FINITE ELEMENT METHODS: AN INTRODUCTION
CONTENTS INTRODUCTION BACKGROUND APPLICATIONS OF FINITE ELEMENTS REFERENCES DIRECT APPROACH FOR DISCRETE SYSTEMS DESCRIBING THE BEHAVIOR OF A SINGLE BAR ELEMENT EQUATIONS FOR A SYSTEM EQUATIONS FOR ASSEMBLY BOUNDARY CONDITIONS AND SYSTEM SOLUTION APPLICATIONS TO OTHER LINEAR SYSTEMS TWO-DIMENSIONAL TRUSS SYSTEMS TRANSFORMATION LAW THREE-DIMENSIONAL TRUSS SYSTEMS PROBLEMS STRONG AND WEAK FORMS FOR ONE-DIMENSIONAL PROBLEMS THE STRONG FORM IN ONE-DIMENSIONAL PROBLEMS THE STRONG FORM FOR AN AXIALLY LOADED ELASTIC BAR THE STRONG FORM FOR HEAT CONDUCTION IN ONE DIMENSION DIFFUSION IN ONE DIMENSION THE WEAK FORM IN ONE DIMENSION CONTINUITY THE EQUIVALENCE BETWEEN THE WEAK AND STRONG FORMS ONE-DIMENSIONAL STRESS ANALYSIS WITH ARBITRARY BOUNDARY CONDITIONS STRONG FORM FOR ONE-DIMENSIONAL STRESS ANALYSIS WEAK FORM FOR ONE-DIMENSIONAL STRESS ANALYSIS ONE-DIMENSIONAL HEAT CONDUCTION WITH ARBITRARY BOUNDARY CONDITIONS STRONG FORM FOR HEAT CONDUCTION IN ONE DIMENSION WITH ARBITRARY BOUNDARY CONDITIONS WEAK FORM FOR HEAT CONDUCTION IN ONE DIMENSION WITH ARBITRARY BOUNDARY CONDITIONS TWO-POINT BOUNDARY VALUE PROBLEM WITH GENERALIZED BOUNDARY CONDITIONS STRONG FORM FOR TWO-POINT BOUNDARY VALUE PROBLEMS WITH GENERALIZED BOUNDARY CONDITIONS WEAK FORM FOR TWO-POINT BOUNDARY VALUE PROBLEMS WITH GENERALIZED BOUNDARY CONDITIONS ADVECTION–DIFFUSION STRONG FORM OF ADVECTION–DIFFUSION EQUATION WEAK FORM OF ADVECTION–DIFFUSION EQUATION MINIMUM POTENTIAL ENERGY 3 10 INTEGRABILITY APPROXIMATION OF TRIAL SOLUTIONS, WEIGHT FUNCTIONS AND GAUSS QUADRATURE FOR ONE-DIMENSIONAL PROBLEMS TWO-NODE LINEAR ELEMENT QUADRATIC ONE-DIMENSIONAL ELEMENT DIRECT CONSTRUCTION OF SHAPE FUNCTIONS IN ONE DIMENSION APPROXIMATION OF THE WEIGHT FUNCTIONS GLOBAL APPROXIMATION AND CONTINUITY GAUSS QUADRATURE REFERENCE FINITE ELEMENT FORMULATION FOR ONE-DIMENSIONAL PROBLEMS DEVELOPMENT OF DISCRETE EQUATION: SIMPLE CASE ELEMENT MATRICES FOR TWO-NODE ELEMENT APPLICATION TO HEAT CONDUCTION AND DIFFUSION PROBLEMS DEVELOPMENT OF DISCRETE EQUATIONS FOR ARBITRARY BOUNDARY CONDITIONS CONVERGENCE OF THE FEM CONVERGENCE BY NUMERICAL EXPERIMENTS CONVERGENCE BY ANALYSIS FEM FOR ADVECTION–DIFFUSION EQUATION STRONG AND WEAK FORMS FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS DIVERGENCE THEOREM AND GREEN’S FORMULA STRONG FORM WEAK FORM THE EQUIVALENCE BETWEEN WEAK AND STRONG FORMS GENERALIZATION TO THREE-DIMENSIONAL PROBLEMS STRONG AND WEAK FORMS OF SCALAR STEADY-STATE ADVECTION–DIFFUSION IN TWO DIMENSIONS APPROXIMATIONS OF TRIAL SOLUTIONS, WEIGHT FUNCTIONS AND GAUSS QUADRATURE FOR MULTIDIMENSIONAL PROBLEMS COMPLETENESS AND CONTINUITY THREE-NODE TRIANGULAR ELEMENT HIGHER ORDER TRIANGULAR ELEMENTS DERIVATIVES OF SHAPE FUNCTIONS FOR THE THREE-NODE TRIANGULAR ELEMENT FOUR-NODE RECTANGULAR ELEMENTS FOUR-NODE QUADRILATERAL ELEMENT CONTINUITY OF ISOPARAMETRIC ELEMENTS DERIVATIVES OF ISOPARAMETRIC SHAPE FUNCTIONS HIGHER ORDER QUADRILATERAL ELEMENTS TRIANGULAR COORDINATES LINEAR TRIANGULAR ELEMENT ISOPARAMETRIC TRIANGULAR ELEMENTS CUBIC ELEMENT TRIANGULAR ELEMENTS BY COLLAPSING QUADRILATERAL ELEMENTS COMPLETENESS OF ISOPARAMETRIC ELEMENTS GAUSS QUADRATURE IN TWO DIMENSIONS INTEGRATION OVER QUADRILATERAL ELEMENTS INTEGRATION OVER TRIANGULAR ELEMENTS THREE-DIMENSIONAL ELEMENTS HEXAHEDRAL ELEMENTS TETRAHEDRAL ELEMENTS FINITE ELEMENT FORMULATION FOR MULTIDIMENSIONAL SCALAR FIELD PROBLEMS FINITE ELEMENT FORMULATION FOR TWO-DIMENSIONAL HEAT CONDUCTION PROBLEMS VERI?CATION AND VALIDATION ADVECTION–DIFFUSION EQUATION FINITE ELEMENT FORMULATION FOR VECTOR FIELD PROBLEMS – LINEAR ELASTICITY LINEAR ELASTICITY KINEMATICS STRESS AND TRACTION EQUILIBRIUM CONSTITUTIVE EQUATION STRONG AND WEAK FORMS FINITE ELEMENT DISCRETIZATION ELEMENT BODY FORCE MATRIX BOUNDARY FORCE MATRIX GENERALIZATION OF BOUNDARY CONDITIONS DISCUSSION LINEAR ELASTICITY EQUATIONS IN THREE DIMENSIONS 10 FINITE ELEMENT FORMULATION FOR BEAMS GOVERNING EQUATIONS OF THE BEAM KINEMATICS OF BEAM STRESS–STRAIN LAW BOUNDARY CONDITIONS STRONG FORM TO WEAK FORM WEAK FORM TO STRONG FORM TRIAL SOLUTION AND WEIGHT FUNCTION APPROXIMATIONS DISCRETE EQUATIONS THEOREM OF MINIMUM POTENTIAL ENERGY REMARKS ON SHELL ELEMENTS 11 COMMERCIAL FINITE ELEMENT PROGRAM ABAQUS TUTORIALS STEADY-STATE HEAT FLOW EXAMPLE PRELIMINARIES CREATING A PART CREATING A MATERIAL DE?NITION DE?NING AND ASSIGNING SECTION PROPERTIES ASSEMBLING THE MODEL CON?GURING THE ANALYSIS APPLYING A BOUNDARY CONDITION AND A LOAD TO THE MODEL MESHING THE MODEL CREATING AND SUBMITTING AN ANALYSIS JOB VIEWING THE ANALYSIS RESULTS SOLVING THE PROBLEM USING QUADRILATERALS RE?NING THE MESH BENDING OF A SHORT CANTILEVER BEAM COPYING THE MODEL MODIFYING THE MATERIAL DE?NITION PLATE WITH A HOLE IN TENSION CREATING A NEW MODEL