An improved local radial basis function method for solving small-strain elasto-plasticity

TL;DR

This paper introduces a novel hybrid radial basis function finite difference method with stabilization for solving small-strain elasto-plasticity, overcoming challenges posed by non-smooth constitutive relations.

Contribution

It presents the first application of a hybrid RBF-FD method with virtual finite difference stencils to elasto-plasticity, improving stability and convergence over existing approaches.

Findings

The new method successfully solves small-strain von Mises elasto-plasticity.

It outperforms legacy RBF-FD approaches in stability and accuracy.

Stability and convergence are demonstrated on benchmark problems.

Abstract

Strong-form meshless methods received much attention in recent years and are being extensively researched and applied to a wide range of problems in science and engineering. However, the solution of elasto-plastic problems has proven to be elusive because of often non-smooth constitutive relations between stress and strain. The novelty in tackling them is the introduction of virtual finite difference stencils to formulate a hybrid radial basis function generated finite difference (RBF-FD) method, which is used to solve smallstrain von Mises elasto-plasticity for the first time by this original approach. The paper further contrasts the new method to two alternative legacy RBF-FD approaches, which fail when applied to this class of problems. The three approaches differ in the discretization of the divergence operator found in the balance equation that acts on the non-smooth stress field. Additionally, an innovative stabilization technique is employed to stabilize boundary conditions and is shown to be essential for any of the approaches to converge successfully. Approaches are assessed on elastic and elasto-plastic benchmarks where admissible ranges of newly introduced free parameters are studied regarding stability, accuracy, and convergence rate.