COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Large-Scale Boundary Element Analysis in Solid Mechanics Using Fast
Multipole Method
Z. H. Yao1*, P. B. Wang1, T. Lei1, H. T. Wang2
1
2
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China
Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing, 100084 China
Email: [email protected]
Abstract Combined with the fast multipole method, the boundary element method become quite efficient to deal with
large-scale engineering and scientific problems. In this paper, the outline of the FMBEM are introduced at first, the
accuracy and efficiency are verified and compared with the conventional BEM, a parallel algorithm of FMBEM for PC
cluster are briefly presented. Some applications are presented, including: 2D and 3D simulation of composite
materials, simulation of 2D elastic solid containing large number of cracks and fatigue crack growth. A new approach
of FMBEM for elasto-plasticity problems is also presented.
Key words: fast multipole boundary element method, parallel algorithm, composite material, crack problems,
elasto-plasticity
INTRODUCTION
Boundary element method has been developed as an efficient numerical method followed the finite element method in
recent several decades, and BEM become the most important complement of the FEM in the field of solid mechanics.
BEM has attractive advantages of high accuracy, dimension reduction, and it is especially suitable to deal with the
problems related to infinite or semi-infinite domain, and the problems related to singularity or high gradients. But the
conventional BEM is not capable to deal with practical complex engineering problems, because the matrix of the
resulted algebraic equation is dense and asymmetric, the operations increase in O(N 3) and the memory requirement
increases in O(N 2), N is the number of unknowns.
In 1980s, Rokhlin and Greengard presented a multipole method with O(N) operations and memory requirement, to
solve the 2D potential problem [1, 2]. In 1997, Greengard and Rokhlin introduced the exponential expansion in
multipole to local translation, which leads to further enhancement of the efficiency [3]. Combined with the fast
multipole method (FMM), the BEM become quite efficient to deal with large-scale engineering and scientific
problems. In 2002, Nishimura reviewed the fast multipole accelerated boundary integral equation methods [4].
In the field of solid mechanics, the algorithms for elasticity problems in terms of Taylor series expansion of the
fundamental solution are reported in literature [5]. Several algorithms of new version FMM for the direct BEM of 3D
elasticity problems based upon Taylor series expansion and spherical harmonic expansion are also available [6]. In
recent years, FMBEM attracted more and more researchers.
In authors’ group the investigations on FMBEM started in 2000, several papers have been published in recent years
[7-12]. Using FMBEM, the large-scale problems of approximated half million DOF can be computed on one PC, and
on an inexpensive PC cluster problems of larger than 5 million DOF can be computed. Considering that the abovementioned numbers of DOF are only DOF on the boundary, in some cases the FMBEM is more capable to compute the
large-scale problems than the FEM.
In this paper, the outline of the FMBEM are introduced at first, the accuracy and efficiency are verified and compared
with the conventional BEM, a parallel algorithm of FMBEM for PC cluster are briefly presented. Some applications
are presented, including: 2D and 3D simulation of composite materials, simulation of 2D elastic solid containing large
number of cracks and fatigue crack growth. A new approach of FMBEM for nonlinear problems, namely,
elastoplasticity problems is also presented.
— 28 —
OUTLINE OF FMBEM
Taken the 2D elasticity problem as example, the boundary integral equation is given by
*
Cαβ ( x ) uβ ( x ) + ∫ Tαβ* ( x, y ) uβ ( y ) dS ( y ) = ∫ Uαβ
( x, y ) tβ ( y ) dS ( y )
S
(1)
S
where x and y denote the source point and field point on the boundary S respectively (Fig. 1), uβ and tβ are
*
*
displacements and tractions on the boundary, the tensors, U αβ
and Tαβ
are the fundamental solutions, and the
coefficient Cαβ is determined by the geometry of the surface S at point x.
Figure 1: Procedures of fast multipole method
1. Multipole expansion Suppose field point y is located in box A whose center is yo. Let SA be the subset of boundary
S in this box. Assume the inequality y0 y ≤ 12 y 0 x is valid. By expanding the fundamental functions into Taylor
series, the first integral in Eq. (1) can be evaluated as
∫ Tαβ (x, y )uβ ( y )dS ( y ) = ∑ f (y x )M ( y )
*
i
0
i
(2)
0
i
SA
where M i is the multipole moment centered at y0 , and fi is a regular function related to the vector y o x .
2. First page Multipole to multipole translation (M2M) The multipole moment can be shifted into a new one
centered at y0′ defined as
uuuur
M j ( y0′ ) = ∑ χ ji y0 y0′ M i ( y0 )
(
i
)
(3)
where χ ji is the coefficient of M2M translation.
uuur
3. Multipole to local translation (M2L) Assume the inequality x0′ x <
translated into local moment centered at x0′ defined as
1
2
uuur
y0′ x is valid, the multipole moment can be
uuuur
Ll ( x0′ ) = ∑ φli y0′ x0′ M i ( y0′ )
i
(
)
(4)
where φli is the coefficient of M2L translation.
4. Local to local translation (L2L) Similar to M2M, the local moment can be shifted from x0′ to x0 as follows:
uuuur
Ll ( x0′ ) = ∑ φli y0′ x0′ M i ( y0′ )
i
(
)
(5)
where ξil is the coefficient of L2L translation.
5. Local expansion The first integral in Eq. (1) can be finally evaluated by
— 29 —
∫ Tαβ (x, y )u β ( y )dS ( y ) = ∑ g (x x )L (x )
*
l
SA
0
l
(6)
0
l
where gl is a regular function only related to x0 x . The second integral in Eq. (1) can be evaluated in a similar
formula.
6. Numerical implementation of fast multipole method The algorithm of fast multipole method consists of 6 steps
which in all form the matrix-vector products.
1) Tree Construction: The fast multipole method utilizes the tree structure, which is constructed hierarchically. The
root of the tree represents a box that contains all the elements and is at level 0 of the tree. The root is divided into 4
equal-sized child-boxes that are at level 1. Each of these boxes is divided into 4 more child-boxes until every box
contains at most some fixed number of elements. The shape of the tree adapts to the distribution of elements because
the tree will have more levels in regions having more elements. The resulted tree is called an adaptive quad-tree. An
example is shown in Fig. 2, where the tree is constructed on the model of a square plate with 100 randomly distributed
inclusions of different sizes.
2) Forming multipole moments of leaves: Leaves are the boxes at the finest levels. For each non-empty leaf, its
multipole moments are formed by elements it contains.
Figure 2: Model of a square matrix with random inclusions (left) and its corresponding tree (right)
3) Upward Stage: For each non-leaf box, the multipole moment is formed by shifting the multipole moment of its
children using Eq. (3) and adding up the four shifted multipole moments. This procedure is repeated up to level 2.
4) Downward Stage: For each box starting from level 2, the multipole moment of boxes in the interaction list, as
depicted in Fig. 3, is transformed to this box's local moment using Eq. (4). Then the local moment is shifted to its
children's one using Eq. (5), and the multipole moment in the interaction list of that children level is transformed using
Eq. (4) and added on it. This procedure is repeated down to the finest level.
Figure 3: Definition of interaction list (Boxes with dashed lines indicate empty boxes)
5) Evaluation of the integrals in Eq. (1): The integrals in Eq. (1) are evaluated in 2 parts. For each source point, the
contributions from elements in the same leaf and neighbor leaves are evaluated directly in the way as conventional
BEM. Contributions from others are evaluated using Eq. (6).
6) Update of the iterative vector: The iterative vector is updated and the next iteration is started from Step 2.
— 30 —
7. New version of fast multipole method The new version of FMM with exponential expansion is developed in the
late 1990s by Greengard and Rokhlin. In procedures of the new version, the dense translation operator from multipole
moments to local moments in Eq. (4) is diagonalized by three new steps.
1) Multipole to exponential shift: In order to obtain a new series of exponential moments by multipole moments in the
same cube.
2) Exponential to exponential shift: In order to transfer exponential moments from one cube containing field points to
another cube containing source points.
3) Exponential to local shift: In order to obtain local moments by exponential moments in the same cube.
Since operations needed in FMM to obtain local moments by multipole moments dominate most computer resources
for 3D problems, successful simplification of these operations makes the new version FMM much effective than the
original one.
ACCURACY AND EFFICIENCY OF FMBEM
To verify the accuracy and efficiency of the FMBEM, a series of test examples have been computed.
1. Accuracy of FMBEM related to the orders of multipole and local expansion A square sheet with a circular hole
in the center under uniform normal displacement along 4 edges as shown in Fig. 4 (plane stress problem) is tested. The
side length l =10cm, the radius of the hole r = 2 cm, prescribed normal displacement un = 0.001 cm. All boundaries are
discretized into 4,720 DOF.
Figure 4: A square sheet with a circular hole in the center under uniform normal displacement along 4 edges
The numerical results of normal displacement at three location of hole boundary, using FMBEM with different orders
of multipole and local expansions, are listed in Table 1, in comparison with conventional BEM. The comparison has
shown that the FMBEM can obtain as accurate results as the conventional BEM provided enough orders of multipole
and local expansions are taken.
Table 1 Numerical results of FMBEM in comparison with conventional BEM
un mm
(θ = 0 ° )
relative error
un mm
(θ = 45°)
relative error
un mm
(θ = 90°)
relative error
Gauss
elimination
−0.009191605
---
−0.009374642
---
−0.009191605
---
FMM (p = 20)
−0.009191627
2×10−6
−0.009374664
2×10−6
−0.009191607
2×10−7
FMM (p = 15)
−0.009191628
2×10−6
−0.009374664
2×10−6
−0.009191620
1×10−6
FMM (p = 10)
−0.009187775
4×10−4
−0.009374608
3×10−6
−0.009187732
4×10−4
FMM (p = 5)
−0.009218128
3×10−3
−0.009401706
2×10−3
−0.009220656
3×10−3
2. Accuracy of large-scale computation using FMBEM The plane strain model of a square with a circular inclusion
in the center (the left of Fig. 5) is periodically repeated to form larger computational models: 2×2 model with 10,880
DOF (the right of Fig. 5) and 30×30 model with 696,000 DOF. The length l =10cm, the radius of the hole r = 2 cm. The
— 31 —
material properties are: Em = 1000 MPa, νm = 0.3 for the matrix, and Ei = 5000 MPa, νI = 0.3 for the inclusions. The
prescribed normal displacements are proportional to the side length of the square: un = 0.002 cm for 2×2 model and
un = 0.03 cm for 30×30 model respectively. In this way the traction and the normal displacement relative to each
inclusion center along the inclusion boundary should be the same for different models. The numerical results (Table 2)
have shown high accuracy of the large-scale computation using FMBEM.
Figure 5: A square with a circular inclusion in the center (left) and the corresponding 2×2 model
Table 2 Numerical results of 2×2 model and 30×30 model
FMM (p = 20)
un (mm)
tn (MPa)
θ = 0°
2×2
− 0.0112337
30×30
− 0.0112334
2×2
0. 552577
30×30
0. 552595
error
2×10−5
2×10−5
θ = 45°
−0.0152808
−0.0152802
0. 547241
0. 547221
error
3×10−5
3×10−5
θ = 90°
−0.0112337
−0.0112334
0. 552578
0. 552595
error
2×10−5
2×10−5
3. Efficiency and convergency of FMBEM The computation time and memory requirement using FMBEM for
different problem scale, are shown in Fig. 6 in comparison with those using conventional BEM. For the FMBEM
computation the order of multipole and local expansions is taken as p = 20, and the tolerance for the iteration
ε = 1×10−5, and for the conventional BEM computation the package LAPACK is applied for the Gauss elimination.
Figure 6: The comparison of computation time and memory requirement between FMBEM and BEM
The convergency of the FMBEM is shown in Fig. 7, the order of multipole and local expansion is taken as p=20. The
results have shown good efficiency and convergency of FMBEM.
— 32 —
Figure 7: The convergency of the computation using FMBEM
APPLICATION OF FMBEM ON SIMULATION OF COMPOSITE MATERIALS
The FMBEM has been applied in the 2D and 3D simulation of composite material and the obtained equivalent material
properties have shown good agreement with the available results in micromechanics. But the computation using
FMBEM can obtain not only the equivalent properties, but also the whole fields of displacements and stresses, which
is useful for the further investigation of the failure process of the composite materials.
1. 2D simulation of long-fiber reinforced composite material Long-fiber reinforced composite material can be
simulated using 2D model of plane strain. Fig. 8 shows two models of square containing 1600 randomly distributed
circular inclusions discretized into 544,000 DOF. Fig. 9 shows the results of equivalent volume modulus obtained
using FMBEM in comparison with that obtained using Mori-Tanaka method.
Figure 8: Models of square containing 1600 circular inclusions, volume fraction = 0.2 (left), 0.4 (right)
Figure 9: The comparison of equivalent volume modulus between FMBEM and Mori-Tanaka method
— 33 —
Both results of FMBEM and Mori-Tanaka method show good agreement. Actually, for the computation of equivalent
modulus of the 2D composites, such large-scale computation is not necessary. In these cases, the models containing
approximately 100 inclusions are large enough.
2. 3D simulation of particle reinforced composite material Fig. 10 shows two models of cube containing 100 and
300 randomly distributed identical spherical inclusions, which are discretized into 187,000 and 372,600 DOF
respectively.
Figure 10: Models of cube containing 100 (left) and 300 (right) spherical inclusions
Fig. 11 shows the results of equivalent volume modulus obtained using FMBEM in comparison with that obtained
using Mori-Tanaka method.
Figure 11: The comparison of equivalent modulus between FMBEM and Mori-Tanaka method
Figure 12: Number of particles versus maximum interfacial normal and shear stresses (MPa)
— 34 —
Furthermore, the computation using FMBEM can obtain the whole fields of displacement and stresses. Fig. 12 shows
the number of particles versus maximum interfacial normal and shear stresses, which is related to the failure of the
composite material.
3. 3D simulation of short-fiber reinforced composite material Fig. 13 shows two models of cube containing 100 short
fibers, the volume fractions of the fiber are 0.03 and 0.1 respectively, the scale of these models is 172,800 DOF.
Figure 13: Models of cube containing 100 short fibers with volume fraction c = 0.03 (left) and c = 0.1 (right)
Fig. 14 shows the results of equivalent volume modulus obtained using FMBEM in comparison with that obtained
using Mori-Tanaka method and with H-S bounds. Most numerical values are within the range of H-S bounds, while a
few points are slightly lower than the H-S lower bound. This difference comes from the numerical errors of BEM, i.e.,
discretization error. Numerical examples show that as the boundary element mesh is further refined, these few points
increase to approach the H-S lower bound. This means that the numerical solution obtained by fast multipole BEM
does not violate the theoretical upper and lower bounds.
Figure 14: The comparison of computation time and memory requirement between FMBEM and BEM
As the aspect ratio of fibers increases, for example aspect ratio equals 11 as shown in Fig. 15, the representative
volume element will not be big enough. In such case larger-scale computation will be required, and parallel
computation is necessary.
Figure 15: Models containing 100 longer fibers with different orientation angle: 0° (left), 30° (middle), 90° (right)
— 35 —
LARGE-SCALE PARALLEL COMPUTATION OF FMBEM
The FMBEM is more difficult to parallelize efficiently than the conventional BEM because of two extra key problems.
One is that the elements are non-uniformly distributed and hence the tree is imbalanced. The other is that the tree is a
connected data structure that is harder to traverse than simple arrays. Therefore some special approach has to be
adopted.
1. Task decomposition by boxes Instead of domain decomposition method (DDM) commonly used in conventional
parallel BEM, box decomposition is adopted in the parallel algorithm of FMBEM. The domain is decomposed while
splitting the tree by the unit of boxes. The boxes are first sorted in a 2D or 3D space order by a sorting algorithm
presented in Ref. [13]. Then each task is assigned with maximal approximately even number of elements by a
parameter called dec_level, which stands for the level the tree is decomposed at. The higher the dec_level is, the more
balanced the decomposition will be. But it cannot be higher than the minimum level of leaves in the tree. Fig. 16
depicts decomposing the model in Fig. 2 into six tasks at dec_level=3. The main advantage of this partition scheme is
that the tasks can be decomposed regardless of different phases such as the inclusion and matrix for particle-reinforced
composite.
Figure 16: Task decomposition boxes
Figure 17: Remote checks in parallel tree traversing (Empty boxes are not returned)
2. Parallelization of the tree traversing The most time-consuming step in fast multipole method is the downward stage.
It is also the most complicated step in parallel formulations since interactions from interaction list and neighbor boxes are
needed. And these boxes may be in the same or other tasks, and may be empty anywhere. So a remote-check procedure is
used as a simple example shown in Fig. 17 for the case of interaction list only, where box marked with X in task i need the
— 36 —
remote boxes in task j. In that case task i sends a checking request to j and a checking result of non-empty boxes are
returned. The overhead of this procedure can be neglected since it costs only a few seconds in a model with tens of
thousands unknowns and also the checking result does not change from iteration to iteration for elastostatics.
3. Accuracy verification To verify the accuracy of FMBEM, two models with 1 inclusion centered at the origin of the
matrix and 343 (7×7×7) inclusions that are periodically distributed are used, as shown in Fig. 18. The edge length of
matrix is 20×20×20(cm) and the inclusion volume fraction is 0.1. Each inclusion is discretized into 392 triangle
piecewise constant elements and 120,000 elements for the outer boundary of matrix. The multipole/local expansion
order is 12 and the residual of GMRES is 1.0×10−7.
Figure 18: Model with periodically distributed inclusions and local coordinate of an inclusion (right)
The relative displacement u x( ) − u x( ) and the traction t x( ) are given in Fig. 19. The slight difference of the last 49
inclusions, which are near the front surface (in +x direction) of the matrix, comes from matrix discretization error. The
results have shown satisfactory accuracy of the parallel FMBEM computation.
1
0
1
1
0
1
Figure 19: The results of relative displacement u x( ) − u x( ) and the traction t x( ) for point 1 and 0 of each inclusion
4. Performance and Efficiency of Parallel FMBEM The following computations are carried out on a 16-node SMP
PC cluster. The nodes are connected via 1,000Mb Ethernet switch hub. Each node owns 2 Pentium IV Xeon processors
and 1 GB memory. C++ is used as the programming language and a message passing interface, LAM/MPI, is used for
communications.
To show the scalability of parallel FMBEM, 6 models with different number of inclusions are used. The edge length of
matrix is 20×20×20 (cm) and the radius of inclusion is 0.4571(cm). The matrix is applied with an axial traction of
100MPa. The inclusions are distributed randomly by using Sobol sequence, as shown in Fig. 20. Total 32 processors are
used for those calculations. The multipole/local expansion order is 12 and the residual of GMRES is 1.0×10−5.
Table 3 demonstrates the performance and memory requirements of the computation. Both time and storage
complexity are approximately proportional to the scale of the problem. For the largest model, the running CPU time is
about 43 hours and approximately 9GB memory is required. Fig. 21 shows the computing time versus numbers of
processors and the speedup for the parallel FMBEM.
— 37 —
Figure 20: Models with different number of inclusions
Table 3 Performance of parallel FMBEM computation
Inclusion number
100
500
1000
2000
3000
4000
Volume fraction
0.005
0.025
0.05
0.1
0.15
0.2
Number of DOF
477,600
948,000
1,536,000
2,712,000
3,888,000
5,064,000
Memory requirement (MB)
1,976
3,070
4,067
5,989
7,543
9,182
Computing time (s)
4,660
15,179
32,136
71,636
103,635
161,190
Figure 21: The computing time versus number of processors (left), and the speedup for parallel FMBEM
5. Application on the simulation of fiber reinforced composites Two fiber shapes, bone-shaped short fiber (as
shown in Fig. 22) and conventional straight short fiber, are simulated and compared. Fig. 23 shows a typical
configuration of RVE model of well-aligned bone-shaped short-fiber reinforced composites. There are 200 fibers in
the RVE, the volume fraction is 0.05. It is discretized into 2,596,800 DOF, and the computing time is approximately 38
hours. Fig. 24 shows the comparison of effective tensile modulus for two types of fiber shapes, and a normalized
histogram with fitted Weibull probability density functions.
— 38 —
Figure 22: Well-aligned bone-shaped Ni-fiber reinforced polyester matrix fiber composites and a fiber model
Figure 23: The BEM model of a RVE of matrix containing 200 randomly dispersed bone-shaped short fibers
Figure 24: Comparison of effective tensile modulus for two types of fiber shapes,
and a normalized histogram with fitted Weibull probability density functions
In literature, we can find an example of the largest RVE with 16,000 CNT fibers using FMBEM, which has a total DOF
of 28,896,000 and takes 34 hours for the computation on a super computer Futitsu Primepower HPC2500 [14]. The
authors’ group uses only inexpensive PC cluster.
SIMULATION OF CRACK AND CRACK GROWTH PROBLEMS USING FMBEM
In authors’ group the FMBEM was used to solve the traction boundary integral equation for 2D crack analysis at first.
And then the FMM based on complex Taylor series expansions is applied to the dual boundary element method
(DBEM) for large-scale crack analysis in linear elastic fracture mechanics. An incremental crack-extension analysis
based on the maximum principal stress criterion and the Paris law is used to simulate the fatigue growth of numerous
cracks in a 2D solid. Furthermore, the FMDBEM is applied to compute the effective in-plane bulk modulus of 2D
solids with thousands of randomly distributed microcracks.
1. FMBEM for simulation of 2D solids containing large numbers of cracks A large number of randomly
distributed cracks in 2D infinite elastic space are simulated using fast multipole BEM. The computational model is
— 39 —
shown in the left of Fig. 25. 3,000 cracks are distributed in a square region, and the number of DOF is 900,000. The
truncation number of Taylor series is 20. Total CPU time cost is 6 hours 9 minutes. The right of Fig. 25 shows the COD
results of a part of the whole region. The program is written in ANSI C++ and implemented on a personal computer
with Pentium IV 1.8GHz and 1GB memory.
Figure 25: 3000 randomly distributed cracks in a square region of 2D infinite space (left)
and COD of cracks in a part of region (right)
2. Verification of the accuracy and efficiency of the FMDBEM Because the conventional boundary elements are
inadequate to meet the square root displacement variation near the crack tip, a spherical crack tip element
incorporating the displacement variation by modification of the shape functions is employed. It is a 3-node
discontinuous element, and its shape functions are shown in Fig. 26. To verify the accuracy of the FMDBEM, a square
sheet with center crack, as shown in Fig. 27, was computed, where 2w = 10 mm, 2a varies form 1 mm to 5 mm, the crack
was discretized into 16 discontinuous quadratic elements and the order of multipole and local expansion was taken as
p = 30. Table 4 shows the normalized SIFs of the center crack with different crack sizes. Compared with the solusions
by Isida [15], the maximum error in the K I* values is only 0.37%. In addition, the K II* values are almost equal to the
theoretical solution 0.
Figure 26: Shape functions of the special crack tip element
Figure 27: A center crack in a square sheet
Table 4 Normalized SIFs for a center crack with different crack sizes
a w
0.1
0.2
0.3
0.4
0.5
K I* (left tip)
1.0150
1.0511
1.1248
1.2122
1.3308
K I* (right tip)
1.0150
1.0511
1.1248
1.2121
1.3306
K I* (Isida)
1.014
1.055
1.123
1.216
1.334
K II* (left tip)
−6.525×10 −8
−2.876×10 −7
−2.245×10 −7
−2.251×10 −7
−1.991×10 −7
K II* (right tip)
6.419×10 −8
2.442×10 −7
2.233×10 −7
2.241×10 −7
2.214×10 −7
— 40 —
Fig. 28 shows the opening displacement versus distance from the left crack tip. Fig. 29 shows the CPU time versus the
number of DOF. All the results have shown high accuracy and efficiency of the FMDBEM.
Figure 28: Opening displacement versus distance from the left crack tip
Figure 29: CPU time versus the number of DOF
3. Fatigue crack growth in 2D elastic solid containing numerous cracks An incremental crack-extension analysis
is applied to simulate the fatigue crack growth. It assumes a piece-wise linear discretization of the crack-growth path.
In each incremental step, the fast multipole DBEM is applied to solve the equation system. The directions and
extensions of crack growth are determined in the post-proceeding process. The details, including special crack tip
elements, evaluation of SIF, determination of the direction and extension of fatigue crack growth and the simulation of
crack coalescence, can be found in Ref. [11].
To simulate the fatigue growth of a crack under mixed-mode deformation, a cruciform plate with an initial edge crack
located at one of the interior corners has been computed. Fig. 30 shows a cruciform plate with an initial edge crack
(left), the crack growth paths obtained by the FM DBEM analysis (center), and that presented by Portela [16] et al
(right).
Figure 30: A cruciform plate with an initial edge crack and the crack growth paths
Fig. 31 shows a rectangular elastic sheet containing 400 irregularly distributed cracks, and Fig. 32 shows the
corresponding fatigue growth paths under horizontal tension.
Figure 31: 400 irregularly distributed cracks in a rectangular sheet
— 41 —
Figure 32: Fatigue growth paths of multiple cracks after 14.85×108 load cycles (left) and 17.40×108 load cycles (right)
4. Computation of effective modulus of 2D microcracked solid The FMDBEM is adopted to directly determine the
effective in-plane bulk modulus of a square sheet containing 4,000 randomly oriented microcracks, as shown in Fig. 33.
Fig. 34 shows the numerical results of the effective in-plane bulk modulus by the FMDBEM in comparison with the
corresponding solutions of various micromechanics methods, with the Poisson’s ratio ν = 0.3. The numerical results
agree well with the estimation of GSCM, DM and Feng-Yu [17].
Figure 33: A sheet containing 4000 microcracks
Figure 34: Effective in-plane bulk modulus versus crack density
Furthermore, the effect of crack non-uniform distribution on effective in-plane bulk modulus is also investigated using
the FMDBEM. This work assumes that the microcracked solid contains some local regions having a crack density ω L
higher or lower than the average crack density ω 0 and analyze the variation of the effective in-plane modulus with ω L
when ω 0 is fixed. Fig. 35 shows a square sheet containing 4 higher crack density region, and Fig. 36 shows the results
of K e / K 0 versus ω L / ω 0 for the case of ω 0 = 0.3 . The results show that the non-uniform distribution of microcracks increases
the effective in-plane bulk modulus of the whole microcracked solid.
Figure 36: K e / K 0 versus ω L / ω 0 ( ω 0 = 0.3 )
Figure 35: A square sheet containing 4 local regions
CONCLUDING REMARKS
Based on the progress of FMBEM, the authors’ group carried out a series of investigation on the applications of
FMBEM in solid mechanics. The investigations on large-scale FMBEM analysis in solid mechanics, including 2D and
— 42 —
3D elasticity and 2D fracture problems, have shown its attractive advantages, high accuracy and efficiency.
Combining with FMM the boundary element method become suitable to deal with large-scale practical engineering
and scientific problems.
The first author has been involved in the research on boundary element methods since 1979. The BEM is regarded as an
important complement of the widely-applied FEM, but if the BEM is only capable to obtain the same results as obtained
by FEM, such complement was not necessary. For the complement it is important to do something, which FEM could not
do, or do something significantly better than FEM. The development of FMBEM have shown good prospects at this
aspect. FMBEM have been successfully applied in the field of MEMS design and electro-magnetic field analysis. In the
field of solid mechanics, the most important thing is to develop practical applications of FMBEM.
The further investigation in authors’ group is concentrated on several topics, including FMBEM of elasto-plasticity
problems, thin structure problems, dynamic and coupling problems.
Acknowledgements
Financial support for the projects from the National Natural Science Foundation of China, under grant No. 10172053,
10472051 is gratefully acknowledged.
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