A Quasi-Conforming Embedded Reproducing Kernel Particle Method for Heterogeneous Materials

TL;DR

This paper introduces a novel meshfree quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials, improving accuracy and stability without conformal meshing or penalty parameters.

Contribution

The paper develops a new QCE-RKPM that uses quasi-conforming quadtree subdivision and variationally consistent correction to enhance accuracy and convergence in heterogeneous material modeling.

Findings

Achieves optimal convergence rates with VC correction.

Avoids conformal meshing and penalty parameters.

Demonstrates improved stability and accuracy in examples.

Abstract

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.