摘要
The material point method (MPM) suffers from poor accuracy and suboptimal convergence rates compared to other numerical methods due to the under-integration of the weak form; the locations of material points with respect to the background grid are suboptimal in performing numerical quadrature. Although this approach enables the MPM to model large deformation efficiently, it also results in the loss of Galerkin exactness in the variational equation and possible stress oscillation due to the cell-crossing instability. This paper introduces a novel MPM formulation that employs the reproducing kernel approximation to overcome the cell-crossing instability due to the higher-order continuity employed. The reproducing kernel method also ensures completeness in the approximation. In addition, this paper implements a variationally consistent material point integration scheme into the MPM framework to address the issue of Galerkin exactness, which is shown to recover theoretical convergence and increase the robustness of the formulation. Numerical examples demonstrate that the proposed method recovers optimal accuracy and stability compared to the conventional approaches and removes spurious pressure oscillation. The F-bar stabilization method of overcoming pressure instability is then coupled with the presented formulation to demonstrate its ability to accurately model incompressible materials.