Nonlinear algebraic multigrid for constrained solid mechanics problems using Trilinos

TL;DR: This paper focuses on a variational nonlinear multigrid approach that adopts the smoothed aggregation algebraic multi grid method to generate a hierachy of coarse grids in a purely algebraic manner.

Abstract: The application of the finite element method to nonlinear solid mechanics problems results in the neccessity to repeatedly solve a lar ge 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 multi grid 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 dif ferencing 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.