The Application of the Multigrid Method to the Finite Element Solution of Solid Mechanics Problems

Abstract

A multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems. The method is applied to some simple two and three dimensional problems to evaluate its strengths and weaknesses. The usefulness of the method is demonstrated by solving some large three dimensional problems of practical interest. When conditions of near incompressibility are encountered, the multigrid method performs poorly due to a combination of a reduction in the smoothing effect of the Gauss-Seidel relaxation method and coarse mesh locking. These problems can be partially cured by using the Jacobi preconditioned conjugate gradient method to smooth the error, and assembling the coarse mesh stiffness matrices using a reduced integration scheme. It is also found that the bending behavior of the linear brick and quadrilateral elements used in this thesis slow the convergence of the multigrid method. This effect also causes nonuniform meshes to yield computation times that are not proportional to the problem size; however, the linear dependence can be recovered by increasing the refinement of the finite element meshes. It is demonstrated that reduced integration techniques become less effective in relieving the stiffness of the coarse mesh for nonuniform meshes as the problem size is increased. The solution of a well-conditioned three dimensional test problem shows that the multigrid algorithm requires far less computational effort than a direct method, and that its performance is comparable to that of the Jacobi preconditioned conjugate gradient method. The usefulness of the multigrid method is demonstrated by applying it to the finite element solution of two solid mechanics problems of engineering interest: the elastostatic state near a three dimensional edge crack, and the relationship between the average offset and the stress drop for two and three dimensional faults in a half-space. The features of the solution to these problems are extensively discussed. It is found that the multigrid method is faster than the Jacobi preconditioned conjugate gradient method when applied to these practical problems. The investigations described in this thesis reveal some interesting features of the performance of the multigrid method when it is applied to the finite element solution of solid mechanics problems. In particular, the storage requirements of the method are linearly proportional to the problem size. The constant of proportionality depends only on the dimension of the problem. The solution times of the multigrid method are found to be linearly proportional to the problem size if uniform meshes are used. However, this is not true for most of the problems that are solved with nonuniform meshes. The constant of proportionality in the relationship between the problem size and the solution time depends on the particular problem under consideration.