A Matrix-Free Newton–Krylov Parallel Implicit Implementation of the Absolute Nodal Coordinate Formulation

Abstract

This paper sets out to demonstrate three things: (i) implicit integration with absolute nodal coordinate formulation (ANCF) is effective in handling very stiff systems when an accurate computation of the sensitivity matrix is part of the solution sequence, (ii) parallel computing can provide a vehicle for ANCF to tackle very large kinematically constrained problems with millions of degrees of freedom and produce results in a matter of seconds, and (iii) large systems of equations associated with implicit integration can be solved in parallel by relying on an iterative approach that avoids costly matrix factorizations, which would be prohibitively expensive and memory intensive. For (iii), the approach adopted relies on a Krylov–subspace method that is invoked in the Newton stage at each time step of the numerical solution process. The proposed approach is validated against a commercial package and several simple systems for which analytical solutions are available. A set of numerical experiments demonstrates the scaling of the parallel solution method and provides insights in relation to the size of ANCF problems that are tractable using graphics processing unit (GPU) parallel computing and implicit numerical integration.