Popular eigensolvers such as block-Lanczos require repeated inversion of an eigenmatrix. This is a bottleneck in large-scale modal problems with millions of degrees of freedom. On the other hand, the classic Rayleigh–Ritz conjugate gradient method only requires a matrix-vector multiplication, and is therefore potentially scalable to such problems. However, as is well-known, the Rayleigh–Ritz has serious numerical deficiencies, and has largely been abandoned by the finite-element community. In this paper, we address these deficiencies through subspace augmentation, and consider a subspace augmented Rayleigh–Ritz conjugate gradient method (SaRCG). SaRCG is numerically stable and does not entail explicit inversion. As a specific application, we consider the modal analysis of geometrically complex structures discretized via nonconforming voxels. The resulting large-scale eigenproblems are then solved via SaRCG. The voxelization structure is also exploited to render the underlying matrix-vector multiplication assembly-free. The implementation of SaRCG on multicore central processing units (CPUs) and graphics-programmable units (GPUs) is discussed, followed by numerical experiments and case-studies.

References

References
1.
Hernandez
,
V.
,
Roman
,
J. E.
,
Tomas
,
A.
, and
Vidal
,
V.
,
2009
, “
A Survey of Software for Sparse Eigenvalue Problems
,”
Universidad Politecnica de Valencia
,
Valencia, Spain
, SLEPc Technical Report STR-6, http://www.grycap.upv.es/slepc
2.
Arbenz
,
P.
,
Hetmaniuk
,
U. L.
,
Lehoucq
,
R. B.
, and
Tuminaro
,
R. S.
,
2005
, “
A Comparison of Eigensolvers for Large-scale 3D Modal Analysis Using AMG-Preconditioned Iterative Methods
,”
Int. J. Numer. Methods Eng.
,
64
(
2
), pp.
204
236
.10.1002/nme.1365
3.
Saad
,
Y.
,
2011
,
Numerical Methods for Large Eigenvalue Problems
,
2nd ed.
,
Manchester University Press
, Manchester, UK.
4.
Grimes
,
R. G.
,
Lewis
,
J. G.
, and
Simon
,
H. D.
,
1994
, “
A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems
,”
SIAM J. Matrix Anal. Appl.
,
15
(
1
), pp.
228
272
.10.1137/S0895479888151111
5.
Sorensen
,
D. C.
,
2002
, “
Numerical Methods for Large Eigenvalue Problems
,”
Acta Numerica
,
11
, pp.
519
584
.10.1017/S0962492902000089
6.
Golub
,
G. H.
, and
Ye
,
Q.
,
2002
, “
An Inverse Free Preconditioned Krylov Subspace Method for Symmetric Generalized Eigenvalue Problems
,”
SIAM J. Sci. Comput.
,
24
(
1
), pp.
312
334
.10.1137/S1064827500382579
7.
Bergamaschi
,
L.
,
Martínez
,
Á.
, and
Pini
,
G.
,
2006
, “
Parallel Preconditioned Conjugate Gradient Optimization of the Rayleigh Quotient for the Solution of Sparse Eigenproblems
,”
Appl. Math. Comput.
,
175
(
2
), pp.
1694
1715
.10.1016/j.amc.2005.09.015
8.
Knyazev
,
A. V.
,
2001
, “
Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method
,”
SIAM J. Sci. Comput.
,
23
(
2
), pp.
517
541
.10.1137/S1064827500366124
9.
Sleijpen
,
G. L. G.
, and
Van der Vorst
,
H. A.
,
1996
, “
A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems
,”
SIAM J. Matrix Anal. Appl.
,
17
(
2
), pp.
401
425
.10.1137/S0895479894270427
10.
Ipsen
,
I. C. F.
,
1997
, “
Computing an Eigenvector With Inverse Iteration
,”
SIAM Rev.
,
39
(
2
), pp.
254
291
.10.1137/S0036144596300773
11.
Jang
,
H.-J.
,
2001
, “
Preconditioned Conjugate Gradient Method for Large Generalized Eigenproblems
,”
Trends Math. Inf. Center Math. Sci.
,
4
(
2
), pp.
103
109
.
12.
Feng
,
Y. T.
, and
Owen
,
D. R. J.
,
1996
, “
Conjugate Gradient Methods for Solving the Smallest Eigenpair of Large Symmetric Eigenvalue Problems
,”
Int. J. Numer. Methods Eng.
,
39
(
13
), pp.
2209
2230
.10.1002/(SICI)1097-0207(19960715)39:13<2209::AID-NME951>3.0.CO;2-R
13.
Wright
,
J.
, and
Nocedal
,
S.
,
2006
,
Numerical Optimization
,
Springer Science + Business Media
,
New York
.
14.
Yang
,
H.
,
1993
, “
Conjugate Gradient Methods for the Rayleigh Quotient Minimization of Generalized Eigenvalue Problems
,”
Computing
,
51
(
1
), pp.
79
94
.10.1007/BF02243830
15.
Duster
,
A.
,
Parvizian
,
J.
,
Yang
,
Z.
, and
Rank
,
E.
,
2008
, “
The Finite Cell Method for 3D Problems of Solid Mechanics
,”
Comput. Methods Appl. Mech. Eng.
,
197
, pp.
3768
3782
.10.1016/j.cma.2008.02.036
16.
Karabassi
,
E. A.
,
Papaioannou
,
G.
, and
Theoharis
,
T.
,
1999
, “
A Fast Depth-Buffer-Based Voxelization Algorithm
,”
J. Graphics Tools
,
4
(
4
), pp.
5
10
.10.1080/10867651.1999.10487510
17.
Zienkiewicz
,
O. C.
,
2005
,
The Finite Element Method for Solid and Structural Mechanics
,
Elsevier
,
New York
.
18.
Taiebat
,
H. H.
, and
Carter
,
J. P.
,
2001
,
Three-Dimensional Non-Conforming Elements
,
Centre for Geotechnical Research, The University of Sydney
,
Sydney
, p.
R808
.
19.
Augarde
,
C. E.
,
Ramage
,
A.
, and
Staudacher
,
J.
,
2006
, “
An Element-Based Displacement Preconditioner for Linear Elasticity Problems
,”
Comput. Struct.
,
84
(
31–32
), pp.
2306
2315
.10.1016/j.compstruc.2006.08.057
20.
NVIDIA Corporation
,
2008
,
NVIDIA CUDA: Compute Unified Device Architecture, Programming Guide
,
NVIDIA Corporation
,
Santa Clara
.
21.
SolidWorks
,
2005
, “
SolidWorks
,” www.solidworks.com
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