FINITE ELEMENT MODELING FOR STRESS ANALYSIS
CONTENTS VARIATIONAL FORMULATIONS AND FINITE ELEMENT METHODS CLASSICAL METHODS MODEL PROBLEMS AND ELEMENTARY PROPERTIES OF SOME FUNCTIONAL SPACES EIGENVALUE PROBLEMS DUALITY METHODS GENERALITIES EXAMPLES FOR SYMMETRIC PROBLEMS DUALITY METHODS FOR NON SYMMETRIC BILINEAR FORMS MIXED EIGENVALUE PROBLEMS DOMAIN DECOMPOSITION METHODS, HYBRID METHODS MODI?ED VARIATIONAL FORMULATIONS AUGMENTED FORMULATIONS PERTURBED FORMULATIONS BIBLIOGRAPHICAL REMARKS FUNCTION SPACES AND FINITE ELEMENT APPROXIMATIONS PROPERTIES OF THE SPACES H M ?/, H DIVI ?/, AND H CURLI ?/ BASIC PROPERTIES PROPERTIES RELATIVE TO A PARTITION OF ? PROPERTIES RELATIVE TO A CHANGE OF VARIABLES DE RHAM DIAGRAM FINITE ELEMENT APPROXIMATIONS OF H 1 ?/ AND H 2 ?/ CONFORMING METHODS EXPLICIT BASIS FUNCTIONS ON TRIANGLES AND TETRAHEDRA NONCONFORMING METHODS QUADRILATERAL FINITE ELEMENTS ON NON AF?NE MESHES QUADRILATERAL APPROXIMATION OF SCALAR FUNCTIONS NON POLYNOMIAL APPROXIMATIONS SCALING ARGUMENTS SIMPLICIAL APPROXIMATIONS OF H DIVI ?/ AND H CURLI ?/ SIMPLICIAL APPROXIMATIONS OF H DIVI ?/ SIMPLICIAL APPROXIMATION OF H CURLI ?/ APPROXIMATIONS OF H DIVI K/ ON RECTANGLES AND CUBES RAVIART-THOMAS ELEMENTS ON RECTANGLES AND CUBES OTHER APPROXIMATIONS OF H DIVI K/ ON RECTANGLES OTHER APPROXIMATIONS OF H DIVI K/ ON CUBES APPROXIMATIONS OF H CURLI K/ ON CUBES INTERPOLATION OPERATOR AND ERROR ESTIMATES APPROXIMATIONS OF H DIVI K/ APPROXIMATION SPACES FOR H DIVI ?/ APPROXIMATIONS OF H CURLI ?/ APPROXIMATION SPACES FOR H CURLI ?/ QUADRILATERAL AND HEXAHEDRAL APPROXIMATION OF VECTOR-VALUED FUNCTIONS IN H DIVI ?/ AND H CURLI ?/ DISCRETE EXACT SEQUENCES EXPLICIT BASIS FUNCTIONS FOR H DIVI K/ AND H CURLI K/ ON TRIANGLES AND TETRAHEDRA BASIS FUNCTIONS FOR H DIVI K/: THE TWO-DIMENSIONAL CASE BASIS FUNCTIONS FOR H DIVI K/: THE THREE-DIMENSIONAL CASE BASIS FUNCTIONS FOR H CURLI K/: THE TWO-DIMENSIONAL CASE BASIS FUNCTIONS FOR H CURLI K/: THE THREE-DIMENSIONAL CASE CONCLUDING REMARKS ALGEBRAIC ASPECTS OF SADDLE POINT PROBLEMS NOTATION, AND BASIC RESULTS IN LINEAR ALGEBRA BASIC DE?NITIONS SUBSPACES ORTHOGONAL SUBSPACES ORTHOGONAL PROJECTIONS BASIC RESULTS RESTRICTIONS OF OPERATORS EXISTENCE AND UNIQUENESS OF SOLUTIONS: THE SOLVABILITY PROBLEM A PRELIMINARY DISCUSSION THE NECESSARY AND SUF?CIENT CONDITION SUF?CIENT CONDITIONS EXAMPLES COMPOSITE MATRICES THE SOLVABILITY PROBLEM FOR PERTURBED MATRICES PRELIMINARY RESULTS MAIN RESULTS STABILITY ASSUMPTIONS ON THE NORMS THE INF-SUP CONDITION FOR THE MATRIX B: AN ELEMENTARY DISCUSSION THE INF-SUP CONDITION AND THE SINGULAR VALUES THE CASE OF A ELLIPTIC ON THE WHOLE SPACE THE CASE OF A ELLIPTIC ON THE KERNEL OF B THE CASE OF A SATISFYING AN INF-SUP ON THE KERNEL OF B ADDITIONAL RESULTS SOME NECESSARY CONDITIONS THE CASE OF B NOT SURJECTIVE MODI?CATION OF THE PROBLEM SOME SPECIAL CASES STABILITY OF PERTURBED MATRICES THE BASIC ESTIMATE THE SYMMETRIC CASE FOR PERTURBED MATRICES SADDLE POINT PROBLEMS IN HILBERT SPACES REMINDERS ON HILBERT SPACES SCALAR PRODUCTS, NORMS, COMPLETENESS CLOSED SUBSPACES AND DENSE SUBSPACES ORTHOGONALITY CONTINUOUS LINEAR OPERATORS, DUAL SPACES, POLAR SPACES BILINEAR FORMS AND ASSOCIATED OPERATORS; TRANSPOSED OPERATORS DUAL SPACES OF LINEAR SUBSPACES IDENTI?CATION OF A SPACE WITH ITS DUAL SPACE RESTRICTIONS OF OPERATORS TO CLOSED SUBSPACES QUOTIENT SPACES EXISTENCE AND UNIQUENESS OF SOLUTIONS MIXED FORMULATIONS IN HILBERT SPACES STABILITY CONSTANTS AND INF-SUP CONDITIONS THE MAIN RESULT THE CASE OF IMB ¤ Q0 EXISTENCE AND UNIQUENESS FOR PERTURBED PROBLEMS REGULAR PERTURBATIONS SINGULAR PERTURBATIONS APPROXIMATION OF SADDLE POINT PROBLEMS THE BASIC ASSUMPTIONS THE DISCRETE OPERATORS ERROR ESTIMATES FOR FINITE DIMENSIONAL APPROXIMATIONS DISCRETE STABILITY AND ERROR ESTIMATES ADDITIONAL ERROR ESTIMATES FOR THE BASIC PROBLEM VARIANTS OF ERROR ESTIMATES A SIMPLE EXAMPLE AN IMPORTANT EXAMPLE: THE PRESSURE IN THE HOMOGENEOUS STOKES PROBLEM THE CASE OF KERBT H ¤ F0G THE CASE OF KERBT H KERBT THE CASE OF KERBT H ª KERBT THE CASE OF ?H OR Q?H GOING TO ZERO THE INF-SUP CONDITION: CRITERIA SOME LINGUISTIC CONSIDERATIONS GENERAL CONSIDERATIONS THE INF-SUP CONDITION AND THE B-COMPATIBLE INTERPOLATION OPERATOR ?H CONSTRUCTION OF ?H AN ALTERNATIVE STRATEGY: SWITCHING NORMS EXTENSIONS OF ERROR ESTIMATES PERTURBED PROBLEMS PENALTY METHODS DUAL ERROR ESTIMATES NUMERICAL PROPERTIES OF THE DISCRETE PROBLEM THE MATRIX FORM OF THE DISCRETE PROBLEM AND IF THE INF-SUP CONDITION DOES NOT HOLD? SOLUTION METHODS COMPLEMENTS: STABILISATION METHODS, EIGENVALUE PROBLEMS AN ABSTRACT FRAMEWORK FOR STABILISED METHODS STABILISING TERMS STABILITY CONDITIONS FOR AUGMENTED FORMULATIONS DISCRETISATIONS OF AUGMENTED FORMULATIONS STABILISING WITH THE “ELEMENT-WISE EQUATIONS” OTHER STABILISATIONS GENERAL STABILITY CONDITIONS STABILITY OF DISCRETISED FORMULATIONS MINIMAL STABILISATIONS ANOTHER FORM OF MINIMAL STABILISATION ENHANCED STRAIN METHODS SOME CLASSICAL RESULTS EIGENVALUE PROBLEMS IN MIXED FORM SPECIAL RESULTS FOR PROBLEMS OF TYPE F; 0/ AND 0; G/ EIGENVALUE PROBLEMS OF THE TYPE F; 0/ EIGENVALUE PROBLEMS OF THE FORM 0; G/ MIXED METHODS FOR ELLIPTIC PROBLEMS NON-STANDARD METHODS FOR DIRICHLET’S PROBLEM DESCRIPTION OF THE PROBLEM MIXED FINITE ELEMENT METHODS FOR DIRICHLET’S PROBLEM EIGENVALUE PROBLEM FOR THE MIXED FORMULATION PRIMAL HYBRID METHODS PRIMAL MACRO-HYBRID METHODS AND DOMAIN DECOMPOSITIONS DUAL HYBRID METHODS NUMERICAL SOLUTIONS PRELIMINARIES INTER-ELEMENT MULTIPLIERS A BRIEF ANALYSIS OF THE COMPUTATIONAL EFFORT ERROR ANALYSIS FOR THE MULTIPLIER ERROR ESTIMATES IN OTHER NORMS APPLICATION TO AN EQUATION ARISING FROM SEMICONDUCTOR THEORY USING ANISOTROPIC MESHES RELATIONS WITH FINITE VOLUME METHODS THE ONE AND TWO-DIMENSIONAL CASES THE TWO-DIMENSIONAL CASE THE THREE-DIMENSIONAL CASE NONCONFORMING METHODS: A TRAP TO AVOID AUGMENTED FORMULATIONS (GALERKIN LEAST SQUARES METHODS) A POSTERIORI ERROR ESTIMATES INCOMPRESSIBLE MATERIALS AND FLOW PROBLEMS INTRODUCTION THE STOKES PROBLEM AS A MIXED PROBLEM MIXED FORMULATION SOME EXAMPLES OF FAILURE AND EMPIRICAL CURES CONTINUOUS PRESSURE: THE P 1 P1 ELEMENT APPROXIMATION BUILDING A B-COMPATIBLE OPERATOR: THE SIMPLEST STABLE ELEMENTS BUILDING A B-COMPATIBLE OPERATOR A STABLE CASE: THE MINI ELEMENT ANOTHER STABLE APPROXIMATION: THE BI-DIMENSIONAL P 2 P0 ELEMENT THE NONCONFORMING P 1 P0 APPROXIMATION OTHER TECHNIQUES FOR CHECKING THE INF-SUP CONDITION PROJECTION ONTO CONSTANTS VERF¨URTH’S TRICK SPACE AND DOMAIN DECOMPOSITION TECHNIQUES MACRO-ELEMENT TECHNIQUE MAKING USE OF THE INTERNAL DEGREES OF FREEDOM TWO-DIMENSIONAL STABLE ELEMENTS CONTINUOUS PRESSURE ELEMENTS DISCONTINUOUS PRESSURE ELEMENTS QUADRILATERAL ELEMENTS, QK PK1 ELEMENTS THREE-DIMENSIONAL STABLE ELEMENTS CONTINUOUS PRESSURE 3-D ELEMENTS DISCONTINUOUS PRESSURE 3-D ELEMENTS P K PK1 SCHEMES AND GENERALISED HOOD–TAYLOR ELEMENTS DISCONTINUOUS PRESSURE P K PK1 ELEMENTS GENERALISED HOOD–TAYLOR ELEMENTS OTHER DEVELOPMENTS FOR DIVERGENCE-FREE STOKES APPROXIMATION AND MASS CONSERVATION EXACTLY DIVERGENCE-FREE STOKES ELEMENTS, DISCONTINUOUS GALERKIN METHODS STOKES ELEMENTS ALLOWING FOR ELEMENT-WISE MASS CONSERVATION SPURIOUS PRESSURE MODES LIVING WITH SPURIOUS PRESSURE MODES: PARTIAL CONVERGENCE THE BILINEAR VELOCITY-CONSTANT PRESSURE Q1 P0 ELEMENT NEARLY INCOMPRESSIBLE ELASTICITY, REDUCED INTEGRATION METHODS AND RELATION WITH PENALTY METHODS VARIATIONAL FORMULATIONS AND ADMISSIBLE DISCRETISATIONS REDUCED INTEGRATION METHODS EFFECTS OF INEXACT INTEGRATION OTHER STABILISATION PROCEDURES AUGMENTED METHOD FOR THE STOKES PROBLEM MINIMAL STABILISATIONS FOR STOKES CONCLUDING REMARKS: CHOICE OF ELEMENTS CHOICE OF ELEMENTS COMPLEMENTS ON ELASTICITY PROBLEMS CONTINUOUS FORMULATION OF STRESS METHODS NUMERICAL APPROXIMATIONS OF STRESS FORMULATIONS RELAXED SYMMETRY TENSORS, TENSORIAL NOTATION AND RESULTS ON SYMMETRY CONTINUOUS FORMULATION OF THE RELAXED SYMMETRY APPROACH NUMERICAL APPROXIMATION OF RELAXED-SYMMETRY FORMULATIONS SOME FAMILIES OF METHODS WITH REDUCED SYMMETRY METHODS BASED ON STOKES ELEMENTS STABILISATION BY H CURL/ BUBBLES TWO EXAMPLES METHODS BASED ON THE PROPERTIES OF ? LOOSING THE INCLUSION OF KERNEL: STABILISED METHODS COMPLEMENTS ON PLATE PROBLEMS A MIXED FOURTH-ORDER PROBLEM THE ! BIHARMONIC PROBLEM EIGENVALUES OF THE BIHARMONIC PROBLEM DUAL HYBRID METHODS FOR PLATE BENDING PROBLEMS MIXED METHODS FOR LINEAR THIN PLATES MODERATELY THICK PLATES THE MATHEMATICAL FORMULATION MIXED FORMULATION OF THE MINDLIN-REISSNER MODEL A DECOMPOSITION PRINCIPLE AND THE STOKES CONNECTION DISCRETISATION OF THE PROBLEM CONTINUOUS PRESSURE APPROXIMATIONS MIXED FINITE ELEMENTS FOR ELECTROMAGNETIC PROBLEMS USEFUL RESULTS ABOUT THE SPACE H CURLI ?/, ITS BOUNDARY TRACES, AND THE DE RHAM COMPLEX THE DE RHAM COMPLEX AND THE HELMHOLTZ DECOMPOSITION WHEN ? IS SIMPLY CONNECTED THE FRIEDRICHS INEQUALITY EXTENSION TO MORE GENERAL TOPOLOGIES H CURLI ?/ IN TWO SPACE DIMENSIONS THE TIME HARMONIC MAXWELL SYSTEM MAXWELL’S EIGENVALUE PROBLEM ANALYSIS OF THE TIME HARMONIC MAXWELL SYSTEM APPROXIMATION OF THE TIME HARMONIC MAXWELL EQUATIONS APPROXIMATION OF THE MAXWELL EIGENVALUE PROBLEM ANALYSIS OF THE TWO-DIMENSIONAL CASE DISCRETE COMPACTNESS PROPERTY NODAL FINITE ELEMENTS EDGE FINITE ELEMENTS ENFORCING THE DIVERGENCE-FREE CONDITION BY A PENALTY METHOD SOME REMARKS ON EXTERIOR CALCULUS CONCLUDING REMARKS REFERENCES