Adaptive finite elements for viscoelastic deformation problems

Abstract

This thesis is concerned with the theoretical and computational aspects of generating solutions to problems involving materials with fading memory, known as viscoelastic materials. Viscoelastic materials can be loosely described as those whose current stress configuration depends on their recent past. Viscoelastic constitutive laws for stress typically take the form of a sum of an instantaneous response term and an integral over their past responses. Such laws are called hereditary integral constitutive laws.

The main purpose of this study is to analyse adaptive finite element algorithms for the numerical solution of the quasistatic equations governing the small displacement of a viscoelastic body subjected to prescribed body forces and tractions. Such algorithms for the hereditary integral formulation have appeared in the literature. However the approach here is to consider an equivalent formulation based on the introduction of a set of unobservable interval variables. In the linear viscoelastic case we exploit the structure of the quasistatic problem to remove the displacement from the equations governing the internal variables. This results in an elliptic problem with right hand side dependent on the internal variables, and a separate independent system of ordinary differential equations in a Hilbert space.

We consider a continuous in space and time Galerkin finite element approximation to the reformulated problem for which we derive optimal order a priori error estimates. We then apply the techniques of the theory of adaptive finite element methods for elliptic boundary value problems and ordinary differential equations, deriving reliable and efficient a posteriori error estimates and detailing adaptive algorithms. We consider the idea of splitting the error into space and time portions and present results regarding a splitting for space time projections. The ideas for splitting the error in projections is applied to the finite element approximation and a further set of a posteriori error estimates derived. Numerical studies confirm the theoretical properties of all of the estimators and we show how they can be used to drive adaptive in space and time solution algorithms.

We consider the extension of our results for the linear case to the constitutively nonlinear case. A model problem is formulated and the general techniques for dealing with a posterior error estimation for nonlinear space time problems are considered.