The Bugfree Blog ;-): Approximation, Finite Elements and Isogeometric Analysis (with sample Matlab implementation)

As the first post, I think it might be good to provide some documents that I have wrote and that might be of help to someone. This is the thesis I wrote for my degree in Computer Engineering, keeping it in my disk is useless, so be it my first post! :-)

Abstract

The thesis begins with a short introduction to elliptic, parabolic and hyperbolic Partial Differential Equations and precedes with a presentation of the Finite Element Method. Some important models for the representation of curves, surfaces and solids are explored before introducing a generalization of the Finite Element Method, namely Isogeometric Analysis. Algorithms for computation and numerical examples are presented.

Content

An efficient way of describing the behaviour of a system (e.g. natural systems) is to use a mathematical model. Among the mathematical models, differential equations (briefly described in Section 1.3) are commonly used when trying to relate functions to their rates of change. Examples of processes described by differential equations are constructions, weather, flow of liquids (e.g. flow of blood in human veins), deformation of solid bodies, heat transfer, chemical reactions, electromagnetism etc... Differential equations can be grouped in Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). ODEs are equations in which the unknown function is a function of a single variable; vice versa in PDEs the unknown function is a function of multiple variables. The manipulation of models is an important task for people to control, predict and understand the processes they describe.

Unfortunately, we are not able to find exact solutions for many of these types of equations, and sometimes we don’t even know whether the solution exists or not or whether it is unique or not. However, approximation methods exist, and one of these is the Finite Element Method (FEM). FEM is currently one of the most efficient way of finding approximations for PDEs for which we’re not able to find an exact solution. The idea of FEM is to divide the domain of the problem in finite elements where a finite set of polynomial basis functions is defined, and which create the basis for the space where the approximated solution is searched. This way, the problem is transformed into a discrete problem, as the solution is expressed through a finite number of unknown parameters. A key concept in this process is the choice of the finite set of polynomials used in the approximation: until recent times, mostly simple shape functions were used, so as to simplify implementation and usage. During the last years, the usage of higher-order shape functions has been reconsidered, due to their superior approximation capabilities. However, employing such more complex shape functions requires a better understanding of the underlying mathematics. Chapter 2 reports a brief discussion of finite elements and of the main concepts of FEM.

A new development of FEM has been recently proposed in [3, 18] (but already proposed before in [15]) and named Isogeometric Analysis (IGA). This can be considered an improvement which generalizes the concepts of FEM through a different definition of the basis functions using CAD basis functions, whose description is reported in Chapter 3.

This new idea is described in Chapter 4. Examples of usage of IGA are given in Chapter 5.