The two-level FETI method for static and dynamic plate problems I: An optimal iterative solver for biharmonic systems. (English) Zbl 0964.74062
Summary: We present a Lagrange multiplier based substructuring method for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems. The proposed method is essentially an extension of the FETI domain decomposition algorithm to fourth-order problems. The main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the precoditioned conjugate projected gradient iterations. This results in a two-level FETI substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures, and grows at most polylogarithmically with the number of elements per substructure. These theoretically proven optimal convergence properties of the new FETI method are numerically demonstrated for several finite element static and transient plate bending problems. The two-level iterative solver presented in this paper is applicable to a large family of biharmonic time-independent as well as time-dependent systems. It is also extendible to shell problems.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74K20 | Plates |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
Keywords:
biharmonic systems; Lagrange multipliers; finite element discretization; plate bending; two-level FETI substructuring method; optimal convergence; two-level iterative solverReferences:
| [1] | Gupta, A.; Kumar, V., Optimally scalable parallel sparse Cholesky factorization, (Bailey, D.; etal., Parallel Processing for Scientific Computing (1995), SIAM), 442-447 · Zbl 0836.65038 |
| [2] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1983), Johns Hopkins: Johns Hopkins Baltimore, Maryland · Zbl 0559.65011 |
| [3] | ANSYS PowerSolver, USACM-Net Digest (Jul. 27, 1995) |
| [4] | Cook, R. D., Concepts and Applications of Finite Element Analysis (1974), John Wiley |
| [5] | Keyes, D. E.; Xu, J., (Proc. Seventh Int. Conf. on Domain Decomposition. Proc. Seventh Int. Conf. on Domain Decomposition, Contemporary Mathematics, 180 (1994)) |
| [6] | Bourgat, J. F.; Glowinski, R.; Le Tallec, P.; Vidrascu, M., Variational formulation and algorithm for trace operator in domain decomposition calculations, (Glowinski, R.; Golub, G. H.; Meurant, G. A.; Periaux, J., First Int. Symp. Domain Decomposition Methods for Partial Differential Equations (1988), SIAM: SIAM Philadelphia) · Zbl 0684.65094 |
| [7] | Morice, P., Transonic computations by a multidomain technique with potential and Euler solvers, (Zienep, J.; Ortel, H., IUTAM Symposium. IUTAM Symposium, Symposium Transsonicum III (1989)) |
| [8] | Farhat, C., A Lagrange multiplier based divide and conquer finite element algorithm, J. Comput. Syst. Engrg., 2, 149-156 (1991) |
| [9] | Farhat, C., A saddle-point principle domain decomposition method for the solution of \(f\) solid mechanics problems, (Keyes, D. E.; Chan, T. F.; Meurant, G. A.; Scroggs, J. S.; Voigt, R. G., Proc. Fifth SIAM Conference on Domain Decomposition Methods for Partial Differential Equations (1991), SIAM), 271-292 · Zbl 0778.73067 |
| [10] | Farhat, C.; Roux, F. X., A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Methods Engrg., 32, 1205-1227 (1991) · Zbl 0758.65075 |
| [11] | Farhat, C.; Roux, F. X., An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems, SIAM J. Sci. Stat. Comput., 13, 379-396 (1992) · Zbl 0746.65086 |
| [12] | Mandel, J., Balancing domain decomposition, Comm. Appl. Numer. Methods, 9, 233-241 (1993) · Zbl 0796.65126 |
| [13] | LeTallec, P.; Mandel, J.; Vidrascu, M., Balancing domain decomposition for plates, (Keyes, D. E.; Xu, J., Proc. Seventh Int. Conf. on Domain Decomposition. Proc. Seventh Int. Conf. on Domain Decomposition, Contemporary Mathematics, 180 (1994)), 15-524 · Zbl 0816.73061 |
| [14] | Farhat, C.; Mandel, J.; Roux, F. X., Optimal convergence properties of the FETI domain decomposition method, Comput. Methods Appl. Mech. Engrg., 115, 367-388 (1994) |
| [15] | Farhat, C.; Roux, F. X., Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2, 1-124 (1994) · Zbl 0805.73062 |
| [16] | Glowinski, R.; Wheeler, M. F., Domain decomposition and mixed finite element methods for elliptic problems, (Glowinski, R.; Golub, G. H.; Meurant, G. A.; Périaux, J., Proc. First Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (1988), SIAM), 144-172 · Zbl 0661.65105 |
| [17] | Roux, F. X., Acceleration of the outer conjugate gradient by reorthogonalization for a domain decomposition method with Lagrange multipliers, (Chan, T. F.; Glowinski, R.; Périaux, J.; Widlund, O., Third Int. Symp. of Domain Decomposition Methods for Partial Differential Equations (1990), SIAM), 314-321 · Zbl 0706.65118 |
| [18] | Mandel, J.; Tezaur, R., On the convergence of a substructuring method with Lagrange multipliers, (UCD/CCM Report 33 (1994), Center for Mathematics, University of Colorado at Denver), Also submitted to Numerische Mathematik · Zbl 0880.65087 |
| [19] | Farhat, C.; Géradin, M., On a component mode synthesis method and its application to incompatible substructures, Comput. Struct., 51, 459-473 (1994) · Zbl 0900.73338 |
| [20] | Farhat, C.; Chen, P. S.; Mandel, J., A scalable Lagrange multiplier based domain decomposition method for implicit tine-dependent problems, Int. J. Numer. Methods Engrg., 38, 3831-3854 (1995) · Zbl 0844.73077 |
| [21] | Farhat, C.; Rixen, D., A new coarsening operator for the optimal preconditioning of the dual and primal domain decomposition methods: application to problems with severe coefficient jumps, (Proc. Copper Mountain Conference on Multigrid Methods (1995), Copper Mountain: Copper Mountain Colorado), April 3-7 |
| [22] | Farhat, C.; Crivelli, L.; Roux, F. X., Extending substructure based iterative solvers to multiple load and repeated analyses, Comput. Methods Appl. Mech. Engrg., 117, 195-209 (1994) · Zbl 0851.73059 |
| [23] | Farhat, C.; Chen, P. S., Tailoring domain decomposition methods for efficient parallel coarse grid solution and for systems with many right hand sides, Contemp. Math., 180, 401-406 (1994) · Zbl 0817.65123 |
| [24] | Farhat, C., Lecture notes on recent advances in iterative algorithms for solving systems and eigenvalue problems (March 22-24, 1994), University of Leuven: University of Leuven Belgium |
| [25] | J. Mandel, R. Tezaur and C. Farhat, An optimal Lagrange multiplier based domain decomposition method for plate bending problems, SIAM J. Sci. Stat. Comput., submitted for publication.; J. Mandel, R. Tezaur and C. Farhat, An optimal Lagrange multiplier based domain decomposition method for plate bending problems, SIAM J. Sci. Stat. Comput., submitted for publication. · Zbl 0956.74059 |
| [26] | Zhang, X., Domain decomposition algorithms for the biharmonic Dirichlet problem, (Chan, T. F.; Keyes, D. E.; Meurant, G. A.; Scroggs, J. S., Fifth Int. Symp. on Domain Decomposition Methods for Partial Differential Equations (1992), SIAM), 119-126 · Zbl 0770.65095 |
| [27] | S. Brenner, A two-level additive Schwarz preconditioner for nonconforming plate elements, Numer. Math., in press.; S. Brenner, A two-level additive Schwarz preconditioner for nonconforming plate elements, Numer. Math., in press. · Zbl 0855.73071 |
| [28] | (Gill, P. E.; Murray, W., Numerical Methods for Constrained Optimization (1974), Academic Press: Academic Press London), 132-135 |
| [29] | Farhat, C.; Crivelli, L.; Roux, F. X., A transient FETI methodology for large-scale parallel implicit computations in structural mechanics, Int. J. Numer. Methods Engrg., 37, 1945-1975 (1994) · Zbl 0824.73067 |
| [30] | Timoshenko, S. P.; Woinowsky-Krieger, S., Theory of Plates and Shells (1959), McGraw-Hill: McGraw-Hill New York · Zbl 0114.40801 |
| [31] | Batoz, J. L., An explicit formulation for an efficient trinagular plate bending element, Int. J. Numer. Methods Engrg., 18, 1077-1081 (1982) · Zbl 0487.73087 |
| [32] | Farhat, C.; Chen, P. S.; Roux, F. X., Comput. Methods Appl. Mech. Engrg., 155, 153-179 (1998) · Zbl 1040.74513 |
| [33] | Clough, R. W.; Tocher, J. L., Finite element stiffness matrices for analysis of plate bending, (Proc. 1965 Conf. Matrix Methods Struct. Mech.. Proc. 1965 Conf. Matrix Methods Struct. Mech., Wright-Patterson AFB, Ohio. Proc. 1965 Conf. Matrix Methods Struct. Mech.. Proc. 1965 Conf. Matrix Methods Struct. Mech., Wright-Patterson AFB, Ohio, AFFDL-TR-66-80 (1966)), 515-546 |
| [34] | Hughes, T. J.R., Analysis of transient algorithms with particular reference to stability behavior, (Belytschko, T.; Hughes, T. J.R., Computational Methods for Transient Analysis (1983)), 67-155 · Zbl 0547.73070 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
