Comput Mech (2012) 49:21–33DOI 10.1007/s00466-011-0624-3

ORIGINAL PAPER

Refined h-adaptive finite element procedure for large deformationgeotechnical problems

Mina Kardani · Majidreza Nazem · Andrew J. Abbo ·Daichao Sheng · Scott W. Sloan

Received: 5 November 2010 / Accepted: 26 June 2011 / Published online: 15 July 2011© Springer-Verlag 2011

Abstract Adaptive finite element procedures automaticallyrefine, coarsen, or relocate elements in a finite element meshto obtain a solution with a specified accuracy. Although asignificant amount of research has been devoted to adap-tive finite element analysis, this method has not been widelyapplied to nonlinear geotechnical problems due to their com-plexity. In this paper, the h-adaptive finite element techniqueis employed to solve some complex geotechnical problemsinvolving material nonlinearity and large deformations. Thekey components of h-adaptivity including robust mesh gener-ation algorithms, error estimators and remapping proceduresare discussed. This paper includes a brief literature reviewas well as formulation and implementation details of theh-adaptive technique. Finally, the method is used to solvesome classical geotechnical problems and results are pro-vided to illustrate the performance of the method.

Keywords Adaptivity · Finite elements ·Large deformation · Geomechanics · h-Adaptive refinement

1 Introduction

The finite element method (FEM) is a robust technique forthe analysis of a large number of problems in engineeringwhere analytical solutions cannot be obtained. Efficaciousapplication of the method, however, requires experience anda certain amount of trial and error, particularly when choosingan optimal time and spatial discretisation. Adaptive methodsprovide a means for obtaining more reliable solutions by

M. Kardani (B) · M. Nazem · A. J. Abbo · D. Sheng · S. W. SloanCentre for Geotechnical and Materials Modeling,The University of Newcastle, University Drive, Callaghan,NSW 2308, Australiae-mail: mina.kardani@uon.edu.au

continuously adjusting the discretisation in time and spaceaccording to the current solution. In geotechnical engineer-ing, there is a wide range of nonlinear problems for whichadaptivity can improve the accuracy of numerical solutions.For example, adaptive time stepping for controlling the time(load) discretisation has been used successfully to solve manygeotechnical problems [1,28,29,34,35]. Automatic substep-ping in explicit stress integration, which adjusts the strainincrement size according to the current stress state, is anotherform of adaptivity that has proved to be highly effective forimplementing advanced constitutive models [32,36]. In con-trast, adaptive spatial discretisation has attracted less atten-tion in the field of geomechanics, perhaps because of theprevalence of the small-deformation assumption.

The FEM with adaptive spatial discretisation has beenactively researched over the last three decades. A numberof techniques have been developed, with the most com-mon ones for dealing with large deformation problems beingthe r -adaptive and h-adaptive methods [15]. The r -adaptivemethod adjusts the spatial discretisation, but usually doesnot increase the overall number of elements. A classic exam-ple of this approach is the so-called Arbitrary Lagrangian–Eulerian (ALE) method. Nazem et al. [17] demonstratedthe ability of this method to solve general large deforma-tion problems in geomechanics, while Khoei et al. [12] pre-sented an extended formulation based on the ALE techniquefor large deformation analysis of solid mechanics problems.Although powerful, the r -adaptive method does not directlyaddress problems like localised deformation or stress concen-tration. The h-adaptive method adjusts the spatial discretisa-tion by continuously increasing the mesh density in zoneswhere a more accurate response is expected. This approachis particularly effective in improving the solution accuracyfor problems involving large deformation, localised defor-mation, moving boundaries, post-failure response, steep

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gradients and sharp fronts. Although it is well establishedin solid mechanics, the h-adaptive FEM has attracted lessattention in geomechanics with the work of Hu and Randolph[7,8] being a notable exception. As finite element analysisbecomes increasingly common in geotechnical engineering,with many practical problems involving the complex featureslisted above, there is a pressing need for reliable methods thatcontrol the discretisation error to within a desired limit. Analternative approach to tackling the mesh distortion issue isbased on the so-called meshless methods (e.g. [6,19]). How-ever, the applicability and accuracy of these approaches forgeotechnical problems are not clearly understood.

The h-adaptive FE strategy subdivides the integrationregion into successively smaller sub-regions [10], thuschanging the density of the elements to yield a more accu-rate solution while keeping the element order constant. In thecase of small deformations, where the mesh geometry doesnot change throughout the solution process, the h-adaptivemethod can be used to generate a better mesh at the end ofthe analysis with a repeated analysis being expected toimprove the solution accuracy. In the literature, it is commonto use the h-adaptive method in this way for small defor-mations (e.g. [2,13,24]). In the case of large deformations,where the mesh is continuously updated according to the cur-rent displacements, the h-adaptive method provides a naturalstrategy for controlling the discretisation error as the solu-tion proceeds. The approach is thus well suited to handlingproblems associated with large deformations, such as meshdistortion, but reports of its use in the literature for this pur-pose are scarce. Another advantage is that, for the same levelof accuracy, an h-adaptive solution is usually substantiallycheaper than a conventional solution that is based on a con-stant (perhaps substantially higher) number of elements.

The three main components of an h-adaptive finite ele-ment analysis are (1) error estimation, (2) mesh generation,and (3) convection of the state variables from the old meshto the new mesh. The error estimation procedure attempts tocontrol the spatial discretisation error in the finite elementdomain which, in turn, determines the distribution of the ele-ments in future mesh refinements. Various types of error esti-mators have been proposed in the literature. Residual errorestimators evaluate the error by using the residuals of theapproximate solution [11]. Recovery based error estimators,on the other hand, use a recovered solution instead of theexact solution to measure the error, which is usually definedin terms of some energy norm [40,41]. It is also possible touse an incremental form of error estimation based upon aglobal error measure in the constitutive equations [5]. Thenewer approach of goal-oriented error estimation is based oncontrolling a local quantity of interest, such as a specific dis-placement or stress, instead of a global quantity such as anenergy norm [21,25]. After estimating the error, a criterionis needed to refine and optimise the mesh. Some well known

mesh optimality criteria include Zienkiewicz and Zhu’s (ZZ)procedure [39] which is based on an equal distribution of dis-cretisation error between the elements in the current mesh,Onate and Bugeda’s (OB) method [22] which is based on anequal distribution of the error density, and Li and Bettess’(LB) strategy [15] which is based on equal distribution oferror in each element in the new mesh. It has been proventhat the LB criterion, which is actually an improved form ofthe ZZ criterion, leads to the lowest number of elements andthe minimum number of degrees of freedom for a given accu-racy [4]. An alternative remeshing criterion can be obtainedby combining the LB and the OB strategies [4].

After assessing the error, a new mesh is generatedaccording to the refinement criterion. Then, two differentapproaches are available to continue the analysis. One isto restart the analysis with the new mesh from time zero.This avoids the cumbersome process of transferring vari-ables from one mesh to another, but is only appropriate forsmall deformations. For large deformations, the new meshrepresents the current deformed configuration and cannot beused as the initial undeformed configuration, thus making arestart of the analysis infeasible (e.g., in penetration problemssuch as those described by Sheng et al. [27]). An alternativestrategy, which is known as remapping or convection, is totransfer the history-dependent variables and displacementsfrom the old mesh to the new mesh. Different approaches forremapping are available in the literature such as global leastsquare projection [20], element-based transfer which uses anextrapolation algorithm based on shape functions, and patch-based transfer or least-square approximation techniques [2].By remapping state variables, the analysis continues withthe new mesh for the next load increment. Thus, the totalnumber of load increments is not increased, but with a costof remapping the state variables between the old and newmeshes.

Various h-adaptive finite element strategies can be devisedby combining different forms of error estimators, mesh gen-erators, and remapping techniques. For example, Hu andRandolph [7] proposed an h-adaptive finite element proce-dure for geotechnical problems in which the error is estimatedusing the Superconvergent Patch Recovery (SPR) techniqueto recover the strains at Gauss points. The mesh refinementprocedure is based on a minimum element size criterion. Totransfer the stresses and soil properties from the old mesh tothe new mesh two interpolation strategies, a modified formof the unique element method and the stress-SPR method,were applied. This method has been used to analyse thebearing capacity of strip and circular foundations on non-homogeneous soil where the strength increases linearly withdepth. The strategy proposed by [7] is based upon small strainanalysis where the effect of rigid body rotations is neglected.

As mentioned above, the h-adaptive method has beenused mostly for small deformation problems with only a few

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studies focusing on large deformations mainly on other areasrather than geomechanics [9,23]. The key aim of previouswork was to improve the accuracy of the solution, with lit-tle attention being devoted to important issues such as meshdistortion [14]. In this paper, a new and robust h-adaptiveprocedure is developed to solve nonlinear problems in geo-mechanics, with a particular focus on large deformations. Inthis approach, the error is estimated using a global energynorm, and a robust Delaunay triangulation algorithm is thenemployed to generate a refined mesh. Finally, all the statevariables and displacement are transferred from the old meshto the new mesh using the SPR technique. Moreover, unlikemany published works in the area [7], the adaptivity conceptemployed in this study does not only include the mesh optimi-sation and refinement but also covers other important aspectsof adaptivity such as automatic adaptive time stepping andadaptive stress-integration with error control. The ability ofthis h-adaptive FEM to handle complex large deformationproblems is illustrated by solving some classical problemssuch as the bearing capacity of footings, cavity expansionand a biaxial test.

2 h-Adaptive FEM

Among others, the Updated-Lagrangian (UL) method is acommon strategy to solve problems involved with largedeformations. In the UL method the material particles areconnected to a computational grid. The spatial position ofthis computational grid is updated according to incremen-tal displacements at the end of each time step. Potentially,mesh distortion and entanglement of elements can occurin the material mesh due to continuous change in geome-try, leading to a negative Jacobian of finite elements or asignificant inaccuracy in the solution. Mesh distortion andoverlapping of elements in zones with high stress and strainconcentration is almost inevitable if the finite element mesh isfine enough to achieve acceptable results. This phenomenoncan be observed while solving a wide range of geotechnicalproblems such as the bearing capacity of relatively soft soilsunder a footing, the penetration of objects into soil layers,the lateral movements of pipelines on seabeds, the pull-outcapacity of anchors embedded in soft soils, and slope sta-bility analysis. The h-adaptive finite element technique caneliminate the mesh distortion by, if necessary, generating anundistorted new mesh at the end of the UL analysis.

The h-adaptive procedure generates a sequence of approx-imate meshes which are aimed at either converging to aproper mesh gradually or eliminating mesh distortion in thefinite element domain. In the adaptive FEM developed in thisstudy the analysis starts with an UL step using a relativelycoarse mesh over the domain of the problem. In this step,an adaptive implicit time-stepping [34] is used to solve the

nonlinear global equations, during which an automaticexplicit stress-integrator solves the nonlinear differentialconstitutive equations due to material nonlinearity. Afterequilibrium is reached, the nodal coordinates are updatedaccording to the incremental displacements and the error inthe finite element mesh is evaluated by an error estimator. Ifthe error satisfies an acceptable accuracy, the analysis is con-tinued using the same mesh in the next increment. Otherwise,the new area of each element is calculated according to theestimated error and a new mesh is generated. All state vari-ables are then transferred from the old mesh to the new meshand the analysis is performed using the new mesh in the nextincrement. This transfer process occurs at integration pointsas well as nodal points. In the following sections, the keycomponents of the h-adaptive procedure developed, includ-ing the UL method, the error estimator, the mesh generationand remapping approach are explained in more detail.

2.1 The UL method

In the UL method it is assumed that the solution at time tis known and the main goal of the new increment is to findthe state variables which satisfy equilibrium at tine t + �t.The matrix form of the global equilibrium equation is usuallyderived from the principle of virtual displacements which, inits weak form, states that for any virtual displacement δuequilibrium is achieved provided∫

V

σi jδεi j dV =∫

V

δui bi dV +∫

S

δui fi d S (1)

where σ denotes the Cauchy stress tensor, δε is the variationof strain due to virtual displacement u, b is the body force,and f is the surface load acting on area S of volume V . Lin-earisation of the principle of virtual displacement providesthe equation of equilibrium for a solid in the following matrixform

KepU = Fext (2)

where K is the tangent stiffness matrix, U denotes the dis-placement vector, and Fext is a vector of external forces.The solution of the equilibrium equation (2) requires a step-by-step integration scheme in an explicit or implicit form.Explicit methods, such as the central difference method, cir-cumvent the factorization of the global matrix equation butare only conditionally stable, i.e., the size of the time stepsmust be smaller than a critical value. Implicit methods, onthe other hand, are unconditionally stable but usually haveto be combined with another procedure such as the Newton–Raphson method. In this study we adopt the time-steppingscheme with automatic subincrementation and error controlproposed by Sloan and Abbo [34].

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It is notable that in a large deformation analysis theconstitutive equations must be objective, i.e. rigid bodymotion must not induce any strains in the material. In manyimplementations of h-adaptivity, a formulation based on mul-tiplicative decomposition of the deformation gradient is used,together with an additive decomposition of the deformationrate which involves the definition of an intermediate stress-free configuration (e.g. [23]). This formulation originatedfrom the micromechanical observations of metal crystals, butits application to geomaterials is yet to be investigated deeply.To address the principle of objectivity we use the Jaumannstress rate, σ∇ J , in the stress–strain relations according to

dσ∇ Ji j = Cep

i jkl · dεkl (3)

where Cep denotes the elastoplastic constitutive matrix. Forevery iteration in which plastic yielding occurs, the nonlineardifferential equation in (3) needs to be solved at each inte-gration point inside the mesh. An explicit automatic stress-integration scheme with error control was developed by Sloan[32] and refined by Sloan et al. [36] for this purpose. Nazemet al. [18] successfully extended this scheme to large defor-mation problems of solid mechanics and demonstrated that itis slightly more efficient to apply rigid body corrections dur-ing integration of the constitutive equations. This adaptivestress-integration scheme is adopted in this study.

2.2 Error estimator

Since an exact solution is usually not available, an approxi-mate error assessment is used to guide the mesh refinement.During the error assessment, the error in each element iscalculated and compared with a prescribed limit, thus iden-tifying parts of the domain where the discretisation needsrefinement. The error estimation method implemented inthe present work is based on that of Boroomand andZienkiewicz [2], and uses the energy norm for nonlinearelasto-plasticity. The error in an increment is calculated using

‖e‖ =⎡⎣

∫

�

|(σ − σ )T (�ε −�ε)|d�⎤⎦

12

(4)

where �ε and �ε represent the exact and the finite elementstrain increments, σ and σ denote the exact and finite elementapproximation of the stresses, and� is the problem domain.To estimate the error, the exact stress and strain fields arereplaced by their recovered values, σ ∗ and �ε∗, which arecalculated using the SPR procedure. This means that Eq. (4)

becomes

‖e∗‖ =⎡⎣

∫

�

|(σ ∗ − σ )T (�ε∗ −�ε)|d�⎤⎦

1/2

=[

nel∑i=1

‖e∗el‖i

]1/2

(5)

where nel is the total number of elements and ‖e∗el‖ repre-

sents the estimated error in each element defined by

‖e∗el‖ =

ngp∑i=1

wi (σ∗i − σi )

T(�ε∗i −�εi ) (6)

In the above, ngp is the total number of Gauss points in anelement andwi is the standard Gauss quadrature weight. Therelative error in the solution based on the incremental energynorm can be obtained by

η = ‖e∗‖E

× 100 (7)

where

E =⎛⎝ nel∑

j=1

ngp∑i=1

wiσ∗Tji �ε

∗j i

⎞⎠

1/2

(8)

This error is then compared with a prescribed accuracy, ηaccording to

η ≤ η (9)

If the condition (9) is satisfied, no further refinement isneeded. Otherwise, we assume that the error is equally dis-tributed over the elements and use the LB criterion to obtainthe new element area:

Anew =(

ηE

‖e∗i ‖√N

)1/p

× Aold (10)

in which N represents the total number of elements in the oldmesh and p is the polynomial order of the approximation.Since this method estimates the error using a global quantity,the energy norm, it cannot be used to control the error in localquantities of interest, such as a specific component of stressor strain.

2.3 Mesh generation

Mesh generation algorithms fall into two broad catego-ries: structured and unstructured. A structured mesh gener-ator produces elements with regular connectivity that canbe expressed as a two- or three-dimensional array. In thisapproach the nodes are located directly using a variety ofmethods including specified functions, simple algorithmssuch as the Laplacian operator, or by manually decomposingthe problem domain into simple patches. Unstructured mesh

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Comput Mech (2012) 49:21–33 25

generators, on the other hand, are often boundary based andtypically require some form of boundary discretisation to bespecified. In every step of the element generation, the geom-etry of the unmeshed region is assessed and the boundarydefinition is preserved. Unstructured meshes are character-ized by irregular connectivity, and are efficient in the sensethat they permit elements to be highly concentrated in zoneswhere they are needed. For 2D applications the Delaunaytriangulation (DT) (see, e.g. [33]) is probably the most pop-ular unstructured mesh generation method. A key feature ofthis method is that the circumcircle/circumsphere of everytriangle/tetrahedron does not contain any other nodal pointsof the triangulation. Moreover, in 2D it can be shown that theDT maximises the minimum angle in any triangle, therebyguaranteeing high quality grids for computational work.

In this paper the mesh generation is based on a programnamed Triangle [31] which takes advantage of various exist-ing Delaunay triangulation algorithms to generate high qual-ity meshes for two-dimensional domains. The program is afast and robust tool to construct Delaunay triangulations, con-strained Delaunay triangulations and Voronoi diagrams. Inthis code, Ruppert’s Delaunay refinement algorithm is usedto check that no angle in any triangle is allowed to be smallerthan 20◦. The availability of Triangle’s source code made itpossible to apply new changes to the code. In this work, sig-nificant improvements have been made to Triangle, to makeit compatible with the requirements of this study. For exam-ple, keeping track of the previous stage of the mesh for therefinement procedure, dealing with problems involving largedeformation by constantly changing the domain, constrain-ing the minimum element area, and introducing regions withdifferent maximum and minimum areas. Also, the order ofthe generated nodes is optimised to minimise the memoryallocation and CPU time, with mesh distortion being avoidedduring the refinement procedure.

2.4 Remapping

After generating the new mesh based on the estimated error,all state variables at Gauss points and nodal points need to betransferred from the old mesh to the new mesh. Nodal vari-ables can be transferred from the old nodes to the new nodesby a direct interpolation. If the new coordinates of a nodematch the old coordinates of a node the displacements remainunchanged, but if a new node is generated on a segment witha prescribed displacement it will inherit the characteristics ofthat segment. The displacements of all other new nodes arefound simply by using the displacement shape functions ofthe surrounding old element.

Transferring the information from old Gauss points to thenew Gauss points is less straightforward, as it can lead to aloss of equilibrium and violation of the yield condition. Inthis work, the super convergent patch recovery technique

[40,41] is used to compute the nodal quantities of inter-est (such as stresses). This technique has been successfullyused with h-adaptive as well as r -adaptive FEMs [7,17], andassumes that the quantities in a patch can be modelled usinga polynomial of the same order as the displacements. Thus,for two-dimensional quadratic elements, the stresses over apatch may be written as

σ = P · a (11)

where

P = [1, x, y, x2, xy, y2] (12)

and

a = [a1, a2, a3, a4, a5, a6]T (13)

In Eq. (12), normalised coordinates are used instead ofglobal coordinates to avoid ill-conditioning of equations forhigher order elements. The normalised coordinates in a two-dimensional patch, x∗

i and y∗i , can be written as

x∗i = −1 + 2

xi − xmin

xmax − xmin (14)y∗

i = −1 + 2yi − ymin

ymax − ymin

in which xmin and xmax represent the maximum and mini-mum values of the x-coordinates in the patch, respectively,and ymax and ymin are defined similarly for the y-coordinates.A least squares fit is then used to find the unknown values ofa by minimising

F(a) =m∑

i=1

(σ (x∗i , y∗

i )− P(x∗i , y∗

i ) · a)2 (15)

in which σ(x∗i , y∗

i ) are the stress values at Gauss points andm is the number of Gauss points in a patch. This defines aaccording to

a = A−1b (16)

where

A =m∑

i=1

PT (x∗i , y∗

i )P(x∗i , y∗

i ) (17)

b =m∑

i=1

PT (x∗i , y∗

i )σ (x∗i , y∗

i ) (18)

The stresses at any node or Gauss point in the new meshare obtained by substituting the normalised coordinates inEq. (11) for the surrounding patch in the old mesh. As thetopology of the mesh can change, it is possible to find morethan one enclosed patch for a specific Gauss point or node.In this case, the final value of the stresses is set equal to themean of the calculated stresses in each patch.

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26 Comput Mech (2012) 49:21–33

Generate a coarse mesh by Triangle

Use the UL method to solve the increment and

achieve equilibrium

Estimate the error

Acceptability criterion is satisfied?

Stop

Yes Calculate the new

size of each element

Generate a new mesh with new sizes of elements

Transfer all the variables from old mesh to new mesh.

No

Last increment

Increase the increment

No

Yes

Allocate memory for all state variables in new mesh.

Free the memory allocated for old mesh.

Check equilibrium and plasticity consistency condition, conduct further iterations if necessary.

Fig. 1 Procedure for h-adaptive finite element analysis

2.5 Implementation of the h-adaptive algorithm

The components of the h-adaptive FEM described in theabove sections have been implemented into SNAC, a finiteelement program developed over the past two decades by thegeotechnical research group at the University of Newcastle,Australia. The algorithm is shown schematically in Fig. 1.

In the h-adaptive procedure, the analysis starts with Tri-angle to generate a coarse mesh on the domain based uponthe user information. It is possible to define the maximum

area of elements in each region. In the data given by the user,the domain of the problem is defined by a combination ofpoints and segments. Each point represents the intersectionof two segments and a segment can be either a straight line ora curve. The domain of the problem may consist of severalregions.

In the first increment, the global nonlinear equations aresolved to find the incremental displacements and conse-quent incremental strains. The constitutive equations arethen integrated to obtain the stresses. After finding thestresses, the analysis continues by calculating the inter-nal forces and checking equilibrium. The error in the cur-rent mesh is then calculated based on the error in theglobal energy norm and the program checks if this erroris acceptable. For all elements where the estimated erroris above a specified tolerance, a new area is assigned tominimise the error. The old mesh is then refined accordingto the new element sizes. After generating the new mesh,all the state parameters are transferred from the old meshto the new mesh. Since the topology of the new mesh ispotentially different to the topology of the old mesh, thecode must store the information of two meshes in mem-ory simultaneously, demanding a precise memory reallo-cation routine. This information must include the totalnumber of nodes and their spatial coordinates, the num-ber of elements and their connectivity, the boundaryconditions, the applied loading, and all state variables at theGauss points.

For elastoplastic materials, the remapped state variablesmay violate the equilibrium and yield conditions in the newmesh. Ideally, transferring the state variables from an oldmesh to a new mesh should not cause any straining and henceequilibrium should still be satisfied. Moreover, the remappedstresses must lie inside or on the yield surface. Unfortunately,there is no simple way to satisfy these two conditions simul-taneously and the development of a rigorous algorithm forremapping remains a challenge. In this work, if a stress pointis found to lie outside the yield surface after remapping, it isprojected back to the yield surface using the stress correctionprocedure described in Sloan et al. [36]. After these correc-tions are executed, equilibrium is enforced by conductingadditional iterations at the global level (if needed).

In some stages of the process it may be observed that fur-ther refinement may not improve the results and also lead tomesh distortion due to the generation of very small elements.To avoid this problem, a minimum element area is prescribedfor every region inside the meshed domain. This guaranteesthat the area of each individual element will not be smallerthan a minimum value, even if the associated error estimatorindicates otherwise. In addition, the solution algorithm canterminate the mesh refinement at any time-step, thus allow-ing the analysis to be continued with the last generated meshfor the remaining load increments.

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Comput Mech (2012) 49:21–33 27

Soil Layer

G=150.5 kPa

= su =8.66 kPa

Correcting Layer G = 66.66 kPa

v = 0.25

r0 Soil Layer Correcting layer C

entr

e lin

e

60r0 60r0

Fig. 2 The cavity expansion problem (geometry not to scale)

3 Numerical examples

In this section, the h-adaptive FEM described above is used tosolve some classical geotechnical problems. Problems con-sidered include the expansion of a cylindrical cavity, a biax-ial test, and the bearing capacity of a flexible as well as arigid footing. All problems have been solved using 6-nodetriangular elements with six integration points and analysedassuming large deformation.

3.1 Elastoplastic cylindrical cavity expansion

The cavity expansion problem in a Tresca soil is one of thefew cases that can be solved analytically using finite-strainplasticity, and is thus very useful for validating finite elementformulations. The internal radius of the cavity is denoted asa0 while the outer radius is set to 60r0. To simulate the effectsof an infinite medium, a correcting elastic layer is added tothe soil layer [3]. Figure 2 represents the boundary condi-tions, material properties, and the finite element mesh of thecavity used in this analysis. In Fig. 2, G, su, and v representthe shear modulus, undrained shear strength, and Poisson’sratio of the soil, respectively. A total prescribed radial dis-placement of magnitude 6r0 is applied over 200 equal timeincrements. The analytical solution according to Yu [38] is:

Ψ

su= 1 + ln

[G

su

(1 − r2

0

r2

)+ r2

0

r2

](19)

where Ψ represents the internal pressure of the cavity andr is the current internal radius of the cavity. The numericalsolution for the normalised internal pressure of the cavityversus the normalised radial displacement is compared withthe analytical solution in Fig. 3. Good agreement betweenthe analytical and the numerical solutions is observed.

3.2 Biaxial test

In this example, a biaxial test is simulated using the Mohr–Coulomb criterion with a non-associated flow rule. The prob-lem is solved using two different material properties. First auniform soil layer is considered for the entire domain. Then,localised failure is initiated by defining weaker propertiesfor a small area in the soil specimen. The geometry of the

r/r0

Ψ/ s

u

1

1.5

2

2.5

3

3.5

4

4.5

1 2 3 4 5 6 7

h-adaptive FE solution

Analytical solution

Fig. 3 Normalised internal pressure of cavity versus normalised radialdisplacement

Smooth boundary

2B

B

E′ = 1000 kPa c′ = 5.5 kPa φ′ = 10° ψ ′ = 5°

Smooth boundary

Soil properties:

E′ = 1000 kPa c′ = 5 kPa φ′ = 10° ψ ′ = 5°

Weak zone:

Weak zone

(only in 2nd analysis)

0.06B

0.2B

Fig. 4 Biaxial test (Sect. 3.2)

problem, material properties and boundary conditions areshown in Fig. 4. The soil is modelled using a rounded Mohr–Coulomb model proposed by Sheng et al. [30]. In Fig. 4,E ′, c′, φ′, ψ ′ and B represent the drained Young’s modulus,the drained cohesion, the drained friction angle of the soil,the dilation angle and the width of the specimen, respectively.

A total axial strain of 7% is applied vertically on the spec-imen, in 100 equal increments. Figure 5 shows the initialmeshes, the meshes at the increments 10 and 20 for the uni-form soil properties, and the meshes at the increments 20and 40 for the non-uniform soil properties with a minimumelement area of 0.00016B2. For comparison, the problemwas reanalysed by using a very fine fixed mesh with 7,763elements and 15,768 nodes considering both uniform and

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2) After 10 remeshing (731 elements)

(a) Uniform soil properties

1) Coarse mesh (80 elements) 3) After 20 remeshing (1658 elements)

1) Coarse mesh (47 elements)

(b) Non-uniform soil properties

2) After 20 remeshing (2337 3) After 40 remeshing (2792 elements)

Fig. 5 Mesh results of the biaxial tests

non-uniform soil properties. The applied pressure normal-ized by c′ is plotted versus the axial strain in Fig. 6 for both theadaptive mesh and the fixed fine mesh in each analysis. Obvi-ously, analysis using a fixed mesh does not locate the shearband. However as shown in Fig. 7, the h-adaptive methodcan predict the location and formation of the shear band inthis problem. To demonstrate the effect of prescribing largerminimum element areas on the accuracy of the solution, bothproblems are also analysed with minimum area of 0.0032B2

and 0.00032B2. As shown in Fig. 6, reducing the size of theminimum element area increases the accuracy of the solution.Moreover, the load–deflection curve does not change for thevalues of minimum element area less than about 0.00016B2,which implies that no further mesh refinement is necessarywhen an element size reaches this value. Also, by increasingthe number of elements in a very fine structured mesh, thesolution remains unchanged and convergence is evident.

It is interesting to note that the continuous mesh refinementleads to a thinner and thinner shear band. However, the load–displacement curves shown in Fig. 6 indicate convergence toa unique solution, i.e., the load deflection behaviour does notchange significantly with the density of the initial mesh aslong as the analysis begins with a reasonably coarse mesh.Runesson et al. [26] showed that introducing some soften-ing to the associated model, or using a non-associated flowrule, would also permit bifurcation. In general, a more robustalternative for tackling this localisation problem would be tocombine the proposed adaptive method with an appropriateconstitutive model for the localisation [16,37].

3.3 Bearing capacity of soil under a strip footing

In the third example, a strip footing on an undrained soil layerof Tresca material is considered. Only half of the footing

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Comput Mech (2012) 49:21–33 29

(b) Non-uniform soil properties

0

0.5

1

1.5

2

2.5

3

3.5

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Adaptive, minimum area 0.00016BxB,0.00032BxB

Very fine structured mesh

Adaptive, minimum area 0.0032BxB

Axial strain

App

lied

pres

sure

/ c´

(a) Uniform soil properties

0

0.5

1

1.5

2

2.5

3

3.5

Adaptive, minimum area 0.00016BxB,0.00032BxB

Very fine structured mesh

Adaptive (minimum area 0.0032BxB)

Axial strain

App

lied

pres

sure

/ c´

Fig. 6 Load–displacement response of the biaxial test

is analysed due to symmetry, and the domain, its boundaryconditions and material properties are shown in Fig. 8. Thefooting is assumed to be either flexible or rigid.

To model a flexible load, a vertical pressure 6.5su isapplied to the footing in 200 equal time steps. The domainis analysed using two different finite element meshes: a veryfine mesh and a relatively coarse mesh. The coarse mesh hasonly 299 elements and a maximum element area of 0.25B2,while the fine mesh has 3,713 elements and a maximum ele-ment area of 0.02B2. Table 1 provides the minimum elementarea, number of nodes, number of elements and the consumedCPU time in each analysis. The topology of the domain is keptconstant in the fine mesh, while the coarse mesh is refinedcontinuously using the h-adaptive technique. The numberof nodes and the elements in the h-adaptive analysis pro-vided in Table 1 correspond to those at the end of analysis.The h-adaptive solution was found using a total of 32 meshrefinements. The evolution of the adaptive mesh shown inFig. 9 indicates the finite element meshes at the beginningof the analysis and after 10, 25 and 30 mesh refinements,respectively. This figure clearly shows that the h-adaptivemethod is able to predict the occurrence and location of theshear band under the footing successfully. Figure 9 plots theapplied pressure on the footing, normalised with respect to su ,versus the vertical displacement normalised with respect tothe half width of the footing. The results presented in Fig. 10are based upon the assumption of large deformations, i.e., thelimit pressure does not necessarily converge to Prandtl’s plas-ticity solution, (2 + π)su . The two load–displacement plots

Fig. 7 Final deformedh-adaptive meshes for biaxialtest

(a) uniform soil properties (2964 elements and 6001 nodes)

(b) Non-uniform soil properties (3363 elements and 6802 nodes)

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30 Comput Mech (2012) 49:21–33

Smoo

th

Rough

Smoo

th

Centre line

B

6B

8B

G/su=33.5

u =

u = 0°

Fig. 8 Footing problem

coincide perfectly with each other, while the h-adaptiveanalysis is approximately 14 times faster than the analysisusing the fixed fine mesh (Table 1). In another attempt tosolve this example, a minimum element area of 0.01B2 wasselected and no mesh refinement was permitted. As a result, anew fine mesh with 15,168 nodes and 7,467 elements leadto almost an identical load–displacement curve as shown

Table 1 Results for analysis of flexible strip footing

Analysis Minimum Number Number CPUelement of nodes of elements time (s)area (B2)

Very fine 0.02 7,604 3,713 8,013

H -adaptive

Initial 0.25 648 299 589

Final 0.002 2,132 1,025

in Fig. 10. On the other hand, a fixed mesh with less than1,000 elements would lead to significantly different load–displacement response. Figure 11 shows the final deformedmesh at the end of analysis. Again, the localised failure mech-anism is well captured.

For the case of a rigid footing, a vertical displacement of0.5B is applied to the footing in 100 equal steps. The problemis analysed as a large deformation formulation, with someintermediate meshes being shown in Fig. 12. The final resultis achieved after 32 remeshing steps. For comparison, theproblem is also analysed by a small deformation assump-tion where the configuration remains unchanged during the

(a) Initial mesh (299 elements) (b) After 10 remeshes (437 elements)

(c) After 25 remeshes (539 elements) (d) After 30 remeshes (888 elements)

Fig. 9 Finite element mesh during analysis of flexible strip footing

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Comput Mech (2012) 49:21–33 31

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25

Fine structured mesh

h-adaptive analysis

Vertical displacement / B

App

lied

p res

sure

/ s u

Fig. 10 Load–displacement response of flexible footing

analysis. Figure 13 plots the applied pressure on the footing,normalised with respect to su , versus the vertical displace-ment, normalised with respect to the half width of the footing.To study the effect of minimum element area, the analysis isalso performed using minimum element areas of 0.001B2

and 0.05B2. As shown in Fig. 13, values of the minimumelement area less than 0.002B2 do not increase the accuracyof the solution.

The small deformation analysis results in a final appliedpressure of 5.18su which is 0.8% above Prandtl’s plasticitysolution of (2 +π)su . The final deformed mesh at the end ofh-adaptive large deformation analysis is shown in Fig. 14.

Fig. 11 Final deformed meshfor flexible strip footing

(a) Initial mesh (300 elements) (b) After 10 remeshes (1103 elements)

(c) After 15 remeshes (1730 elements) (d) After 20 remeshes (2143 elements)

Fig. 12 Finite element meshes during analysis of rigid strip footing

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32 Comput Mech (2012) 49:21–33

Fig. 13 Load–displacementresponse of rigid strip footing

0

1

2

3

4

5

6

7

8

9

0 0.1 0.2 0.3 0.4 0.5 0.6

Adaptive large deformation (Min area 0.001BxB,0.002BxB)Adaptive small deformationPrandtl solution, small deformationAdaptive large deformation (Min area 0.05BxB)

Vertical displacement / B

App

lied

pres

sure

/ s u

Fig. 14 Final deformed meshfor rigid footing

4 Conclusion

The h-adaptive FEM presented here is able to solve a widerange of nonlinear geotechnical problems efficiently andaccurately. It is particularly effective in capturing localisedfailure and shear bands, which are difficult to predict usingfixed grids. The problems analysed in this paper involve largedeformations. This h-adaptive technique also seems to bequite efficient in dealing with problems involving large defor-mation, where mesh distortion can affect the accuracy of thesolution. In terms of efficiency, the h-adaptive technique isable to produce accurate results with considerably less num-bers of elements and nodes and significantly less computa-tional time.

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