Interpolation-based reproducing kernel particle method

TL;DR

This paper introduces an interpolation-based implementation of the reproducing kernel particle method (RKPM) within open-source finite element software, achieving high accuracy and efficiency for complex PDEs and multi-material problems.

Contribution

It extends interpolation-based methods to RKPM, enabling easier implementation and high-order accuracy in solving PDEs within FEM software.

Findings

Error convergence rates match classic RKPM with high-order quadrature.

Successfully applied to higher-order PDEs like the biharmonic problem.

Achieves similar accuracy with reduced computational cost compared to traditional methods.

Abstract

Meshfree methods, including the reproducing kernel particle method (RKPM), have been widely used within the computational mechanics community to model physical phenomena in materials undergoing large deformations or extreme topology changes. RKPM shape functions and their derivatives cannot be accurately integrated with the Gauss-quadrature methods widely employed for the finite element method (FEM) and typically require sophisticated nodal integration techniques, preventing them from easily being implemented in existing FEM software. Interpolation-based methods have been developed to address similar problems with isogeometric and immersed boundary methods, allowing these techniques to be implemented within open-source finite element software. With interpolation-based methods, background basis functions are represented as linear combinations of Lagrange polynomial foreground basis functions defined upon a boundary-conforming foreground mesh. This work extends the applications of interpolation-based methods to implement RKPM within open-source finite element software. Interpolation-based RKPM is applied to several PDEs, and error convergence rates are equivalent to classic RKPM integrated using high-order Gauss-quadrature schemes. The interpolation-based method is able to exploit the continuity of the RKPM basis to solve higher-order PDEs, demonstrated through the biharmonic problem. The method is extended to multi-material problems through Heaviside enrichment schemes, using local foreground refinement to reduce geometric integration error and achieve high-order accuracy. The computational cost of interpolation-based RKPM is similar to the smoothed gradient nodal integration schemes, offering significant savings over Gauss-quadrature-based meshfree methods while enabling easy implementation within existing finite element software.