Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems

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) meth- ods are used to solve di cult problems with small (.1) average element aspect ratios. Incomplete Cholesky (ILLT) factorizations based on a drop tolerance parameter are used to form the preconditioning matrices. Var- ious new techniques known as reduction techniques are examined. Combinations of these reduction techniques result in highly e ective 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. Manteu elʹs method of perturbing the sti ness matrix to ensure positive pivots occur during factorization is used, and its e ects on the convergence of the preconditioned system are discussed. Copyright