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Graham, E.; Forsyth, P. A.

Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems. (English) Zbl 0939.74065

Int. J. Numer. Methods Eng. 44, No. 1, 77-99 (1999).
Summary: 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 \((\ll 1)\) average element aspect ratios. Incomplete Cholesky \((\text{ILL}^T)\) 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 geometries with up to 5 0000 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.

Cited in 4 Documents

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs

Keywords:

three-dimensional linear elasticity; Manteuffel’s method of perturbing stiffness matrix; convergence of preconditioned system; structural analysis; triquadratic basis functions; hexahedral elements
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