Sanchez-Palencia / Millet / Bechet Singular Problems in Shell Theory

Computing and Asymptotics
1. Auflage 2010
ISBN: 978-3-642-13815-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 54, 266 Seiten

Reihe: Lecture Notes in Applied and Computational Mechanics

This book deals with various aspects in relation with thin shell theory: general geometric formalism of shell theory, analysis of singularities, numerical computing of thin shell problems, mathematical considerations on boundary values problems.

Autoren/Hrsg.

Sanchez-Palencia, Evariste

Millet, Olivier

Bechet, Fabien

Inhaltsverzeichnis

1;Title;1
2;Contents;5
3;Notations;11
4;Introduction;14
4.1;Non-inhibited (or non-geometrically rigid) middle surface;18
4.2;Inhibited (or geometrically rigid) hyperbolic shells;19
4.3;Inhibited (or geometrically rigid) parabolic shells;20
4.4;Well-inhibited elliptic shells (fixed or clamped all along the boundary);20
4.5;Ill-inhibited elliptic shells (fixed or clamped along a part of the boundary, free by the rest);20
5;Geometric Formalism of Shell Theory;25
5.1;Introduction;25
5.2;Recall on Surface Theory;25
5.2.1;Mapping - Covariant Basis;25
5.2.2;First Fundamental Form of the Surface S - Contravariant Basis;26
5.2.3;Second Fundamental Form;27
5.3;Classification of Surfaces;28
5.4;Differentiation on the Surface S;30
5.5;Surface Rigidity;33
5.5.1;Deformation of a Surface;33
5.5.2;The Rigidity System and its Characteristic Curves;34
5.5.3;Handling Systems of Equations with Various Orders: Indices of Equations and Unknowns;37
5.6;The Koiter Shell Model;38
5.7;The Limit Membrane Model;41
5.7.1;The Membrane Model;41
5.7.2;The System of Membrane Tension;42
5.7.3;Back to the Membrane System;43
6;Singularities and Boundary Layers in Thin Elastic Shell Theory;45
6.1;Introduction;45
6.2;Geometrically Rigid Surfaces;46
6.2.1;Inextensional Displacements;46
6.2.2;Examples of Geometrically Rigid Surface;47
6.3;Limit Behavior of Koiter Model;49
6.3.1;The Limit Membrane Problem;49
6.3.2;Boundary Layers and Singularities;50
6.3.3;Convergence to the Membrane Model in the Inhibited Case;50
6.3.4;A More General Result of Convergence;52
6.3.5;Convergence to the Pure Bending Model in the Non-inhibited Case;55
6.4;Complements on Nagdhi Model and its Limits;57
6.5;Reduction of the Membrane System to One PDE for Each Component of the Displacement;59
6.5.1;Case of the Normal Displacement u3;60
6.5.2;Tangential Displacements u1 and u2;61
6.6;Structure of the Displacement Singularities when the Loading Is Singular along a Curve;62
6.6.1;Singularity along a Non-characteristic Line;65
6.6.2;Singularity along a Characteristic Line;68
6.6.3;Summary of the Results;73
6.7;Pseudo-reflections for Hyperbolic Shells;75
6.8;Thickness of the Layers;75
6.8.1;Case of a Layer along a Non-characteristic Line;76
6.8.2;Case of a Layer along a Characteristic Line;77
6.9;Conclusion;79
7;Anisotropic Error Estimates in the Layers;81
7.1;Introduction;81
7.2;Estimate for Galerkin Approximation in Singular Perturbation and Penalty Problems;82
7.2.1;Degradation of the Estimate in a Singular Perturbation Problem;84
7.2.2;Degradation of the Estimate in a Penalty Problem;84
7.3;Interpolation Error for Isotropic Meshes in Layers;85
7.3.1;The Basic F. E. Interpolation Error Estimate;85
7.3.2;Case of a Layer: Interpolation Error for Isotropic Meshes;86
7.4;Interpolation Error for Anisotropic Meshes in Layers;88
7.5;Galerkin Error Estimates in a Layer;90
7.6;First Remarks on Approximations in Layers;92
7.7;Estimates for Significant Entities in the Layer: Local Locking in Layers;94
7.8;Conclusion;97
8;Numerical Simulation with Anisotropic Adaptive Mesh;99
8.1;Introduction;99
8.2;Review on the Numerical Locking;100
8.2.1;Introduction;100
8.2.2;Locking in the Non-inhibited Case (Classical Locking Associated with a Limit Constraint);100
8.2.3;Locking in the Inhibited Case (Singular Perturbations);105
8.3;Shell Element and Associated Discrete Problem;106
8.3.1;The Shell Element D.K.T.;107
8.3.2;Discretization of Naghdi Model;108
8.3.3;Adaptive Mesh Strategy: BAMG;110
8.3.4;Coupling BAMG-MODULEF for Shell Computations;112
8.4;Membrane and Bending Energies Computation with MODULEF;113
8.4.1;Implementation Procedure in MODULEF;113
8.4.2;Validation on Simple Examples;114
8.5;Conclusion;117
9;Singularities of Parabolic Inhibited Shells;118
9.1;Introduction;118
9.2;Study of the Singularities and of Their Propagation;119
9.2.1;Singularity along a Characteristic Line;120
9.2.2;Singularity along a Non-characteristic Line;122
9.3;Example of a Half-Cylinder;125
9.3.1;Geometric Description of the Cylinder;125
9.3.2;Constitutive Law;127
9.3.3;Loading and Boundary Conditions;127
9.4;Numerical Simulations with Anisotropic Adaptive Mesh;135
9.4.1;Remark for the Interpretation of the Numerical Results in Terms of Singularities;136
9.4.2;Convergence of the Adaptive Mesh Procedure;137
9.4.3;Computing the Displacements;138
9.4.4;Influence of the Relative Thickness e;140
9.4.5;Localization of Membrane and Bending Energies;142
9.5;Comparison between Uniform and Adapted Meshes;144
9.6;Numerical Study of Singularities on Non-characteristic Lines;146
9.7;Singularity along a Boundary;147
9.7.1;Theoretical Considerations;148
9.7.2;Numerical Simulations;148
9.8;Singularities due to the Shape of the Domain;153
9.8.1;Conclusion;155
10;Singularities of Hyperbolic Inhibited Shells;157
10.1;Introduction;157
10.2;The Limit Problem for a Hyperbolic Inhibited Shell;157
10.2.1;Example of a Hyperbolic Paraboloid;158
10.2.2;Singularities of the Displacements due to a Loading Singular on the Line y1 = 0;159
10.2.3;Three Cases of Loading;161
10.2.4;The Singularities of the Resulting Displacements;164
10.3;Numerical Computations Using Adaptive Meshes;164
10.3.1;Numerical Results for Loading A;164
10.3.2;Results for the Loading B;168
10.3.3;Results for the Loading C;171
10.4;Some Examples Including Pseudo-reflections;173
10.4.1;Reflection of a Characteristic Layer;173
10.4.2;Reflection of a Non-characteristic Layer;175
10.4.3;Reflection of a Characteristic Layer when the Loading “Touches” the Non-characteristic Boundary;178
10.5;Conclusion;180
11;Singularities of Elliptic Well-Inhibited Shells;181
11.1;Introduction;181
11.2;Existence of Logarithmic Point Singularities at the Corners of the Loading Domain;181
11.2.1;Model Problem of Second Order;183
11.2.2;The Membrane Problem .2u3 = C4 f3(.);185
11.2.3;Particular Case when the Logarithmic Point Singularity Vanishes;187
11.2.4;Existence Condition of a Logarithmic Singularity;187
11.3;Example of an Elliptic Paraboloid;191
11.3.1;Geometric Properties;192
11.3.2;Numerical Results;193
11.3.3;Mesh Adaptation;194
11.3.4;Thickness of the Internal Layer along y1 = 0.5;197
11.3.5;The Logarithmic Singularity at the Corner;199
11.3.6;Membrane and Bending Energies;202
11.4;Conclusion;203
12;Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems;205
12.1;Introduction;205
12.2;The Cauchy Problem for Equations and Systems;206
12.2.1;Generalities;206
12.2.2;Role of the Characteristics;207
12.2.3;Normal Form of a Hyperbolic System: Riemann Invariants;209
12.2.4;Elliptic Equations or Systems;211
12.3;Boundary Value Problems for Elliptic Equations and Systems;214
12.3.1;Regularity of the Solution;214
12.3.2;The Shapiro–Lopatinskii Condition;216
12.4;The Shapiro–Lopatinskii Condition and the Membrane Problem;217
12.5;Sensitive Problems;220
12.5.1;Elliptic Shell Clamped by a Part G0 of the Boundary and Free by the Rest G1;220
12.5.2;Qualitative Description of the Solution of Sensitive Problems;222
12.5.3;Heuristic Treatment of the Problem;224
12.6;Conclusion;226
13;Numerical Simulations for Sensitive Shells;228
13.1;Introduction;228
13.2;First Examples of Numerical Computations for Sensitive Problems (Ill-Inhibited Shells);229
13.3;Asymptotic Process when e Tends to Zero;231
13.4;Influence of the Free Edge Length;234
13.5;Energy Repartition in Sensitive Problems;237
13.6;Influence of the Loading Domain;238
13.7;Conclusion;241
14;Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems);243
14.1;Examples of Partially Non-inhibited Shells;244
14.1.1;First Case: a = 0 and ß = 0.25;244
14.1.2;Second Case: a = 0.25 and ß = 0.25;246
14.2;Propagation of Singularities in the Partially Non-inhibited Regions;248
14.2.1;Loading Applied in the Inhibited Area;248
14.2.2;Loading Domain Tangent to the Non-inhibited Area;251
14.2.3;Loading Partially Applied in the Non-inhibited Area;252
14.3;Conclusion;253
15;References;254
16;Characteristics of the Membrane System;260
17;Reduced Membrane and Koiter Equations;262
17.1;Membrane Problem;262
17.1.1;Case of the Normal Displacement u3;263
17.1.2;Reduced Equation for the Tangential Displacements u1 and u2;266
17.2;Koiter Problem;267
18;Index;269