Finite Element Method

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.

• Introduction  10.1 The finite element method  10.1.1 General remarks 10.1.2 Historical evolution of the method  20.1.3 State of the art 30.2 Object and organization of the book 30.2.1 Teaching the finite element method  30.2.2 Objectives of the book 40.2.3 Organization of the book 40.3 Numerical modeling approach  60.3.1 General aspects 60.3.2 Physical model 70.3.3 Mathematical model 90.3.4 Numerical model 100.3.5 Computer model 13Bibliography 16Conference proceedings 17Monographs 18Periodicals 19Chapter 1. Approximations with finite elements 211.0 Introduction 211.1 General remarks 211.1.1 Nodal approximation 211.1.2 Approximations with finite elements 281.2 Geometrical definition of the elements 331.2.1 Geometrical nodes 331.2.2 Rules for the partition of a domain into elements  331.2.3 Shapes of some classical elements 351.2.4 Reference elements 371.2.5 Shapes of some classical reference elements 411.2.6 Node and element definition tables 441.3 Approximation based on a reference element 451.3.1 Expression of the approximate function u(x)  451.3.2 Properties of approximate function u(x) 491.4 Construction of functions N (ξ ) and N (ξ ) 541.4.1 General method of construction 541.4.2 Algebraic properties of functions N and N 591.5 Transformation of derivation operators 611.5.1 General remarks 611.5.2 First derivatives 621.5.3 Second derivatives 651.5.4 Singularity of the Jacobian matrix 681.6 Computation of functions N, their derivatives and the Jacobian matrix 721.6.1 General remarks 721.6.2 Explicit forms for N 731.7 Approximation errors on an element 751.7.1 Notions of approximation errors 751.7.2 Error evaluation technique  801.7.3 Improving the precision of approximation 831.8 Example of application: rainfall problem  89Bibliography 95Chapter 2. Various types of elements 972.0 Introduction 972.1 List of the elements presented in this chapter 972.2 One-dimensional elements 992.2.1 Linear element (two nodes, C0) 992.2.2 High-precision Lagrangian elements: (continuity C0)  1012.2.3 High-precision Hermite elements 1052.2.4 General elements 1092.3 Triangular elements (two dimensions) 1112.3.1 Systems of coordinates 1112.3.2 Linear element (triangle, three nodes, C0)  1132.3.3 High-precision Lagrangian elements (continuity C0) 1152.3.4 High-precision Hermite elements 1232.4 Quadrilateral elements (two dimensions)  1272.4.1 Systems of coordinates 1272.4.2 Bilinear element (quadrilateral, 4 nodes, C0) 1282.4.3 High-precision Lagrangian elements 1292.4.4 High-precision Hermite element 1342.5 Tetrahedral elements (three dimensions) 1372.5.1 Systems of coordinates 1372.5.2 Linear element (tetrahedron, four nodes, C0)  1392.5.3 High-precision Lagrangian elements (continuity C0) 1402.5.4 High-precision Hermite elements 1422.6 Hexahedric elements (three dimensions)  1432.6.1 Trilinear element (hexahedron, eight nodes, C0)  1432.6.2 High-precision Lagrangian elements (continuity C0) 1442.6.3 High-precision Hermite elements 1502.7 Prismatic elements (three dimensions) 1502.7.1 Element with six nodes (prism, six nodes, C0)  1502.7.2 Element with 15 nodes (prism, 15 nodes, C0)  1512.8 Pyramidal element (three dimensions) 1522.8.1 Element with five nodes 1522.9 Other elements 1532.9.1 Approximation of vectorial values 1532.9.2 Modifications of the elements 1552.9.3 Elements with a variable number of nodes 1562.9.4 Superparametric elements 1582.9.5 Infinite elements 158Bibliography 160Chapter 3. Integral formulation  1613.0 Introduction  1613.1 Classification of physical systems 1633.1.1 Discrete and continuous systems 1633.1.2 Equilibrium, eigenvalue and propagation problems 1643.2 Weighted residual method 1723.2.1 Residuals 1723.2.2 Integral forms 1733.3 Integral transformations 1743.3.1 Integration by parts 1743.3.2 Weak integral form 1773.3.3 Construction of additional integral forms 1793.4 Functionals 1823.4.1 First variation 1823.4.2 Functional associated with an integral form  1833.4.3 Stationarity principle 1873.4.4 Lagrange multipliers and additional functionals 1883.5 Discretization of integral forms 1943.5.1 Discretization of W 1943.5.2 Approximation of the functions u 1973.5.3 Choice of the weighting functions ψ 1983.5.4 Discretization of a functional (Ritz method) 2053.5.5 Properties of the systems of equations 2083.6 List of PDEs and weak expressions 2093.6.1 Scalar field problems 2103.6.2 Solid mechanics 2133.6.3 Fluid mechanics 217Bibliography 229Chapter 4. Matrix presentation of the finite element method 2314.0 Introduction 2314.1 The finite element method 2314.1.1 Finite element approach 2314.1.2 Conditions for convergence of the solution 2434.1.3 Patch test  2564.2 Discretized elementary integral forms We 2644.2.1 Matrix expression of We 2644.2.2 Case of a nonlinear operator L 2674.2.3 Integral form We on the reference element 2694.2.4 A few classic forms of We and of elementary matrices 2744.3 Techniques for calculating elementary matrices 2744.3.1 Explicit calculation for a triangular element (Poisson’s equation) 2744.3.2 Explicit calculation for a quadrangular element (Poisson’s equation) 2794.3.3 Organization of the calculation of the elementary matrices by numerical integration 2804.3.4 Calculation of the elementary matrices: linear problems 2824.4 Assembly of the global discretized form W 2974.4.1 Assembly by expansion of the elementary matrices 2984.4.2 Assembly in structural mechanics 3034.5 Technique of assembly 3054.5.1 Stages of assembly 3054.5.2 Rules of assembly 3054.5.3 Example of a subprogram for assembly 3074.5.4 Construction of the localization table LOCE  3084.6 Properties of global matrices  3104.6.1 Band structure 3104.6.2 Symmetry  3144.6.3 Storage methods 3144.7 Global system of equations 3184.7.1 Expression of the system of equations 3184.7.2 Introduction of the boundary conditions 3184.7.3 Reactions 3214.7.4 Transformation of variables 3214.7.5 Linear relations between variables 3234.8 Example of application: Poisson’s equation  3244.9 Some concepts about convergence, stability and error calculation 3294.9.1 Notations 3294.9.2 Properties of the exact solution 3304.9.3 Properties of the solution obtained by the finite element method  3314.9.4 Stability and locking 3344.9.5 One-dimensional exact finite elements 337Bibliography  343Chapter 5. Numerical Methods 3455.0 Introduction 3455.1 Numerical integration 3465.1.1 Introduction 3465.1.2 One-dimensional numerical integration 3485.1.3 Two-dimensional numerical integration 3605.1.4 Numerical integration in three dimensions 3685.1.5 Precision of integration 3725.1.6 Choice of number of integration points 3755.1.7 Numerical integration codes 3795.2 Solving systems of linear equations 3845.2.1 Introduction 3845.2.2 Gaussian elimination method 3855.2.3 Decomposition 3915.2.4 Adaptation of algorithm (5.44) to the case of a matrix stored bythe skyline method 3995.3 Solution of nonlinear systems 4045.3.1 Introduction 4045.3.2 Substitution method 4075.3.3 Newton–Raphson method  4115.3.4 Incremental (or step-by-step) method  4205.3.5 Changing of independent variables  4215.3.6 Solution strategy 4245.3.7 Convergence of an iterative method 4265.4 Resolution of unsteady systems 4295.4.1 Introduction 4295.4.2 Direct integration methods for first-order systems 4315.4.3 Modal superposition method for first-order systems 4635.4.4 Methods for direct integration of second-order systems 4665.4.5 Modal superposition method for second-order systems 4765.5 Methods for calculating the eigenvalues and eigenvectors 4805.5.1 Introduction 4805.5.2 Recap of some properties of eigenvalue problems 4815.5.3 Methods for calculating the eigenvalues 488Bibliography 502 Chapter 6. Programming technique 5056.0 Introduction 5056.1 Functional blocks of a finite element program 5066.2 Description of a typical problem 5076.3 General programs 5086.3.1 Possibilities of general programs 5086.3.2 Modularity 5116.4 Description of the finite element code 5126.4.1 Introduction 5126.4.2 General organization 5136.4.3 Description of tables and variables 5176.5 Library of elementary finite element method programs 5216.5.1 Functional blocks 5216.5.2 List of thermal elements 5306.5.3 List of elastic elements 5386.5.4 List of elements for fluid mechanics  5456.6 Examples of application 5496.6.1 Heat transfer problems 5506.6.2 Planar elastic problems 5586.6.3 Fluid flow problems 566Appendix. Sparse solver 5777.0 Introduction  5777.1 Methodology of the sparse solver 5787.1.1 Assembly of matrices in sparse form: row-by-row format 5797.1.2 Permutation using the “minimum degree” algorithm 5847.1.3 Modified column–column storage format 5877.1.4 Symbolic factorization 5897.1.5 Numerical factorization 5907.1.6 Solution of the system by descent/ascent 5927.2 Numerical examples 593Bibliography 595Index 597