Differential Geometry: Theory and Applications (Contemporary Applied Mathematics)

Contents......Page 8
Preface......Page 6
Introduction......Page 10
1 Three-dimensional differential geometry Outline......Page 11
1.1 Curvilinear coordinates......Page 12
1.2 Metric tensor......Page 15
1.3 Volumes, areas, and lengths in curvilinear coordinates......Page 18
1.4 Covariant derivatives of a vector field......Page 21
1.5 Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor......Page 26
1.6 Existence of an immersion defined on an open set in R3 with a prescribed metric tensor......Page 28
1.7 Uniqueness up to isometries of immersions with the same metric tensor......Page 40
1.8 Continuity of an immersion as a function of its metric tensor......Page 48
2 Differential geometry of surfaces Out line......Page 52
2.1 Curvilinear coordinates on a surface......Page 55
2.2 First fundamental form......Page 56
2.3 Areas and lengths on a surface......Page 61
2.4 Second fundamental form; curvature on a surface......Page 62
2.5 Principal curvatures; Gaussian curvature......Page 68
2.6 Covariant derivatives of a vector field defined on a surface; the Gauss and Weingarten formulas......Page 73
2.7 Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations; Gauss' Theorema Ggregium......Page 77
2.8 Existence of a surface with prescribed first and second fundamental forms......Page 81
2.9 Uniqueness up to proper isometries of surfaces with the same fundamental forms......Page 92
2.10 Continuity of a surface as a function of its fundamental forms......Page 97
References......Page 99
Introduction......Page 103
1 Three-dimensional theory Outline......Page 104
1.1 Notation, definitions, and some basic formulas......Page 105
1.2 Equations of equilibrium......Page 107
1.3 Constitutive equations of elastic materials......Page 111
1.4 The equations of nonlinear and linearized three-dimensional elasticity......Page 115
1.5 A fundamental lemma of J.L. Lions......Page 117
1.6 Existence theory in linearized three-dimensional elasticity......Page 118
1.7 Existence theory in nonlinear three-dimensional elasticity by the implicit function theorem......Page 123
1.8 Existence theory in nonlinear three-dimensional elasticity by the minimization of energy (John Ball’s approach)......Page 126
2 Two-dimensional theory Outline......Page 130
2.1 A quick review of the differential geometry of surfaces in R3......Page 132
2.2 Geometry of a shell......Page 134
2.3 The three-dimensional shell equations......Page 137
2.4 The two-dimensional approach to shell theory......Page 139
2.5 Nonlinear shell models obtained by r-convergence......Page 141
2.6 Linear shell models obtained by asymptotic analysis......Page 151
2.7 The nonlinear Koiter shell model......Page 156
2.8 The linear Koiter shell model......Page 159
2.9 Korn’s inequalities on a surface......Page 167
2.10 Existence, uniqueness, and regularity of the solution to the linear Koiter shell model......Page 180
References......Page 188
1 Introduction......Page 194
2.1 Discretizations of classical shell models......Page 197
2.2 General shell elements......Page 200
3 Computational reliability issues for thin shells......Page 205
3.1 Asymptotic behaviours of shell models......Page 206
3.2 Asymptotic reliability of shell finite elements......Page 212
3.3 Guidelines for assessing and improving the reli- ability of shell finite elements......Page 219
References......Page 227
Introduction......Page 231
1.1 Triangulations and meshes......Page 237
1.1.1 Numerical simulations......Page 240
1.1.3 Mesh adaptation......Page 241
1.2 Notion of metric tensor......Page 242
1.2.1 Metric, scalar product and distance......Page 243
1.2.2 Metric decomposition......Page 245
1.2.3 Geometric representation......Page 246
1.2.4 Metric intersection......Page 247
1.2.5 Metric interpolation......Page 248
1.3 A differential geometry primer......Page 249
1.3.1 The first fundamental form; area......Page 250
1.3.2 The second fundamental form; curvatures......Page 252
2 A geometric error estimate......Page 255
2.1.1 Interpolation error in L2 norm and H1 seminorm......Page 256
2.1.2 PI elements in one dimension......Page 259
2.1.3 Lagrange PI elements in two dimensions......Page 261
2.2.1 Anisotropic formulation of the interpolation error......Page 265
2.2.3 Metric construction......Page 267
2.2.4 Evaluation of the Hessian matrix......Page 268
2.2.5 An error estimate for CFD problems......Page 272
3 Mesh adaptation using a geometric error estimate......Page 273
3.1.1 The Delaunay triangulation......Page 274
3.1.2 Constrained triangulation......Page 275
3.1.4 Creation and insertion of internal points......Page 277
3.2.1 Problem statement: the concept of geometric mesh......Page 279
3.2.2 Local deformation of a surface......Page 283
3.2.3 Local curvature of a surface......Page 284
3.2.4 Evaluation of the intrinsic properties of a discrete sur- face......Page 285
4.1 An academic example......Page 288
4.2 Curvature-driven evolution flows......Page 290
4.3 A CFD example in two dimensions......Page 293
4.5 Mesh adaptation in three dimensions......Page 296
References......Page 299