A new incremental method of computing the limit load in deformation plasticity models

TL;DR

This paper introduces a novel incremental method for numerically estimating the limit load in deformation plasticity models, utilizing an inverse mapping approach to provide guaranteed bounds and convergence guarantees.

Contribution

The paper presents a new incremental procedure based on inverse mapping, offering guaranteed lower bounds for the limit load and convergence analysis as mesh size decreases.

Findings

The method provides guaranteed lower bounds for the limit load.

Numerical tests demonstrate the method's practical efficiency.

The approach converges to the true limit load as mesh refinement increases.

Abstract

The aim of this paper is to introduce a new incremental procedure that can be used for numerical evaluation of the limit load. Existing incremental type methods are based on parametrization of the energy by the loading parameter $\zeta\in[0,\zeta_{lim})$, where $\zeta_{lim}$ is generally unknown. In the new method, the incremental procedure is operated in terms of an inverse mapping and the respective parameter $\alpha$ is changing in the interval $(0,+\infty)$. Theoretically, in each step of this algorithm, we obtain a guaranteed lower bound of $\zeta_{lim}$. Reduction of the problem to a finite element subspace associated with a mesh $\mathcal T_h$ generates computable bound $\zeta_{lim,h}$. Under certain assumptions, we prove that $\zeta_{lim,h}$ tends to $\zeta_{lim}$ as $h\rightarrow0_+$. Numerical tests confirm practical efficiency of the suggested method.