Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

TL;DR

The paper addresses small-strain elasto-plastic deformation under plane strain using a meshless, strong-form RBF-FD discretization with Picard iteration. It couples an elastic Navier–Cauchy solve with a local return-mapping correction for plasticity, solved incrementally on scattered nodes without meshing; Polyharmonic Splines augmented with monomials yield stable operator weights. Validation on a thick-walled cylinder under various loading regimes shows strong agreement with analytical solutions and Abaqus FEM, including elastic, perfectly plastic, and linearly hardening cases, and demonstrates robustness to irregular geometries. The work highlights meshless methods as a competitive alternative for complex geometries, offering straightforward implementation, potential hp/adaptivity, and scalable performance in elasto-plastic simulations.

Abstract

This paper deals with a numerical analysis of plastic deformation under various conditions, utilizing Radial Basis Function (RBF) approximation. The focus is on the elasto-plastic von Mises problem under plane-strain assumption. Elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress surpasses the yield stress, corrections are applied locally through a return mapping algorithm. The non-linear deformation problem in the plastic domain is solved using the Picard iteration. The solutions for the Navier-Cauchy equation are computed using the Radial Basis Function-Generated Finite Differences (RBF-FD) meshless method using only scattered nodes in a strong form. Verification of the method is performed through the analysis of an internally pressurized thick-walled cylinder subjected to varying loading conditions. These conditions induce states of elastic expansion, perfectly-plastic yielding, and plastic yielding with linear hardening. The results are benchmarked against analytical solutions and traditional Finite Element Method (FEM) solutions. The paper also showcases the robustness of this approach by solving case of thick-walled cylinder with cut-outs. The results affirm that the RBF-FD method produces results comparable to those obtained through FEM, while offering substantial benefits in managing complex geometries without the necessity for conventional meshing, along with other benefits of meshless methods.

Figures (15)

• Figure 1: Sketch of stress and strain relation as a result of an uniaxial tension test.
• Figure 2: Schematic representation of a pressurized thick-walled pipe expansion.
• Figure 3: Assessment of the present solution of the purely elastic case. The scattered plot is displaying the relative error in all nodes.
• Figure 4: An example of scattered nodes using two different random seeds (left), spatial distribution of the displacement magnitude error (middle) and convergence of the $\ell^2\mathrm{-norm}$ of the displacement vector for the linear-elastic case (right).
• Figure 5: Sparsity of final matrices (top row) and spectrum (bottom row) for three different discretization qualities.