Implicit Time Integration in the Generalized Interpolation Material Point Method for Finite Deformation Hyperelasticity

Abstract

In this article, a novel implicit time-integration scheme for the generalized interpolation material point method (GIMP) is discussed. The implicit algorithm discussed in this article is an extension of the algorithm introduced for the material point method (MPM) by Sulsky and Kaul [1]. In this article, the numerical solution employs the Newton-Krylov technique coupled with the conjugate gradient (CG) method to solve the governing equations in a robust, matrix-free fashion. In implicit MPM, the solution procedure needs to be re-ordered in terms of momentum if the algorithm has to solve large deformation problems [1]. This is a direct consequence of the cell-crossing issues in MPM. The solution procedure in this case uses the generalized minimum residual method to solve the resulting non-symmetric equations. However, GIMP is specifically formulated to overcome these particle cell-crossing issues in MPM; hence, a specialized procedure for large deformation is not required. The proposed implicit algorithm for GIMP is benchmarked for hyperelasticity problems by comparison with finite element analysis (FEA) and exact solution.