MIXED AND HYBRID FINITE ELEMENT METHODS
CONTENTS 1 INTRODUCTION 11 CONSERVATION LAWS FOR A CONTINUUM MEDIUM 111 CONSERVATION OF MASS 112 CONSERVATION OF MOMENTUM 113 CONSERVATION OF ENERGY 114 CONSTITUTIVE RELATIONS 12 GOVERNING EQUATIONS IN TERMS OF PRIMITIVE VARIABLES 121 VECTOR FORM 122 COMPONENT FORM IN CARTESIAN COORDINATES 123 COMPONENT FORM IN CYLINDRICAL COORDINATES 124 SUMMARY 13 SPECIES TRANSPORT EQUATIONS 14 GOVERNING EQUATIONS IN TRANSLATING AND ROTATING FRAMES OF REFERENCE 15 BOUNDARY AND INITIAL CONDITIONS 151 GENERAL BOUNDARY CONDITIONS 152 FREE BOUNDARY CONDITIONS 153 MOVING INTERFACE CONDITIONS 154 PHASE CHANGE CONDITIONS 16 GOVERNING EQUATIONS FOR FLOWS THROUGH POROUS MEDIA 17 GOVERNING EQUATIONS IN CONSERVATION FORM EXERCISES REFERENCES 2 DISCONTINUOUS FINITE ELEMENT PROCEDURES 21 THE CONCEPT OF DISCONTINUOUS FINITE ELEMENTS 211 WEAKLY IMPOSED CROSS-ELEMENT CONTINUITY 212 NUMERICAL BOUNDARY FLUXES FOR DISCONTINUITY 213 BOUNDARY CONSTRAINT MINIMIZATION 214 TREATMENT OF DISCONTINUITY FOR NON-CONSERVATIVE SYSTEMS 215 TRANSIENT PROBLEMS 22 DISCONTINUOUS FINITE ELEMENT FORMULATION 221 INTEGRAL FORMULATION 222 TIME INTEGRATION 23 SOLUTION PROCEDURES 24 ADVANTAGES AND DISADVANTAGES OF DISCONTINUOUS FINITE ELEMENT FORMULATIONS 241 ADVANTAGES 242 DISADVANTAGES 25 EXAMPLES 3 SHAPE FUNCTIONS AND ELEMENTAL CALCULATIONS 31 SHAPE FUNCTIONS 311 1-D SHAPE FUNCTIONS 312 2-D SHAPE FUNCTIONS 3121 TRIANGULAR ELEMENTS 3122 QUADRILATERAL ELEMENTS 313 3-D SHAPE FUNCTIONS 3131 TETRAHEDRAL ELEMENTS 3132 HEXAHEDRAL ELEMENTS 32 CONSTRUCTION OF SPECIAL ELEMENTS 321 NON-STANDARD ELEMENTS 322 CONSTRUCTION OF ELEMENT SHAPE FUNCTIONS BY NODE COLLAPSING 323 SPECTRAL ELEMENTS 33 HIERARCHICAL SHAPE FUNCTIONS 331 1-D HIERARCHICAL CORRECTION 332 CANONICAL SQUARE AND CUBIC ELEMENTS 333 TRIANGULAR AND TETRAHEDRAL ELEMENTS 334 OBTAINING HIERARCHICAL ELEMENTS THROUGH COORDINATE TRANSFORMATIONS 335 ORTHOGONAL MASS MATRIX CONSTRUCTION 34 INTERPOLATION ERROR ANALYSIS 341 HILBERT SPACE AND VARIOUS ERROR MEASURES 342 INTERPOLATION ERROR ANALYSIS FOR 1-D ELEMENTS 343 INTERPOLATION ERROR ANALYSIS FOR 2-D/3-D ELEMENTS 35 NUMERICAL INTEGRATION 351 1-D NUMERICAL INTEGRATION 352 2-D AND 3-D NUMERICAL INTEGRATION 353 INTEGRATION FOR TRIANGULAR AND TETRAHEDRAL ELEMENTS 36 ELEMENTAL CALCULATIONS 361 DOMAIN CALCULATIONS 362 BOUNDARY CALCULATIONS 4 CONDUCTION HEAT TRANSFER AND POTENTIAL FLOWS 41 1-D STEADY STATE HEAT CONDUCTION 42 STEADY STATE HEAT CONDUCTION IN MULTIDIMENSIONS 43 1-D TRANSIENT HEAT CONDUCTION 431 ALTERNATING UPWINDING SCHEME 432 CENTRAL FLUXES 433 UNIFIED REPRESENTATION 434 NUMERICAL IMPLEMENTATION 435 RUNGEKUTTA TIME INTEGRATION 436 COMPUTATIONAL PROCEDURES 44 TRANSIENT HEAT CONDUCTION IN MULTIDIMENSIONS 45 POTENTIAL FLOWS AND FLOWS IN POROUS MEDIA 46 SELECTION OF NUMERICAL FLUXES 461 STABILITY FOR STEADY STATEPROBLEMS 4611 STABILITY AND NUMERICAL FLUXES 4612 DISCONTINUOUS AND MIXED FINITE ELEMENT FORMULATIONS 462 STABILITY FOR TIME DEPENDENT PROBLEMS 4621 NUMERICAL FLUXES FOR TRANSIENT PROBLEMS 4622 STABILITY ANALYSIS USING MATRIX 463 FOURIER ANALYSIS 5 CONVECTION-DOMINATED PROBLEMS 51 PURE CONVECTION PROBLEMS 511 1-D PURE CONVECTION 5111 METHOD OF CHARACTERISTICS 5112 DISCONTINUOUS FINITE ELEMENT FORMULATION 512 PURE CONVECTION IN MULTIDIMENSIONS 513 STABILITY ANALYSIS 5131 L2-STABILITY – INTEGRAL ANALYSIS 5132 L2-STABILITY – DISCRETIZED ANALYSIS 5133 FOURIER ANALYSIS 52 STEADY STATE CONVECTION-DIFFUSION 521 1-D PROBLEM 522 ORIGIN OF OSCILLATORY STABILITY 523 STEADY CONVECTION-DIFFUSION IN MULTIDIMENSIONS 53 TRANSIENT CONVECTION-DIFFUSION 531 MULTIDIMENSIONAL PROBLEM 532 STABILITY ANALYSIS 5321 L2-STABILITY – INTEGRAL ANALYSIS 5322 L2-STABILITY – DISCRETIZED ANALYSIS 5323 FOURIER ANALYSIS 54 NONLINEAR PROBLEMS 541 1-D INVISCID BURGERS’ EQUATION 5411 BASIC CONSIDERATIONS 5412 DISCONTINUOUS FINITE ELEMENT FORMULATION 542 MULTIDIMENSIONAL INVISCID BURGERS’ EQUATION 5421 DISCONTINUOUS FINITE ELEMENT FORMULATION 5422 CHARACTERISTIC DECOMPOSITION 543 HIGHER ORDER APPROXIMATIONS AND TVD FORMULATIONS 5431 CONCEPT OF TOTAL VARIATION DIMINISHING 5432 FLUX LIMITERS 5433 SLOPE LIMITERS 5434 TVD-RUNGEKUTTA SCHEMES 55 VISCOUS BURGERS’ EQUATIONS 551 1-D BURGERS’ EQUATION 552 2-D VISCOUS BURGERS’ EQUATION 6 INCOMPRESSIBLE FLOWS 61 PRIMITIVE VARIABLE APPROACH 62 FRACTIONAL STEP (PROJECTION) APPROACH 63 VORTICITY AND STREAM FUNCTION APPROACH 64 COUPLED FLOW AND HEAT TRANSFER 7 COMPRESSIBLE FLUID FLOWS 71 1-D COMPRESSIBLE FLOWS 711 GOVERNING EQUATIONS 712 BASIC PROPERTIES OF THE EULER EQUATIONS 713 THE RANKINE–HUGONIOT CONDITIONS 714 1-D RIEMANN SOLVER – EXACT SOLUTION 715 1-D RIEMANN SOLVER – APPROXIMATE SOLUTION 716 DISCONTINUOUS FINITE ELEMENT FORMULATION 717 LOW ORDER (FINITE VOLUME) APPROXIMATIONS 7171 THE GODUNOV SCHEME 7172 THE ROE SCHEME 718 HIGH ORDER TVD APPROXIMATIONS 719 NUMERICAL EXAMPLES 72 MULTIDIMENSIONAL INVISCID COMPRESSIBLE FLOWS 721 GOVERNING EQUATIONS 722 BASIC PROPERTIES OF THE SPLIT 3-D EULER EQUATIONS 723 DISCONTINUOUS FINITE ELEMENT FORMULATION 73 MULTIDIMENSIONAL COMPRESSIBLE VISCOUS FLOWS 74 ALE FORMULATION 741 ALE KINEMATIC DESCRIPTION 742 CONSERVATION OF MASS 743 CONSERVATION OF MOMENTUM 744 CONSERVATION OF ENERGY 745 SUMMARY OF ALE EQUATIONS 746 CONSTITUTIVE RELATIONS 747 ALE DESCRIPTION OF COMPRESSIBLE FLOWS 748 DISCONTINUOUS FINITE ELEMENT FORMULATION 8 EXTERNAL RADIATIVE HEAT TRANSFER 81 INTEGRAL EQUATION FOR SURFACE RADIATION EXCHANGES 811 GOVERNING EQUATION 812 KERNEL FUNCTIONS 82 DISCONTINUOUS GALERKIN FINITE ELEMENT FORMULATION 83 SHADOWING ALGORITHMS 831 SHADOWING ALGORITHM FOR 2-D GEOMETRY 832 SHADOWING ALGORITHM FOR AXISYMMETRIC CONFIGURATIONS 833 SHADOWING ALGORITHM FOR 3-D GEOMETRY 84 COUPLING WITH OTHER HEAT TRANSFER CALCULATIONS 841 DIRECT COUPLING 842 ITERATIVE COUPLING 85 NUMERICAL EXAMPLES 9 RADIATIVE TRANSFER IN PARTICIPATING MEDIA 91 GOVERNING EQUATION AND BOUNDARY CONDITIONS 911 RADIATIVE TRANSFER EQUATION 912 BOUNDARY CONDITIONS 92 APPROXIMATION METHODS 921 THE DISCRETE ORDINATE METHOD 922 THE SPHERICAL HARMONICS METHOD 93 DISCONTINUOUS FINITE ELEMENT FORMULATION 94 NUMERICAL IMPLEMENTATION 941 2-D CALCULATIONS 942 3-D CALCULATIONS 943 INTEGRATION OF THE SOURCE TERM 9431 THE EMITTING CONTRIBUTION 9432 THE SCATTERING CONTRIBUTION 95 RADIATION IN SYSTEMS OF AXISYMMETRY 951 GOVERNING EQUATION IN CYLINDRICAL COORDINATES 952 VOLUME INTEGRATION 953 SURFACE INTEGRATION OVER *P 954 INTEGRATION OVER *M 955 MAPPING 956 TREATMENT OF THE EMITTING AND SCATTERING TERM USE OF RTE FOR EXTERNAL RADIATION CALCULATIONS 97 COUPLING OF THE DISCONTINUOUS METHOD WITH OTHER METHODS 98 CONSTANT ELEMENT APPROXIMATION 10 FREE AND MOVING BOUNDARY PROBLEMS 101 FREE AND MOVING BOUNDARIES 102 BASIC RELATIONS FOR A CURVED SURFACE 1021 DESCRIPTION OF A SURFACE 1022 DIFFERENTIAL AND INTEGRAL RELATIONS FOR CURVED SURFACES 103 PHYSICAL CONSTRAINTS AT A MOVING BOUNDARY 1031 KINEMATIC CONDITIONS AT A MOVING BOUNDARY 1032 STRESS CONDITION AT A MOVING INTERFACE 1033 THERMAL CONDITIONS AT A MOVING INTERFACE 104 MOVING GRIDS VS FIXED GRIDS FOR NUMERICAL SOLUTIONS 105 MOVING GRID METHODS 1051 MOVING BOUNDARIES BETWEEN FLUIDS 1052 MOVING PHASE BOUNDARIES 10521 SOLIDLIQUID PHASE TRANSITION 10522 LIQUIDVAPOR PHASE TRANSITION 106 FIXED GRID METHODS 1061 VOLUME OF FLUID METHOD 10611 STRUCTURED MESH 10612 UNSTRUCTURED MESH 1062 THE MARKER-AND-CELL METHOD 1063 THE LEVEL SET METHOD 1064 FIXED GRID METHODS FOR PHASE CHANGE PROBLEMS 10641 FLOW-BASED METHODS 10642 ENTHALPY-BASED METHODS 107 PHASE FIELD MODELING OF MOVING BOUNDARIES 1071 BASIC IDEAS OF PHASE FIELD MODELS 1072 GOVERNING EQUATIONS FOR INTERFACIAL PHENOMENA 1073 DISCONTINUOUS FINITE ELEMENT FORMULATION 1074 PHASE FIELD MODELING OF MICROSTRUCTURE EVOLUTION 10741 GOVERNING EQUATIONS 10742 DISCONTINUOUS FORMULATION 10743 NUMERICAL EXAMPLES 1075 FLOW AND ORIENTATION EFFECTS ON MICROSTRUCTURE EVOLUTION 10751 FLOW EFFECTS ON MICROSTRUCTURE EVOLUTION 10752 MICROSTRUCTURE EVOLUTION DURING POLYCYSTALLINE SOLIDIFICATION 11 MICRO AND NANO SCALE FLUID FLOW AND HEAT TRANSFER 111 MICRO SCALE HEAT CONDUCTION 1111 TWO-TEMPERATURE EQUATIONS 1112 PHONON SCATTERING EQUATION 1113 PHONON RADIATIVE TRANSFER EQUATION 112 DISCONTINUOUS FINITE ELEMENT FORMULATION 113 MICRO AND NANO FLUID FLOW AND HEAT TRANSFER 114 THE BOLTZMANN TRANSPORT EQUATION AND NUMERICAL SOLUTION 1141 THE BOLTZMANN INTEGRAL-DIFFERENTIAL EQUATION 1142 NUMERICAL SOLUTION OF THE BOLTZMANN TRANSPORT EQUATION 115 THE BOLTZMANBGK EQUATION AND NUMERICAL SOLUTION 1151 THE BOLTZMANNBGK EQUATION 1152 DISCONTINUOUS FINITE ELEMENT FORMULATION 116 THE LATTICE BOLTZMAN EQUATION AND NUMERICAL SOLUTION 1161 DERIVATION OF THE LATTICE BOLTZMANN EQUATION 1162 BOUNDARY CONDITIONS 11621 BOUNCE BACK BOUNDARY CONDITIONS (NO SLIP) 11622 SYMMETRIC BOUNDARY CONDITIONS (FREE SLIP) 11623 INFLOW AND OUTFLOW BOUNDARY CONDITIONS (NO GRADIENT) 11624 FORCE FIELD CONDITIONS 11625 MOVING WALL CONDITIONS 1163 DISCONTINUOUS FINITE ELEMENT FORMULATION 12 FLUID FLOW AND HEAT TRANSFER IN ELECTROMAGNETIC FIELDS 121 MAXWELL EQUATIONS AND BOUNDARY CONDITIONS 1211 MAXWELL EQUATIONS 1212 BOUNDARY CONDITIONS 12121 INTERFACE BOUNDARY CONDITION 12122 PERFECT CONDUCTING SURFACE 12123 IMPEDANCE CONDITION 12124 SOMMERFIELD RADIATION CONDITION 12125 SYMMETRY BOUNDARY CONDITION 122 MAXWELL STRESSES AND ENERGY SOURCES 123 DISCONTINUOUS FORMULATION OF THE MAXWELL EQUATIONS 1231 SOLUTION IN TIME DOMAIN 1232 SOLUTION IN FREQUENCY DOMAIN 1233 SOLUTION IN OTHER FORMS 124 ELECTROOSMOTIC FLOWS 1241 GOVERNING EQUATIONS 1242 DISCONTINUOUS FINITE ELEMENT FORMULATION 125 MICROWAVE HEATING 126 ELECTRICALLY DEFORMED FREE SURFACES 127 COMPRESSIBLE FLOWS IN MAGNETIC FIELDS