aug-

mented Lagrangian method is complicated and not numerically

economical; and (3) a major concern in contact problems is

satisfying normal contact constraints, the augmented

Lagrangian method is used only for normal contacts. The aug-

mented Lagrangian approach uses penalty stiffness but itera-

tively updates the contact traction to impose the contact

constraints with a specified precision. The main features of

this method are:

� No additional equations are required.

� Large penalty values are not required, avoiding the ill con-

ditioning of the stiffness matrices. However, if the initial

penalty number is too small, many iterations are required.

� The constraints are satisfied within a user-defined required

tolerance.

� The algorithm can be used effectively for applications

where the contact pressures become very large in compar-

ison with the material elastic parameters.

4.1 Calculation of normal contact force

When a point-to-face contact occurs, the normal spring with a

stiffness of Pn is introduced into the formulation to return

the point to the surface along the shortest distance. When

using the augmented Lagrangian method, the normal contact

force at the contact point (�n) can be accurately approximated

Figure 4. Edge-to-face contact type.

Figure 5. Vertex-to-face contact type.

Figure 6. Representation of normal and shear contacts in 3-D.



by iteratively calculating the Lagrange multiplier ��n. A first-

order updated value for ��n can be written as:

�n � � �n knþ1¼ � �n kn

þ Pndn ð10Þ

where the penalty number, Pn, can be variable and does not

have to be a very large number as in the penalty method. In

Equation (10), � �n kn is the Lagrangian multiplier at the

kthn iteration and ��n knþ1

is the updated Lagrange multiplier

(Mohammadi 2003), and dn is the shortest (normal) distance

from the contact point to the contact face.

As shown in Figure 7, let P1ðx1; y1; z1Þ be a vertex of block i

before the displacement increment, and the polygon P2P3 :::Pm

is the contact face, which is a face of block j. Let ðxi; yi; ziÞ and

ðui; vi;wiÞ be the coordinates and displacement increments,

respectively, of the vertices Pi ði ¼ 1; 2; ::: ;mÞ, and let

P¢i ði ¼ 1; 2; ::: ;mÞ be the respective vertices after the displace-

ment increments are applied. The normal distance between the

P¢1 and the contact face P¢

2P¢3 :::P

¢m, dn, is given by:

dn ¼~n :P¢2P¢

1 ð11Þ

where ~n is the unit vector pointing out of the block that is

normal to the contact face P2P3 :::Pm.

If dn< 0, the point P1 will penetrate the contact face, which

means a penetration takes place.

Let

u1

v1

w1

8<:

9=; ¼ ½Tiðx1; y1; z1Þ� : fDig ð12Þ

and

ul

vl

wl

8<:

9=; ¼ ½Tjðxl; yl; zlÞ� : fDjg; l ¼ 2; 3; 4 ð13Þ

Hence, P¢2P¢

1 can be written with the following form:

P¢2P¢

1 ¼x1 þ u1

y1 þ v1

z1 þ w1

8<:

9=;�

x2 þ u2

y2 þ v2

z2 þ w2

8<:

9=;¼

x1 � x2

y1 � y2

z1 � z2

8<:

9=;þ

u1

v1

w1

8<:

9=;�

u2

v2

w2

8<:

9=;

ð14Þ

Let

fBg ¼x1 � x2

y1 � y2

z1 � z2

8<:

9=; ð15Þ

and using Equations (12) and (13), we have:

P¢2P¢

1 ¼ fBg þ ½Tiðx1; y1; z1Þ� : fDig� ½Tjðx2; y2; z2Þ� : fDjg ð16Þ

and~n can be written as:

~n ¼ P¢2P¢

3 · P¢2P¢

4

P¢2P¢

3 · P¢2P¢

4

�� ð17Þ

Assuming displacements of the block in a time step are small,

we have:

P¢2P¢

3 · P¢2P¢

4

�� @ P2P3 · P2P4j j

¼~i ~j ~k

x3 � x2 y3 � y2 z3 � z2

x4 � x2 y4 � y2 z4 � z2

��

�� ¼ A ð18Þ

and

P2¢P3¢ · P2¢P4¢

¼~i ~j ~k

ðx3 þ u3Þ � ðx2 þ u2Þ ðy3 þ v3Þ � ðy2 þ v2Þ ðz3 þ w3Þ � ðz2 þ w2Þðx4 þ u4Þ � ðx2 þ u2Þ ðy4 þ v4Þ � ðy2 þ v2Þ ðz4 þ w4Þ � ðz2 þ w2Þ

��

��¼ðy3 þ v3Þ � ðy2 þ v2Þ ðz3 þ w3Þ � ðz2 þ w2Þðy4 þ v4Þ � ðy2 þ v2Þ ðz4 þ w4Þ � ðz2 þ w2Þ

��~i

þðz3 þ w3Þ � ðz2 þ w2Þ ðx3 þ u3Þ � ðx2 þ u2Þðz4 þ w4Þ � ðz2 þ w2Þ ðx4 þ u4Þ � ðx2 þ u2Þ

��~j

þðx3 þ u3Þ � ðx2 þ u2Þ ðy3 þ v3Þ � ðy2 þ v2Þðx4 þ u4Þ � ðx2 þ u2Þ ðy4 þ v4Þ � ðy2 þ v2Þ

��~k

¼ n1~iþ n2

~jþ n3~k

ð19Þ

that:

1P′

Block i

Block i

Contact face of block j

1P

2P

3P 4P

5P

6P

mP

Figure 7. Point-to-face contact.



n1 ¼1 y2 þ v2 z2 þ w2

1 y3 þ v3 z3 þ w3

1 y4 þ v4 z4 þ w4

��

��

¼1 y2 z2

1 y3 z3

1 y4 z4

��

��þ

1 y2 w2

1 y3 w3

1 y4 w4

��

��þ

1 v2 z2

1 v3 z3

1 v4 z4

��

��þ

1 v2 w2

1 v3 w3

1 v4 w4

��

��¼ n1ð1Þ þ n1ð2Þ þ n1ð3Þ þ n1ð4Þ

ð20Þ

n2 ¼1 z2 þ w2 x2 þ u2

1 z3 þ w3 x3 þ u3

1 z4 þ w4 x4 þ u4

��

��¼

1 z2 x2

1 z3 x3

1 z4 x4

��

��

þ1 z2 u2

1 z3 u3

1 z4 u4

��

��þ

1 w2 x2

1 w3 x3

1 w4 x4

��

��þ

1 w2 u2

1 w3 u3

1 w4 u4

��


ð21Þ

n3 ¼1 x2 þ u2 y2 þ v2

1 x3 þ u3 y3 þ v3

1 x4 þ u4 y4 þ v4

��

��¼

1 x2 y2

1 x3 y3

1 x4 y4

��

��þ

1 x2 v2

1 x3 v3

1 x4 v4

��

��

þ1 u2 y2

1 u3 y3

1 u4 y4

��

��þ

1 u2 v2

1 u3 v3

1 u4 v4

��


ð22Þ

Assuming the displacements of block j in a time step are small,

the second, third and fourth terms of Equations (20--22) can be

ignored. Therefore:

~n ¼ 1

An1; n2; n3h i @

1

An1ð1Þ; n2ð1Þ; n3ð1Þh i ð23Þ

From Equations (10), (16) and (23):

dn ¼1

An1ð1Þ; n2ð1Þ; n3ð1Þh i : fBg þ ½Tiðx1; y1; z1Þ� : fDigf

�½Tjðx2; y2; z2Þ� : fDjgg ð24ÞLet

G ¼ 1

An1ð1Þ; n2ð1Þ; n3ð1Þh i : fBg ð25Þ

½Hi� ¼1

An1ð1Þ; n2ð1Þ; n3ð1Þh i : ½Tiðx1; y1; z1Þ� ð26Þ

½Qj� ¼1

An1ð1Þ; n2ð1Þ; n3ð1Þh i : ½Tjðx2; y2; z2Þ� ð27Þ

Therefore:

dn ¼ Gþ ½Hi� fDig � ½Qj� fDjg ð28Þ

At the kthn iteration, the potential energy of the normal spring is

given by:

�n ¼ ��nkndn þ

1

2Pnd2

n

¼ ��nknGþ ½Hi� fDig � ½Qj� fDjg� �

þ 1

2Pn Gþ ½Hi� fDig � ½Qj� fDjg� �

: Gþ ½Hi� fDig � ½Qj� fDjg� �

ð29Þ

This equation consists of two components. The first component

is the strain energy resulting from the iteration Lagrange multi-

plier ��nkn, and the penalty constraint creates the second.

Expanding the right side of Equation (29) and minimising �n

by taking derivatives, four 12·12 stiffness submatrices and two

12·1 force submatrices are obtained in the global equilibrium

equation (Equation (3)).

The derivatives of �n:

krs ¼@2�n

@dri @dsi

rs ¼ 1; 2; ::: ; 12 ð30Þ

form a 12 · 12 submatrix which is added to the submatrix ½Kii�in Equation (3):

Pn ½Hi�T ½Hi� ! ½Kii� ð31Þ


krs ¼@2�n

@dri @dsj

rs ¼ 1; 2; ::: ; 12 ð32Þ

form a 12·12 submatrix which is added to the submatrix ½Kij� inEquation (3):

Pn ½Hi�T ½Qj� ! ½Kij� ð33Þ




krs ¼@2�n

@drj @dsi

r; s ¼ 1; 2; ::: ; 12 ð34Þ

form a 12·12 submatrix which is added to the submatrix ½Kji� inEquation (3):

Pn ½Qj�T ½Hi� ! ½Kji� ð35Þ


krs ¼@2�n

@drj @dsj

r; s ¼ 1; 2; ::: ; 12 ð36Þ

form a 12·12 submatrix which is added to the submatrix ½Kjj� inEquation (3):

Pn ½Qj�T ½Qj� ! ½Kjj� ð37Þ

The derivatives of �n at 0:

fri ¼ �@�nð0Þ@dri

r ¼ 1; 2; ::: ; 12 ð38Þ

form a 12·1 submatrix which is added to the submatrix ½Fi� inEquation (3):

� ð��nknþ Pn :GÞ ½Hi� ! ½Fi� ð39Þ

The derivatives of �n at 0:

frj ¼ �@�nð0Þ@dri

r ¼ 1; 2; ::: ; 12 ð40Þ

form a 12·1 submatrix which is added to the submatrix ½Fj� inEquation (3):

� ð��nknþ Pn :GÞ ½Qj� ! ½Fj� ð41Þ

The final exact contact forces can always be obtained by the

iterative method even with small initial values of the penalty

number.

4.2 Calculation of shear contact force

Assume the point P0ðx0; y0; z0Þ denotes the projection of P1 on the

contact plane and P¢0 represents this point after the displacement

increment (Figure 8). The shear displacement is calculated as:

ds ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP¢

0P¢1j j2�d2

n

qð42Þ

The potential energy of the shear spring is given by:

�sc ¼1

2Psd

2s ¼

1

2Ps P¢

0P¢1

�� 2�d2n

�

¼ 1

2Ps

ðx1 þ u1Þ � ðx0 þ u0Þðy1 þ v1Þ � ðy0 þ v0Þðz1 þ w1Þ � ðz0 þ w0Þ

264

375

T

:

ðx1 þ u1Þ � ðx0 þ u0Þðy1 þ v1Þ � ðy0 þ v0Þðz1 þ w1Þ � ðz0 þ w0Þ

264

375� d2

n

0B@

1CA

ð43Þ

WherePs is the stiffness of the shear spring. Let

u1

v1

w1

8<:

9=; ¼ ½Tiðx1; y1; z1Þ� : fDig ð44Þ

and

u0

v0

w0

8<:

9=; ¼ ½Tjðx0; y0; z0Þ� : fDjg ð45Þ

and using Equation (28), we have:

�sc ¼1

2Ps

x1 � x0

y1 � y0

z1 � z0

264

375

T

þfDigT ½Ti�T � fDjgT ½Tj�T

0B@

1CA

·x1 � x0

y1 � y0

z1 � z0

264

375þ ½Ti� fDig � ½Tj� fDjg

0B@

1CA

� 1

2Ps Gþ ½Hi� fDig � ½Qj� fDjg� 2

ð46Þ

By expanding and minimising the potential energy �sc, the

following matrices can be added to the submatrices

½Kii�; ½Kij�; ½Kji�; and ½Kjj� in the global stiffness matrix:

0P′

Block i

1P

Contact face of block j

2P

3P 4P

5P

6PmP

0PL

1P′

Block i

Figure 8. Illustration of 3-D contact.



½Kii� ¼ Ps½Ti�T ½Ti� � Ps½Hi�T ½Hi�

½Kij� ¼ �Ps½Ti�T ½Tj� � Ps½Hi�T ½Qj�

½Kji� ¼ �Ps½Tj�T ½Ti� � Ps½Qj�T ½Hi�

½Kjj� ¼ Ps½Tj�T ½Tj� � Ps½Qj�T ½Qj� ð47Þ

And the vectors ½Fi� and ½Fj� are calculated as follows and then

added to the global force vector:

½Fi� ¼ �Ps½Ti�Tx1 � x0

y1 � y0

z1 � z0

24

35þ Ps G ½Hi�T

½Fj� ¼ Ps½Tj�Tx1 � x0

y1 � y0

z1 � z0

24

35þ Ps G ½Qj�T ð48Þ

4.3 Calculation of frictional force

When the state of the contact is sliding, a pair of equal and

opposite frictional forces parallel to the sliding direction is

applied on the contact face at the points P1 and P0. The magni-

tudes and directions of the frictional forces are obtained from a

previous iteration. Coulomb’s law is used to evaluate the dis-

location movements of the block interfaces. When the shear

force F conforms to:

F � N tanð�Þ þ C ð49Þ

in the above formula, N is the normal force on the contact

boundary and � and C are the friction angle and cohesion,

respectively. The frictional force is calculated from the normal

contact compressive force from the previous step (Jiang and

Yeung 2004):

F ¼ �n tanð�Þ ð50Þ

where �n is taken from the previous step. Let L be the direction

of the frictional force (Figure 8) and n̂ be the unit vector

pointing out of the block in a direction normal to the contact

face. We have:

L ¼ P¢0P¢

1 � hP¢0P¢

1; n̂i: n̂ ð51Þ

Therefore, the potential energy of the friction force is given by:

�f ¼ F:ðd:LLj j Þ ð52Þ

where

d ¼ ½u1 v1 w1� � ½u0 v0 w0�¼ fDig T :½Tiðx1; y1; z1Þ� T � fDjg T :½Tjðx0; y0; z0Þ� T

therefore:

�f ¼ F:ð½Di�T ½M� � ½Dj�T ½N�Þ ð53Þ

where

½M� ¼ 1

Lj j ½Tiðx1; y1; z1Þ�T LT ð54Þ

and

½N� ¼ 1

Lj j ½Tjðx1; y1; z1Þ�T LT ð55Þ

By minimising the potential energy, we have:

½Fi� ¼ �F M½ � ð56Þ

½Fj� ¼ F N½ � ð57Þ

which are added to the global force vector.

4.4 Residual forces in augmented Lagrangian method

From physical point of view the Lagrange multiplier, �n, repre-

sents the stiffness of the normal contact between two blocks and

the penalty number, Pn, represents the stiffness of the normal

contact spring, and the final exact contact forces can always be

obtained by the iterative method. As it is mentioned by Lin et al.

(1996) the precision of the solution depends on residual forces

that are produced during the iterative calculations of contact

forces. From a physical point of view, the residual forces are the

unbalanced forces between external and internal forces. The

criterion for convergence is based on the L2 norm of the

residual forces that is (Lin et al. 1996):

½K� ½Dk� � ½K� ½Dk�1�k k½K� ½Dk�1�k k Tol

� �ð58Þ

The tolerance, Tol, is a positive number that is specified by the

user. Lin et al. (1996) reported that a Tol value ranging between

0 (corresponding to no contact) and 0.1 will give excellent

results. Algorithm 1 lists the complete DDA algorithm using

the augmented Lagrangian method.



Algorithm 1. Discontinuous deformation analysis program

algorithm

1: Initialise data

2: For each ti, i=1, . . ., n time steps do

3: Find nearest point between probable blocks in contact

4: Find vertices falling in the tolerance

5: Find contact points

6: Compute type of contact with regard to number of falling

points in tolerance

7: Assemble stiffness matrix

8: Integrate

9: Repeat {open-close iteration}

10: Initialise �n1 ¼ 0

11: For each iteration kn

12: Solve for displacements (½K� ½D� ¼ ½F�)13: Compute penalty forces (f c

n ¼ ��n ¼ Pn · dnmax)

14: Check for convergence:

14.1: If ��n Tol1ð Þ and½K� ½Dk ��½K� ½Dk�1�k k

½K� ½Dk�1�k k Tol2

�Then

14.2: Goto 18

14.3: End If

15: update Lagrange multipliers �nknþ1¼ �nkn

þ �n

� 16: Goto 11

17: End For

18: Until no-tension, no-penetration

19: Update vertices positions

20: Update blocks stresses

21: End For

5. Examples

The algorithm described in the previous sections has been pro-

grammed in VC++. To investigate it, three examples are pre-

sented and the results are compared with the results obtained by

using the penalty method that is used in the original 3-D DDA.

5.1 Sliding of a block along an inclined plane

This example simulates the sliding of a block along an inclined

plane at an angle � to the horizontal direction with friction

angle � (Figure 9).

Under the action of gravitational force, the displacement s of the

block is determined analytically as a function of time t given as:

sðtÞ ¼ 1

2g sin�� cos� tan�ð Þ t2 ð59Þ

The inclination of the modelled plane is 20 and the density,

Young’s modulus and Poisson’s ratio for both blocks are

2:6 · 103 kg=m3, 5 GPa and 0.25, respectively. The maximum

time increment for each time step is 0.01 s.

Once the problem was solved with � ¼ 0. The accumulated

displacements are calculated up to 5 s. A comparison between

the analytical solution in Equation (59) and 3-D DDA results

using the classic penalty method and the augmented Lagrangian

method for different values of the stiffness of the normal con-

tact spring is shown in Figures 10 and 11, respectively. Figure

12 shows the displacements of a sliding block in X and Z

directions for the penalty method.

The deformation of the block system, using the augmented

Lagrangian method to enforce the contact interface, after 0 s, 3

s, 4 s and 5 s for P = 50 MN/m is shown in Figure 13. It is clear

that no block interpenetration occurs here even though the

penalty number is low. Figure 14 and Figure 15 show the

deformation of the block system, using the classic penalty

method, after 0 s, 3 s, 4 s and 5 s for P = 500 MN/m and P =

50 MN/m, respectively. It indicates that a small penalty number

is unable to enforce the interpenetration constraint.

Again the example was solved with � ¼ 10 and low value of

normal contact stiffness (Pn ¼ 50 MN=m). Figure 16 shows a

comparison between the analytical solution and 3-D DDA

results using the classic penalty method and the augmented

Figure 9. A single block sliding along an inclined plane.

–5

5

15

25

35

45

55

65

0 1 2 3 4 5Time (s)

Dis

pla

cem

ent (m

)

Analytical DDA (P = 50GN/m)

DDA (P = 500MN/m) DDA (P = 50MN/m)

Figure 10. Comparison between the analytical solution and 3-D DDA resultsusing the penalty method.



0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5

Time (s)

Dis

pla

ce

me

nt (m

)Analytical

DDA (P = 50GN/m)

DDA (P = 500MN/m)

DDA (P = 50MN/m)

Figure 11. Comparison between the analytical solution and 3-D DDA resultsusing the augmented Lagrangian method.

–5

5

15

25

35

45

55

65

0 10 20 30 40

X Direction Displacement (m)

Z D

irection D

ispla

cem

ent (m

) DDA (P = 50GN/m)

DDA (P = 500MN/m)

DDA (P = 50MN/m)

Figure 12. Sliding block displacements in X and Z directions using the penaltymethod.

Figure 13. The deformation of the block system, using the augmentedLagrangian method, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 50 MN/m.

Figure 14. The deformation of the block system, using the classic penaltymethod, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 500 MN/m.

Figure 15. The deformation of the block system, using the classic penaltymethod, after (a) 0, (b) 3, (c) 4 and (d) 5 s for P = 50 MN/m.

0

10

20

30

40

50

60

70

0 1 2 3 4 5

Time (s)

Dis

pla

ce

me

nt

(m)

Analytical

DDA (Penalty Method)

DDA (Augmented Lagrangin Method)

Figure 16. Comparison between the analytical solution and 3-D DDA resultsusing the augmented Lagrangian method.



Lagrangian method. It can be seen that unlike the penalty

method, the augmented Lagrangian method is able to solve

the problem very well with low values of penalty numbers.

5.2 Block sliding under two forces

As the second example, a two-block system as shown in Figure 17

is considered. The two blocks are hexahedral, with the fixed

bottom block having dimensions of 5 m · 3:5 m · 2:5 m and

the top block having dimensions of 1 m · 1 m · 1 m. The top

block is subjected to two horizontal forces F1 and F2

ðF1 ¼ 500 N; F2 ¼ 250 NÞ. F1 acts in the x-direction at the

centre of a face parallel to the y--z plane, F2 acts in the negative

y-direction at the centre of a face parallel to the x--z plane. The

density of the blocks is � ¼ 2:0 t=m3

and Young’s modulus and

Poisson’s ratio for both blocks are 100 MPa and 0.3, respectively.

In this example, Pn ¼ 100 MN=m is assumed. The analytical

solution for displacement S as a function of time t is given by:

S ¼ 1

2at2 ¼ 1

2

F

m

� �t2 ð60Þ

where F is the force acted to the block and m is the mass of the

block.

Figure 18 shows time-dependent total displacements up to 4.5 s.

This figure shows that the analytical solutions well agree with the

results computed using the augmented Lagrangian method, but

using the classic penalty method the results are not in agreement

with the theoretical solution. Figures 19 and 20 show the deforma-

tion of the blocks, using the classic penalty method, after 0 s, 2 s,

3.5 s and 4.5 s, respectively. It indicates that a small penalty

number is unable to enforce the interpenetration constraint.

5.3 Block falling

As shown in Figure 21a, this case involves a block falling in

which the inclined plane angle is 20. The block falls freely

initially and then bounces down the slope. The penalty number

Figure 17. Initial configuration of a two-block system.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4

Time (s)

Dis

pla

cem

ent (m

)

Analytical

Augmented 3D DDA

Penalty 3D DDA

Figure 18. Time-dependent total displacements using the classic penalty andthe augmented Lagrangian methods.

Figure 19. The deformation of the block system, using the augmentedLagrangian method after (a) 0, (b) 2, (c) 3.5 and (d) 4.5 s.

Figure 20. The deformation of the block system, using the penalty methodafter (a) 0, (b) 2, (c) 3.5 and (d) 4.5 s.



is assumed low in this example. Figures 21b and 21c show

results of the 3-D DDA using the classic penalty method and

the augmented Lagrangian method, respectively. They indicate

that using the classic penalty method, a small penalty number is

unable to enforce the interpenetration constraint.

Figure 21. (a) Initial configuration of block falling example. (b) Results of3-D DDA using the penalty method. (c) Results of 3-D DDA using theaugmented Lagrangian method.

Figure 22. Initial configuration of a five-block system.

Figure 23. The deformation of the block system, using the augmentedLagrangian method after (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and(d) 5000 steps (Pn = 40 MN/m).



5.4 A five-block system

This example includes a row of four blocks on one block that fall

over. As the blocks fall, many simultaneous contacts can occur

between them. The values for the elastic modulus, Poisson’s ratio

and mass density for each block are E = 3 GPa, v = 0.2, and � =

2700 kg/m3, respectively. The friction angle is 10 and the max-

imum displacement ratio allowed and the maximum time increment

for each time step are 0.1 and 0.001s, respectively. Initial config-

uration of the example is shown in Figure 22. The example was

solved for 5000 steps with two pairs of low values of contact

stiffness (Pn ¼ 40 MN=m and Pn ¼ 20 MN=m ). The deforma-

tions of the block system using the augmented Lagrangian method

and the penalty method for the first case are shown in Figures 23

and 24, respectively, and for the second case are shown in Figures

25 and 26, respectively. As can be seen in the figures, the penalty

method is unable to enforce the interpenetration constraint, but

using the augmented Lagrangian method the problem can be solved

very well even with small values of penalty numbers. This example

shows the efficiency of the proposed contact algorithm as well.

Figure 24. The deformation of the block system, using the penalty methodafter (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and (d) 5000 steps (Pn = 40MN/m).

Figure 25. The deformation of the block system, using the augmentedLagrangian method after (a) 2000 steps, (b) 3000 steps, (c) 4000 steps and (d)5000 steps (Pn = 20 MN/m).



6. Conclusions

In this paper, a new algorithm to detect and calculate the 3-D

point-to-face contacts is presented and the related contact formulas

for the normal spring submatrices are derived in detail, using the

augmented Lagrangian method. The success and accuracy of the

algorithm are demonstrated through several examples involving

two or more blocks. The results presented show that the newly

developed 3-D DDA, involving the 3-D formulations and the

point-to-face contact searching algorithm, correctly simulates the

behaviour of blocks in contact with each other in the 3-D domain

successfully. Several conclusions come out from this study:

� The proposed algorithm is a simple and efficient method

and it can be easily coded into a computer program.

� Unlike the ‘Common-Plane’ method used in the present 3-D

DDA, an iterative algorithm in each time step to obtain the

contact plane is not required in this approach.

� In this new method, the contact mechanics computation does

not need to project vertices and simply uses only coordinates

of block vertices, unlike Jiang and Yeung’s (2004) approach.

� In the existing 3-D DDA, the accuracy of the contact solution

depends highly on the choice of the penalty number and the

optimal number cannot be explicitly found beforehand. These

limitations are overcome by using the augmented Lagrangian

method that is used for normal contacts in this research.

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