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FEM/BEM NOTES

Professor Peter Hunter
p.hunter@auckland.ac.nz

Associate Professor Andrew Pullan
a.pullan@auckland.ac.nz

Department of Engineering Science The University of Auckland New Zealand February 21, 2001

c Copyright 1997 : Department of Engineering Science

Contents
1 Finite Element Basis Functions 1.1 Representing a One-Dimensional Field . 1.2 Linear Basis Functions . .. . . . . . . 1.3 Basis Functions as Weighting Functions 1.4 Quadratic Basis Functions . . . . . . . 1.5 Two- and Three-Dimensional Elements 1.6 Higher Order Continuity . . . . . . . . 1.7 Triangular Elements . . . . . . . . . . . 1.8 Curvilinear Coordinate Systems . . . . 1.9 CMISS Examples . . . . . . . . . . . . 1 1 2 4 7 7 10 14 16 19 21 21 22 22 23 24 25 27 27 27 28 29 30 32 34 35 37 40 41 4141 41 43

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2 Steady-State Heat Conduction 2.1 One-Dimensional Steady-State Heat Conduction . . . . . 2.1.1 Integral equation . . . . . . . . . . . . . . . . . . 2.1.2 Integration by parts . . . . . . . . . . . . . . . . . 2.1.3 Finite element approximation . . . . . . . . . . . 2.1.4 Element integrals . . . . . .. . . . . . . . . . . . 2.1.5 Assembly . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Boundary conditions . . . . . . . . . . . . . . . . 2.1.7 Solution . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 2.2 An x-Dependent Source Term . . . . . . . . . . . . . . . 2.3 The Galerkin Weight Function Revisited . . . . . . . . . . 2.4 Two andThree-Dimensional Steady-State Heat Conduction 2.5 Basis Functions - Element Discretisation . . . . . . . . . . 2.6 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Assemble Global Equations . . . . . . . . . . . . . . . . . 2.8 Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . 2.9 CMISS Examples . . . . . . . . . . . . . . . . . . . . . . 3 The Boundary Element Method3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 The Dirac-Delta Function and Fundamental Solutions 3.2.1 Dirac-Delta function . . . . . . . . . . . . . 3.2.2 Fundamental solutions . . . . . . . . . . . .

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ii

CONTENTS

3.3 3.4 3.5 3.6 3.7 3.8

3.9 3.10 3.11 3.12

3.13 3.14 3.15 3.16

3.17 4

The Two-Dimensional Boundary Element Method . . . . . . . . . Numerical Solution Procedures for the Boundary Integral Equation Numerical Evaluation of Coefficient Integrals . . . . . . . . . . . The Three-DimensionalBoundary Element Method . . . . . . . . A Comparison of the FE and BE Methods . . . . . . . . . . . . . More on Numerical Integration . . . . . . . . . . . . . . . . . . . 3.8.1 Logarithmic quadrature and other special schemes . . . . 3.8.2 Special solutions . . . . . . . . . . . . . . . . . . . . . . The Boundary Element Method Applied to other Elliptic PDEs . . Solution of Matrix Equations . . ....
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