A variationally consistent reproducing kernel enhanced material point method and its applications to incompressible materials
Journal
Computational Mechanics
Journal Volume
73
Journal Issue
3
Start Page
599
End Page
618
ISSN
01787675
14320924
Date Issued
2024
Author(s)
Rodriguez, Cameron
Abstract
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.
Subjects
F-bar Stabilization
Incompressible Material
Material Point Method
Reproducing Kernel Approximation
Variationally Consistent Integration
Convergence Of Numerical Methods
Galerkin Methods
Integration
Consistent Integrations
Convergence Rates
F-bar Stabilization
Incompressible Material
Its Applications
Kernel Approximation
Material Point Methods
Reproducing Kernel
Reproducing Kernel Approximation
Variationally Consistent Integration
Stabilization
Publisher
Springer Science and Business Media Deutschland GmbH
Type
journal article