The increasing demands of large-scale complex solid mechanics simulations are placing greater emphasis on the challenges associated with the efficient solution of the set of nonlinear equations. Here, the solution of large systems of equations with material and geometric nonlinearities in a parallel framework is addressed.
Instead of applying widely used Newton- or Newton-Krylov type methods that involve the derivation of a stiffness matrix and a sequence of linear solves, the presented work details the implementation of a (nearly) matrix-free nonlinear algebraic multigrid algorithm applied to solve this set of nonlinear equations.
As a basic iterative method, a nonlinear conjugate gradient algorithm (nlnCG) using the Polak-Ribiere formula and a secant method for the stepsize is applied. It has the advantage of being a completely matrix-free method eliminating the need to form a stiffness matrix.
Preconditioned nonlinear CG is then applied as a smoother/coarse solver in a classical full approximation scheme (FAS) nonlinear multigrid cycle. Transfer operators are constructed by an aggregation approach operating on the graph of the fine level problem. The preconditioners to the nlnCG on all levels are obtained by multicolor finite differencing and are chosen to be either simple Jacobi or a direct solve on the coarsest level.
For Jacobi-preconditioned nonlinear CG, only the main diagonal of a Jacobian matrix needs to be formed involving a distance-1 graph coloring algorithm and an inexpensive modified colored finite difference scheme.
The algorithm is implemented within Sandia National Laboratories' freely available parallel 'Trilinos' linear algebra framework and makes use of it's smoothed aggregation multigrid library 'ML' and it's nonlinear solver library 'NOX'. The outline of the algorithm and implementation are given together with examples demonstrating the advantages of this new approach. Several variants of the algorithm will be discussed and compared.