Research Papers

Assembly-Free Large-Scale Modal Analysis on the Graphics-Programmable Unit

[+] Author and Article Information
Krishnan Suresh

e-mail: suresh@engr.wisc.edu
Department of Mechanical Engineering,
UW-Madison, Madison, WI 53706

Contributed by the Computers and Information Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received July 11, 2012; final manuscript received December 1, 2012; published online January 7, 2013. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 13(1), 011003 (Jan 07, 2013) (7 pages) Paper No: JCISE-12-1108; doi: 10.1115/1.4023168 History: Received July 11, 2012; Revised December 01, 2012

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.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


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
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. [CrossRef]
Saad, Y., 2011, Numerical Methods for Large Eigenvalue Problems, 2nd ed., Manchester University Press, Manchester, UK.
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. [CrossRef]
Sorensen, D. C., 2002, “Numerical Methods for Large Eigenvalue Problems,” Acta Numerica, 11, pp. 519–584. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Ipsen, I. C. F., 1997, “Computing an Eigenvector With Inverse Iteration,” SIAM Rev., 39(2), pp. 254–291. [CrossRef]
Jang, H.-J., 2001, “Preconditioned Conjugate Gradient Method for Large Generalized Eigenproblems,” Trends Math. Inf. Center Math. Sci., 4(2), pp. 103–109.
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. [CrossRef]
Wright, J., and Nocedal, S., 2006, Numerical Optimization, Springer Science + Business Media, New York.
Yang, H., 1993, “Conjugate Gradient Methods for the Rayleigh Quotient Minimization of Generalized Eigenvalue Problems,” Computing, 51(1), pp. 79–94. [CrossRef]
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. [CrossRef]
Karabassi, E. A., Papaioannou, G., and Theoharis, T., 1999, “A Fast Depth-Buffer-Based Voxelization Algorithm,” J. Graphics Tools, 4(4), pp. 5–10. [CrossRef]
Zienkiewicz, O. C., 2005, The Finite Element Method for Solid and Structural Mechanics, Elsevier, New York.
Taiebat, H. H., and Carter, J. P., 2001, Three-Dimensional Non-Conforming Elements, Centre for Geotechnical Research, The University of Sydney, Sydney, p. R808.
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. [CrossRef]
NVIDIA Corporation, 2008, NVIDIA CUDA: Compute Unified Device Architecture, Programming Guide, NVIDIA Corporation, Santa Clara.
SolidWorks, 2005, “SolidWorks,” www.solidworks.com


Grahic Jump Location
Fig. 1

A thin gear housing whose eigenspectrum is desired

Grahic Jump Location
Fig. 2

Complex thin-structures require a fine-mesh

Grahic Jump Location
Fig. 3

Brute-force voxelization of the structure

Grahic Jump Location
Fig. 4

First four modes computed via SolidWorks

Grahic Jump Location
Fig. 5

Nonconforming voxelization of a beam

Grahic Jump Location
Fig. 6

First eigenmode of a connecting-rod at 529.6 Hz

Grahic Jump Location
Fig. 7

(a) A knuckle component, (b) conforming mesh, and (c) voxel-mesh

Grahic Jump Location
Fig. 8

The first five modes computed via SolidWorks

Grahic Jump Location
Fig. 9

The first five modes computed via proposed method

Grahic Jump Location
Fig. 10

The first mode of the structure in Fig. 1



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In