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Oden J.T. — Finite Elements: An Introduction (Vol. 1)

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Название: Finite Elements: An Introduction (Vol. 1)

Автор: Oden J.T.

Аннотация:

Our purpose in writing this book is to provide the undergraduate student of engineering and science with a concise introduction to finite element methods — one that will give a reader, equipped with little more than calculus, some matrix algebra, and ordinary differential equations, a clear idea of what the finite element method is, how it works, why it makes sense, and how to use it to solve problems of interest to him. We imposed on ourselves three constraints that we felt were of fundamental importance in designing a text of this type.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1981

Количество страниц: 258

Добавлена в каталог: 16.04.2010

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Предметный указатель

$H(^{h}_{0})$       11 16
${C}^{1}$ basis functions      233—237
${H}^{1}$       55—56 43 147 223—224 244
${H}^{1}$ norm      57
${H}^{1}_{0}$       7—11 16 20 23 56—58 60 241
${H}^{2}$       231
${H}^{2}_{0}$       235—236
${H}^{h}$       58 224 232 235
${H}^{h}_{0}$       16 23 39
A-posteriori estimates      36
A-priori estimates      36
Accuracy, of finite element approximation      36—39 189 197 see “Errors”
Admissible functions      see “Test function”
Admissible functions, for one-dimensional problems      7
Admissible functions, for three-dimensional problems      223
Admissible functions, for two-dimensional problems      57 147
Admissible functions, fourth-order problRems      231—232
Approximation, best Galerkin      14—15
Approximation, interpretation of      31—36
Approximation, of systems      241
Approximation, of three-dimensional problems      224
Approximation, of time-dependent problems      249
Approximation, of two-dimensional problems      162 165
Approximation, of two-point problems by finite elements      73—76
Area coordinates      201—205
Area coordinates, and nonlinear maps      202
Area coordinates, definition      201
Backward difference scheme      251
Banded matrix      26
Banded matrix, elimination of      118—121
Banded matrix, storage of      118—121 209
Bandwidth, definition of      119
Basis functions, ${C}^{1}$ -type      232 234 236—237
Basis functions, for Galerkin's method      11—15 59
Basis functions, for model two-point problem      15—23
Basis functions, global      33 67 69 157 234
Basis functions, Hermite      233—237
Basis functions, Lagrange polynomial      67 195 204
Basis functions, on rectangles      159 236—237
Basis functions, on triangles      154 157 237
Basis functions, piecewise-constant      22
Basis functions, piecewise-linear      17 20 25 30 38 67—68 85 154 157
Basis functions, piecewise-polynomial      16
Basis functions, piecewise-quadratic      23
Beam bending      238—239
Bicubic rectangle      237
Biharmonic equation      235—236 239
Body forces      44
Boundaries, parametric definition of      141—142 193
Boundaries, smoothness of      132
Boundary condition data, input of      104 212—223
Boundary conditions, calculation and flowchart of      216
Boundary conditions, clamped plate      235
Boundary conditions, compatability of      229—230
Boundary conditions, convective      216
Boundary conditions, essential      43 49 55—56 60 81 138 140 223 231—232
Boundary conditions, for biharmonic equation      235
Boundary conditions, for fourth-order problems      229—230
Boundary conditions, for general two-point problems      42 79—84
Boundary conditions, in CODE2      218—229
Boundary conditions, inCODEl      107—109
Boundary conditions, natural (definition)      49 138 146 231—232
Boundary integrals, computation of      191—192
Boundary, restriction of shape functions      191
Brick element      226
Capacitance matrix      see “Mass matrix”
Chain rule, in area coordinates      205
Chain rule, in shape function transformation      189
Charge      4
Classical boundary-value problem      3 8 40—52 62 224 228—229 235 240
Compatability condition for, data in two-point problems      83 141 228—230
Complete polynomials, integration error for      160—161 205—208
Completeness, and rate of convergence      197—198
Conservation principle      43 47 49—51 83 136 138 141 246—247
Conservation principle, of electric flux      44
Conservation principle, of energy      44
Conservation principle, of linear momentum      44 242—243 247
Conservation principle, of mass      44
Constitutive equation      43 44 46 49 136 140 243 247
Constitutive equation, Stokes' law      44
Convergence in an ${H}_{0}^{1}$ sense      11
Convergence rates      36—37 189 197 see “Errors”
Coordinate function, derivatives      189 201
Coulomb's law      44
Cube element      225—227
Cubic triangle      240
Curvilinear coordinates, from master element map      176—177
Darcy's law      44
Data, management of      90
Data, of a boundary-value problem      1 140 229—230
Deformations, calculation of      127—130
Deformations, of an elastic bar      44
Degrees-of-freedom      12 15 59 235—237
Degrees-of-freedom, definition of      12
Degrees-of-freedom, for beam element      238
Degrees-of-freedom, for fourth-order problems      235—237
Degrees-of-freedom, symbolism      236—237
Derivatives, directional      133
Derivatives, of element transformation      183
Diffusion      247
Dirac delta      2 4 46 48 84 144 167 232
Divergence theorem      136 137 143 145
Divergence, definition of      134
Elastic modulus      246
Elasticity      242 244 246—247
Electrostatics      44
Element calculations, flow chart of      30
Element calculations, for ODE system      242
Element calculations, general      187—194
Element calculations, transformation to master element      177
Element calculations, two-dimensional      175—121
Element flux vector      166
Element matrices, assembly of      91
Element matrices, calculation of      26 74—76 91 187 190
Element matrices, for time dependent problems      251
Element transformations, in three dimensions      225—226
Element transformations, in two dimensions      175—187
Energy, conservation of      247
Energy, first variation of      61—64
Energy, functional      60—64
Energy, norm      36 38 57 114
Energy, second variation of      61—64
Energy, strain      32—33 36
entropy      247
Equilibrium equations      243
Errors, analysis of      37—38 57
Errors, asymptotic estimates of      35
Errors, estimates      35—38 70
Errors, estimates for two-point problems      84—85
Errors, estimates, a-posteriori      36
Errors, estimates, a-priori      36
Errors, in Galerkin approximation      14
Errors, interpolation      67
Errors, log-log-plots of      37
Errors, measures      38
Errors, pointwise      38 124
Errors, quadrature      190
Errors, residual      53
Essential boundary condition      see “Boundary conditions”
Evolution problems      see “Time dependent problems”
Extrapolation      33—34
Finite element mesh, definition      16
Finite element method, definition      1
Finite element, definition      1 16
Flow through pourous media      44 247
Fluid flow      44 242
Flux      43 45—46 48—49 73 74 76—77 80 247
Flux, electric      44
Flux, flow rate      44
Flux, heat flux      44

Flux, jump in      46
Flux, shear stress      44
Forward difference scheme      250
Fourier series      10
Fourier's law      44 247
Fourth-order problems      222 228—240
Fourth-order problems, essential boundary conditions      231—232
Fourth-order problems, natural boundary conditions      231—232
Fourth-order problems, two-dimensional      235—239
Fourth-order problems, two-point problem      228—235
Fully discrete method      250
Functional, definition      60
Galerkin approximation, finite element subspace for      23 58
Galerkin approximation, for the general one-dimensional boundary value problem      58—60 73
Galerkin approximation, for the model problem      10—15
Galerkin's method, definition      12
Gauss' theorem      see “Divergence theorem”
Gaussian elimination      109—111
Gaussian quadrature      23 95 190 206
Global matrices      105—107 228
Gradient, definition of      133
Green — Gauss Theorem      148
Green's formula      143 148
Heat conduction      44 247
Heat sources      44
Hermite interpolation      233—237
Hooke's law      see “Constitutive equation”
Hyperbolic equation      248
Ill-posed problem      48 55 230—238
Implicit time integration      251—252
Initial-value problems      41 248—252
Integrals, coordinate transformation of      190
Integrals, on boundaries      191—192
Integration, exact for tetrahedra      228
Integration, exact for triangles      207—208
Integration, points, definition of      93
Integration, rules for one dimension      75 91—95
Integration, rules for quadrilaterals      199
Integration, rules for triangles      205
Interpolant, finite element      22 72
Interpolant, of data      75
Interpolation, error      67
Interpolation, error, rate of convergence      161
Interpolation, Hermite      233—235
Interpolation, higher-order      64
Interpolation, Lagrange      65 67
Interpolation, of one-dimensional problems      64—73
Interpolation, of problem data      191
Interpolation, piecewise-linear      21
Inverse map ${T}^{-1}_{e}$       179
Invertibility of map      179 182
Isoparametric elements      193
Isotropic      136
Jacobian, for boundary integrals      191—192
Jacobian, geometric interpretation of      181—182 201—202
Jacobian, invertibility condition on      179 182—186 194
Jump conditions      46 48 62 73 78 139—140 148—149
Lagrange, basis functions      67
Lagrange, families      65 67 72—73 157 159
Lagrange, interpolation      65 67 72
Lagrange, triangles      203—204
Laplaces equation      142
Laplacian operator      143 236
Line source      149 167
Linear momentum, conservation of      242—243 2
Linear space      10
Load, vector      23 26 28 62 76 85 164 225
Load, vector for Galerkin's method      12 59
Map ${T}_{e}$ , construction of      179—187
Map ${T}_{e}$ , criteria for      179—181
Map ${T}_{e}$ , curved elements      207
Map ${T}_{e}$ , invertibility conditions      185
Map ${T}_{e}$ , using finite element shape functions      180
Mass matrix      250
Master element, calculations on      188—193
Master element, in one dimension      65
Master element, in three dimensions      225—226
Master element, in two dimensions      175—179
Master element, integration      190
Master element, map      176—177 235
Master element, shape functions      188
Master element, square      181
Master element, triangle      181 200
Material, modulus      43 45 47 136
Material, modulus-dielectric permittivity, permeability, thermal conductivity, viscosity, Young's modulus of elasticity      44
Material, properties      89—99 104
Mesh, construction by master elements      175—178
Mesh, finite element      16 23 72—73 77 150 223
Mesh, parameter h      36
Mesh, refinement      21 36 125
Midstep time integration      252
Mixed boundary conditions      83
Modulus of elasticity      44 127 242
Natural boundary conditions, definition of      49 138 146 231—232
Natural boundary conditions, for beam bending      239
Natural boundary conditions, for fourth order problems      231—232
Natural boundary conditions, in one dimension      80
Natural boundary conditions, in three dimensions      223
Natural boundary conditions, in two dimensions      141
Natural boundary conditions, relation to conservation principle      139—140
Neumann, boundary conditions      82
Neumann, problem      82 85
Nodal, displacements      168
Nodal, points      16 23 72
Nodal, values      33
Norm, ${H}^{1}-$       57
Norm, definitions of      35
Norm, energy      35 38 57
Norm, maximum      35—36
Norm, mean-square      35—36 38
Numerical integration      see “Integration”
Orthogonality      14 38
Parabolic equations      247
Parametric map      193
Pascal's triangle      157 204
Penalty method      121—123
Plane, strain      127
Plane, stress      127 242—246
Plate bending      235
Point loads      see “Dirac delta”
Point source      see “Dirac delta”
Poisson's ratio      242 246
Post processing      88 100-11 210—225
Principle, conservation      see “Conservation principle”
Principle, of superposition      41
Principle, of virtual work      244 246
Quadratic element      196—197
Quadrature      see “Integration”
Quadrature, error      190
Quadrature, Gauss      see “Gaussian quadrature”
Quadrature, points      198—199 205—206
Quadrature, rules for quadrilaterals      198—199
Quadrature, rules for triangles      205—206
Quadrature, weights      190 192
Quadrilateral elements      195—199 207
Quintic triangle      237
Rate of convergence      36—37 84—85
Rate of convergence, of interpolation error      161
Refinement of mesh      21 36 125
Residual, for fourth order problems      229
Residual, for the model problem      5
Residual, for time dependent problems      248
Residual, of the differential equation      6 53 144
Self-adjoint, boundary-value problem      26
Self-adjoint, operator      63—64
Semi-discrete method      249—250
Shape functions, calculation      95—96 217 219
Shape functions, definition      26
Shape functions, derivative calculation      95—96 189 216—227
Shape functions, for one dimension      26—27 65—68 72—74

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