Estimating errors in quantities of interest in the case of hyperelastic membrane deformation

Abstract

There are many mathematical and engineering methods, problems and experiments which make use of the finite element method. For any given use of the finite element method we get an approximate solution and we usually wish to have some indication of the accuracy in the approximation. In the case when the calculation is done to estimate a quantity of interest the indication of the accuracy is concerned with estimating the difference between the unknown exact value and the finite element approximation. With a means of estimating the error, this can sometimes be used to determine how to improve the accuracy by repeating the computation with a finer mesh. A large part of this thesis is concerned with a set-up of this type with the physical problem described in a weak form and with the error in the estimate of the quantity of interest given in terms of a function which solves a related dual problem. We consider this in the case of modelling the large deformation of thin incompressible isotropic hyperelastic sheets under pressure loading. We assume throughout that the thin sheet can be modelled as a membrane, which gives us a two dimensional description of a three dimensional deformation and this simplifies further to a one space dimensional description in the axisymmetric case when we use cylindrical polar coordinates. In the general case we consider the deformation under quasi-static conditions and in the axisymmetric case we consider both quasi-static conditions and dynamic conditions, which involves the full equations of motion, which gives three different problems. In all the three problems we describe how to get the finite element solution, we describe associated dual problems, we describe how to solve these dual problems and we consider using the dual solutions in error estimation. There is hence a common framework. The details however vary considerably and much of the thesis is in describing each case.