Evaluation of a Geometric and an Algebraic Multigrid Method on 3D Finite Element Problems in Solid Mechanics

Mark Adams

531 Soda Hall University of California, Berkeley Berkeley CA 94720


Abstract

The availability of large high performance computers is providing scientists and engineers with the opportunity to simulate a variety of complex physical systems with ever more accuracy and thereby exploit the advantages of computer simulations over laboratory experiments. The finite element method is widely used for these simulations. The finite element method requires that one or several linearized system of sparse unstructured algebraic equations (the stiffness matrix) be solved at each time step when implicit time integration is used. The linear system solves become the computational bottleneck (once the simulation has been setup and before the results are interpreted) as the scale of problems increases. Direct solution methods have been (and still are) popular as they are dependable, though the asymptotic complexity of direct methods, or any fixed level method, is high in comparison to optimal iterative methods (ie, multigrid). Multigrid has been a popular method for solving finite element (and finite difference) systems on regular grids for over 30 years; the application of multigrid to unstructured problems is, however, not well understood and has been an active area of research in recent years. We are aware of two categories of promising unstructured multigrid methods: 1) ``geometric'' methods that use standard finite element coarse grid function spaces (and hence have a concrete geometric interpretation), and 2) rigid body mode coarse grid space (``algebraic'') methods which we call algebraic as the coarse grid function spaces can in principle be deduced from the stiffness matrix or its component element stiffness matrices and the method that we use uses the stiffness matrix to modify the initial rigid body mode coarse grid spaces. This paper evaluates the effectiveness of two promising unstructured multigrid methods (one geometric and one algebraic) on several challenging unstructured problems in solid mechanics with up to 56 million degrees of freedom.