The gradient flow structure of an extended Maxwell viscoelastic model and a structure-preserving finite element scheme

TL;DR

This paper explores the mathematical structure of an extended Maxwell viscoelastic model, demonstrating its gradient flow property, and develops a structure-preserving finite element scheme with proven stability, validated through numerical examples of creep and stress relaxation.

Contribution

It introduces a novel gradient flow framework for the extended Maxwell model and proposes a structure-preserving finite element scheme with proven stability.

Findings

The model exhibits a gradient flow property with respect to viscoelastic energy.

The finite element scheme is stable and energy-preserving.

Numerical examples illustrate creep deformation and stress relaxation effects.

Abstract

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.