Iterative methods for p-version finite elements: preconditioning thin solids

Abstract We present new preconditioning strategy for thin 3D p -version elements used to model plate and shell structures as well as the microstructure of composite materials. The strategy is incorporated in an adaptive preconditioner suitable for large real-world problems. The method is shown to perform well for several large real world problems, including a model of the skin of an aircraft with over 1.6 million degrees of freedom, and a highly accurate model of a 28 ply laminated fiber plate.

[1]  Olof B. Widlund,et al.  A Polylogarithmic Bound for an Iterative Substructuring Method for Spectral Elements in Three Dimensions , 1996 .

[2]  Jan Mandel,et al.  Iterative solvers by substructuring for the p -version finite element method , 1990 .

[3]  Manolis Papadrakakis.  Solving Large-scale Problems in Mechanics , 1993 .

[4]  Olof B. Widlund,et al.  Domain Decomposition Algorithms with Small Overlap , 1992, SIAM J. Sci. Comput..

[5]  M. Papadrakakis.  Solving Large-Scale Problems in Mechanics: The Development and Application of Computational Solution Methods , 1993 .

[6]  Yusheng Feng,et al.  Parallel Domain Decomposition Solver for Adaptive hp Finite Element Methods , 1997 .

[7]  Jan Mandel,et al.  Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions , 1990 .

[8]  Ivo Babuška,et al.  The Problem of Selecting the Shape Functions for a p-Type Finite Element , 1989 .

[9]  Barry F. Smith.  A domain decomposition algorithm for elliptic problems in three dimensions , 1990 .

[10]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[11]  Harvey J. Greenberg,et al.  The use of the optimal partition in a linear programming solution for postoptimal analysis , 1994, Oper. Res. Lett..

[12]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[13]  Philippe G. Ciarlet,et al.  Plates And Junctions In Elastic Multi-Structures , 1990 .

[14]  Petter E. Bjørstad,et al.  On the spectra of sums of orthogonal projections with applications to parallel computing , 1991 .

[15]  Some Schwarz Algorithms for the P-Version Finite Element Method , 1992 .

[16]  Anthony T. Patera,et al.  Spectral element multigrid. I. Formulation and numerical results , 1987 .

[17]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[19]  Jacques Periaux,et al.  On Domain Decomposition Methods , 1988 .

[20]  I. Babuska,et al.  Efficient preconditioning for the p -version finite element method in two dimensions , 1991 .

[21]  Jan Mandel,et al.  An iterative solver for p-version finite elements in three dimensions , 1994 .

[22]  William F. Mitchell,et al.  Optimal Multilevel Iterative Methods for Adaptive Grids , 1992, SIAM J. Sci. Comput..

[23]  P. Carnevali,et al.  Adaptive solution strategy for solving large systems of p‐type finite element equations , 1992 .

[24]  S. Payne.  The fundamental theorem ofq-clan geometry , 1996 .

[25]  Gene H. Golub,et al.  Matrix computations , 1983 .

[26]  Vijay Sonnad,et al.  Multilevel solution method for the p-version of finite elements , 1989 .

[27]  Jan Mandel,et al.  Domain decomposition preconditioning for p-version finite elements with high aspect rations , 1991 .

[28]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[29]  T. Manteuffel,et al.  FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS : PART II , 1994 .

[30]  J. Pitkäranta.  The problem of membrane locking in finite element analysis of cylindrical shells , 1992 .

[31]  Vijay Sonnad,et al.  A comparison of preconditioned iterative techniques using rapid operator application against direct solution methods , 1991 .