Geometric integrators for multiplicative viscoplasticity: analysis of error accumulation

TL;DR

This paper analyzes geometric integrators for multiplicative viscoplasticity, demonstrating their high accuracy and stability when the inelastic incompressibility constraint is exactly preserved, and highlighting issues when it is violated.

Contribution

It provides a mathematical and numerical analysis of geometric integrators, emphasizing the importance of preserving inelastic incompressibility for stability and convergence in viscoplasticity models.

Findings

Exact preservation of incompressibility ensures exponential stability.

Violating incompressibility leads to loss of stability and poor convergence.

Geometric integrators outperform non-preserving methods in accuracy.

Abstract

The inelastic incompressibility is a typical feature of metal plasticity/viscoplasticity. Over the last decade, there has been a great amount of research related to construction of numerical integration algorithms which exactly preserve this geometric property. In this paper we examine, both numerically and mathematically, the excellent accuracy and convergence characteristics of such geometric integrators. In terms of a classical model of finite viscoplasticity, we illustrate the notion of exponential stability of the exact solution. We show that this property enables the construction of effective and stable numerical algorithms, if incompressibility is exactly satisfied. On the other hand, if the incompressibility constraint is violated, spurious degrees of freedom are introduced. This results in the loss of the exponential stability and a dramatic deterioration of convergence behavior.