Large-scale finite element analysis (FEA) with millions of degrees of freedom (DOF) is becoming commonplace in solid mechanics. The primary computational bottleneck in such problems is the solution of large linear systems of equations. In this paper, we propose an assembly-free version of the deflated conjugate gradient (DCG) for solving such equations, where neither the stiffness matrix nor the deflation matrix is assembled. While assembly-free FEA is a well-known concept, the novelty pursued in this paper is the use of assembly-free deflation. The resulting implementation is particularly well suited for large-scale problems and can be easily ported to multicore central processing unit (CPU) and graphics-programmable unit (GPU) architectures. For demonstration, we show that one can solve a 50 × 106 degree of freedom system on a single GPU card, equipped with 3 GB of memory. The second contribution is an extension of the “rigid-body agglomeration” concept used in DCG to a “curvature-sensitive agglomeration.” The latter exploits classic plate and beam theories for efficient deflation of highly ill-conditioned problems arising from thin structures.

References

References
1.
Golub
,
G. H.
,
1996
,
Matrix Computations
,
Johns Hopkins
,
Balitmore, MD
.
2.
Aubry
,
R.
,
Mut
,
F.
,
Dey
,
S.
, and
Lohner
,
R.
,
2011
, “
Deflated Preconditioned Conjugate Gradient Solvers for Linear Elasticity
,”
Int. J. Numer. Methods Eng.
,
88
(
11
), pp.
1112
1127
.10.1002/nme.3209
3.
2012
, “
ANSYS 13. ANSYS
,” www.ansys.com
4.
Efendiev
,
Y.
, and
Hou
,
T. Y.
,
2009
,
Multiscale Finite Element Methods: Theory and Applications
, Vol.
4
.,
Springer
,
New York
.
5.
Arbenz
,
P.
,
van Lenthe
,
G. H.
,
Mennel
,
U.
,
Müller
,
R.
, and
Sala
,
M.
,
2008
, “
A Scalable Multi-Level Preconditioner for Matrix-Free μ-Finite Element Analysis of Human Bone Structures
,”
Int. J. Numer. Methods Eng.
,
73
(
7
), pp.
927
947
.10.1002/nme.2101
6.
Suresh
,
K.
, and
Yadav
,
P.
,
2012
, “
Large-Scale Modal Analysis on Multi-Core Architectures
,”
ASME
Paper No. DETC2012-70281. 10.1115/DETC2012-70281
7.
Saad
,
Y.
,
2003
,
Iterative Methods for Sparse Linear Systems
,
SIAM
,
New Delhi, India
.
8.
Williams
,
S.
,
Oliker
,
L.
,
Vuduc
,
R.
,
Shalf
,
J.
,
Yelick
,
K.
, and
Demmel
,
J.
,
2007
, “
Optimization of Sparse Matrix–Vector Multiplication on Emerging Multicore Platforms
,”
Proceedings ACM/IEEE Conference on Supercomputing
, Reno, NV, Nov. 10–16.
9.
Bell
,
N.
,
2008
, “
Efficient Sparse Matrix–Vector Multiplication on CUDA
,” NVIDIA, Technical Report No. NVR-2008-004.
10.
Yang
,
X.
,
Parthasarathy
,
S.
, and
Sadayappan
,
P.
,
2011
, “
Fast Sparse Matrix–Vector Multiplication on GPUs: Implications for Graph Mining
,”
37th International Conference on Very Large Data Bases
, Seattle, WA, Aug. 29–Sept. 3.10.14778/1938545.1938548
11.
Akbariyeh
,
A.
,
Carrigan
,
T. J.
,
Dennis
,
B. H.
,
Chan
,
W. S.
,
Wang
,
B. P.
, and
Lawrence
,
K. L.
,
2012
, “
Application of GPU-Based Computing to Large Scale Finite Element Analysis of Three-Dimensional Structures
,”
Proceedings of the Eighth International Conference on Engineering Computational Technology
, Stirlingshire, UK, Paper No. 610.4203/ccp.100.6.
12.
Adams
,
M.
,
2002
, “
Evaluation of Three Unstructured Multigrid Methods on 3D Finite Element Problems in Solid Mechanics
,”
Int. J. Numer. Methods Eng.
,
55
(
2
), pp.
519
534
.10.1002/nme.506
13.
Benzi
,
M.
, and
Tuma
,
M.
,
2003
, “
A Robust Incomplete Factorization Preconditioner for Positive Definite Matrices
,”
Numer. Linear Algebra Appl.
,
10
(
5,6
), pp.
385
400
.10.1002/nla.320
14.
Benzi
,
M.
,
2002
, “
Preconditioning Techniques for Large Linear Systems: A Survey
,”
J. Comput. Phys.
,
182
(
2
), pp.
418
477
.10.1006/jcph.2002.7176
15.
Briggs
,
W. L.
,
Henson
,
V. E.
, and
McCormick
,
S. F.
,
2000
,
A Multigrid Tutorial
.
SIAM
,
New Delhi, India
.
16.
Wesseling
,
P.
, “
Geometric Multigrid With Applications to Computational Fluid Dynamics
,”
J. Comput. Appl. Math.
,
128
(
1
), pp.
311
334
.10.1016/S0377-0427(00)00517-3
17.
Griebel
,
M.
,
Oeltz
,
D.
, and
Schweitzer
,
M. A.
,
2003
, “
An Algebraic Multigrid Method for Linear Elasticity
,”
SIAM J. Sci. Comput.
,
25
(
2
), pp.
385
407
.10.1137/S1064827502407810
18.
Karer
,
E.
, and
Kraus
,
J. K.
,
2010
, “
Algebraic Multigrid for Finite Element Elasticity Equations: Determination of Nodal Dependence via Edge-Matrices and Two-Level Convergence
,”
Int. J. Numer. Methods Eng.
,
83
(
5
), pp.
642
670
10.1002/nme.2853.
19.
Ruge
,
J.
, and
Brandt
,
A.
,
1988
, “
A Multigrid Approach for Elasticity Problems on ‘Thin’ Domains
,”
Multigrid Methods: Theory, Applications, and Supercomputing
, Vol.
110
,
S. F.
McCormick
, ed.,
Marcel Dekker Inc.
,
New York
, pp.
541
555
.
20.
Jonsthovel
,
T. B.
,
van Gijzen
,
M. B.
,
MacLachlan
,
S.
,
Vuik
,
C.
, and
Scarpas
,
A.
,
2012
, “
Comparison of the Deflated Preconditioned Conjugate Gradient Method and Algebraic Multigrid for Composite Materials
,”
Comput. Mech.
,
50
(
3
), pp.
321
333
.10.1007/s00466-011-0661-y
21.
Mishra
,
V.
, and
Suresh
,
K.
,
2009
, “
Efficient Analysis of 3D Plates via Algebraic Reduction
,”
ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE2009)
, San Diego, CA, July 19–23, Vol.
2
, pp.
75
82
.
22.
Mishra
,
V.
, and
Suresh
,
K.
,
2012
, “
A Dual-Representation Strategy for the Virtual Assembly of Thin Deformable Objects
,”
Virtual Reality
,
16
(
1
), pp.
3
14
.10.1007/s10055-009-0143-0
23.
Saad
,
Y.
,
Yeung
,
M.
,
Erhel
,
J.
, and
Guyomarc'h
,
F.
,
2000
, “
A Deflated Version of the Conjugate Gradient Algorithm
,”
SIAM J. Sci. Comput.
,
21
(
5
), pp.
1909
1926
.10.1137/S1064829598339761
24.
Baker
,
A. H.
,
Kolev
,
T. V.
, and
Yank
,
U. M.
,
2010
, “
Improving Algebraic Multigrid Interpolation Operators for Linear Elasticity Problems
,”
Numer. Linear Algebra Appl.
,
17
(
2,3
), pp.
495
517
10.1002/nla.688.
25.
Timoshenko
,
S.
, and
Krieger
,
S. W.
,
1959
,
Theory of Plates and Shells
,
McGraw-Hill Book Company
,
New York
.
26.
Hughes
,
T. J. R.
,
Levit
,
I.
, and
Winget
,
J.
,
1983
, “
An Element-by-Element Solution Algorithm for Problems of Structural and Solid Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
36
(
2
), pp.
241
254
.10.1016/0045-7825(83)90115-9
27.
Michopoulos
,
J.
,
Hermanson
,
J. C.
,
Iliopoulos
,
A. P.
,
Lambrakos
,
S. G.
, and
Furukawa
,
T.
,
2011
, “
Data-Driven Design Optimization for Composite Material Characterization
,”
ASME J. Comput. Inf. Sci. Eng.
,
11
(
2
), p.
021009
.10.1115/1.3595561
28.
NVIDIA Corporation,
2008
,
NVIDIA CUDA: Compute Unified Device Architecture, Programming Guide
,
Santa Clara
,
CA
.
29.
“OpenMP.org,” Accessed May 4, 2014, http://openmp.org/wp/
30.
Suresh
,
K.
,
2013
, “
Efficient Generation of Large-Scale Pareto-Optimal Topologies
,”
Struct. Multidiscip. Optim.
,
47
(
1
), pp.
49
61
.10.1007/s00158-012-0807-3
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