An Adaptive Multigrid FE-Method Application to Elasto-Plastic Wave Propagation

摘要

The solution of linear systems of equations is the most time-consuming part in large-scale implicit FE computations of wave propagation problems. Traditionally, direct solvers have been used but recent developments of iterative solvers and precondition techniques may impose a change. In particular, preconditioning by multigrid seems to be favorable for finite element (FE) applications since it has a natural interpretation and substantially improves the rate of convergence of the conventional iterative solvers.The multigrid preconditioner uses a sequence of grids on which the fine-grid residual forces are prolongated to coarser grids and computed corrections are interpolated back to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time and at low memory requirements.The proposed adaptive multigrid FE-method combines adaptivity and multigrid solution strategy in a sophisticated manner. The sequence of computational grids is successively refined (adapted) and generated according to the guidance of a posteriori error estimates until the solution fulfils a predefined accuracy specification. In contrast to standard adaptive procedures where rejected grids are deleted, the adaptive multigrid algorithm uses previous solution and generated grids to speed up the solution process.A refinement strategy based on element splitting and introduction of hanging nodes requires special care since the constraint equations of the hanging nodes are incorporated in the system by usage of Lagrange multipliers. The approach leads to indefinite systems and hence a special preconditioner that enforces the constraint equations to be fulfilled while iterating has been developed.The paper presents results using the adaptive multigrid procedure on an elasto-plastic wave propagation problem.