Nonlinear algebraic multigrid for constrained solid mechanics problems using Trilinos

The application of the finite element method to nonlinear solid mechanics problems results in the neccessity to repeatedly solve a large nonlinear set of equations. In this paper we limit ourself to problems arising in constrained solid mechanics problems. It is common to apply some variant of Newton’s method or a Newton– Krylov method to such problems. Often, an analytic Jacobian matrix is formed and used in the above mentioned methods. However, if no analytic Jacobian is given, Newton methods might not be the method of choice. Here, we focus on a variational nonlinear multigrid approach that adopts the smoothed aggregation algebraic multigrid method to generate a hierachy of coarse grids in a purely algebraic manner. We use preconditioned nonlinear conjugent gradient methods and/or quasi–Newton methods as nonlinear smoothers on fine and coarse grids. In addition we discuss the possibility to augment this basic algorithm with an automatically generated Jacobian by applying a block colored finite differencing scheme. After outlining the fundamental algorithms we give some examples and provide documentation for the parallel implementation of the described method within the Trilinos framework. Acknowledgment This work was partially funded by the Department of Energy Office of Science MICS program at Sandia National Laboratory. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United Stated Department of Energy’s National Nuclear Security Administration under contract DE– AC04–94AL85000. M.W. Gee, R.S. Tuminaro 2 (page intentionally left blank) M.W. Gee, R.S. Tuminaro 3