An efficient material point method framework based on the affine matrix

The material point method (MPM) has proven to be an effective numerical approach for simulating extreme events involving large deformations. Although numerous MPM variants have been proposed to enhance the capability of MPM, its application is currently limited by issues of computational efficiency and stability. This study presents an innovative affine matrix-based MPM scheme (AM-MPM) that can substantially promote the computational efficiency of MPM while maintaining exceptional stability. The proposed scheme is featured by its use of the affine matrix as a first-order approximation of the velocity gradient in the subsequent step and the direct computation of associated strain/stress for particles. This approach reduces the information interpolation between particles and grid nodes from two times to a single time, significantly leveraging computational efficiency. The proposed MPM scheme is firstly validated by the quasi-static compaction under self-weight and then applied to various classical problems involving large deformations, including biaxial compression test, column collapse, the impact of two rings, and the twisting of a 3D elastic column. The performance of the proposed scheme in terms of accuracy and efficiency is also discussed in detail.