Multigrid Preconditioner for Unstructured Nonlinear 3D FE Models
Publication: Journal of Engineering Mechanics
Volume 125, Issue 2
Abstract
Multigrid and multigrid-preconditioned conjugate-gradient solution techniques applicable for unstructured 3D finite-element models that may involve sharp discontinuities in material properties, multiple element types, and contact nonlinearities are developed. Their development is driven by the desire to efficiently solve models of rigid pavement systems that require explicit modeling of spatially varying and discontinuous material properties, bending elements meshed with solid elements, and separation between the slab and subgrade. General definitions for restriction and interpolation operators applicable to models composed of multiple, displacement-based isoparametric finite-element types are proposed. Related operations are used to generate coarse mesh element properties at integration points, allowing coarse-level coefficient matrices to be computed by a simple assembly of element stiffness matrices. The proposed strategy is shown to be effective on problems involving spatially varying material properties, even in the presence of large variations within coarse mesh elements. Techniques for solving problems with nodal contact nonlinearities using the proposed multigrid methods are also described. The performance of the multigrid methods is assessed for model problems incorporating irregular meshes and spatially varying material properties, and for a model of two rigid pavement slabs subjected to thermal and axle loading that incorporates nodal contact conditions and both solid and bending elements.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Ashby, S., and Falgout, R. ( 1996). “A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations.” Nuclear Sci. and Engrg., 124, 145–159.
2.
Bathe, K.-J. ( 1996). Finite element procedures. Prentice-Hall, Englewood Cliffs, N.J.
3.
Brandt, A. ( 1977). Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation, 31(138), 333–390.
4.
Channakeshava, C., Barzegar, F., and Voyiadjis, G. (1993). “Nonlinear FE analysis of plain concrete pavement with doweled joints.”J. Transp. Engrg., 119(5), 763–781.
5.
Darter, M., Hall, K., and Kuo, C.-M. ( 1995). “Support under Portland cement concrete pavements.” NCHRP report No. 372 National Cooperative Highway Research Program, Washington, D.C.
6.
Davids, W., and Turkiyyah, G. (1997). “Development of embedded bending member to model dowel action.”J. Struct. Engrg., 123(10), 1312–1320.
7.
Demmel J. W., Gilbert, J. R., and Li, X. S. ( 1997). SuperLU users' guide. Comp. Sci. Div., University of California, Berkeley, Berkeley, Calif.
8.
Dracopoulos, M., and Crisfield, M. ( 1995). “Coarse/fine mesh preconditioners for the iterative solution of finite element problems.” Int. J. Numer. Methods in Engrg., 38, 3297–3313.
9.
Farhat, C., and Sobh, N. ( 1989). “A coarse/fine mesh preconditioner for very ill-conditioned finite element problems.” Int. J. Numer. Methods in Engrg., 28, 1715–1723.
10.
Fish, J., Pan, L., Belsky, V., and Gomaa, S. ( 1996). “Unstructured multigrid method for shells.” Int. J. Numer. Methods in Engrg., 39, 1181–1197.
11.
Goodman, J. E., and O'Rourke, J. ( 1997). Handbook of discrete and computational geometry. CRC, Boca Raton, Fla.
12.
Jesperson, D. ( 1984). “Multigrid methods for partial differential equations.” MAA studies in mathematics, G. Golub, ed., Vol. 24, Mathematical Assoc. of America, Washington, D.C., 270–317.
13.
Kcvara, M. ( 1993). “An adaptive multigrid technique for three-dimensional elasticity.” Int. J. Numer. Methods in Engrg., 36, 1703–1716.
14.
Kuo, C.-M., Hall, K., and Darter, M. ( 1996). “Three-dimensional finite element model for analysis of concrete pavement support.” Transp. Res. Rec. 1505, TRB, National Research Council, Washington, D.C., 119–127.
15.
Li, X. S. ( 1996). Sparse Gaussian elimination on high performance computers. Comp. Sci. Div., University of California, Berkeley, Berkeley, Calif.
16.
Löhner, R., and Morgan, K. ( 1987). “An unstructured multigrid method for elliptic problems.” Int. J. Numer. Methods in Engrg., 24, 101–115.
17.
Meza, J., and Tuminaro, R. ( 1996). “A multigrid preconditioner for the semiconductor equations.” SIAM J., 17(1), 118–132.
18.
Mitchell, S., and Vavasis, S. ( 1992). “Quality mesh generation in three dimensions.” Proc., ACM Computational Geometry Conf., Assoc. for Computing Machinery, New York, 212–221.
19.
Muttin, F., and Chenot, J.-L. ( 1995). “On a conjugate-gradient two-grid method for three-dimensional elasticity.” Engrg. Computations, 12, 3–20.
20.
Oosterlee, C., and Washio, Y. ( 1998). An evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed equations. SIAM J., 19(1), 87–110.
21.
Parsons, I., and Hall, J. ( 1990a). “The multigrid method in solid mechanics: Part I—algorithm description and behavior.” Int. J. Numer. Methods in Engrg., 29, 719–737.
22.
Parsons, I., and Hall, J. ( 1990b). “The multigrid method in solid mechanics: Part II—practical applications.” Int. J. Numer. Methods in Engrg., 29, 739–753.
23.
Rider, W., and Knoll, D. ( 1997). “Solving nonlinear heat conduction problems with multigrid preconditioned Newton-Krylov methods.” Tech. Rep. LA-UR-97-2929, Los Alamos National Lab, Los Alamos, N.M.
24.
Saad, Y. ( 1996). Iterative methods for sparse linear systems. PWS Publishing, Boston.
25.
Zienkiewicz, O., and Taylor, R. ( 1994). The finite element method—volume I. McGraw-Hill, New York.
26.
Zienkiewicz, O., Vilotte, J., Toyoshima, S., and Nakazawa, S. ( 1985). “Iterative methods for constrained and mixed approximation. An Inexpensive Improvement of FEM Performance.” Computational Methods of Appl. Mech. and Engrg., 51, 3–29.
Information & Authors
Information
Published In
History
Published online: Feb 1, 1999
Published in print: Feb 1999
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.