Adaptive finite elements for viscoelastic deformation problems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.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.EPSRC; Japanese Society for the Promotion of Science(JSPS

Adaptive finite element approximations for elliptic problems using regularized forcing data

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two dimensional space and sub-optimal in three dimensional space. Numerical simulations are provided to confirm our findings.Comment: 28 pages, 6 Figure

Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems

We propose and implement an automatic hp-adaptive refinement algorithm for the Stokes model problem. In this work, the strategy is based on the earlier work done by Dörfler at al. in 2007 for the Poisson problem. Similar to any other adaptivity approach, an a posteriori estimator is needed to control the error in areas with high residuals. We define a family of residual-based estimators Ŋva a € [0; 1] for the hp-adaptive finite element approximation of the exact solution. Moreover, we show the reliability and efficiency of the estimators Ŋva. Finally, numerical examples illustrate the exponential convergence rate of the hp-AFEM in comparison with the h-AFEM. In many applications, such as analysis of fluid flows in our case, we are not interested in computing the solution itself, but instead the aim is finding a good approximation for some functional of interest. In these cases, the idea is to develop some a posteriori error estimates to generate a sequence of h- or hp-adaptive grids that minimize the error in our goal functional with respect to the problem size. In this work, we apply local averaging interpolation operators such as Scott-Zhang and Clément type operators to formulate the dual weight of our proposed goal-oriented error estimator. This idea was recently used in an application to the Poisson problem. We extend those results to saddle-point problems and provide a dual-weighted goal estimator for each cell. The reliability of the goal estimator is proved and numerical examples demonstrate the performance of the locally defined dual-weighted goal-estimator in terms of reliability, efficiency, and convergence. Another important aspect of this research is providing a goal-oriented adaptive finite element method for symmetric second-order linear elliptic problems. We prove that the product of primal and dual estimators, which is a reliable upper bound for the error in the goal functional, decays at the optimal rate. The results reported in the numerical experiments confirm the quasi-optimality behavior of our goal-oriented algorithm