Finite Element Method

This book offers an in-depth presentation of the finite element method, aimed at engineers, students and researchers in applied sciences. The description of the method is presented in such a way as to be usable in any domain of application. The level of mathematical expertise required is limited to differential and matrix calculus. The various stages necessary for the implementation of the method are clearly identified, with a chapter given over to each one: approximation, construction of the integral forms, matrix organization, solution of the algebraic systems and architecture of programs. The final chapter lays the foundations for a general program, written in Matlab, which can be used to solve problems that are linear or otherwise, stationary or transient, presented in relation to applications stemming from the domains of structural mechanics, fluid mechanics and heat transfer.

Auteur

Gouri Dhatt obtained his D.SC in 1968 from Laval
University on Numerical Modelling. Since 1968 he has been working
as Professor of Engineering at Laval University Quebec, University
of Technology Compiegne and INSA Rouen. He is co-author of various
books on Finite Elements and its applications.

Emmanuel Lefrançois is currently an associate
professor at the University of Technology of Compiègne (UTC).
Areas of expertise concern computer sciences for multiphysics
applications and essentially for fluid-structure interactions.
Teaching areas concern the computer sciences (finite element) and
fluid mechanics.

Gilbert Touzot is Emeritus Professor. Since 1967 he has
been working in computational méchanics at Université du
Québec, Université de Compiègne and National
Institute of Applied Sciences Rouen. He is presently président
of the french national digital University of Technology - UNIT.

Contenu

Introduction 1

0.1 The finite element method 1

0.1.1 General remarks 1

0.1.2 Historical evolution of the method 2

0.1.3 State of the art 3

0.2 Object and organization of the book 3

0.2.1 Teaching the finite element method 3

0.2.2 Objectives of the book 4

0.2.3 Organization of the book 4

0.3 Numerical modeling approach 6

0.3.1 General aspects 6

0.3.2 Physical model 7

0.3.3 Mathematical model 9

0.3.4 Numerical model 10

0.3.5 Computer model 13

Bibliography 16

Conference proceedings 17

Monographs 18

Periodicals 19

Chapter 1. Approximations with finite elements 21

1.0 Introduction 21

1.1 General remarks 21

1.1.1 Nodal approximation 21

1.1.2 Approximations with finite elements 28

1.2 Geometrical definition of the elements 33

1.2.1 Geometrical nodes 33

1.2.2 Rules for the partition of a domain into elements 33

1.2.3 Shapes of some classical elements 35

1.2.4 Reference elements 37

1.2.5 Shapes of some classical reference elements 41

1.2.6 Node and element definition tables 44

1.3 Approximation based on a reference element 45

1.3.1 Expression of the approximate function u(x) 45

1.3.2 Properties of approximate function u(x) 49

1.4 Construction of functions N ( ) and N ( ) 54

1.4.1 General method of construction 54

1.4.2 Algebraic properties of functions N and N 59

1.5 Transformation of derivation operators 61

1.5.1 General remarks 61

1.5.2 First derivatives 62

1.5.3 Second derivatives 65

1.5.4 Singularity of the Jacobian matrix 68

1.6 Computation of functions N, their derivatives and the Jacobian matrix 72

1.6.1 General remarks 72

1.6.2 Explicit forms for N 73

1.7 Approximation errors on an element 75

1.7.1 Notions of approximation errors 75

1.7.2 Error evaluation technique 80

1.7.3 Improving the precision of approximation 83

1.8 Example of application: rainfall problem 89

Bibliography 95

Chapter 2. Various types of elements 97

2.0 Introduction 97

2.1 List of the elements presented in this chapter 97

2.2 One-dimensional elements 99

2.2.1 Linear element (two nodes, C0) 99

2.2.2 High-precision Lagrangian elements: (continuity C0) 101

2.2.3 High-precision Hermite elements 105

2.2.4 General elements 109

2.3 Triangular elements (two dimensions) 111

2.3.1 Systems of coordinates 111

2.3.2 Linear element (triangle, three nodes, C0) 113

2.3.3 High-precision Lagrangian elements (continuity C0) 115

2.3.4 High-precision Hermite elements 123

2.4 Quadrilateral elements (two dimensions) 127

2.4.1 Systems of coordinates 127

2.4.2 Bilinear element (quadrilateral, 4 nodes, C0) 128

2.4.3 High-precision Lagrangian elements 129

2.4.4 High-precision Hermite element 134

2.5 Tetrahedral elements (three dimensions) 137

2.5.1 Systems of coordinates 137

2.5.2 Linear element (tetrahedron, four nodes, C0) 139

2.5.3 High-precision Lagrangian elements (continuity C0) 140

2.5.4 High-precision Hermite elements 142

2.6 Hexahedric elements (three dimensions) 143

2.6.1 Trilinear element (hexahedron, eight nodes, C0) 143

2.6.2 High-precision Lagrangian elements (continuity C0) 144

2.6.3 High-precision Hermite elements 150

2.7 Prismatic elements (three dimensions) 150

2.7.1 Element with six nodes (prism, six nodes, C0) 150

2.7.2 Element with 15 nodes (prism, 15 nodes, C0) 151

2.8 Pyramidal element (three dimensions) 152

2.8.1 Element with five nodes 152

2.9 Other elements 153

2.9.1 Approximation of vectorial values 153 2.9.2 Modifications o...