Anisotropic Local Maximum-Entropy Shape Functions for Strain Localization in MPM
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The Material Point Method (MPM) is well suited for large-deformation solid mechanics problems, but its accuracy can degrade in the presence of strain localization due to cell-crossing errors, mesh dependency, and tensile instabilities. These issues become particularly severe when modeling failure in elasto-plastic materials, where narrow shear bands dominate the response. This work investigates the use of Local Maximum-Entropy (LME) shape functions within MPM as a robust meshfree alternative to conventional interpolation schemes. While LME-MPM has shown improved accuracy for elastic problems, its performance in localized elasto-plastic deformation remains largely unexplored. To address this, we propose an anisotropic extension of LME, in which the locality parameter is defined by a deformation-dependent tensor. This anisotropic support adapts to the evolving strain field, enhancing resolution along shear bands while reducing spurious interactions across them. Numerical examples demonstrate improved capture of strain localization, reduced mesh sensitivity, and enhanced robustness in failure simulations, making the proposed formulation a promising tool for modeling localized deformation in geomaterials and other elasto-plastic solids.