Error Estimates for Finite Element Approximations of Viscoelastic Dynamics: The Generalized Maxwell Model

TL;DR

This paper develops and analyzes a space–time finite element method for dynamic linear viscoelasticity modeled by the generalized Maxwell model. By formulating a continuous Galerkin scheme in time and space and deriving an error representation via a discrete dual problem, the authors obtain a priori error estimates in both the end-time energy norm and the end-time $L^2$ norm, with the crucial property that constants depend only linearly on the end time $T$. A key contribution is the elimination of viscoelastic internal variables to yield a reduced system with the same computational complexity as purely elastic dynamics, while maintaining accuracy through a simple reconstruction of viscoelastic states. Numerical results on manufactured and realistic problems, including a radial shaft seal, corroborate the theoretical rates and demonstrate energy conservation and practical applicability of the method.

Abstract

We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in $L^2$ norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

Figures (11)

• Figure 1: Convergence of model problem end time error in energy norm with respect to the mesh size $h$. For the finite element approximation, standard Lagrange $P_3$ elements on tetrahedra were used ($p=3$).
• Figure 2: Convergence of model problem end time error in energy norm with respect to the timestep $k$. For the finite element approximation, standard Lagrange $P_3$ elements on tetrahedra were used ($p=3$).
• Figure 3: Convergence of model problem end time $L^2(\Omega)$-error with respect to the mesh size $h$. For the finite element approximation, standard Lagrange $P_3$ elements on tetrahedra were used ($p=3$).
• Figure 4: Convergence of model problem end time $L^2(\Omega)$-error with respect to the timestep $k$. For the finite element approximation, standard Lagrange $P_3$ elements on tetrahedra were used ($p=3$).
• Figure 5: Initial state of the deformed unit cube in the conservation law experiment with the color indicating the von Mises stress. Displacements on part of the lid are initially prescribed to $\boldsymbol{u}=(0,0,0.20)$ and then released.

Theorems & Definitions (5)

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