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Preconditioning methods for very ill‐conditioned three‐dimensional linear elasticity problems
Author(s) -
Graham E.,
Forsyth P. A.
Publication year - 1999
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19990110)44:1<77::aid-nme493>3.0.co;2-0
Subject(s) - cholesky decomposition , conjugate gradient method , finite element method , linear system , mathematics , linear elasticity , mathematical optimization , stiffness matrix , condition number , stiffness , elasticity (physics) , system of linear equations , factorization , degrees of freedom (physics and chemistry) , reduction (mathematics) , linear equation , matrix (chemical analysis) , incomplete cholesky factorization , algorithm , mathematical analysis , geometry , structural engineering , engineering , eigenvalues and eigenvectors , materials science , composite material , physics , quantum mechanics
Finite element models of linear elasticity arise in many application areas of structural analysis. Solving the resulting system of equations accounts for a large portion of the total cost for large, three‐dimensional models, for which direct methods can be prohibitively expensive. Preconditioned Conjugate Gradient (PCG) methods are used to solve difficult problems with small (≪1) average element aspect ratios. Incomplete Cholesky (ILL T ) factorizations based on a drop tolerance parameter are used to form the preconditioning matrices. Various new techniques known as reduction techniques are examined. Combinations of these reduction techniques result in highly effective preconditioners for problems with very poor aspect ratios. Standard and hierarchical triquadratic basis functions are used on hexahedral elements, and test problems comprising a variety of geo‐metries with up to 50 000 degrees of freedom are considered. Manteuffel’s method of perturbing the stiffness matrix to ensure positive pivots occur during factorization is used, and its effects on the convergence of the preconditioned system are discussed. Copyright © 1999 John Wiley & Sons, Ltd.