Geometric integrators for multiplicative viscoplasticity: Analysis of error accumulation

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 property. In this paper we examine, both numerically and mathematically, the excellent accuracy and convergence characteristics of such integrators. In order to simplify the considerations, we consider strain-driven processes without hardening effects.In terms of a classical model of multiplicative 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 the incompressibility is exactly satisfied. On the other hand, if the incompressibility constraint is violated, spurious degrees of freedom are introduced. In general, this results in the loss of the exponential stability and a dramatic deterioration of convergence of numerical methods.