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Large-Scale Geotechnical Finite Element Analysis on Desktop PCs
Analyse par éléments finis de problèmes géotechniques de grandes dimensions sur ordinateur de
bureau
Chaudhary K.B.
GeoSoft Pte Ltd, 2 Kaki Bukit Place, #03-00 Tritech Building, Singapore 416180
Phoon K.K.
Department of Civil & Environmental Engineering, National University of Singapore, Block E1A, #07-03, 1 Engineering
Drive 2, Singapore 117576
Toh K.C.
Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
ABSTRACT: With the development of new hardware and software technologies, the trend of using three-dimensional finite element
analysis in geotechnical engineering is growing recently. However, the solution of realistic large-scale problems still demands a
significant amount of computational time and resources. The computational time can be even longer for the ill-conditioned systems
when the stiffness of different elements differs by several orders of magnitude. In this paper, we demonstrate how the recent
development of block diagonal preconditioning has effectively reduced the computational time of iterative solvers so that large-scale
finite element analysis can be performed in a reasonable time on Desktop PCs using GeoFEA.
RÉSUMÉ : Avec le développement des nouvelles technologies matérielles et logicielles, l'analyse tridimensionnelle par éléments finis
en géotechnique est de plus en plus utilisée. Cependant, la solution de problèmes de grandes dimensions réels exige toujours une
quantité importante de temps de calcul et de ressources. Le temps de calcul peut être encore plus long pour les systèmes mal posés,
lorsque la raideur des différents éléments diffère de plusieurs ordres de grandeur. Dans cet article, nous montrons comment le
développement récent de préconditionnement diagonale par blocs a permis de réduire le temps de calcul des solveurs itératifs de sorte
que l'analyse de problème de grandes dimensions par éléments finis peut être effectuée dans un délai raisonnable sur les ordinateurs
de bureau utilisant GeoFEA.
KEYWORDS: Large-scale finite element analysis, iterative solvers, preconditioning, GeoFEA, Desktop PC.
1 INTRODUCTION
With the advancement of new hardware and software
technologies (sophisticated finite element programs), fairly
large-scale analyses are within the reach of geotechnical design
offices and the emphasis of designs and analyses has been
shifting from simple or empirical approaches to large-scale
three-dimensional (3D) finite element modelling. 3D analysis is
also useful in understanding the complex soil-structure
interation problems. However, significant amount of time and
large memory requirement for storage are the major challenges
for 3D analysis because a large number of finite elements are
required to represent the problem realistically. The resulting
system of equations has, in general, the form:
Ax b
(1)
where
N N
A
is known as coefficient matrix,
N
x
is the
vector of unknowns,
is the force vector.
N
is the
dimension of the linear system, that is, the degrees of freedom
(DOFs) of the discretized mesh. Solution of this linear system
(Eq. 1) is one of the most expensive computational parts in
finite element analysis. For large linear systesms, Krylov
subspace iterative method is popularly used to solve (Cipra
2000) them because of smaller memory requirement than direct
solvers. However, for Krylov subspace iterative methods to be
successful or efficient, preconditioning plays an important role.
N
b
In geotechnical engineering, consolidation is a general
phenomenon, for which the coefficient matrix
A
can be severely
ill-conditioned (Chan et al. 2001, Ferronato et al. 2001, Lee et
al. 2002). Some effective preconditioners have been proposed in
the past decade for Biot’s (Biot 1941) consolidation equations;
see, for example, Gambolati et al. (2011), Chen and Li (2011)
for a brief review. Besides consolidation equations, highly
heterogeneous soil profile or soil-structure interaction problems
can further exaggerate the numerical instability of the solution.
The recently proposed block diagonal preconditioners
(Chaudhary et al., 2011, 2012) have shown to have effectively
mitigated the ill-conditioning issues due to significant constrasts
in stiffness as well as hydraulic conductivity of the materials in
such problems.
This paper discusses the feasibility of 3D analysis with the
implementation of these latest developments in preconditioned
iterative solvers in GeoFEA, a commercial software package
(
). The results and how geometric
idealizations can sometimes lead to erroneous results will be
elaborated through using a case study of a basement excavation
in Singapore.
1 PRECONDITIONERS
The finite element discretization of the Biot’s coupled
consolidations equations is usually expressed in 2×2 block
linear system (Smith and Griffiths, 1997):
T
t
f
K B u
Cp
B C p
(2)
where
K
is solid stiffness matrix,
C
is fluid stiffness matrix,
B
is
displacement-pore pressure coupling matrix,
u
is displacement
increment,
p
is excess pore pressure increment,
f
is nodal
load increment, and
p
t
is nodal pore-pressure at current time
step. Chaudhary et al. (2012) observed that the performance of
existing preconditioners based on above 2×2 block form of the
coefficient matrix may deteriorate significantly for the problems
with significant contrasts in material properties, such as in soil-
structure interaction problems. They proposed to partition the
solid stiffness matrix
K
such that the coefficient matrix
A
takes
a 3×3 block form, which has more flexibility to construct
