Singular Problems in Shell Theory: Computing and Asymptotics (Lecture Notes in Applied and Computational Mechanics, 54)

Lecture Notes in Applied and Computational Mechanics Volume 54 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. D...

Author: Evariste Sanchez-Palencia | Olivier Millet | Fabien Bechet

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Lecture Notes in Applied and Computational Mechanics Volume 54 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes of this series found on our homepage: springer.com Vol. 54: Sanchez-Palencia, E., Millet, O., Béchet, F. Singular Problems in Shell Theory 265 p. 2010 [978-3-642-13814-0]

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Vol. 39: Lucchesi, M., Padovani, C., Pasquinelli, G., Zani, N. Masonry Constructions: Mechanical Models and Numerical Applications 176 p. 2008 [978-3-540-79110-2]

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Vol. 37: Chaudhary, H., Saha, S.K. Dynamics and Balancing of Multibody Systems 200 p. 2008 [978-3-540-78178-3]

Vol. 48: Su, Z.; Ye, L. Identification of Damage Using Lamb Waves 346 p. 2009 [978-1-84882-783-7]

Vol. 36: Leine, R.I.; van de Wouw, N. Stability and Convergence of Mechanical Systems with Unilateral Constraints 250 p. 2008 [978-3-540-76974-3]

Vol. 47: Studer, C. Numerics of Unilateral Contacts and Friction 191 p. 2009 [978-3-642-01099-6]

Vol. 35: Acary, V.; Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics 545 p. 2008 [978-3-540-75391-9]

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Vol. 34: Flores, P.; Ambrósio, J.; Pimenta Claro, J.C.; Lankarani Hamid M. Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies 186 p. 2008 [978-3-540-74359-0

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Spectral Method in Multiaxial Random Fatigue 146 p. 2007 [978-3-540-73822-0] Vol. 32: Bardzokas, D.I.; Filshtinsky, M.L.; Filshtinsky, L.A. (Eds.) Mathematical Methods in Electro-Magneto-Elasticity 530 p. 2007 [978-3-540-71030-1] Vol. 31: Lehmann, L. (Ed.) Wave Propagation in Infinite Domains 186 p. 2007 [978-3-540-71108-7]

Singular Problems in Shell Theory Computing and Asymptotics

Evariste Sanchez-Palencia, Olivier Millet, Fabien Béchet

123

Prof. Evariste Sanchez-Palencia Institut Jean Le Rond d’Alembert 4 place Jussieu 75252 Paris Cedex 05 France E-mail: [email protected]

Dr. Fabien Bechet Metz University LPMM Ile du Saulcy 57045 Metz Cedex 01 France E-mail: [email protected]

Prof. Olivier Millet La Rochelle University LEPTIAB Avenue Michel Crépeau 17000 La Rochelle France E-mail: [email protected]

ISBN: 978-3-642-13814-0

e-ISBN: 978-3-642-13815-7

DOI 10.1007/ 978-3-642-13815-7 Lecture Notes in Applied and Computational Mechanics

ISSN 1613-7736 e-ISSN 1860-0816

Library of Congress Control Number: 2010928271 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 9876543210 springer.com

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

1

Geometric Formalism of Shell Theory . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Recall on Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Mapping - Covariant Basis . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 First Fundamental Form of the Surface S - Contravariant Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classification of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Differentiation on the Surface S . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Surface Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Deformation of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 The Rigidity System and Its Characteristic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Handling Systems of Equations with Various Orders: Indices of Equations and Unknowns . . . . . . . . 1.6 The Koiter Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Limit Membrane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 The Membrane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 The System of Membrane Tension . . . . . . . . . . . . . . . . . 1.7.3 Back to the Membrane System . . . . . . . . . . . . . . . . . . . .

13 13 13 13

25 26 29 29 30 31

Singularities and Boundary Layers in Thin Elastic Shell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Geometrically Rigid Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Inextensional Displacements . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Examples of Geometrically Rigid Surface . . . . . . . . . . .

33 33 34 34 35

14 15 16 18 21 21 22

VI

Contents

2.3 Limit 2.3.1 2.3.2 2.3.3

2.4 2.5

2.6

2.7 2.8

2.9 3

4

Behavior of Koiter Model . . . . . . . . . . . . . . . . . . . . . . . . The Limit Membrane Problem . . . . . . . . . . . . . . . . . . . . Boundary Layers and Singularities . . . . . . . . . . . . . . . . . Convergence to the Membrane Model in the Inhibited Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 A More General Result of Convergence . . . . . . . . . . . . . 2.3.5 Convergence to the Pure Bending Model in the Non-inhibited Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complements on Nagdhi Model and its Limits . . . . . . . . . . . . Reduction of the Membrane System to one PDE for Each Component of the Displacement . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Case of the Normal Displacement u3 . . . . . . . . . . . . . . . 2.5.2 Tangential Displacements u1 and u2 . . . . . . . . . . . . . . . Structure of the Displacement Singularities when the Loading is Singular along a Curve . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Singularity along a Non-characteristic Line . . . . . . . . . 2.6.2 Singularity along a Characteristic Line . . . . . . . . . . . . . 2.6.3 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . Pseudo-reflections for Hyperbolic Shells . . . . . . . . . . . . . . . . . . Thickness of the Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Case of a Layer along a Non-characteristic Line . . . . . 2.8.2 Case of a Layer along a Characteristic Line . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anisotropic Error Estimates in the Layers . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Estimate for Galerkin Approximation in Singular Perturbation and Penalty Problems . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Degradation of the Estimate in a Singular Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Degradation of the Estimate in a Penalty Problem . . . 3.3 Interpolation Error for Isotropic Meshes in Layers . . . . . . . . . 3.3.1 The Basic F. E. Interpolation Error Estimate . . . . . . . 3.3.2 Case of a Layer: Interpolation Error for Isotropic Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Interpolation Error for Anisotropic Meshes in Layers . . . . . . . 3.5 Galerkin Error Estimates in a Layer . . . . . . . . . . . . . . . . . . . . . 3.6 First Remarks on Approximations in Layers . . . . . . . . . . . . . . 3.7 Estimates for Significant Entities in the Layer: Local Locking in Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation with Anisotropic Adaptive Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 38 38 40 43 45 47 48 49 50 53 56 61 63 63 64 65 67 69 69 70 72 72 73 73 74 76 78 80 82 85

87 87

Contents

4.2 Review on the Numerical Locking . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Locking in the Non-inhibited Case (Classical Locking Associated with a Limit Constraint) . . . . . . . 4.2.3 Locking in the Inhibited Case (Singular Perturbations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Shell Element and Associated Discrete Problem . . . . . . . . . . . 4.3.1 The Shell Element D.K.T. . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Discretization of Naghdi Model . . . . . . . . . . . . . . . . . . . 4.3.3 Adaptive Mesh Strategy: BAMG . . . . . . . . . . . . . . . . . . 4.3.4 Coupling BAMG-MODULEF for Shell Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Membrane and Bending Energies Computation with MODULEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Implementation Procedure in MODULEF . . . . . . . . . . 4.4.2 Validation on Simple Examples . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Singularities of Parabolic Inhibited Shells . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Study of the Singularities and of their Propagation . . . . . . . . 5.2.1 Singularity along a Characteristic Line . . . . . . . . . . . . . 5.2.2 Singularity along a Non-characteristic Line . . . . . . . . . 5.3 Example of a Half-Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Geometric Description of the Cylinder . . . . . . . . . . . . . 5.3.2 Constitutive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Loading and Boundary Conditions . . . . . . . . . . . . . . . . . 5.3.4 Singularities of the Displacements . . . . . . . . . . . . . . . . . 5.4 Numerical Simulations with Anisotropic Adaptive Mesh . . . . 5.4.1 Remark for the Interpretation of the Numerical Results in Terms of Singularities . . . . . . . . . . . . . . . . . . 5.4.2 Convergence of the Adaptive Mesh Procedure . . . . . . . 5.4.3 Computing the Displacements . . . . . . . . . . . . . . . . . . . . . 5.4.4 Influence of the Relative Thickness ε . . . . . . . . . . . . . . . 5.4.5 Localization of Membrane and Bending Energies . . . . 5.5 Comparison between Uniform and Adapted Meshes . . . . . . . . 5.6 Numerical Study of Singularities on Non-characteristic Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Singularity along a Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Singularities due to the Shape of the Domain . . . . . . . . . . . . . 5.8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII

88 88 88 93 94 95 96 98 100 101 101 102 105 107 107 108 109 111 114 114 116 116 119 124 125 126 127 129 131 133 135 136 137 137 142 144

VIII

6

7

8

Contents

Singularities of Hyperbolic Inhibited Shells . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Limit Problem for a Hyperbolic Inhibited Shell . . . . . . . . 6.2.1 Example of a Hyperbolic Paraboloid . . . . . . . . . . . . . . . 6.2.2 Singularities of the Displacements due to a Loading Singular on the Line y 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Three Cases of Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Singularities of the Resulting Displacements . . . . 6.3 Numerical Computations Using Adaptive Meshes . . . . . . . . . . 6.3.1 Numerical Results for Loading A . . . . . . . . . . . . . . . . . . 6.3.2 Results for the Loading B . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Results for the Loading C . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Some Examples Including Pseudo-reflections . . . . . . . . . . . . . . 6.4.1 Reflection of a Characteristic Layer . . . . . . . . . . . . . . . . 6.4.2 Reflection of a Non-characteristic Layer . . . . . . . . . . . . 6.4.3 Reflection of a Characteristic Layer when the Loading “Touches” The Non-characteristic Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singularities of Elliptic Well-Inhibited Shells . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Existence of Logarithmic Point Singularities at the Corners of the Loading Domain . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Model Problem of Second Order . . . . . . . . . . . . . . . . . . 7.2.2 The Membrane Problem Δ2 u3 = C4 f 3 (θ) . . . . . . . . . 7.2.3 Particular Case when the Logarithmic Point Singularity Vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Existence Condition of a Logarithmic Singularity . . . . 7.3 Example of an Elliptic Paraboloid . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Geometric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Mesh Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Thickness of the Internal Layer along y 1 = 0.5 . . . . . . 7.3.5 The Logarithmic Singularity at the Corner . . . . . . . . . 7.3.6 Membrane and Bending Energies . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Cauchy Problem for Equations and Systems . . . . . . . . . . 8.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Role of the Characteristics . . . . . . . . . . . . . . . . . . . . . . . .

147 147 147 148 149 151 154 154 154 158 161 163 163 165

168 170 171 171 171 173 175 177 177 181 182 183 184 187 189 192 193 195 195 196 196 197

Contents

8.3

8.4 8.5

8.6 9

IX

8.2.3 Normal Form of a Hyperbolic System: Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Elliptic Equations or Systems . . . . . . . . . . . . . . . . . . . . . Boundary Value Problems for Elliptic Equations and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Regularity of the Solution . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Shapiro–Lopatinskii Condition . . . . . . . . . . . . . . . . The Shapiro–Lopatinskii Condition and the Membrane Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Elliptic Shell Clamped by a Part Γ0 of the Boundary and Free by the Rest Γ1 . . . . . . . . . . . . . . . . 8.5.2 Qualitative Description of the Solution of Sensitive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Heuristic Treatment of the Problem . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Numerical Simulations for Sensitive Shells . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 First Examples of Numerical Computations for Sensitive Problems (Ill-Inhibited Shells) . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Asymptotic Process when ε Tends to Zero . . . . . . . . . . . . . . . . 9.4 Influence of the Free Edge Length . . . . . . . . . . . . . . . . . . . . . . . 9.5 Energy Repartition in Sensitive Problems . . . . . . . . . . . . . . . . . 9.6 Influence of the Loading Domain . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems) . . . . . . . . . . . . . . . . . . . . . 10.1 Examples of Partially Non-inhibited Shells . . . . . . . . . . . . . . . . 10.1.1 First Case: α = 0 and β = 0.25 . . . . . . . . . . . . . . . . . . . . 10.1.2 Second Case: α = 0.25 and β = 0.25 . . . . . . . . . . . . . . . 10.2 Propagation of Singularities in the Partially Non-inhibited Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Loading Applied in the Inhibited Area . . . . . . . . . . . . . 10.2.2 Loading Domain Tangent to the Non-inhibited Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Loading Partially Applied in the Non-inhibited Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 201 204 204 206 207 210 210 212 214 216 219 219 220 222 225 228 229 232 235 236 236 238 240 240 243 244 245

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 APPENDICES A

Characteristics of the Membrane System . . . . . . . . . . . . . . . . 253

X

B

Contents

Reduced Membrane and Koiter Equations . . . . . . . . . . . . . . . B.1 Membrane Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Case of the Normal Displacement u3 . . . . . . . . . . . . . . . B.1.2 Reduced Equation for the Tangential Displacements u1 and u2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Koiter Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 256 259 260

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Notations

latine subscripts (i, j, k . . . )

vary from 1 to 3

greek subscripts (α, β, γ . . . )

vary from 1 to 2

ε

thickness of the shell

η

layer thickness

S

middle surface of the shell

∂S

boundary of S

Ω

domain of R2

(Ω, Ψ )

local mapping

Tp S

tangent plane to S at point p

(y 1 , y 2 )

local system of coordinates

(a1 , a2 )

covariant basis of tangent plane

(a1 , a2 )

contravariant basis of tangent plane

N = a3 = a3

unit normal vector to S

aαβ , aαβ

covariant or contravariant components of metric tensor

a = det(aαβ )

determinent of metric tensor

XII

Notations

bαβ , bβα

covariant or mixed components of second fundamental form

uε

displacement solution of Koiter model for a shell with a thickness ε

u0 or u

limit solution of membrane model for ε = 0

uα , u α

covariant or contravariant components of the tangent displacement u

u3 = u3

normal displacement

γαβ , γαβ

covariant or mixed components of membrane strain tensor

ραβ , ρβα

covariant or mixed components of curvature variation tensor

θα

components of rotation vector of the normal a3

T αβ

contravariant components of membrane stress tensor

Aαβγδ

contravariant components of elastic stiffness tensor

Bαβγδ

covariant components of compliance elastic tensor

G

subspace of inextentional displacements

δαβ

Kronecker symbol

λ Γαβ

Christoffel symbols

∂α or

∂ ∂y α

derivative with respect to y α

∂αi or

∂ ∂(y α )i

ith order derivative with respect to y α

Dα uβ or uβ |α

covariant derivative of uβ with respect to y α

Δ

Laplacian operator

V

space of admissible displacements

V

.V

dual space of V natural norm on V

Notations

XIII

v

test displacement in V

Vh

sequence of subspaces of V of finite dimension

uεh

discrete approximation of the solution uε on Vh

am (·, ·)

bilinear form of membrane energy

ab (·, ·)

bilinear form of bending energy

Vm

completion space of V with the norm induced by am (., .)

f

applied loading

fα

tangential component of loading f

f3

normal component of loading f

f (.)

linear form (representing the work of external forces)

., .V ×V

duality product in V × V

δ

Dirac distribution

δ,δ ,δ

successive derivatives of Dirac distribution

H

Heaviside step function

C 0 (Ω)

space of continuous functions on Ω

C k (Ω)

space of continuous differentiable functions up to order k on Ω

L2 (Ω)

space of square-summable functions on Ω

H k (Ω)

Sobolev space of order k

H −k (Ω)

Sobolev space of negative order −k (dual space of H0k (Ω))

E

Young modulus

ν

Poisson coefficient

Introduction

Thin shells are three-dimensional structures with a dimension (the thickness) small with respect to the two others. Such thin structures are widely used in automobile and aviation industries, or in civil engineering, because they provide an important stiffness, due to their curvature, with a small weight.

Fig. 0.1. Airbus A380

Fig. 0.2. Hemispherical roof (Marseille, France)

One of the challenges is often to reduce the weight (and consequently the thickness) of the shells, preserving their stiffness. So that it is essential to have accurate models for thin and even very thin shells1 , and to be able to compute the displacements resulting from a given loading. In particular, singularities leading to fractures in some cases must be absolutely predicted a priori and of course avoided (see Fig. 0.3 for example). Since the pioneering models of Novozhilov-Donnell [81] and Koiter [65][66], numerous works have been devoted to establish linear and non linear elastic shell model using direct or surfacic approaches [18][25][100]. More recently, the asymptotic methods [87] have been used, to try to justify rigorously, from the three-dimensional equations, the shell models obtained by direct approaches relying on a priori assumption, and to construct new models [54][55]. This way, 1

Very thin shells are present in certain domains of industry, as plastic films for packaging or for electronics, streched sails, or even very thin metal sheets obtained by drawing.

E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 1–11. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

2

Introduction

Fig. 0.3. Example of a propagated singularity leading to a fracture of the gas pipeline (Belgium, July 2004)

numerous plate and shell models have been justified in linear and non-linear elasticity [30, 31, 32, 33, 41, 73, 74, 75, 76, 77, 78, 79, 80, 88, 91, 92, 93]. Asymptotic approaches were also extended to the justification of linear and non-linear elastic beams and thin-walled beam models [34, 49, 50, 51, 52, 56, 57, 86, 99]. This book focuses on the study of singularities in linear elastic shell theory which appear for very small thicknesses. We choose to use the linear Koiter shell model [65][66] for the theoretical and numerical study of the singularities that will be carried out. The Koiter shell model is one of the most currently used for numerical computations because it contains both membrane and bending effects coupled at different order of magnitudes: the membrane energy is proportional to the relative thickness ε, whereas the bending energy is proportional to ε3 . It is important to quote that the use of other shell models (with transverse shear for example) would lead to the same results for very small thicknesses (that are the cases considered in this book) and to the same singularities. Obviously, as we will see in the sequel, when the singularities appearing at the limit (for very small values of ε) are completely developed, moderate displacements may result from the level of applied forces considered (in particular for sensitive problems) according to the boundary conditions. In such cases, even if we are at the limit of the domain of validity of the linear Koiter model, we focus on the process of formation and evolution of the singularities, and not on accurate computations of the bending displacements. Therefore, the use of the linear Koiter shell model for the numerical simulations that will be performed is relevant. It would always be possible, as the problem is linear, to consider weaker loadings leading to displacements (essentially bending displacement) more in adequation with the linear range. This would not modify the qualitative analysis developed in this book. On the other hand, for the numerical computation of Koiter model that will be performed thoughout this book, we will use the DKT finite element for shells. Of course, other choices of finite elements would be possible and relevant (MITC, element with transverse shear, etc.). However, the results obtained for very thin shells would be the same, as the theoretical limit and the resulting singularities do not depend on the finite element used.

Introduction

3

Main features of deformation of very thin shells are not widely known, and there is a certain misunderstanding in this concern. Everybody knows that numerical locking is a serious drawback when computing shells, but some think that it is an imperfection of the shell model or of the finite element employed. In fact, it is inherent to the peculiarities of the deformation for very small ε. There is no magic finite element or model avoiding locking in any situation. We must live (and compute!) with it. But a little knowledge of that peculiarities helps very much to improve computations and to have an idea of their reliability. The situation, as we shall explain later, is very much alike as in computing fluid flows. As a rule, “almost any” shell model and finite element works fairly well for moderately small ε. Difficulties arise for very small ε, and benchmarks should be used in that case. For very small ε, the main feature of shells is that the membrane rigidity is proportional to ε, whereas bending rigidity is proportional to ε3 . Obviously, the natural trend of the shell is to perform (if possible!) pure bendings, i.e. displacements not modifying lengths of the middle surface, as a paper sheet that deforms into a developable surface. But (according to the geometric properties of the middle surface and to the fixation of the boundary) this is not always possible. In that case, the shell “must use” its membrane rigidity, which is much larger than the bending one. Obviously, in most applications, this is the favorable situation, and most of the shells used in applied mechanics are in that case. Let us specify the definition of geometrically rigid or inhibited shells (respectively, non-geometrically rigid or non-inhibited shells) that will be used in the sequel. A shell is said to be “geometrically rigid” or “inhibited” when the only existing “inextensional displacements” (which keep invariant the metrics of the middle surface of the shell) are zero, i.e. the pure bendings are inhibited. Obviously, as we will see, the existence of such inextensional displacements depends not only on the geometric nature of the middle surface (elliptic, parabolic or hyperbolic), but also on the associated boundary conditions [26][88][92]. When a shell is geometrically rigid or inhibited, the Koiter model tends for very small thicknesses to the membrane model (we have a classical singular perturbation problem [69]). Oppositely, when a shell is non-geometrically rigid or non-inhibited, the limit model is the pure bending model, accounting only to the bending effects. In the later case, penalty terms naturally emerge during the asymptotic process, leading to a limit problem in the constrained sub-space of inextensional displacements. It follows that in the inhibited case, the behavior is, in principle, “membranelike”. In that case, on one hand, the solutions of the limit membrane problem (for a zero thickness) are less regular than the solutions of the Koiter model for a strictly positive thickness (in particular the normal displacement u3 ). Thus, for very small thicknesses, classical boundary layers appear near the boundaries. On the other hand, it generally happens that the loading does not satisfy the mathematical condition f ∈ Vm , where f denotes the applied loading (for more

4

Introduction

details and the accurate definition of the space2 Vm , see section 2.3 and in particular (2.6)). As we will see in the sequel, this condition is more or less restrictive according to the geometry of the shell (elliptic, parabolic or hyperbolic)3 . For example, it is not satisfied for hyperbolic shells as soon as the normal loading f 3 has a jump across a regular curve, and for parabolic shells as soon as the gradient of f 3 is discontinuous across a regular curve. In all that cases, when f 3 is not in Vm , the energy of the shell “tends to become infinite” when the thickness ε tends to zero. Therefore, internal layers appear around the discontinuity of the loading. In these layers, the displacements tend (when the thickness approaches zero) to be very singular [96], and both membrane and bending energies are of the same order. Moreover, a very important point is that these internal (and boundary) layers can propagate along the characteristic curves of the shell (for parabolic and hyperbolic shells, elliptic shells having no real characteristics). Generally, the loss of regularity of the displacements leading to several kinds of possible layers is the most important when the shell is not elliptic. For elliptic shells, when a part of the boundary is free, a very peculiar and less known phenomenon appears: large oscillations on the free edge progressively emerge when the thickness decreases to zero [12][48][70][83]. Moreover, the phenomenon is non-local, and this instability progressively spread all over the shell! This is the case of ill-inhibited or sensitive shells4 that will be considered in detail in chapters 8 and 9. The diversity of the limit problems and of the associated singularities existing in shell theory when the thickness ε tends to zero induces serious difficulties for the numerical computation of the Koiter shell model (or of any another shell model). Indeed, as well as the membrane locking is present in most of the situations, the main difficulty is to describe accurately the singularities of the displacements in the layers resulting from a singularity of the loading for very small thicknesses. As in these layers, the resulting displacements vary strongly and anisotropically (essentially in the direction perpendicular to the layers that appear as long lines), and as we do not know a priori the position and propagation of the layers in complex situations, the only way to get accurate numerical results would be to refine strongly the mesh in the whole shell. However, in practice, that would lead to non-reasonable computation times! Therefore, to limit the number of elements (and of degrees of freedom), a solution is to refine strongly the mesh only in the areas and in the directions where the displacements vary the most, i.e. in the layers. However, this is only possible if we use an adaptive (and anisotropic) mesh generator, which re-meshes automatically in 2 3

4

Vm is the completed with the membrane energy norm of the kinematical space V associated to the Koiter model. We recall that a surface is said to be elliptic, parabolic or hyperbolic if the determinant of the second fundamental form is, respectively, positive, null, or negative. See section 1.3 for more details. As we will see later, the case of sensitive shells corresponds mathematically to the non-satisfied condition f ∈ Vm for elliptic shells, as soon as a part of the boundary is free.

Introduction

5

the necessary areas, as in complex situations it is impossible to predict a priori the position of the layers. Let us notice that these kinds of situations are encountered classically in fluid mechanics computations, when the dissipation tends to zero (it plays the role of ε for shells). Whereas the subsonic and supersonic regimes are, respectively, elliptic and hyperbolic, the transonic regime is parabolic. In the hyperbolic regime, classical shock waves appear along the associated characteristics, whose evolution and propagation are in general unknown in complex situations (with propagation and reflection as for hyperbolic shells). Thus, a certain analogy exists between numerical computations in shell theory and in fluid mechanics : in complex situations, only the numerical simulations shall give qualitative (and quantitative in some case) results on the formation and propagation of the singularities (or of shock waves). That is the main reason why we decided in this book to use the two-dimensional anisotropic mesh generator BAMG (coupled with a the finite element software MODULEF5 , see chapter 4) which was developed for numerical computations of shock waves in fluid mechanics [16][17][22][59]. In this context, according to all the complex situations that can be encountered in numerical shell computations, we can define a priori five “main deformation patterns” of thin shells. These five patterns (different patterns shall coexist in different areas of the shell, according to the loading and the boundary conditions) correspond to five large classes of qualitative behaviors of shells, which are a priori identifiable. This scheme is slightly arbitrary, as some ones are not very different (inhibited hyperbolic and parabolic patterns). We present in what follows the five identified patterns, illustrated in each cases, with: - the three-dimensional visualization of the deformed middle surface including the anisotropic adapted mesh generated with BAMG, - the repartition of the bending energy surface density Ebs (represented in the plane of the parameters of the mapping of the middle surface) which describes qualitatively the behavior of the shell near the internal and boundary layers, -the main corresponding properties. Note that all the figures that will be presented are issued from the numerical simulations (performed with the softwares BAMG and MODULEF coupled together) which will be presented in detail in the next chapters. 1- Non-inhibited (or non-geometrically rigid) middle surface Main properties: • Bending dominated. • Large displacements (with respect to the same loading as in the inhibited case) but small deformation (elastic behavior). • Very peculiar (inextensional) displacements inducing numerical locking. • In most cases, only a part of the shell is non-inhibited, the rest behaves as in the corresponding inhibited case. 5

Both developed by INRIA, France.

6

Introduction 1

95 90 80 70 60 50 40 30 20 10 1

0.75 0.5

1 0.5

0.25

y2

0 -0.5 -1

0

-0.25 -0.5

1

-0.75

0.5 0 -0.5 -1 -1

0.5

0

-0.5

1.5

1

-1 -1

-0.5

0

0.5

1

y1

Fig. 0.4. Example of hyperbolic non- Fig. 0.5. Repartition of Ebs for ε = 10−4 geometrically rigid (non-inhibited) shell for ε = 10−4

2- Inhibited (or geometrically rigid) hyperbolic shells Main properties: • Layers along non-characteristic curves: thickness ε1/2 . The order of the singularity of u3 is analogous to that of f 3 . • Layers along characteristic curves (propagated layers): thickness ε1/3 . The order of the singularity of u3 is “two orders larger than” that of f 3 . • Pseudo-reflections of the propagated singularities. • Local locking in the layers along the characteristics. • Mixed (membrane and bending) problem in the layers. Layers appearing as soon f 3 is discontinuous across a regular curve (the condition f 3 ∈ Vm is not satisfied anymore).

10 5 0 -5 -10 -15

60 40 20 0 -20 -40 -60

-60

-40

-20

0

20

40

60

Fig. 0.6. Example of hyperbolic geomet- Fig. 0.7. Repartition of Ebs for ε = 10−4 rically rigid (inhibited) shell for ε = 10−4

Introduction

7

3- Inhibited (or geometrically rigid) parabolic shells Main properties: • Layers along non-characteristic curves: thickness ε1/2 . The order of the singularity of u3 is analogous to that of f 3 . • Layers along characteristic curves (propagated layers): thickness ε1/4 . The order of the singularity of u3 is “four orders larger than” that of f 3 . • Absence of reflections of the propagated singularities. • Local locking in the layers along the characteristics. • Mixed (membrane and bending) problem in the layers. Layers appearing as soon the gradient of f 3 is discontinuous across a regular curve (the condition f 3 ∈ Vm is not satisfied anymore).

Fig. 0.8. Deformed shape of a parabolic Fig. 0.9. Repartition of Ebs for ε = 10−5 shell for ε = 10−5

4- Well-inhibited elliptic shells (fixed or clamped all along the boundary) Main properties: • There are no characteristics. • Singularities along curves bearing singularities of f 3 : thickness ε1/2 . The order of the singularity of u3 is analogous to that of f 3 . • Absence of reflections of the singularities. • Genuine membrane-dominated problem provided f ∈ H −1 × H −1 × L2 , otherwise mixed (membrane and bending) problem (because of the layers). 5- Ill-inhibited elliptic shells (fixed or clamped along a part of the boundary, free by the rest) This is the less well-known case which bears the following main properties: • Oscillations along the free boundary, their amplitude decays exponentially inside the shell, which amounts to some kind of instability. These oscillations are bending-dominated.

8

Introduction

2 1.5 1 0.5 0 -0.5

1 0.5 0 -0.5 -1 -1

-0.5

1

0.5

0

Fig. 0.10. Example of geometrically rigid Fig. 0.11. Bending energy surface density or inhibited elliptic shell for ε = 10−4 for ε = 10−4 (represented in a quarter of the domain)

0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 1 0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 0.12. Example of sensitive elliptic Fig. 0.13. Bending energy surface density shell ε = 10−4 for ε = 10−4

• Whatever the loading is, the deformation pattern is dominated by large oscillations with moderatly small wavelength which depends on ε as log(1/ε) • There are no characteristics. • Singularities along curves bearing singularities of u3 analogous to those of the previous well-inhibited case also occur, but for small ε they are masked by the oscillations. The first chapter of this book is devoted to theoretical recalls on surface theory and surface rigidity, and on the classical formulations of the linear elastic Koiter shell model. In chapter 2, we recall the classical results of convergence (when the thickness ε tends towards zero) of the Koiter shell model to the membrane or to the pure bending model, whether the shell is geometrically rigid or not. The proofs of the convergence results, recalled briefly in their mathematical context, enable us to understand the classical phenomena of boundary or internal layers

Introduction

9

that appear at the limit when ε approaches zero (they correspond to a loss of regularity of the solution). The second part of chapter 2 is devoted to a general theoretical analysis of the singularities appearing in the internal and boundary layers when the applied loading is singular along a curve. In this analysis of singularities, we focus on the action of the normal loading f 3 , leading to the most singular displacements. We begin with establishing a reduced formulation of the membrane system only involving one component of the displacements at the same time. Then, from this reduced formulation particularly well-adapted to a general theoretical study of the singularities, we establish some general results giving the order of the singularities of the displacements from the singularities of the loading. The results obtained, and the propagation (or not) of the singularities, depend not only on the geometric nature of the middle surface of the shell (elliptic, parabolic or hyperbolic), but also on the discontinuity of the loading along a characteristic line (or not) of the middle surface. We prove that the displacements are generally more singular (up to four order more singular) than the normal loading f 3 itself. This comes directly from the reduced formulation (2.55), involving fourth order derivatives of f 3 . This is the reason why singularities of the solutions are much more important in shell theory than in other areas of mechanics and physics. Finally, a similar reduced equation established from the whole Koiter model (including bending effects) enables to recover the classical results existing on the layer thicknesses. In chapter 3, we reveal theoretically the efficiency of anisotropic adaptive meshes for the computation of very thin shell problems including boundary and internal layer phenomena. In particular, we address error estimates for singular perturbation problems (and penalty problems) with layers, including isotropic and anisotropic elements. We prove that the error estimates obtained with an anisotropic mesh are similar to those obtained with an isotropic one containing much more elements. The numerical illustration on practical examples is presented in the next chapters. Chapter 4 contains a general review on the membrane locking phenomenon existing in all the numerical resolutions of shell problems with Finite Element Methods for very small thicknesses. We then present the solutions existing to reduce this phenomenon, and the one chosen for the numerical computations that will be performed: a non-conforming shell element (DKT element of MODULEF) coupled with an adaptive anisotropic mesh generator BAMG which enables to refine strongly the mesh only in the areas where the locking is important (in the layers). The second part of the chapter is devoted to a detailed presentation of the two-dimensional anisotropic mesh generator (BAMG) and of the Finite Element software MODULEF used in the sequel. Chapter 5 is devoted to the theoretical and numerical study of singularities for inhibited parabolic shells. In the first part, a direct integration of the membrane system (possible for the parabolic shells considered) reveals clearly the differences existing between the singularities orders and their propagation, whether the loading is singular along a characteristic line or not. In the particular case

10

Introduction

considered of a constant loading singular on a circular boundary, the resulting singularities are found to have non-classical fractional orders (with respect to the Dirac singularity family). These theoretical calculations are then compared to accurate numerical simulations performed with MODULEF and BAMG coupled together. In chapter 6, we focus on the hyperbolic inhibited shells, whose limit behavior is different from that of parabolic shells. Indeed, for a given loading, the resulting displacements are generally less singular than in the parabolic case, and the singularities may propagate in the two directions of the characteristics (whereas parabolic shells only have one characteristic family). Moreover, we address the phenomenon of pseudo-reflections that can only happen for this kind of shells when a part of the boundary is not a characteristic line. We propose several numerical computations which are compared to the existing theoretical results for the considered singularities. We will see that, as it is very difficult to predict all the pseudo-reflections that may occur, an automatic remeshing procedure reveals to be necessary to perform accurate computations for these kind of problems. Chapter 7 is devoted to well-inhibited shells, fixed along their whole boundaries. When the loading is singular, the singularities of the displacements correspond to those obtained when the singularity of the loading is along a noncharacteristic line (an elliptic shell has no real characteristic line). On the other hand, starting again from the reduced formulation of the membrane problem, we reveal that another kind of singularity may appear: a logarithmic point singularity of the normal displacement at the corners of the loading domain. Numerical computations will illustrate these various kinds of singularities and prove again the reliability of the numerical method. Indeed, the mesh is refined automatically, anisotropically inside the layers but isotropically around the logarithmic singularity. Chapters 8 and 9 are devoted to the theoretical and numerical developments on sensitive (or ill-inhibited) shell problems. In chapter 8, we present theoretical developments about partial differential equations and systems. We show the importance of the characteristic curves and of the boundary conditions on the determinacy of the solutions. In particular, when elliptic problems do not satisfy the Shapiro-Lopatinskii condition, the solutions are very unstable and belong to the very abstract space of analytical functionals, i.e. functionals on spaces of analytic functions. We then prove that elliptic membrane problems with free boundary conditions enter in this framework. Consequently, elliptic shells with free boundaries exhibit some instabilities when the thickness ε tends to zero: the displacements are oscillating in the direction tangent to the free boundary whereas they are exponentially decreasing inwards the domain. These problems are called sensitive. Chapter 9 is devoted to numerical computations for the sensitive problems addressed in chapter 8. We recover numerically very accurately the pathological behavior predicted by the theory. When the thickness ε tends to zero, a complexification phenomenon arises: the resulting displacements are more and more oscillating with an increasing amplitude along the free edges (the bending

Introduction

11

energy concentrates essentially along the free boundary). Moreover, we observe that the wavelength of the phenomenon is proportional log(1/ε), as predicted by the simplified heuristic considerations of chapter 8. The last chapter deals with non-inhibited or “partially” non-inhibited hyperbolic shells, and even with more complex mixed problems: a part of the shell is inhibited whereas the rest is not inhibited. In particular, we will put in a prominent position the very different behavior between non-inhibited shells (all the rotations are allowed) or “partially” non-inhibited shells (only the rotations around one family of characteristics are allowed). For various situations, we observe with the numerical computations that the displacements vary considerably from an inhibited to another non-inhibited or “partially” non-inhibited area, and depend also strongly on the applied loading (on a non-inhibited or an inhibited area). Remark on the lecture of the book 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 which enable to understand the sensitive problems encountered. Therefore, the lecture of this book may not be continuous and the reader who wants to improve his knowledge in a specific area may refer directly to the chapters concerned with6 . In particular: • General theory of shells and the analysis of singularities is mainly concerned with chapters 1 and 2, even if it is also present in chapters 5, 6 and 7 in computing of parabolic, hyperbolic and elliptic shells. So that the reader only interested in theoretical aspects of singularities in thin shell problems may focus on chapters 1 and 2. • The reader interested more particularly by numerical simulations of thin shell problems with anisotropic elements may begin with reading directly chapters 3 and 4. General anisotropic error estimates are addressed in detail in chapter 3, whereas the softwares used for the numerical simulations with anisotropic adaptive meshes are considered in chapter 4. Obviously, all the other chapters (excepting chapter 8) are concerned with numerical simulations for various examples of thin shell problems. • Very particular sensitive shell problems are addressed in chapters 8 and 9. The mathematical aspects are developed in chapter 8 which contains generalities on boundary conditions for equations and systems, and an introduction to sensitive problems. Numerical computing of various sensitive shell problems is concerned with chapter 9. • Geometric and mathematical consideration on non-inhibited (or non geometrically rigid) shells can be found in chapters 1 and 2 (in particular sections 1.3 and 2.3.5). The numerical aspects are presented in chapter 10 on various examples. 6

To this end, the index at the end of the book, even if it is not exhaustive, may be helpful.

1 Geometric Formalism of Shell Theory

1.1

Introduction

Shells are classically described geometrically by a middle surface (a twodimensional surface embedded in R3 ) and a thickness which is generally constant. In this book, we will focus on thin shells whose thickness is small with respect to the characteristic dimension of the middle surface. Moreover, we will restrain our analysis to the linear elastic framework. Thus in this book, the thin elastic shells studied will be described by Koiter shell model, or at the limit by the membrane model, which are two-dimensional models involving only the variables of the middle surface. In this chapter, we will begin with a brief recall on the geometric formalism of surface theory necessary to describe the two-dimensional surfaces imbedded in R3 (local mapping, embedding, local basis), and the variations of scalar, vector or tensor fields defined on such surfaces (derivatives of vector and tensor fields on a surface). Then, we will recall the various formulations existing for the twodimensional Koiter model (variational and local formulations), which will be used in the next chapters.

1.2

Recall on Surface Theory

This section summarizes the main aspects of surface theory that are useful in shell theory. For more information, the reader may refer for instance to [21, 98]. 1.2.1

Mapping - Covariant Basis

Let (O, e1 , e2 , e3 ) be an orthonormal frame of R3 . Let Ω be an open connected domain of R2 (the plane of parameters (y 1 , y 2 )). A surface S is defined by the local mapping (Ω, Ψ ), where Ψ is an injective application, at least C 2 (Ω) in R3 : E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 13–32. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

14


Ψ : Ω ⊂ R2 −→ S ⊂ R3 (y 1 , y 2 ) −→ Ψ (y 1 , y 2 )

(1.1)

p = Ψ(y 1, y 2)

S

TpS

a2 a1

Ψ(y1 , y 2)

e3

y2

e2 1

Ω

y

O

e1

Fig. 1.1. Definition of a surface embedded in R3

For each point p = Ψ (y 1 , y 2 ) of S, we can define two vectors called coordinate vectors which are given by: aα =

∂Ψ (y 1 , y 2 ) ∂y α

(1.2)

where α {1, 2}. For a given mapping, these two vectors are uniquely defined and linearly independent. They define the tangent plane at each point p of S, denoted Tp S. We define then the unit vector perpendicular to the tangent plane: N=

a1 ∧ a2 a1 ∧ a2

(1.3)

Definition 1.2.1. The set (a1 , a2 , N ) is called the covariant basis, defined for each point p of the surface S. 1.2.2

First Fundamental Form of the Surface S - Contravariant Basis

From the vectors of the covariant basis, we defined the metric tensor (aαβ ) of the surface S, corresponding to the mapping (Ω, Ψ ) by its covariant components: aαβ = aα · aβ where · denotes the scalar product of R3 .

(1.4)

1.2 Recall on Surface Theory

15

Considering a regular curve Γ of the surface S, defined in a parametric form (y 1 = y 1 (t), y 2 = y 2 (t)), the infinitesimal length ds of an element of Γ is given by: ds2 = a11 dy 1 dy 1 + 2a12 dy 1 dy 2 + a22 dy 2 dy 2 (1.5) The quadratic form defined by (1.5) is called the first fundamental form of the surface S. The infinitesimal area element dS of S is related to the infinitesimal area element dy 1 dy 2 of Ω by: √ dS = ady 1 dy 2 with a = det(aαβ ) = a11 a22 − (a12 )2 (1.6) Because the covariant basis is generally not orthonormal, one may associate a contravariant basis (a1 , a2 , a3 ) such that: aβ · aα = δαβ

and a3 = N

(1.7)

where δαβ is the Kronecker symbol1 . The expression of the vectors of the covariant basis and of the contravariant basis are linked by the metric tensor: aα = aαβ aβ

(1.8)

aα = aαβ aβ

(1.9)

and reciprocally: where (aαβ ) is the metric tensor associated with the contravariant basis defined by: ⎛ ⎞ a22 −a12 1 ⎠ aαβ = aα · aβ = (aαβ )−1 = ⎝ (1.10) a −a12 a11 We have the trivial relation det(aαβ ) =

1.2.3

1 1 = . det(aαβ ) a

Second Fundamental Form

The second fundamental form of the surface S is directly linked to its curvature. The associated quadratic form writes: b(dy 1 , dy 2 ) = b11 dy 1 dy 1 + 2b12 dy 1 dy 2 + b22 dy 2 dy 2

(1.11)

where bαβ are the covariant components of the curvature tensor defined by: bαβ = bβα = −aα · N,β = N · aα,β

1

β We recall that δα = 1 if α = β and δαβ = 0 if α = β.

(1.12)

16


It can also be written under a contravariant form: bαβ = aαμ aλβ bμλ

(1.13)

Finally, the curvature tensor can be written in a mixed form: α bα β = −a · N,

β

= N · aα , β

(1.14)

which represent the component of the associated endomorphism. These components have a physical meaning, associated with the curvatures.

1.3

Classification of Surfaces

Let us consider a surface, which has a component bβα different from zero. One −→ can define the nature of each point p of S (with Op = Ψ (y 1 , y 2 )) by studying −− → the position of a point p of S, close to the point p (with Op = Ψ (y 1 + dy 1 , y 2 + dy 2 )) compared with the tangent plane at p. Let us call d the algebraic distance between the tangent plane and the point p (see Fig. 1.2).

S p d p

Tp S

Fig. 1.2. Definition of the distance d

The distance d can be written as follows [98]: d = Ψ (y 1 + dy 1 , y 2 + dy 2 ) − Ψ (y 1 , y 2 ) · a3

(1.15)

Developing Ψ (y 1 + dy 1 , y 2 + dy 2 ), it can be shown that d can be written as: d=

1 b11 dy 1 dy 1 + 2b12 dy 1 dy 2 + b22 dy 2 dy 2 ] + . . . 2

(1.16)

where . . . refers to terms that are at least cubic. The directions (dy 1 , dy 2 ) that cancel (1.16) are called asymptotic directions of the surface S at the point p = Ψ (y 1 , y 2 ). The existence and the nature of these directions depend on the nature of the middle surface S at the point p. The nature of the middle surface at the point p depends on the sign of d which is the same as the sign of the determinant of the second fundamental form:

1.3 Classification of Surfaces

17

• If b11 b22 − (b12 )2 > 0, d has a constant sign and the surface stays at the same side of the tangent plane in a neighborhood of the point p. The point p is then an elliptic point of the surface S. The surface has locally two imaginary asymptotic directions. • If b11 b22 − (b12 )2 = 0, there exists locally one direction where d = 0. The tangent plane lies to the surface in p along this direction. The point p is then a parabolic point of the surface S. The direction where d vanishes is an asymptotic direction (double root of the quadratic polynomial d = 0). • If b11 b22 − (b12 )2 < 0, there exist locally two distinct directions where d = 0. The principal curvatures of S at p have a different sign. The tangent plane cut the surface S at p. The surface S is on both the sides of the tangent plane in the neighbourhood of p. The point p is a hyperbolic point of the surface S. The two directions that cancel d are two distinct asymptotic directions (simple roots of the polynomial d = 0). Definition 1.3.1. A surface S is said to be uniformly elliptic, parabolic or hyperbolic when the determinant of the second fundamental form is respectively positive, null, or negative at each point of S. The asymptotic lines of the surface S are the curves tangent to an asymptotic direction at each of their points. These curves satisfy the following equation: b11 dy 1 dy 1 + 2b12 dy 1 dy 2 + b22 dy 2 dy 2 = 0

(1.17)

TpS p

S

Fig. 1.3. Example of an elliptic point - No real asymptotic line

p

Tp S

S

Fig. 1.4. Example of a parabolic point - One family of double asymptotic lines

18


Tp S

p

S

Fig. 1.5. Example of a hyperbolic point - Two families of simple asymptotic lines

Fig. 1.6. Example of a more complex surface of revolution containing elliptic, parabolic and hyperbolic points.

The asymptotic lines at point p are represented (dotted lines) in Figs. 1.3 to 1.5 for the three types of surface (elliptic, parabolic, and hyperbolic). These three examples of elliptic, parabolic, and hyperbolic surfaces are classical generic ones, where all the points of the surface are of the same type. However, this is not always the case, as for the surface of revolution of Fig. 1.6 which contains the three kinds of points (elliptic, parabolic, and hyperbolic). However, these types of more complex surfaces will not be considered in this book.

1.4

Differentiation on the Surface S

The covariant basis (a1 , a2 , N ) and the contravariant basis (a1 , a2 , N ) are generally neither orthogonal nor normalized and they depend on the considered point p of S. Thus, the derivatives of the vectors of these two basis do not vanish. The derivative of a tangent vector is generally outside the tangent plane. It comprises a component along the normal vector, which is linked to the curvature of the middle surface S. We have the following formulae of differentiation (Gauss formula for the in-plane vectors and Weingarten formula for the normal vector) which can be written as:

1.4 Differentiation on the Surface S

19

λ aα,β = Γαβ aλ + bαβ N

(1.18)

α λ = −Γλβ a λ N,α = −bα aλ

(1.19)

aα ,β

+

bα βN

(1.20)

where λ λ Γαβ = Γβα = aλ · aα,β

(1.21)

are the Christoffel symbols associated with the first fundamental form. We generally define the Christoffel symbols associated with the second fundamental form: β 3 Γαβ = bαβ and Γ3α = −bβα

(1.22)

Let u be a vector field defined on the surface S. It can be written either as the covariant basis or as the contravariant basis: u = uα aα + u3 N or u = uα aα + u3 N

(1.23)

Equivalently, any vector field of R3 defined on S can be decomposed into a vector of the tangent plane ut = uα aα and a vector along the normal direction u3 N . In other words, we have R3 = Tp S ⊕ RN . The differentiation on the surface S of the vector field u defined on S needs to define the following notions. Differentiation of a scalar field Let f : S→R p → f (p)

(1.24)

be a scalar field defined on the surface S. We have: ∂f ∂f α = a = f,α aα ∂p ∂y α

(1.25)

∂f ∂f is a linear form on Ω. Its transposed is a vector called gradient ∂p ∂p of f (denoted gradf ). We have: Thus

gradf = f,α aαβ aβ

(1.26)

Differentiation of a tangent vector field By generalizing the previous definition, we have: ∂ut ∂ut α = a = ut,α aα ∂p ∂y α

(1.27)

Since ut = uβ aβ , we obtain the differentiation formula of a tangent vector using (1.18) to (1.20):

20


∂ut = uα |β aα .aβ + (uα bαβ )N.aβ ∂p

(1.28)

Thus, the derivative of a tangent vector is not necessary in the tangent plane of the surface S at p. It contains a term in the direction normal to the tangent plane, which corresponds to the second term of (1.28). ∂ut The in-plane component of , projection onto the tangent plane TpS at p ∂p ˆ t ∂u is called covariant derivative of ut and is denoted . We have: ∂p ˆ t ∂u ∂ut =Π = uα |β aα .aβ ∂p ∂p

(1.29)

where Π denotes the orthogonal projection operator onto Tp S. Thus, the covariant derivative of a tangent vector ut is an endomorphism of Tp S whose components are defined by:

⇐⇒

α λ u α | β = uα ,β + Γλβ u

(1.30)

λ uα |β = uα,β − Γαβ uλ

(1.31)

with uα |β = aαγ uγ |β

(1.32)

Differentiation of a vector field of R3 Finally, let us generalize the previous definition to a vector field of R3 which writes: u = ut + u3 N (1.33) According to the previous definitions, the differentiation

∂u of the vector field ∂p

u defined on S is given by: ∂u α β α β = u |β − bα β u3 aα a + u3,β + bβ uα N.a ∂p

(1.34)

with the complementary relations: uα |β = aαλ uλ |β

α and bα β uα = bαβ u

(1.35)

In what follows, we will use the simplified notation Dα uβ = uα |β . Differentiation of a tensor field Let T be a second-order tensor field defined on the surface S. The tensor T can be represented either by its contravariant components: T = T αβ aα ⊗ aβ

(1.36)

1.5 Surface Rigidity

21

or by its mixed components2 T = Tβα aα aβ . In what follows, to write the equilibrium equations of shells, we need to define the following differentiation operator for T which is a generalization of what was presented for vectors: β α T αβ |γ = ∂γ T αβ + Γγλ T αλ + Γγλ T λβ

(1.37)

In the sequel, to simplify the notations, we will write Dγ T αβ = T αβ |γ whenever no confusion is possible.

1.5 1.5.1

Surface Rigidity Deformation of a Surface

When the surface S is subjected to a solicitation (a loading for instance), it ˜ The mapping of S˜ writes: deforms into a new surface S. Ψ˜ (y 1 , y 2 ) = Ψ (y 1 , y 2 ) + u(y 1 , y 2 )

(1.38)

where u(y 1 , y 2 ) is the displacement undergone by the point p = Ψ (y 1 , y 2 ) of S (see Fig. 1.7).

˜ 1, y 2 ) p˜ = Ψ(y

p = Ψ(y 1, y 2 ) u y1 , y2

Ψ(y 1, y 2) ˜ 1, y 2 ) Ψ(y y = y1, y 2

Ω Fig. 1.7. Deformation of a surface

We can now compute the new metric tensor a ˜αβ and the new tensor of curva˜ ˜ ture bαβ associated with the deformed surface S. To characterize the deformation of the surface subjected to the displacement u, we define the tensors of metric variation γαβ (u) and the tensor of curvature variation ραβ (u) as follows: 2

Writing T with its covariant components does not interest us here.

22


γαβ =

1 ∗ (˜ a − aαβ ) 2 αβ

ραβ = ˜b∗αβ − bαβ

(1.39)

(1.40)

where a ˜∗αβ and ˜b∗αβ are the pulled-back in the initial configuration S of the components of the tensors a ˜∗αβ and ˜bαβ which are defined in the deformed con˜ One can find (in [42, 88] for instance) a detailed study on the figuration S. subject, where the expressions of γαβ and ραβ are obtained, either by computing the three-dimensional variation of the metrics in the thickness of the shell, or by a study of the variations of aαβ and bαβ . Because the displacements are supposed to be small, the linearization of the expressions (1.39) and (1.40) leads to (see [15] for instance): γαβ (u) =

1 (Dα uβ + Dβ uα ) − bαβ u3 2

λ ραβ (u) = −∂α ∂β u3 + Γαβ ∂λ u3 + bλα bλβ u3 − Dα bλβ uλ − bλα Dβ uλ

(1.41) (1.42)

with Dα uβ = uβ |α . It has to be noticed that in linearized theory (hypothesis of infinitesimal displacements), which is the case in this book, the knowledge of γαβ (u) and ραβ (u) is enough to characterize completely the deformation of a surface S subjected to a displacement field u. Indeed, it is proved by the following Koiter theorem which is a linearized version of the fundamental theorem of surface theory: Theorem 1.5.1. Let u(p) be a displacement field applied to a surface S, whose points are defined by p = Ψ (y 1 , y 2 ). If γαβ (u) = ραβ (u) = 0 ∀ α, β ∈ {1, 2}, u is then a rigid body displacement. 1.5.2

The Rigidity System and its Characteristic Curves

The rigidity of a surface is directly linked to the existence of non-trivial displacements which can deform the surface without modifying its metrics called “inextensional displacements” and such inextensional displacements are solution of the the rigidity system: γαβ (u) = 0 (1.43) In this section, we will study only the characteristic lines of the rigidity system (1.43), but not the complete solutions which depend obviously on the associated boundary conditions. This will be done later, in section 2.2 of chapter 2. Let us now consider the rigidity system (1.43) in some region of the plane of the parameters Ω. The associated boundary conditions, not necessary here, will be introduced later. The rigidity system is a system of three equations with three unknowns: the three displacements (u1 , u2 , u3 ). Explicitly, it can be written as:


⎧ D1 u1 − b11 u3 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ D2 u2 − b22 u3 = 0 ⎪ ⎪ ⎪ ⎪ ⎩1 (D1 u2 + D2 u1 ) − b12 u3 = 0 2

23

(1.44)

where the expression of the covariant derivatives Dα are given by λ Dα uβ = uα,β − Γαβ uλ

with uα,β = ∂β uα . In terms of displacements, we equivalently have: ⎧ 1 2 u1,1 − Γ11 u1 − Γ11 u2 − b11 u3 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ 1 2 u2,2 − Γ22 u1 − Γ22 u2 − b22 u3 = 0 ⎪ ⎪ ⎪ ⎪ ⎩1 1 2 (u1,2 + u2,1 ) − Γ12 u1 − Γ12 u2 − b12 u3 = 0 2

(1.45)

Clearly, uα (α = 1, 2) and u3 play very different roles, as uα appears with derivatives, whereas u3 only appears in an “algebraic” form (without derivatives). In fact, the rigidity system (1.45) is “essentially” a system of two equations of first order in uα (α = 1, 2). Indeed, we may eliminate u3 provided one of the curvatures b11 , b22 or b12 is different from zero. For instance, let us assume in the sequel that b12 = 0. We may now eliminate u3 from the third equation, and the rigidity system becomes: ⎧ b11 b11 ⎪ ⎪ (D − D )u − D u =0 ⎪ ⎨ 1 2b12 2 1 2b12 1 2 (1.46) ⎪ ⎪ b b ⎪ ⎩ − 22 D2 u1 + (D2 − 22 D1 )u2 = 0 2b12 2b12 which constitutes two equations of the first order in u1 and u2 . The eliminated unknown u3 is then given by: u3 =

1 (D1 u2 + D2 u1 ) 2b12

Under the form (1.46), we observe that the terms of higher order of differentiation (of order one) are the term ∂α of the covariant derivative Dα , whereas the terms λ involving the Christoffel symbols Γαβ uλ are of lower order (of order zero of differentiation). Let us now determine the directions of the characteristics of system (1.46). They are the families of curves orthogonal to the elementary vector ζ = (ζ1 , ζ2 ) defined on Ω as follows. First, let us consider in (1.46) only the higher order terms, which is equivalent to replace the covariant derivatives Dα by the classical derivative ∂α . Then, we replace ∂α by the numerical parameter ζα . At this stage, the system is formally an algebraic system for the unknowns u1

24


and u2 , linear and homogeneous. To have non-zero solutions, the determinant of the system must vanish. This gives an equation in ζ1 , ζ2 (the characteristic equation) whose solutions are the directions orthogonal to the characteristic curves. We may consider this as a definition of the characteristic curves; we refer to section 8.2 later on for their mathematical properties (see also Remark 1.5.1 in this respect). In the case of system (1.46), this gives: b b11 ζ1 − 11 ζ2 − ζ1 2b 2b 12 12 (1.47) =0 b b 22 − 22 ζ ζ2 − ζ1 1 2b12 2b12 Multiplying by 2b12 , we get: −b22 (ζ1 )2 + 2b12 ζ1 ζ2 − b11 (ζ2 )2 = 0

(1.48)

which, as we know, defines the directions of the vectors ζ = (ζ1 , ζ2 ) which are orthogonal to the characteristic curve. Thus, the direction of the characteristic curves are given by the family of tangent vectors dy = (dy 1 , dy 2 ) = (−ζ2 , ζ1 ) which are solutions of b11 dy 1 dy 1 + 2b12 dy 1 dy 2 + b22 dy 2 dy 2 = 0 Obviously, this equation corresponds to (1.17) giving the direction of the asymptotic curves of the surface S. Accordingly, the characteristic lines of the rigidity system coincide with the asymptotic lines of the middle surface S of the shell. System (1.46) is a system of two equations of first order. It is said that the “total order” of the system is 2. It corresponds to the order of homogeneity in ζ of equation (1.48). Classically, the system is said to be hyperbolic, parabolic, or elliptic when the two characteristic directions are real and different, real coincident (double) or complex conjugate, respectively. Moreover, we just proved that the type of system (1.46) is the same that of the type of surface (see section 1.3). Remark 1.5.1. When the coefficient of the system (1.46) are constant, the characteristics are two families of parallel straight lines. Moreover, in that case, the directions ζ are those for which the system has solutions of the form: 1

2

u1 (y 1 , y 2 ) = v1 ei(ζ1 y +ζ2 y ) 1 2 u2 (y 1 , y 2 ) = v2 ei(ζ1 y +ζ2 y )

(1.49)

with v1 and v2 constant (note that the factor i or any other in the exponent disappears, as equation (1.48) is homogeneous). When ζ is real, i.e when there are solutions with ζ1 and ζ2 real, this amounts to solutions which are sinusoidal in the direction ζ (with a wave length of 2π/|ζ|) and constant in the perpendicular direction. Obviously, we may take |ζ| very large. The corresponding solutions (1.49) are then smooth (constant) along the characteristic lines, but very rough across them.


25

Remark 1.5.2. The considerations of the previous remark hold true in some kind of “asymptotically local” sense in the general case of systems with variable coefficients and having lower order terms. Indeed, in a small region around a point, the coefficient may be considered as constant. However, when searching solutions of the form (1.49) with large |ζ| (or equivalently with small wave length), equation (1.48) has also terms of degree 1 and 0 in ζ coming from derivatives of order 1 and 0 in the system. But that supplementary terms in (1.48) are negligible for large |ζ| and we recover the same result. Therefore, even in the general case of variable coefficients, characteristics appear as lines of smoothness of the solutions, whereas oscillation and roughness may appear across then. 1.5.3

Handling Systems of Equations with Various Orders: Indices of Equations and Unknowns

Coming back to the rigidity system (1.43), it is in some case useful to define the characteristics directly, without eliminating u3 . Indeed, explicit elimination may be difficult, or even impossible, in certain regions. The principal difficulty in this concern is the definition of “terms of higher order differentiation”. Indeed, λ we saw that the term in Γαβ , involving uα (but not u3 ) are “of lower order”. Oppositely, u3 itself (without any differentiation) is relevant (see the role of bαβ in (1.46)). A way to take into account such differences between the various equations and variables is to define indices (s1 , s2 , s3 ) for the equations and (t1 , t2 , t3 ) for the unknowns so that the “high order terms” (often denoted “principal terms”) are in equation j the terms where the unknown k appears by its derivative of order sj + tk . For instance, in (1.45), with the equations written in that order and with the unknowns in the order (u1 , u2 , u3 ), we may take (0, 0, 0) and (1, 1, 0) as indices of equations and unknowns. It means that the “principal terms” in any of the equation are the first-order derivatives of u1 , u2 , and u3 itself (derivative λ of order 0). This amounts exactly to neglect the Γαβ , not the bαβ . Then, the characteristics are defined as before, replacing the derivatives ∂α by ζα in the principal terms, and imposing that the discriminant of the system vanishes. Obviously, this is formally analogous to search for solutions of the form (1.49) for (u1 , u2 , u3 ). In the case of the system (1.45), this gives: ζ1 0 −b11 0 ζ2 −b22 = b22 ζ12 − b12 ζ1 ζ2 + b11 ζ22 (1.50) 2 2 1 1 ζ2 ζ1 −b12 2

2

which is exactly the same equation as (1.48). The “total order of the system” is the degree in ζ1 , ζ2 of the equation (1.50). It is 2 in our example, and in general it is total order of the system = sj + tk j

k

26


Remark 1.5.3. The definition of the indices sj and tk for a system is slightly ambiguous. Indeed, the result is exactly the same adding an integer m to the indices sj and subtracting m from the tk . For instance, in the rigidity system (1.46), we may take as indices of equations (−2, −2, −2) and (3, 3, 2) as indices of unknowns. Other choices leads to non-significant results. For instance, with the choice of (0, 0, 0) and (1, 1, 1) as indices of equations and unknowns respectively, the expressions of the characteristics are still given by (1.50) replacing bαβ with zero. The obtained equation is 0 = 0 which means that any curve is a characteristic!

1.6

The Koiter Shell Model

In this section, we will present some recalls on the theory of thin elastic shells, and mainly on the Koiter model. In the next chapter, we will present the main results existing on the asymptotic behavior of the Koiter model when the relative thickness of the shell tends to zero.

fˆ ε ε Sε = S × − , 2 2 γ1

S

ε

Γ1 = γ1 × − ε2 , ε2 γ0

Γ0 = γ0 × − ε2 , ε2 Fig. 1.8. Considered shell

Let us consider a shell having a middle surface S and a thickness ε. It occupies the domain ε ε Sε = S × − , 2 2 of R3 in its initial configuration. It is made up of an isotropic linear elastic material. The aim of the mechanical problem is to find the displacement field uε of the shell subjected to a loading fˆ and satisfying the boundary conditions. Obviously, the solution uε of the Koiter model depends on the thickness ε of the shell. In all the problems considered, we suppose that the forces are small enough to stay in the framework of the linear elasticity.

1.6 The Koiter Shell Model

27

The boundary γ = ∂S of S is classically decomposed in two parts, γ0 and γ1 which form a partition of γ. The lateral boundary of the shell Sε is defined by Γ = γ×] − 2ε , 2ε [. On this lateral boundary, two types of boundary conditions are generally imposed: kinematic ones on Γ0 = γ0 ×] − 2ε , 2ε [ and static ones on Γ1 = γ1 ×] − 2ε , 2ε [ (see Fig. 1.8). The two-dimensional Koiter model is established from the linear elastic threedimensional problem due to several hypothesis: • the Reissner–Mindlin hypothesis: the normal strain vanishes in the direction normal to the shell (there is no thickness variation during the deformation). • the compression stress in the direction normal to the shell vanishes. These two hypothesis led to a Nagdhi type model which includes transversal shear. If the shell is thin, transversal shear can be neglected, which constitutes the Kirchhoff–Love kinematical hypothesis: • a point situated on a normal (at a point p) to the initial middle surface remains on the normal to the deformed surface. These three hypothesis allow us to establish the two-dimensional model of Koiter from the three-dimensional equations of the linear elasticity and after an integration over the thickness. When considering that the volume loadings are constant in the thickness, which is itself constant, the displacement uε , solution of the problem, is given by the Koiter model (see [15, 33, 88] for instance): Find uε V, such that, ∀ v V : ε3 αβλμ ε αβλμ ε ε A γλμ (u )γαβ (v)dS + A ρλμ (u )ραβ (v)dS = fî vi dS 12 S S S (1.51) 1 1 2 with V = v = (v1 , v2 , v3 ) ∈ H (Ω)×H (Ω)×H (Ω) ; v satisfying the kinematic boundary conditions . The coefficients Aαβλμ represent the coefficients of the linear elastic isotropic constitutive law. They are fourth-order tensors. Their expressions are given by (see [15]): E 2ν αβ λμ αβλμ αλ βμ αμ βλ A = a a +a a + a a (1.52) 2(1 + ν) 1−ν where ν and E are, respectively, the Poisson’s ratio and the Young’s modulus. These coefficients allow to link the membrane stress tensor to the membrane strain tensor, and the bending moments tensor to the curvature variation tensor with the following relations: T αβ (uε ) = Aαβλμ γλμ (uε )

(1.53)

28


1 αβλμ A ρλμ (uε ) 12 They satisfy the usual conditions of symmetry M αβ (uε ) =

(1.54)

Aαβλμ = Aβαλμ = Aλμαβ

(1.55)

and positivity ∃C > 0

such that

Aαβλμ γλμ γαβ ≥ C

2 γαβ

(1.56)

(α,β)

We can also define the compliance coefficients Bαβλμ which satisfy the inverse relations: γαβ = Bαβλμ T λμ (1.57) ραβ = 12Bαβλμ M λμ

(1.58)

The left-hand side of the variational formulation (1.51) comprises two parts. The first one is the bilinear form of membrane energy, proportional to the thickness ε, whereas the second one, the bilinear form of bending energy, proportional to the cube of the thickness ε3 . The bending rigidity is consequently much weaker than the membrane one when ε is small. To address the asymptotic process of the Koiter model when ε 0, we rewrite it under the following form P(ε): ⎧ ⎨ F ind uε in V such that P(ε) (1.59) ⎩ am (uε , v) + ε2 ab (uε , v) = f, v ∀v ∈ V

where am (uε , v) =

Aαβλμ γλμ (uε )γαβ (v)dS

(1.60)

S

denotes the bilinear form of membrane energy and 1 ab (uε , v) = Aαβλμ ρλμ (uε )ραβ (v)dS 12 S

(1.61)

the bilinear form of bending energy. Moreover, we set f, v = f i vi dS S

with fˆ = εf

(1.62)

where ·, · denotes the duality product between V and V . Remark 1.6.1. For ε > 0, the existence and uniqueness of the solution of P(ε) is showed with the Lax–Milgram theorem [14]. This theorem relies on the continuity and the coerciveness of the left hand side of (1.59). The coerciveness of this bilinear form was proved in [14] due to Korn’s inequalities on a surface.

1.7 The Limit Membrane Model

29

The problem P(ε) can be written under the following local form which is a system composed of three coupled PDE’s (see for instance section VIII.3.1 of [88]): ⎧ β αβ 2 αγ γ αβ β ⎪ ⎨ −Dα T − ε bγ Dα M + Dγ (bα M ) = f f or β = 1, 2 (1.63) ⎪ ⎩ −bαβ T αβ + ε2 Dα Dβ M αβ − bα bβδ M αβ = f 3 δ where T αβ and M αβ are the tensors of membrane stresses and bending moments defined (1.53) and (1.54). This system (1.63) is associated with kinematical boundary conditions on (uε1 , uε2 , uε3 ), and also free edge or imposed loadings conditions which concern T αβ and M αβ .

1.7 1.7.1

The Limit Membrane Model The Membrane Model

The Koiter shell model (1.59) obviously depends on the thickness ε of the shell. In the next chapter, we will study in details the limit of the Koiter variational formulation (1.59), or equivalently of the associated local formulation (1.63), when the thickness ε decreases and tends towards zero. Even if in the reality, the thickness never attempts the value zero, for very small thicknesses, according to the factor ε2 multiplying the bending bilinear form ab (·, ·), the behavior of the shell is given by the limit membrane model corresponding formally to set ε = 0 in Koiter formulation (1.59). Of course, passing to the limit in Koiter formulation (1.59) is not so simple, and necessitates to establish convergence results in appropriate spaces which depend on the geometric rigidity of the middle surface. Moreover, it generally leads to a loss of regularity for the bending displacement uε3 (corresponding to the appearing of boundary layers). This will be studied in detail in the next chapter. In this section, we only study formally the local formulation of the membrane system and the associated characteristics. Passing to the limit for ε = 0 in the problem P(ε) given by (1.59), or equivalently in the associated local formulation (1.63), we obtain the membrane problem P(0) which writes in the local formulation: ⎧ ⎨ −Dα T αβ (u0 ) = f β f or β = 1, 2 (1.64) ⎩ −bαβ T αβ (u0 ) = f 3 In the sequel of this chapter, when no confusion arises, the solution u0 of the limit membrane model (corresponding to ε = 0) will be noted more simply u, whereas the solution of the Koiter model will still be noted uε . We will prove in that follows, that the limit membrane system (1.64) is respectively elliptic, parabolic or hyperbolic when the middle surface S is respectively elliptic, parabolic, or hyperbolic itself.

30

1.7.2


The System of Membrane Tension

The membrane system (1.64) is obviously equivalent to the three following coupled equations : ⎧ −D1 T 11 − D2 T 12 = f 1 ⎪ ⎪ ⎪ ⎪ ⎨ −D1 T 12 − D2 T 22 = f 2 (1.65) ⎪ ⎪ ⎪ ⎪ ⎩ −b11 T 11 − b22 T 22 − 2b12 T 12 = f 3 It is apparent that the three unknowns play analogous roles. On the contrary, concerning the equations, it is clear that the third equation is different from the first two ones which are symmetrical in the indices 1 and 2. Moreover, the high-order terms in the first two equations of (1.65) are clearly λ in those of order 1; terms of order of differentiation zero (involving the Γαβ the covariant derivatives) are of lower order. But in the third equation of the membrane system (1.65), the membrane tension T αβ without differentiation are clearly relevant. Accordingly, in the framework of section 1.5.3, we shall try the same indices for the three unknowns (T 11 , T 22, T 12 ), and indices of the form (λ + 1, λ + 1, λ) for the three equations. It is easily seen that the good choice is (0, 0, 0) for the indices of unknowns and (1, 1, 0) for equations, or any analogous ones according to remark 1.5.3. Considering only the principal order terms of differentiation in the membrane system (1.65), which is equivalent to replace the covariant derivatives Dα by the classical derivatives ∂α , we obtain the following characteristic equation of the membrane tensions: −ζ1 0 −ζ2 0 −ζ1 −ζ2 = 0 (1.66) −b11 −b22 −2b12 which leads to

b22 ζ12 − 2b12 ζ1 ζ2 + b11 ζ22 = 0

(1.67)

We obtain the same characteristic equation as (1.48) giving the directions ζ = (ζ1 , ζ2 ) orthogonal to the characteristics of the system (1.65) of membrane tensions. The conclusions are obviously the same: the characteristics of the membrane system involving the membrane tensions coincide with the asymptotic curves of S. The system (1.65) is of the same type (hyperbolic, parabolic, or elliptic) as the points of the middle surface. Remark 1.7.1. As in section 1.5.2, the above result may also be obtained by reducing (1.65) to a system of two unknowns, after eliminating one of them from the third equation, when one of the bαβ does not vanish. The reason why the characteristics of (1.65) are the same of (1.44) is that both systems are adjoint each other. The verification of this point is immediate by

1.7 The Limit Membrane Model

31

integrating by parts. A first simple verification may be done using only the principal order terms of differentiation (the only which are concerned with the characteristics). Moreover, we shall neglect the boundary conditions, as we only deal with equations and not boundary value problems. We obviously have3 : T 11 (∂1 u1 − b11 u3 ) + T 22 (∂2 u2 − b22 u3 ) Ω +2T 12 12 (∂2 u1 + ∂1 u2 ) − b12 u3 dy 1 dy 2 =

(−∂1 T 11 − ∂2 T 12 )u1 + (−∂1 T 12 − ∂2 T 22 )u2

+(−b11 T 11 − b22 T 22 − 2b12T 12 )u3 dy 1 dy 2

Ω

which may be formally written as: T αβ γαβ (u)dy 1 dy 2 = γ ∗i (T )ui dy 1 dy 2 = γ ∗ (T ).udy 1 dy 2 Ω

Ω

Ω

∗

where γ (T ) denotes the triplet formed by the left hand side of (1.65), and γ ∗i (T ) the ith equation of (1.65). Moreover, when taking the complete system with lower order terms of differentiation, the same property holds true, with integration on the surface S and on its boundary (instead of Ω and its boundary). The complete expression, including boundary terms, would be: αβ ∗ T γαβ (u)dS − γ (T ).udS = nα T αβ uβ dγ S

S

∂S

where S is the middle surface of the shell, dS the area element, γ the length element of its boundary ∂S, and n the unit external normal to ∂S in the tangent plane to S. 1.7.3

Back to the Membrane System

A similar analysis, leading as we will see to the same result, can be also performed directly from the membrane system written in terms of displacements, where (u1 , u2 , u3 ) are the three unknowns. This is equivalent to the membrane tension T αβ in terms of u in the system of tensions (1.65). We obtain: ⎧ −D1 T 11 (u) − D2 T 12 (u) = f 1 ⎪ ⎪ ⎪ ⎪ ⎨ −D1 T 12 (u) − D2 T 22 (u) = f 2 (1.68) ⎪ ⎪ ⎪ ⎪ ⎩ −b11 T 11 (u) − b22 T 22 (u) − 2b12 T 12 (u) = f 3 3

Note that the factor 2 in front of T 12 accounts for the fact that α and β take the values 1, 2, so that both T 12 and T 21 appear, with equal values.

32

with


⎧ αβ αβλμ γλμ (u) ⎪ ⎨ T (u) = A

(1.69) ⎪ ⎩ γαβ (u) = 1 (Dα uβ + Dβ uα ) − bαβ u3 2 To study the type of this system, we should define as previously the corresponding indices of equations and unknowns. As we should see later, this is not absolutely necessary, as we may use jointly the results of sections 1.5.3 and 1.7.2. Clearly, we should take the index on the third equation of system (1.68) a unit lower than the two first ones. Analogously, in order to consider terms bαβ u3 at the same level as the first-order derivatives of uα , the index of u3 should be a unit lower than those of uα . Moreover, the two first equations should be of order two in uα . This gives immediately the indices (1, 1, 0) for both equations and unknowns (with the ambiguity of the remark 1.5.3). This gives the principal orders ⎛ ⎞ 221 ⎝ 2 2 1⎠ (1.70) 110 for the membrane system (1.68)–(1.69). It clearly appears that u3 itself in the third equation is of principal order (as the second-order derivatives of uα and the first-order ones of u3 in the first two equations for instance). It is clearly seen that the principal order system is obtained by disregarding the Christoffel λ symbols Γαβ , i.e. considering covariant differentiation as ordinary differentiation. A direct computation of the equation of characteristics (an indirect simplified proof will be given in appendix A) shows that it is equivalent to:

−b22 ζ12 + 2b12 ζ1 ζ2 − b11 ζ22

2

=0

(1.71)

so that the characteristics are again the asymptotic curves of the surface, but counted twice. Therefore, at elliptic points of the surface S, the characteristics of the membrane system (1.68) are two (double) complex conjugated. At hyperbolic points of the surface S, the characteristics are the asymptotic curves of S, each one with multiplicity two. Finally, at the parabolic points, they are the (unique) asymptotic curve of S with multiplicity four.

2 Singularities and Boundary Layers in Thin Elastic Shell Theory

2.1

Introduction

The aim of this chapter is to study the asymptotic behavior of the Koiter shell model when the relative thickness ε tends to zero, and the singularities of the displacements, resulting from a singular loading, which appear in the internal and boundary layers. As the Koiter model contains membrane and bending effects coupled at different orders, its limit behavior1 is very different according whether the shell is geometrically rigid or not. If the shell is non-geometrically rigid (or equivalently non-inhibited), the asymptotic process2 corresponds to a penalty problem (see for instance [88], section VI.1.3) whose limit problem is the pure bending model [91]. However, as the case of non-inhibited shells is not the main aim of this book, it will only be considered in section 2.3.5 and in the last chapter (where numerical simulations of non-inhibited shells are performed). In this book, we will mainly focus on geometrically rigid (or inhibited) shells where the limit of the asymptotic process is the membrane model. The pathological case of ill-inhibited or sensitive shells will also be considered in chapter 9. In this chapter, we begin with some recalls on surface rigidity, on the asymptotic behavior of Koiter shell model, and on the associated boundary layers and singularities which appear when the thickness of the shell approaches zero. In the second part, we focus on a general theoretical analysis of the singularities resulting from a singular loading. To this end, we begin by establishing a reduced membrane formulation particularly well adapted to the theoretical and analytical analysis of the singularities which appear in boundary and internal layers (section 2.5). The approach developed is largely inspired by microlocal analysis techniques (see [43]) adapted to our specific shell problems. It enables to establish general results on the singularity orders of the displacements resulting from a singularity of the loading (result 2.6.1 of section 2.6). Finally, in the 1 2

For ε = 0. When the thickness ε tends to zero.


34


last section 2.8, this analysis is generalized to Koiter model. From the reduced formulation obtained, we recover easily the main results existing for the layer thicknesses.

2.2 2.2.1

Geometrically Rigid Surfaces Inextensional Displacements

Whenever it is possible, the shell deforms by inextensional displacements, i.e. displacements leaving the metrics unchanged. This type of displacements cancels the bilinear form am (·, ·) of the variational formulation (1.59) of the Koiter model. Indeed, the bending rigidity is much weaker when the relative thickness ε is small. Consequently, the displacements which tend to minimize the deformation energy will be, whenever it is possible, pure bending displacements. For a given shell, the existence of such displacements depends on the geometric nature of the middle surface S (elliptic, hyperbolic, parabolic), on the boundary conditions and also on the presence of folds. It is independent of the considered material. We recall here some classical definitions which will be useful in the sequel. A shell is said geometrically rigid or equivalently inhibited (i.e. the pure bending displacements are inhibited), if it cannot deform by inextensional displacements which let invariant the metrics of the middle surface. Oppositely, it is said to be non-geometrically rigid or equivalently non-inhibited. Thus, we can define G, subset of inextensional displacements as follows: Definition 2.2.1. The subset G of inextensional displacements or pure bendings is defined by G = {v V ; am (v, v) = 0} or equivalently G = {v V ; γαβ (v) = 0} (2.1) These two definitions are equivalent since the coefficients Aαβλμ of the constitutive law enjoy the property of positivity (see (1.56)). Obviously, this is equivalent to saying that am (v, v)1/2 defines a norm on V . Definition 2.2.2. The surface S, associated to the boundary conditions, is said geometrically rigid or inhibited if G = {0}. If G contains non-vanishing displacements, S is said to be non-geometrically rigid or non-inhibited. Moreover, an inhibited surface S is said well-inhibited if there exists a constant C > 0 such that am (v, v) ≥ C ||v1 ||21 + ||v2 ||21 + ||v3 ||20 ∀v ∈ V (2.2) where ||vi ||1 and ||vi ||0 denote, respectively, the classical norm of vi in H 1 (Ω) or in L2 (Ω). Obviously, S is said to be “not-well-inhibited” or “ill-inhibited” when (2.2) is not satisfied.

2.2 Geometrically Rigid Surfaces

35

Remark 2.2.1. Inequality (2.2) has important consequences, mainly concerned with vibration (spectral theory), which is not handled in this book. Indeed, it is easily seen that (2.2) is equivalent to an analogous inequality with only L2 (Ω) norms. Indeed, once we have coerciveness in L2 (Ω) on v3 , the rest follows from Korn’s inequalities. So, (2.2) amounts to continuous invertibility of the membrane problem in L2 (Ω). When (2.2) is not satisfied, there exist eigen-vibrations of the membrane problem with frequencies as small as desired. This implies some kind of instability of the solution of the static problem (whose frequency is 0), and the solutions are in spaces larger than the usual ones. We shall see all along this book many examples of such situation. The specific properties of such “instability” depend on the geometric properties of the surface and its fixation conditions. As a matter of fact, the only “well-inhibited” surfaces are the elliptic ones when fixed (or clamped) all along their boundaries. For these matters, the reader may see [88] chap X for instance. These different rigidity properties (geometrically rigid or not) are very important in the study of the asymptotic behavior of the Koiter model which is the subject of the next section. 2.2.2

Examples of Geometrically Rigid Surface

According to the considerations of surface rigidity of chapter 1, it is clear that the geometrically rigid (or inhibited) character of a surface is directly linked to its geometrical nature (elliptic, hyperbolic or parabolic), and to the associated boundary conditions (fixed or free). An elliptic shell fixed along its boundary is geometrically rigid or inhibited (see Fig. 2.1). Indeed, the conditions u1 = u2 = 0 amount to Cauchy conditions for the rigidity system (u3 may be eliminated) which enjoys uniqueness on the whole domain (see section 8.2.4 later). However, as we will see more in details in chapters 8 and 9, the behavior of the shell is completely different if it is fixed the whole shell is inhibited

fixed or clamped

Fig. 2.1. Example of an inhibited elliptic surface

36


0000 1111 11111 00000 0000 000001111 11111 0000 000001111 11111 0000 000001111 11111 0000 1111 0000 1111 0000 1111 00000000 11111111 00 11 00000000 11111111 0 1 00 11 00000000 11111111 0 1 00 11 00000000 11111111

asymptotic curves

S

fixed or clampled part

inhibited zone

Fig. 2.2. Example of an inhibited hyperbolic surface asymptotic lines

inhibited zone

fixed or clamped

Fig. 2.3. Example of an inhibited parabolic surface

all along its lateral boundary, or only along a part of it (and free elsewhere). In the first case, the shell is well-inhibited and classical properties follow, whereas in the latter case, it is “ill-inhibited” and a pathological behavior emerges for small values of the thickness ε. These more complex and less classical sensitive problems will be considered in detail in chapter 9. A hyperbolic shell fixed along a part of its boundary will be geometrically rigid (or inhibited) in the domain of determination of the characteristics coming from the fixed boundary (see Fig. 2.2). It will be non-geometrically rigid (or non-inhibited) elsewhere. For more details, the reader shall refer to section 8.2. Finally, a parabolic shell fixed along a part of its boundary will be geometrically rigid or inhibited in the domain defined by the generators3 lying on the clamped boundary (see Fig. 2.3). It will be non-geometrically rigid (or noninhibited) elsewhere (see also chapter 5). 3

Which are double characteristics for parabolic surfaces.

2.3 Limit Behavior of Koiter Model

37

The reader is invited to construct variants of Figs. 2.2 and 2.3. In particular, hyperbolic or parabolic shells fixed all along the boundary are inhibited, but this condition is obviously not necessary.

2.3 2.3.1

Limit Behavior of Koiter Model The Limit Membrane Problem

In the sequel, we will essentially focus on the case of geometrically rigid or inhibited shells corresponding to G = {0}. When the shell is inhibited, we enter in the framework of singular perturbation problems. Indeed, the highest order derivatives of the Koiter model (1.59) are contained in ab (·, ·) which vanishes at the limit because of the factor ε2 . The limit problem P(0) then writes4 : ⎧ ⎨ F ind u0 Vm , such that, ∀ v Vm : P(0) (2.3) ⎩ am (u0 , v) = f, v In the best cases, we have Vm = H01 (Ω)×H01 (Ω)×L2 (Ω) for a fixed or a clamped shell. The problem (2.3) has a unique solution in Vm , provided f ∈ Vm , if the shell is well-inhibited (in that case am (·, ·) is coercive on Vm ). It has to be noted that the weak formulation (1.59) of the Koiter model is posed in H01 (Ω)×H01 (Ω)×H02 (Ω) when the shell is fixed on all its boundary. Oppositely, the limit problem P(0) is posed at best in H01 (Ω) × H01 (Ω) × L2 (Ω) when the shell is well-inhibited. It is then impossible to consider boundary conditions for u03 . Indeed, as u03 ∈ L2 (Ω), u03 is consequently undefined on the boundary of Ω (trace theorem). This classical phenomenon corresponds to the arising of boundary layers along the whole boundary of the shell when ε 0. This loss of regularity of uε3 when ε tends to zero is even more important when the shell is not well-inhibited. We recall that the problem P(0) defined by (2.3) is called the membrane problem because only the membrane part of the Koiter model is taken into account. It can be written under the local formulation called the membrane system: ⎧ ⎨ −Dα T αβ (u0 ) = f β (2.4) ⎩ −bαβ T αβ (u0 ) = f 3 with the associated kinematics boundary conditions for u01 and u02 on γ0 , and the free edge conditions on γ1 . We recall that the derivatives Dα T αβ of the components T αβ of the tensor T are defined by (1.37). We have seen in the last chapter that whereas the operator associated to am + ε2 ab is always elliptic (which implies that the problem P(ε) is elliptic), 4

Where Vm is the completion of V with the norm defined by (2.6). See section 2.3.3 for more details.

38


that associated to am is of the same nature as the middle surface S. Thus, the limit membrane problem P(0) is, respectively, elliptic, parabolic or hyperbolic when the middle surface S is, respectively, elliptic, parabolic or hyperbolic (see sections 1.5.2 and 1.7). 2.3.2

Boundary Layers and Singularities

Like in most singular perturbation problems, the solution u0 of the limit problem is not as smooth as the solution uε , since u0 is the solution of a lower order problem. For small values of ε, the behavior of the shell is close to that described by the membrane problem P(0), which does not include bending effects. Even if the limit solution u0 is not smooth, uε is smooth because uε ∈ V . When ε becomes very small, there appears some boundary or internal layer phenomena at the neighborhood of the zones where u0 is singular. Moreover, the normal component u03 is at best in L2 (Ω). This implies a loss of regularity of u03 which cannot satisfy the boundary conditions anymore at the limit for ε = 0. Then, a boundary layer progressively appears along the boundary of the shell when ε decreases. When ε 0, it may also appear as internal layers: • along the curves where f is singular; • along the curves where the curvature is singular; • along the characteristic curves tangent to one the two previous curves. In this book, a large part will be devoted to the study of internal layers due to a singular loading. That is why, in section 2.6, we will try to develop a general theory of singularities, based on a micro-local analysis, which will enable us to deduce the singularity orders of the three displacements from the singularity order of the loading (mainly the normal loading f 3 ), and this with respect to the geometry of the shell. 2.3.3

Convergence to the Membrane Model in the Inhibited Case

We recall in this section the classical variational theory of the limit of the Koiter model, which needs the supplementary assumption (2.8) on the loading f . We will see hereafter that this assumption is very restrictive in shell theory, and that the convergence can also be proved provided we have only f ∈ V . Let us consider a geometrically rigid or inhibited shell. Therefore, we have: am (v, v) = 0, Thus

v∈V ⇒ v=0

||v||Vm = am (v, v)1/2

(2.5)

(2.6)

is a norm on V . We denote Vm the completion of V with this norm. The space Vm obviously is larger than V . Consequently, the associated dual spaces satisfy: Vm ⊃ V ⇔ Vm ⊂ V

(2.7)


39

When the shell is inhibited, we have the following classical result of convergence of the solution uε of the problem P(ε) to the solution u0 of the limit membrane problem (see for instance [48][88][94] for the detailed proof). Theorem 2.3.1. Under the hypothesis (2.5), if f ∈ Vm , i.e. if there exists a constant C such that |f, v| ≤ Cam (v, v)1/2

∀v ∈ V,

(2.8)

the solution uε of the problem P(ε) is such that uε −→ u0 in Vm strongly, ε0

(2.9)

where u0 is the solution of the limit problem P(0) posed in Vm : ⎧ ⎨ F ind u0 ∈ Vm , such that, ∀ v ∈ Vm : ⎩

(2.10)

am (u0 , v) = f, v

where ·, · denotes the duality product between V and V . Proof We recall here only the main steps of the classical proof of this result which can be found in [94]. We classically first establish the weak convergence, and then the strong one. Let us start from the variational formulation (1.59) of the Koiter model P(ε) (we consider that the solution uε depends explicitly on the thickness ε): ⎧ ⎨ F ind uε in V such that P(ε) (2.11) ⎩ am (uε , v) + ε2 ab (uε , v) = f, v ∀v ∈ V

where the loading f ∈ V is given. Taking v = uε in (2.11) and using (2.8), as ab (uε , uε ) is always positive, we have: am (uε , uε ) ≤ am (uε , uε ) + ε2 ab (uε , uε ) = f, uε ≤ Cam (uε , uε )1/2

(2.12)

so that we deduce that: 1

uεVm = am (uε , uε ) 2 ≤ C

(2.13)

Moreover, replacing (2.13) in the right hand side of (2.12), we also have: ε2 ab (uε , uε ) ≤ C 2

(2.14)

From (2.13), we deduce that there exists a sub-sequence still noted uε such that: uε → u∗ weakly in Vm

(2.15)

40


for some u∗ ∈ Vm . Let us now fix v ∈ V in (2.11) and pass to the limit for ε 0. From (2.14), we have: 1 1 |ε2 ab (uε , v)| ≤ ε2 ab (uε , uε ) 2 ab (v, v) 2 ≤ εC˜

(2.16)

1 2

where C˜ = Cab (v, v) is a constant for v fixed in V . So that, passing at the limit in (2.11) and using (2.16), we obtain that u∗ is solution of: am (u∗ , v) = f, v ∀v ∈ V

(2.17)

Finally, by continuity with respect to v ∈ V , equation (2.17) also holds true for any v ∈ Vm (Vm is the completion of V with the norm (2.6)). Consequently, u∗ is the unique solution of the limit problem (2.10), so that (2.15) gives (2.9) in the weak topology of Vm . For the strong convergence, the reader can refer to [94]. Remark 2.3.1. The hypothesis f ∈ Vm is the necessary and sufficient condition for the energy of the solution am (uε , uε ) + ε2 ab (uε , uε )

(2.18)

to remain bounded as ε 0. When f is in V but not in Vm , the energy tends to infinity as ε 0 (see [48]). Moreover, when f ∈ Vm , as the limit problem (2.12) has a solution u0 , the corresponding γαβ and T αβ are elements of L2 (Ω). It then follows easily that the “tensions system” −Dα T αβ = f β (2.19) −bαβ T αβ = f 3 has a solution in L2 (Ω) satisfying the “natural boundary conditions” (T αβ nβ = 0 on the free boundaries). It follows from the two previous remarks that if the tensions system has no solution in L2 (Ω) satisfying the “natural boundary conditions”, then f is not in Vm and the energy of the solutions tends to infinity. 2.3.4

A More General Result of Convergence

The convergence of the Koiter model to the membrane model is established in theorem 2.3.1 under the hypothesis (2.8). Moreover, the limit membrane problem P(0) is well-posed in Vm if f ∈ Vm . However, the hypothesis (2.8) is very restrictive: very usual loadings (even C ∞ ) may be out of Vm . When the shell is well-inhibited (see definition 2.2.2), Vm is the smallest possible. Thus, the dual space Vm = H −1 × H −1 × L2 is the largest possible and contains usual loadings. Oppositely, in ill-inhibited shells, instabilities may appear when ε 0 (see [48][70]). This is the case of “sensitive shells” for which the space Vm is so small that it does not contain the test functions of the distribution space. These problems are said to be sensitive problems and will be studied in detail in chapters 8 and 9.


41

There exists a more general result of convergence in the inhibited case, due to D. Caillerie [20] (see also [72]), where the convergence of the Koiter model to the membrane model is proved with only f ∈ V . This assumption is clearly much weaker than the one f ∈ Vm of theorem 2.3.1. However, this result and its proof are not widely known; it leads to a limit problem which is not in a variational framework. We recall here the main steps of the proof of this result, which is detailed in [20][72]. First, let Am and Ab be the continuous operators of V into V associated with the bilinear forms am (·, ·) and ab (·, ·) by: Am u, v = am (u, v), Ab u, v = ab (u, v),

∀(u, v) ∈ V × V

(2.20)

where we recall that ·, · denotes the duality product between V and V . Thus, Koiter variational formulation (2.11) becomes: Am uε + ε2 Ab uε = f

(2.21)

We then have the following lemma:

Lemma 2.1. The operator Am is surjective and its range is dense in V . Proof The detailed proof of this lemma can be found in [20][72]. We only recall here the main steps. Let Am u = Am v. Taking the duality product with u − v, we obtain: Am (u − v), u − v = am (u − v, u − v) = 0 As the shell is geometrically rigid or inhibited, it follows from (2.5) that u = v, whence the first assertion. To prove the second assertion of lemma 2.1, let R(Am ) be the range of Am . Let u be an element of its polar set5 . We then have: Am v, u = 0 ∀v ∈ V Taking v = u it follows that am (u, u) = 0 and then u = 0 according to (2.5) as before. This ends the proof of lemma 2.1. It then appears that Am is an one to one mapping of V into its range R(Am ), which is a dense subset in V . Let us now define on V a norm by: vVA = Am vV

(2.22)

Obviously, V is not complete for the norm (2.22). But Am defines an isomorphism between V (with the norm (2.22)) and R(Am ) (with the norm .V ). Automatically, Am has an extension by continuity which is an isomorphism between the completions of both spaces. Denoting by Am the extended operator and by VA the completion of V with the norm (2.22), Am is an isomorphism 5

Which belongs to the orthogonal of Am in the sense of duality.

42


between VA and V (which is the completion of R(Am ) with the norm of V according to lemma 2.1). Then equation (2.21) may be written as well: Am uε + ε2 Ab uε = f

(2.23)

In order to pass to the limit when ε 0, the classical way consists in obtaining an a priori energy estimate of uε by taking the duality product of (2.23) with uε . But such a way needs a hypothesis of boundedness of the functional f with respect to the limit form am (·, ·), and this does not work for any f ∈ V . In the general case, we shall follow an idea of [20], which consists in proving that the term ε2 Ab uε tends to zero in V . Passing it to the right hand side of (2.23), the latter tends obviously to f in V . Taking then advantage of the property that Am is an isomorphism, we automatically prove the existence of a limit uε in VA . More precisely, we have the following theorem: Theorem 2.3.2. Under the hypothesis (2.5), there exists a unique element u0 in VA such that: Am u 0 = f (2.24) Moreover

uε −→ u0 strongly in VA ε0

(2.25)

where uε ∈ V is the solution of the limit problem (2.21). Proof The first part of the proof (existence of a unique solution of (2.24)) follows from the fact that Am defines an isomorphism of VA on V . For the convergence (2.25), we shall only prove it in the weak topology of VA . For the proof of the strong convergence, refer to [20][72]. Taking v = ε2 uε in Koiter formulation (2.11), we have: ε−2 am (ε2 uε , ε2 uε ) + ab (ε2 uε , ε2 uε ) = f, ε2 uε

(2.26)

and then as am (·, ·) is definite positive on V , we have: am (ε2 uε , ε2 uε ) + ab (ε2 uε , ε2 uε ) ≤ f V ε2 uε V

(2.27)

But the expression in the left hand side is equivalent to the square of the norm of ε2 uε in V , so that we have: ε2 uε V ≤ C

(2.28)

for some C, provided f is bounded in V . It follows that, after extraction of a subsequence (we shall see later that it is the whole sequence): ε2 uε → u∗ weakly in V for some u∗ ∈ V .

(2.29)


43

On the other hand, multiplying (2.21) by ε2 , we get: Am ε2 uε + ε2 Ab ε2 uε = ε2 f

(2.30)

We may now pass to the limit when ε 0 in the weak topology of V . Indeed, Am is continuous from the weak topology of V into the weak topology of V . 2 ε Moreover, (2.28) shows that ε Ab u remains bounded in V , so that: Am u∗ = 0

(2.31)

which implies (as Am is surjective) that: u∗ = 0

(2.32)

Coming back to (2.21), as Ab is continuous from the weak topology of V into the weak topology of V , it follows from (2.29) and (2.32) that: Ab ε2 uε → 0 weakly in V

(2.33)

We then have from (2.21) and (2.24): Am uε = f − ε2 Ab uε → f = Am u0 weakly in V

(2.34)

Finally, as Am is an isomorphism of VA on V , we have uε −→ u0 weakly in VA ε0

which proves the weak convergence (2.9) of theorem 2.3.2. For the strong convergence, see [20]. It should be emphasized that theorem 2.3.2 holds true without special hy pothesis on f , apart from the obvious one f ∈ V . The limit u0 ∈ VA is the solution of the abstract problem (2.24) which is not a variational one. The classical (variational) theory of the limit needs the supplementary hypothesis (2.8) on f which is very restrictive, as already mentioned. 2.3.5

Convergence to the Pure Bending Model in the Non-inhibited Case

Even if the case of non-geometrically rigid shells is not the main aim of this book, it will be considered in chapter 10. So that we recall here the main result existing in this case. We shall consider non-geometrically rigid or non-inhibited shells, where the property: γαβ (v) = 0 ⇒ v = 0 does not hold. Then the subspace G of inextensional displacements defined by (2.1), which is the kernel of the membrane bilinear form am (·, ·), is a closed subspace of V . Indeed, the limit in the topology of V of element of G obviously belongs to G. Consequently, G is a Hilbert space of it own, with the norm of V .

44


But, as am (v, v) = 0 in G, we see that the norm of V in G is equivalent to 1 ab (v, v) 2 . As a consequence, the problem ⎧ ⎨ F ind u0 ∈ G, such that, (P)b (2.35) ⎩ ab (u0 , v) = f, v ∀ v ∈ G is well posed and has a unique solution. Moreover, it is evident that the fact that the limit membrane form am (·, ·) in the Koiter formulation (2.11) vanishes on G implies some kind of weakness in G. In other words, we have a very important loss of rigidity of the shell: for the same applied forces, the corresponding solution will be very large. Therefore, in order to conserve a finite limit when ε tends to zero, we shall consider much lower forces ε2 f instead of f . Thus, Koiter formulation (2.11) becomes: ⎧ ⎨ F ind uε in V such that P(ε) (2.36) ⎩ −2 ε am (uε , v) + ab (uε , v) = f, v ∀v ∈ V We then have the following theorem of convergence:

Theorem 2.3.3. If f ∈ V , the solution uε of the problem P(ε) defined by (2.36) satisfies: uε → u0 strongly in V u0 is the solution of the pure bending model (2.35). Proof We recall only here the main steps of the classical proof of theorem 2.3.3 which can be found in [94]. Once again, we classically begin with establishing the weak convergence, and then the strong one. Taking v = uε in (2.36), we get: ε−2 am (uε , uε ) + ab (uε , uε ) = f, uε ≤ Cuε V

(2.37)

for some C. As the left hand side is coercive for any value of ε (see remark 1.6.1), and satisfies: ε−2 am (uε , uε ) + ab (uε , uε ) ≥ αuε 2V for some α > 0, it follows that: αuε V ≤ C

(2.38)

and coming back to the right hand side of (2.37), we have: ε−2 am (uε , uε ) ≤ C

(2.39)

2.4 Complements on Nagdhi Model and its Limits

45

Moreover, from (2.38), there exists a subsequence still noted uε such that: uε → u∗ weakly in V

(2.40)

∗

for some u ∈ V . Moreover, for a fixed v ∈ V , we have: am (u∗ , v) = lim am (uε , v) ε0

(2.41)

and from (2.39), we obtain: 1

1

|am (uε , v)| ≤ a(uε , uε ) 2 a(v, v) 2 ≤ Cε → 0

(2.42)

Passing to the limit for ε 0, we obtain that: am (u∗ , v) = 0

(2.43)

for any v ∈ V . Moreover, taking v = u∗ in (2.43), we obtain that u∗ ∈ G. Finally, considering v ∈ G in (2.36) we have: ab (uε , v) = f, v

∀v ∈ G

(2.44)

and passing to the limit with (2.40), we obtain: ab (v ∗ , v) = f, v

∀v ∈ G

(2.45)

which proves that u∗ = u0 , where u0 is the solution of (2.35). This proves that (2.40) is the desired convergence in the weak topology. For the proof of the strong convergence, see [94].

2.4

Complements on Nagdhi Model and its Limits

Before addressing in the next section an accurate study of the singularities arising for very thin shells, a question naturally emerges: do the results obtained (singularity orders, propagation or not) depend on the shell model considered, i.e. the Koiter model in our concerns. In this section we shall prove that if we consider the more complex Nagdhi shell model (including transverse shear), its limits when ε 0 being identical as those of Koiter model, the resulting singularities will be also identical. We shall not come back in details on the establishing of the Nagdhi shell model, but only start from its classical formulation. For more details, the reader can refer to [15]. In the Nagdhi model, the kinematics of the shell is described by the displacements ui and the (small, in the linearized sense) angles θλ turned by the normal fiber. The unknowns are the three components of u and the two components of θ. In addition to the membrane strain tensor γαβ (u) defined by (1.41), we define: ⎧ ⎨ ϕα (u) = ∂α u3 + bλα uλ (2.46) ⎩ χαβ (u, θ) = 12 [Dβ θα + Dα θβ − bλα dλβ (u) − bλβ dλα (u)]

46


where dλβ (u) = Dβ uλ − bλβ u3 . The energy bilinear form of Naghdi model is then am (u, u) + as (u, θ; u, θ) + ε2 ab (u, θ; u, θ)

(2.47)

where am and ab are the membrane and bending forms (1.60)-(1.61), the last one with χαβ instead of the ραβ , whereas as is the “shear energy form” given by: as (u, θ; u, θ) = B αβ [ϕα (u) + θα ][ϕβ (u) + θβ ]dS (2.48) for certain coefficients B αβ satisfying a positivity condition (the shear energy is positive). An important point is that, when we have χαβ (u, θ) = ραβ (u), then as (u, θ; u, θ) = 0. In other words, when θ = −ϕ(u), i.e. when the normal fiber turns remaining normal to the deformed surface, the bilinear forms of the Nagdhi model coincide with that of the Koiter model (the transverse shear vanishes). The limits of the corresponding Nagdhi variational problem shall be addressed in the same way as for the Koiter model (sections 2.3.3 and 2.3.5) with exactly the same limits. More precisely, in the inhibited case, proceeding exactly as in section 2.3.3, we see that the limit problem is described by ⎧ ⎨ F ind (u0 , θ0 ) ∈ Vm such that f or any (v, ζ) ∈ Vm : (2.49) ⎩ am (u0 , v) + as (u0 , θ0 ; v, ζ) =< f, v > which lives in an appropriate space Vm . Taking in particular v = 0 and arbitrary ζ we get B αβ [ϕα (u0 ) + θα0 ]ζdS = 0

(2.50)

for any ζ, so that B αβ [ϕα (u0 ) + θα0 ] = 0, and by the positivity of the B αβ it follows that θ 0 = −ϕ(u0 ). Then, as the term in as vanishes, u0 is precisely the solution of (2.10). As a result, denoting by uε , θε the solution of the Nagdhi problem, uε converges to u0 in the strong topology of Vm , where u0 is the solution of (2.10). Moreover, θ 0 = −ϕ(u0 ), i.e. at the limit the normal fiber turns remaining normal to the deformed surface. The non-inhibited problem is slightly more subtle. Obviously, in the present case, the kernel should be defined by = {(u, θ); γαβ (u) = 0, θα = −ϕα (u)}. G

(2.51)

It is easily seen that the “classical” G(u) defined in (2.1) coincides with the set Now, in order to study the limit when ε 0, of the u such that (u, θ) is in G. we proceed as in (2.36), but the “penalty terms” concern am and as instead of only am (so that the corresponding kernel of af is precisely G). instead of G, and with the The limit problem is analogous to (2.35), but in G energy form ab (u0 , θ0 ; u0 , θ0 ) involving χαβ (u0 , θ0 ) instead of ab (u0 , u0 ) involving only ραβ (u0 ). In fact, the two problems are exactly the same; as we pointed out,

2.5 Reduction of the Membrane System to One PDE

47

both energy forms coincide, and the elements involved are for elements in G the same. As a result, (uε , θε ) converges for ε 0 to (u0 , θ0 ), where u0 is the solution of the limit problem of the Koiter model (in that case the pure bending model (2.35)) and θ0 is such that the Koiter (or Kirchhoff-Love) hypothesis (the normal fiber remains normal under deformation) is satisfied.

2.5

Reduction of the Membrane System to One PDE for Each Component of the Displacement

This section (as well as next one) is devoted to a local study of singularities for solutions of the membrane system. Our approach is largely inspired by micro-local analysis techniques (see for instance [43], though much more formal) and adapted to our specific situations. Taking only account of terms of the leading order of differentiation for each variable in each equation and freezing the coefficients at a point P (i.e. taking constant coefficients with values equal to those of the given problem at point P ), we have a simplified problem allowing reduction to an equation instead of a system. This equation furnishes the directions of the characteristics at P and the leading order of the singularities of the solutions at P . Moreover, in the cases where the singularities propagate along the characteristics (hyperbolic and parabolic cases), the propagation equations involve entities which are not local: lower order terms should be taken into account for a precise description of the propagation. This analysis will be performed for instance in the specific examples of chapters 5 and 6. But, in the present chapter, all the analyses are carried out with only the leading order terms. Obviously, the propagation equations are then only qualitative, preserving orders of magnitude, whereas the very values of the corresponding coefficients along the characteristics are not reached. In the sequel, for the simplicity of the notations, the solution u0 of the limit membrane problem will be noted u. The solution of the Koiter model for ε > 0 will still be noted uε so that no confusion is possible. As our aim is to study the most singular term of the three displacements and the propagation of this singularity, we do not need to compute exactly the resulting displacements. According to the micro-local analysis technique, we only consider the higher order derivatives for each displacement u1 , u2 and u3 in the operators involved. Moreover, we consider constant geometrical coefficients whose value is determined at the point studied. Then, we compute only the most singular term of u1 , u2 and u3 . Considering now the constitutive law (1.53) and the definition (1.41) of the membrane strain tensor γαβ , the membrane system (1.64) writes: ⎧ 1βγ1 ∂β ∂γ u1 − A1βγ2 ∂β ∂γ u2 + A1βγδ bγδ ∂β u3 + · · · = f 1 ⎪ ⎪ −A ⎪ ⎪ ⎨ −A2βγ1∂β ∂γ u1 − A2βγ2 ∂β ∂γ u2 + A2βγδ bγδ ∂β u3 + · · · = f 2 ⎪ ⎪ ⎪ ⎪ ⎩ −A1βγδ bγδ ∂β u1 − A2βγδ bγδ ∂β u2 + Aαβγδ bαβ bγδ u3 + · · · = f 3

(2.52)

48


The terms + . . . contain the derivatives of the constitutive law coefficients Aαβλμ and the Christoffel symbols multiplied by lower order derivatives of the displacements. According to the micro-local analysis (see [43]), these lower order terms6 can be neglected for the local study of singularities. For our study, we can then consider the simplified system (2.52) which can be written under the form: Au = f with A = (Aij ). The matrix A is obtained from (2.52) and writes: ⎛ ⎞ −A1βγ1 ∂β ∂γ −A1βγ2 ∂β ∂γ A1βγδ bγδ ∂β ⎜ ⎟ ⎜ ⎟ 2βγ1 2βγ2 2βγδ ⎟ −A ∂ ∂ −A ∂ ∂ A b ∂ A=⎜ β γ β γ γδ β ⎟ ⎜ ⎝ ⎠ −A1βγδ bγδ ∂β −A2βγδ bγδ ∂β Aαβγδ bαβ bγδ 2.5.1

(2.53)

(2.54)

Case of the Normal Displacement u3

According to the micro-local analysis evoked above, we consider that all the coefficients in the system (2.53) are constant7 . This does not affect the local study of the highest order singularities of each displacement. Applying to the system (2.53) a technique similar to the Cramer’s rule and valid for differential systems with constant coefficients, we get: 1 C 2 C 3 Det(A)u3 = AC 13 f + A23 f + A33 f

where

AC ij

(2.55)

are the cofactors of the matrix A = (Aij ).

Remark 2.5.1. Equation (2.55) is most illuminating for the further study of singularities. Let us focus on the action of f 3 . The equation for u3 is of fourth 3 order, but its right hand side is not f 3 , but AC 33 f ; it means that the effective 3 “data” involve fourth order derivatives of f , i.e. are four steps more singular than f 3 itself. This is the very reason why singularities of the solutions are much more important in shell theory than in other areas of mechanics and physics. As a matter of fact, in most cases, taking account of singularities amounts to a whole description of the solution. In what follows, we consider the particular case of a normal loading f = f 3 e3 , which causes higher singularities than f 1 and f 2 . We then have: f 1 = f 2 = 0 and f 3 = 0

(2.56)

Expression (2.55) reduces to: 3 Det(A)u3 = AC 33 f

6 7

(2.57)

The highest terms of differentiation are not the same for the three displacements and the three equations. Which is the case if the metric tensor aαβ and consequently the coefficients Aαβλμ are supposed constant at least locally.

2.5 Reduction of the Membrane System to One PDE

49

After some calculations which are technical but not difficult (the reader can refer to appendix B for the details of the calculations), we obtain for Det(A) and AC 33 : (2) E3 b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 2 3 2(1 − ν )(1 + ν)a

Det(A) = and

AC 33 =

11 2 (2) E2 a ∂1 + a22 ∂22 + 2a12 ∂1 ∂2 2(1 − ν 2 )(1 + ν)a

(2.58)

(2.59)

We recall that a = det(aαβ ) = a11 a22 − (a12 )2 is the determinant of the metric tensor of the middle surface S. Finally, we get the following PDE for u3 : (2) (2) 3 E b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 u3 = a2 a11 ∂12 + a22 ∂22 + 2a12 ∂1 ∂2 f (2.60) This characterizes the most singular term of the displacement u3 appearing when the loading f 3 is singular. As (aαβ ) is the inverse matrix of (aαβ ), according to (1.10), we get: (2) (2) 3 E b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 u3 = a22 ∂12 + a11 ∂22 − 2a12 ∂1 ∂2 f (2.61) 2.5.2

Tangential Displacements u1 and u2

The method of the previous section is now applied to the tangential displacements u1 and u2 . In order to obtain a PDE for u1 , we start from the relation similar to (2.57): 3 Det(A)u1 = AC (2.62) 31 f where: AC 31

E2 = 4a(1 + ν)2

(2 + J)b11 (a11 )2 +

a a

12 2

b22 ∂13 22 2 22 12 + 2(2 + J)(a ) b12 + 2(2 + J)a a b11 ∂23

11 22

− (2 + 2J)(a )

+ 4(2 + J)a11 a12 b11 + (4(2 + J)(a12 )2 − 2a11 a22 )b12 22 12 −2(2 + J)a a b22 ∂12 ∂2 ((3J + 4)a11 a22 + (6 + 2J)(a12 )2 )b11 22 2 22 12 2 −(2 + J)(a ) b22 + (8 + 4J)a a b12 ∂1 ∂2 where we set J =

2ν . 1−ν

(2.63)

50


In any coordinate system, the operator Ac31 writes: 3 2 2 3 AC 31 = m10 ∂2 + m11 ∂2 ∂1 + m12 ∂2 ∂1 + m13 ∂1

(2.64)

Analogously, for u2 , we obtain the equation: 3 Det(A)u2 = AC 32 f

(2.65)

The expression of AC 32 can be deducted by inverting the subscripts 1 and 2 in the expression of AC 31 and by changing the sign of the result: E2 11 2 11 12 3 C A32 = − 2(2 + J)(a ) b + 2(2 + J)a a b 12 22 ∂1 4a(1 + ν)2 11 22 12 2 22 2 + a a − (2 + 2J)(a ) b11 + (2 + J)b22 (a ) ∂23 + − (2 + J)(a11 )2 b11 + ((3J + 4)a11 a22 + (6 + 2J)(a12 )2 )b22 (2.66) 11 12 +(8 + 4J)a a b12 ∂12 ∂2 + − 2(2 + J)a11 a12 b11 + 4(2 + J)a22 a12 b22 + (4(2 + J)(a12 )2 − 2a11 a22 )b12 ∂1 ∂22 In any coordinate system, the operator AC 32 writes: 3 2 2 3 AC 32 = m20 ∂2 + m21 ∂2 ∂1 + m22 ∂2 ∂1 + m23 ∂1

(2.67)

where the coefficients m2i can be identified from (2.66).

2.6

Structure of the Displacement Singularities when the Loading Is Singular along a Curve

In this part, we address the singularities of the displacements when the loading is purely normal (f1 = f2 = 0 and f 3 = 0). As seen previously (equations (2.57), (2.62) and (2.65)), the membrane system can be reduced to three PDE’s of the type: Lui = Mi f 3 (2.68) where L = a3 det(A) is a 4th order operator given by (2.58). We can write it under the form: L = l0 ∂24 + l1 ∂23 ∂1 + l2 ∂22 ∂12 + l3 ∂2 ∂13 + l4 ∂14

(2.69)

where li can be identified in the expression (2.58) of det(A). We have for instance:

2.6 Structure of the Displacement Singularities

l0 =

E3 b2 2(1 − ν 2 )(1 + ν) 11

51

(2.70)

In (2.68), ui refers to the component i of the displacement u. The rightrd hand side Mi f 3 depends on Mi , equal to a3 AC order 3i . M1 and M2 are then 3 operators which can be written: Mα = mα0 ∂23 + mα1 ∂22 ∂1 + mα2 ∂2 ∂12 + mα3 ∂13 ,

α = 1, 2

(2.71)

In the other hand, M3 is a 4th order operator: M3 = m30 ∂24 + m31 ∂23 ∂1 + m32 ∂22 ∂12 + m33 ∂2 ∂13 + m34 ∂14

(2.72)

For the time being, we consider a shell having an unspecified middle surface S (elliptic, hyperbolic or parabolic). The loading f 3 is supposed to be singular inside the domain along a straight line, for instance, along the line y 2 = 0 (see Fig. 2.4). Obviously, such a line in the plane of the parameters induces a curve on the middle surface of the shell. Moreover, any curve may be written in this form with an appropriate parametrization.

f 3 singular along y 2 = 0

S Ψ y1, y2

e3 2

e2

y

Ω

y1

O

e1

Fig. 2.4. Singularity of the loading in y 2 = 0

Let us now specify the terminology used in the sequel for the study of singularities of functions (or distributions [97]) of a variable x. Definition 2.2. Let S0 (x) be a basic singularity. Then, we have the corresponding chain: . . . S−2 (x), S−1 (x), S0 (x), S1 (x), S2 (x), S3 (x), . . .

(2.73)

d Sk . This chain of singularities must be understood in the sense dx of functions (or distributions) defined up to an additive function (or distribution) with Sk+1 =

52


which is smooth in the neighborhood of x. Thus, we say that S2 (x) is two orders more singular than S0 (x) and S−2 (x) is two orders less singular than S0 (x). An example is: . . . x H(x), H(x), δ(x), δ (x), . . . (2.74) H(.) being the Heaviside jump function and δ(.) the Dirac function, but there are many other ones. Remark 2.6.1. In the sequel, the terminology “n order more or less singular than” (with n ∈ N) will always refer to the definition 2.2 of the chain of singularities considered, even if the basic singularity S0 (x) may differ for each example. Considering that f 3 has a singularity S0 (y 2 ) along the line y 2 = 0, f 3 has the form: f 3 = Φ(y 1 )S0 (y 2 ) (2.75) where Φ(y 1 ) is a smooth function in y 1 . Thus, the right-hand side of (2.68) writes: Mi f 3 = Ψi (y 1 )Sβi (y 2 ) + · · · (2.76) where Ψi (y 1 ) is a function containing derivatives of Φi (y 1 ) (up to the 4th order) and βi is an integer which has to be determined for each displacement ui . We then search for a solution ui of the form: ui (y 1 , y 2 ) = Uiαi (y 1 )Sαi (y 2 ) + Uiαi −1 (y 1 )Sαi −1 (y 2 ) + . . .

(2.77)

The expression (2.77) is a sequence of the chain (2.73), multiplied by the unknown coefficients Uiαi (y 1 ), Uiαi −1 (y 1 ), . . . Replacing (2.76) and (2.77) in (2.68) and identifying in (2.69), we obtain: l0 Uiαi (y 1 )Sαi +4 (y 2 ) + Uiαi −1 (y 1 )Sαi +3 (y 2 ) + . . . (1) +l1 Uiαi (y 1 )Siαi +3 (y 2 ) + . . . (2) +l2 Uiαi (y 1 )Sαi +2 (y 2 ) + . . . (3) +l3 Uiαi (y 1 )Sαi +1 (y 2 ) + . . . (4) +l4 Uiαi (y 1 )Sαi (y 2 ) + . . . = Ψ (y 1 )Sβi (y 2 ) + . . . (2.78) (k)

where Uj

=

∂ k Uj

. ∂y 1 k In what follows, we will see that the fact that the line y 2 = 0 is characteristic or not is very important. The orders of the displacements singularities will be very different and there may even appear propagation of the singularities in some cases.


2.6.1

53

Singularity along a Non-characteristic Line

We consider here the case when y 2 = 0 is not a characteristic8. At each point, the asymptotic lines satisfy equation (1.17). It appears that necessarily b11 = 0 whatever the nature of the shell is. Indeed, if b11 vanishes, y 2 = constant would be an asymptotic direction, according to (1.17). Moreover, we remark that the coefficient l0 never vanishes according to (2.70). In order to determine the order of the singularities of the displacements, we identify the most singular terms in both sides of equation (2.78) (we start from the most singular Sβ , then Sβ−1 , etc). We find that αi + 4 = βi . • Displacement u3 The study of the singularity order of the normal displacement u3 is obtained starting from (2.68) with i = 3. In that case, the coefficient m30 of the righthand site never vanishes. Indeed, we have: m30 =

E2 a11 2(1 − ν 2 )(1 + ν)

(2.79)

with a11 which is strictly positive by definition. Thus: M3 f 3 = m30 Φ(y 1 )S4 (y 2 ) + . . .

(2.80)

and therefore we have, by identification with (2.78), β3 = 4 and α3 = 0. Thus, the highest singularity for the normal displacement u3 is given by: u3 (y 1 , y 2 ) = U30 (y 1 )S0 (y 2 ) + . . . with

U30 (y 1 ) =

m30 Φ(y 1 ) l0

(2.81) (2.82)

which is defined and different from zero since m30 = 0 and l0 = 0. Thus, the normal displacement u3 always has the same singularity as the loading f 3 when this loading is singular along a non-characteristic line. • Displacement u2 In order to determine the most singular terms of the displacement u2 , we perform the same study as that of the displacement u3 , starting again from (2.68) with i = 2. The coefficient m20 in front of ∂23 can vanish. Indeed, it writes: m20 =

E 2 a2 11 22 12 2 22 2 (a a − 2(1 + J)(a ) )b + (2 + J)b (a ) 11 22 4(1 + ν)2

(2.83)

and vanishes in very specific cases, when: 8

In linear theory, we recall that the asymptotic curves of the middle surface S of the shell and the characteristics of the membrane system confuse.

54


b22 =

2(1 + J)(a12 )2 − a11 a22 b11 (2 + J)(a22 )2

(2.84)

In the principal coordinate system for curvatures, this condition becomes: b22 =

ν−1 1 b 2 1

(2.85)

This can only happen for hyperbolic shells (as ν − 1 < 0) in very particular cases which are not considered here. In the sequel, we consider that coefficient m20 does not vanish. Consequently, we have: M2 f 3 = m20 Φ(y 1 )S3 (y 2 ) + . . .

(2.86)

We then obtain that β2 = 3 and α2 = −1, which implies: u2 (y 1 , y 2 ) = U2−1 (y 1 )S−1 (y 2 ) + . . . with

U2−1 (y 1 ) =

m20 Φ(y 1 ) l0

(2.87) (2.88)

Accordingly, in general (except for very particular cases), if the singularity of the normal loading f 3 is not along a characteristic, u2 is one order less singular9 than f 3 . • Displacement u1 The determination of the singularities of u1 leads to a more complex study with subcases. Indeed, we have to distinguish two cases, depending if m10 vanishes or not. According to (2.63), the coefficient m10 writes: m10 =

E 2 a2 E 2 a2 (2 + J)a22 (a22 b12 + a12 b11 ) = (2 + J)a22 b21 2 2(1 + ν) 2(1 + ν)2

(2.89)

Since a > 0 and a22 > 0, m10 vanishes if b21 vanishes. This is the case if the parametrization (y 1 , y 2 ) corresponds to the principal coordinate system for curvature. Thus, two cases are possible: - If b21 = 0 (generic case), we have: M1 f 3 = Ψ1 (y 1 )Sβ1 + · · · = m10 Φ(y 1 )S3 (y 2 ) + . . .

(2.90)

We obtain β1 = 3 and consequently α1 = −1. The displacement u1 is then of the form: u1 (y 1 , y 2 ) = U1−1 (y 1 )S−1 (y 2 ) + . . . (2.91) with

9

U1−1 (y 1 ) =

m10 Φ(y 1 ) l0

In the sense of definition 2.2 of the chain of singularities.

(2.92)


55

The factor U1−1 is always defined because l0 = 0. Thus, in the general case when b21 = 0, the displacement u1 is one order less singular than the normal loading f 3 . - If b21 = 0 (particular case), we have m10 = 0. Two other cases must be considered whether m11 vanishes or not. First, let us recall the expression of m11 deducted from (2.63): E 2 a2 m11 = (3J + 4)a11 a22 + 2(3 + J)(a12 )2 b11 2 4(1 + ν) (2.93) −(2 + J)(a22 )2 b22 + 4(2 + J)a22 a12 b12 It is quite difficult to give a geometric interpretation of the condition m11 = 0. However, if we are in the principal coordinate system for curvatures (which implies m10 = 0), the condition m11 = 0 becomes:

and finally:

(3J + 4)a11 a22 b11 − (2 + J)(a22 )2 b22 = 0

(2.94)

a11 b11 b11 2+J 1 = = = 2 22 a b22 b2 4 + 3J 2+ν

(2.95)

In other words, if (y 1 , y 2 ) corresponds to the parametrization of the principal coordinate system for curvatures, m10 vanishes, and m11 vanishes if b22 = (2 + ν)b11 . This can only occur when the shell is elliptic. Let us come back to the second considered case, when m10 = 0 because b21 = 0. We have two possible cases: - If m11 does not vanish, equation (2.78) becomes: l0 U1α1 (y 1 )Sα1 +4 (y 2 ) + . . . + · · · = m11 Φ(1) (y 1 )S2 (y 2 ) + . . .

(2.96)

We get β1 = 2 and α1 = −2. Thus, the most singular term of the displacement u1 is: u1 (y 1 , y 2 ) = U1−2 (y 1 )S−2 (y 2 ) + . . . (2.97) with U1−2 (y 1 ) =

m11 (1) 1 Φ (y ) l0

(2.98)

In this case, the singularity of the displacement u1 is two orders less singular than the loading f 3 . - Finally, if m11 vanishes, two new cases should be distinguished whether m12 vanishes or not. Since we are in the principal coordinate system, m12 vanishes and equation (2.78) becomes: l0 U1α1 (y 1 )Sα1 +4 (y 2 ) + . . . + · · · = m13 Φ(2) (y 1 )S0 (y 2 ) + . . . (2.99)

56


We have now β1 = 0 and α1 = −4. Thus, the most singular term of the displacement u1 writes: u1 (y 1 , y 2 ) = U1−4 (y 1 )S−4 (y 2 ) + . . .

(2.100)

with U1−4 (y 1 ) =

m13 (3) 1 Φ (y ) l0

(2.101)

In this very particular case, the displacement u1 is four orders less singular than f 3 . We remark that the study of the singularities of the displacements sometimes implies (here for u1 ) to consider different particular cases, which can reveal to be long to treat. However, we want to determine the most singular terms10 of the displacement, which are the most penalizing for the structures. The particular cases considered lead to weaker singularities, so less penalizing. Moreover, they correspond to very particular cases which are rarely encountered in the literature. Remark 2.6.2. If b21 = 0 and if f 3 does not depend on y 1 ( i.e. φ(y 1 ) = constant), according to (2.101), we obtain U1−4 = 0. The continuation of the developments would show that the coefficients U1−n , linked to Φ(n) (y 1 ) (with n > 3) vanish. In this particular case, u1 would not be singular. 2.6.2

Singularity along a Characteristic Line

Contrary to the case of a singularity of the loading along a non-characteristic line, the orders of the singularities of the displacements depend on the nature of the middle surface of the shell, when the singularity of f 3 is along a characteristic line. Since only hyperbolic and parabolic shells have real asymptotic curves, we will distinguish these two types of shells in what follows. Case of a parabolic shell Let us consider a parabolic shell whose characteristic curves are y 2 = constant in the plane of the parameters. The direction dy 2 = 0 cancels the second fundamental form, so that b11 = 0 according to (1.17). The shell being parabolic, we have b11 b22 − b212 = 0, and consequently b12 = 0 and b22 = 0 (the shell is not a plate). We then have l0 = l1 = l2 = l3 = 0 and l4 = 0 according to (2.58). As previously, we consider the case where f 3 has a singularity in S0 (y 2 ) in the neighborhood of y 2 = 0 which corresponds to a characteristic line. Thus, equation (2.78) becomes: (4) l4 Uiαi (y 1 )Sαi (y 2 ) + . . . = Ψ (y 1 )Sβi (y 2 ) + . . . (2.102) 10

In the sense of definition 2.2 of the chain of singularities.


57

This equation allows to determine the orders of the singularities of the displacements u1 , u2 and u3 . Generally, considering the most singular terms, we obtain αi = β i . • Displacement u3 As seen previously, the coefficient m30 never vanishes, whatever the nature of the middle surface S is (see expression (2.79)). We then have: M3 f 3 = m30 Φ(y 1 )S4 + · · ·

(2.103)

which leads to α3 = β3 = 4. Consequently, we always have: u3 (y 1 , y 2 ) = U34 (y 1 )S4 (y 2 ) + . . .

(2.104)

d4 U34 1 m30 (y ) = Φ(y 1 ) 1 4 d(y ) l4

(2.105)

with

Thus, the normal displacement u3 is four orders more singular than the loading f 3 . • Displacement u2 The coefficient m20 vanishes if the condition (2.84) is satisfied. In the parabolic case (b11 = b12 = 0), this last one becomes: a11 a22 − 2(J + 1)(a12 )2 = 0

(2.106)

In the sequel, we suppose that this very particular condition is not verified. This is evident when the parametrization corresponds to the principal coordinate system for curvature11 where a12 = 0 and a11 > 0, a22 > 0. If m20 = 0, the right-hand side of (2.68) becomes: M2 f 3 = m20 Φ(y 1 )S3 (y 2 ) + · · ·

(2.107)

which leads to α2 = β2 = 3. Consequently, the most singular term of the displacement u2 writes: u2 (y 1 , y 2 ) = U23 (y 1 )S3 (y 2 ) + . . .

(2.108)

d4 U23 1 m20 (y ) = Φ(y 1 ) 1 4 d(y ) l4

(2.109)

with

11

In the case of parabolic shells, one of the principal curvature lines confuses with the generators y 2 = const.

58


In that case, the displacement u2 is three orders more singular than the loading f 3 . • Displacement u1 In the parabolic case considered here, since b11 = b12 = 0, we get m10 = 0. The E 2 a2 first non-vanishing coefficient of the right-hand side is then m11 = − 4(1+ν) 2 (2 + 22 2 22 J)(a ) b22 . Since a and b22 are different from zero, we always have m11 = 0 and: M1 f 3 = m11 Φ(1) (y 1 )S2 (y 2 ) (2.110) dΦ . Thus, we find β1 = 2 and α1 = 2, which implies dy 1 that the more singular term for the displacement u1 is given by: where we recall that Φ(1) =

u1 (y 1 , y 2 ) = U12 (y 1 )S2 (y 2 ) + . . .

(2.111)

d4 U12 1 m11 (1) 1 (y ) = Φ (y ) 1 4 d(y ) l4

(2.112)

with

where l4 and m11 are different from zero. Thus, for a parabolic shell, the displacement u1 is two orders more singular than the loading f 3 when f 3 is singular on a characteristic curve of the surface. Case of a hyperbolic shell Let us now consider a hyperbolic shell having characteristics in the direction y 2 = constant. As seen in the parabolic case, the direction dy 2 = 0 cancels the second fundamental form, which implies necessarily that b11 = 0.

y 1 = const

characteristic directions y 2 = const

a1 a2

principal curvatures lines Fig. 2.5. Example of a hyperbolic shell


59

Since the shell is hyperbolic, we have b11 b22 − b212 < 0, and b12 cannot vanish. We have then l0 = l1 = 0 according to (2.58). Equation (2.78) becomes: (2) l2 Uiαi (y 1 )Sαi +2 (y 2 ) + . . . + · · · = Ψ (y 1 )Sβi (y 2 ) + . . . −E 3 b212 = 0. In the case of hyperbolic shells, we have then (1 − ν 2 )(1 + ν) αi + 2 = βi . Thus, displacements are not generally as singular as in the parabolic case. where l2 =

• Displacement u3 As seen previously, m30 is different from zero, so that: M3 f 3 = m30 Φ(y 1 )S4 (y 2 ) + · · ·

(2.113)

We conclude that β3 = 4 and α3 = 2 in the hyperbolic case. We then have: u3 (y 1 , y 2 ) = U32 (y 1 )S2 (y 2 ) + . . .

(2.114)

d2 U34 (y 1 ) m30 = Φ(y 1 ) d(y 1 )2 l4

(2.115)

with

Therefore, the normal displacement u3 is two orders more singular than the loading f 3 for a hyperbolic shell, when the loading f 3 is singular along a characteristic curve. • Displacement u1 E 2 a2 2(2 + J)(a22 )2 b12 = 0 according to (2.63), 4(1 + ν)2 the right-hand side of (2.68) reduces to: Since b11 = 0 and m10 =

M1 f 3 = m10 Φ(y 1 )S3 (y 2 )

(2.116)

which leads to β1 = 3 and α1 = 1. Therefore, the highest singularity for the displacement u1 is given by: u1 (y 1 , y 2 ) = U11 (y 1 )S1 (y 2 ) + . . .

(2.117)

d2 U11 1 m10 (y ) = Φ(y 1 ) d(y 1 )2 l2

(2.118)

with

Thus, the displacement u1 is one order more singular than the loading f 3 .

60


• Displacement u2 According to the expression (2.83) of m20 , this one can vanish if the following condition is satisfied: (2 + J)b22 (a22 )2 = 0 (2.119) - If m20 = 0 (i.e. if b22 = 0), the right-hand side writes: M2 f 3 = m20 Φ(y 1 )S3 (y 2 ) + . . .

(2.120)

and the displacement u2 is: u2 (y 1 , y 2 ) = U21 (y 1 )S1 (y 2 ) + . . .

(2.121)

d2 U21 m20 = Φ(y 1 ) 1 2 d(y ) l2

(2.122)

with

Thus, u2 is one order more singular than f 3 like the displacement u1 . - In the hyperbolic case, if the coordinate system corresponds to the asymptotic lines (which are distinct), we have b11 = b22 = 0 and b12 = 0. Thus, the condition m20 = 0 is automatically verified. The coefficient m21 reduces to m21 =

−E 2 a2 (2(2 + J)(a12 )2 − a11 a22 ) 2(1 + ν)2

(2.123)

which is generally different from zero except for very particular cases when a11 a22 = 2(2 + J)(a12 )2 which are not considered here. If m21 = 0, the right-hand side of (2.68) writes: M2 f 3 = m21 Φ(1) (y 1 )S2 (y 2 ) + · · ·

(2.124)

and the highest singularity of u2 writes, with β2 = 2 and α2 = 0: u2 (y 1 , y 2 ) = U20 (y 1 )S0 (y 2 ) + . . .

(2.125)

d2 U20 (y 1 ) m21 (1) 1 = Φ (y ) 1 2 d(y ) l2

(2.126)

where

Therefore, if the coordinate system corresponds to the asymptotic curves, the singularity of the displacement u2 is of the same order than that of f 3 , whereas u1 is one order more singular.


2.6.3

61

Summary of the Results

Despite the diversity of the studied cases and of the obtained results, depending at the same time on the nature of the middle surface S and on the direction of the singularities of the loading (along a characteristic curve or not), it is possible to exhibit some general results. They allow us to determine a priori the order of the singularities of the resulting displacements u1 , u2 and u3 as soon as the singularity of the loading is known. First, an important difference exists whether the singularity of the loading f 3 is along a characteristic curve or not. This distinction is only relevant in the case of parabolic and hyperbolic shells, an elliptic surface having no real asymptotic curve. • When the normal loading is singular along a characteristic line (i.e. an asymptotic curve of the middle surface), the displacements u1 , u2 and u3 are generally more singular than f 3 . Thus, for instance, if f 3 has a singularity in (y 2 )3 H(y 2 ) which is not very sharp, u3 has a singularity in δ(y 2 ) in the parabolic case and in y 2 H(y 2 ) in the hyperbolic case. Moreover, the singularities propagate along the whole characteristics. Indeed, the results obtained in paragraph 2.6.2 show that in the parabolic case U12 (y 1 ), U23 (y 1 ) and U34 (y 1 ) (respectively, U11 (y 1 ), U21 (y 1 ) and U34 (y 1 ) in the hyperbolic case) are primitives of 4th order of Φ(y 1 ) (respectively of the 2nd order in the hyperbolic case). Thus, even where Φ(y 1 ) = 0 (which is the case out of the loading domain), the factors Uij are generally different from zero. • Oppositely, when the loading f 3 is singular along a curve which is not characteristic (i.e. which is not an asymptotic curve of the middle surface), the factors U1−1 (y 1 ), U2−1 (y 1 ) and U30 (y 1 ) are directly proportional to Φ(y 1 ) (see results of paragraph 2.6.1). They vanish outside the loading domain which implies that there is no propagation of the singularities. All the results obtained in section 2.6 can be summarized in the following result: Result 2.6.1. Let S be a surface of R3 , parametrized by the mapping (Ω, Ψ ) where (y 1 , y 2 ) is a point of Ω ∈ R2 . This surface is subjected to a purely normal loading f 3 , which is singular along the line y 2 = 0. We have the following results12 : 1) If y 2 = 0 is not a characteristic line, • there is no propagation of these singularities. • the normal displacement u3 always has the same singularity as f 3 . • the tangential displacements u1 and u2 are one order less singular than the loading f 3 (except in certain particular cases where they are even less singular). 12

We recall that the terminology “n order more or less singular than” refers to definition 2.2 of the chain of singularities.

62


2) If y 2 = 0 is a characteristic line, the resulting displacements depend on the nature of the middle surface S but in any case, singularities propagate along the line y 2 = 0. i) if the surface S is parabolic: • u3 is always 4 orders more singular than f 3 . • u2 is generally 3 orders more singular than f 3 (except in certain particular cases where it is even less singular). • u1 is generally 2 orders more singular than f 3 (except in certain particular cases where it is even less singular). ii) if the surface S is hyperbolic: • u3 is always 2 orders more singular than f 3 . • u2 is generally 1 order more singular than f 3 (or of the same order if the coordinate system corresponds to that of the characteristic lines). • u1 is always 1 order more singular than f 3 . All these cases can also be synthesized in the following table: Table 2.1. Main results of the singularity orders of the displacements according to the considered case. Properties singularity order of u3 singularity of the tangential displacements propagation

Non Characteristic +0 -1 (or less) no

Characteristic hyperbolic parabolic +2 +4 +1 (or less) yes

+3 (or less) yes

Remark 2.6.3. According to the considerations at the beginning of section 2.5, in the analysis of singularities performed in this chapter (based on some kind of micro-local analysis of the membrane system), we neglected the lower order derivatives of the displacements (involving the Christoffel symbols in the covariant derivatives). This has no consequence on the order of the highest singularities of the displacements obtained; only the expressions of the coefficients Uij (y 1 ) of the singularities are not exact. However, in the case of the parabolic shell which will be considered in chapter 5, according to the parametrization chosen, the Christoffel symbols vanish and the expressions of Uij (y 1 ) are exact. On the other hand, one can also refer to chapter 6 where the calculations are performed accounting to the Christoffel symbols in the hyperbolic case. Remark 2.6.4. When the shell is hyperbolic, it is not easy to interpret geometrically the direction of the displacements u1 and u2 in the coordinate system of the characteristic lines. Indeed, u1 and u2 are covariant components in the dual basis (a1 , a2 ) of the tangent plane, which is orthogonal to the basis (a1 , a2 ). The vectors a1 and a2 are tangent to the characteristic lines y 1 = const and y 2 = const, but are not in general orthogonal. In the parabolic case, where the two families of characteristic lines are confused, the interpretation is easier, if we use the principal coordinate system for curvatures. The local basis (a1 , a2 )

2.8 Thickness of the Layers

63

is orthogonal and one of the principal directions corresponds necessarily to an asymptotic direction (the direction where the curvature vanishes).

2.7

Pseudo-reflections for Hyperbolic Shells

In the particular case of hyperbolic shells, the propagation of singularities enjoys non-classical properties of refraction (incidentally called pseudo-reflections) when arriving to a boundary. The main reason for a non-classical behavior is that the boundary conditions are very different from those of classical Cauchy’s problems of wave propagation, which lead to “classical reflection”. Even these problems are far from completely solved, some general results already have been established in [60, 64]. In particular, the two following rules can be stated (see proposition 8.1 of [64]): - Rule 1 : when f 3 has a singularity of order ψ along a characteristic, u3 bears a singularity of order ψ along that characteristic (see result 2.6.1). It propagates and when arriving to a fixed boundary, the other characteristic issued from that point bears a singularity of order ψ . This pseudo-reflection phenomenon does not occur when the boundary is free (in that case there is no reflection at all). - Rule 2 : when f 3 has a singularity of order ψ along a non-characteristic curve, u3 bears also a singularity of order ψ along it (see result 2.6.1). It has a non-propagating character, but it may arrive to a non-characteristic boundary provided that the support of the given singular loading arrives to it. In that case, the two characteristics issued from that point bear propagating singularities of order ψ or ψ when the boundary is fixed or free, respectively. These rules are sufficient for understanding the examples of pseudo-reflections of singularities which will be studied in section 6.4 of chapter 6 in the case of hyperbolic shells.

2.8

Thickness of the Layers

In this section, we propose to find classical results giving the order of the thicknesses of internal layers and boundary layers according to the nature of the middle surface of the shell (elliptic, parabolic, hyperbolic), and according to the nature of the layer (along a characteristic line or not). Indeed, the thickness of the layer is not the same whether it is along a characteristic line or not. First we begin with establishing a reduced PDE for the normal displacement u3 , from the Koiter model (including the bending terms). This reduced formulation is obtained using a procedure similar to that used for obtaining the reduced membrane model (2.60). We add to (2.52) and to the reduced formulations (2.60) the bending terms in ε2 , or more precisely the highest order terms relevant for this study. In the reduced equation, this term must have the lowest power of ε and the highest differentiation order.

64


Let us write the full system corresponding to the Koiter model (1.51) under the form Au = f . The matrix A writes (details are available in annex B.2): ⎛ ⎞ −A1βγ1 ∂β ∂γ −A1βγ2 ∂β ∂γ A1βγδ bγδ ∂β ⎜ +ε2 (∂ 2 + . . . ) +ε2 (∂ 2 + . . . ) +ε2 (∂ 3 + . . . ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2βγ2 2βγδ ⎜ −A2βγ1 ∂β ∂γ ⎟ −A ∂ ∂ A b ∂ β γ γδ β ⎜ ⎟ (2.127) 2 3 ⎜ +ε2 (∂ 2 + . . . ) +ε2 (∂ 2 + . . . ) ⎟ +ε (∂ + . . . ) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ −A1βγδ bγδ ∂β −A2βγδ bγδ ∂β Aαβγδ bαβ bγδ + ε2 F ⎠ 12 +ε2 (∂ 3 + . . . ) +ε2 (∂ 3 + . . . ) +ε2 (∂ 3 + . . . ) with F = A1111 ∂14 +A2222 ∂24 +(2A1122 +4A1212 )∂12 ∂22 +4A1112 ∂13 ∂2 +4A1222 ∂1 ∂23 and where +ε2 (∂ n + . . . ) denotes other less important bending terms, n being the order of differentiation of these terms. We can see that the most important bending term comes from the multiε2 2 plication of the cofactor AC 33 by the term 12 F of A33 . This term is in ε and th involves differentiations of the 8 order. All the other bending terms contain either lower order differentiations or are in εn with n > 2. In both cases, they are less important, and can be neglected in the following considerations. With this procedure, we obtain a reduced equation of the Koiter model involving only the normal displacement u3 which writes: 2 (4) (2) ε 2 11 2 E a a ∂1 + a22 ∂22 + 2a12 ∂1 ∂2 +(1 + ν) b22 ∂12 + b11 ∂22 −2b12 ∂1 ∂2 + 12 (2) O(ε ) uε3 = a2 (1 + ν) a11 ∂12 + a22 ∂22 + 2a12 ∂1 ∂2 + O(ε2 ) f 3 2

(2.128) For ε = 0 corresponding to the limit membrane model, we find again the expression (2.60) of the reduced membrane model. To calculate the thickness of the layer, we shall write equation (2.128) under the form: 2 (4) (2) ε 1 E a11 ∂12 + a22 ∂22 − 2a12 ∂1 ∂2 +(1 + ν) b22 ∂12 + b11 ∂22 −2b12 ∂1 ∂2 + 12 a2 (2) O(ε ) uε3 = (1 + ν) a22 ∂12 + a11 ∂22 − 2a12 ∂1 ∂2 + O(ε2 ) f 3 2

(2.129) 2.8.1

Case of a Layer along a Non-characteristic Line

When a layer is not along a characteristic curve, its thickness does not depend on the nature of the middle surface. For instance, let us consider a layer in the direction y 2 = constant. As seen previously (section 2.6.1), if this line is not

2.8 Thickness of the Layers

65

characteristic, we have necessarily b11 = 0. We shall perform a classical scaling in the direction perpendicular to the layer: ⎧ 1 1 ⎪ ⎨z = y (y 1 , y 2 ) ⇒ (z 1 , z 2 ) such that (2.130) ⎪ ⎩ z 2 = 1 y2 εα We search now the layer thickness under the form η = εα , α being a positive constant undetermined for the time being. With the new variables (z 1 , z 2 ), equation (2.128) becomes: ! Kb ∂ 8 1 ∂4 2 E ε + . . . + Km 4α + . . . uε3 = ε8α ∂(z 2 )8 ε ∂(z 2 )4 (2.131) (2) (1 + ν) a11 ∂12 + a22 ∂22 − 2a12 ∂1 ∂2 + O(ε2 ) f 3 where + . . . denotes terms containing lower differentiation orders with respect to z 2 , and where Km and Kb are two constants. The aim of the layer is to involve bending terms aside membrane ones, in order to have both membrane and bending effects of the same order inside the layer. Therefore, α has to satisfy: " # ε2 1 = O (2.132) ε8α ε4α which gives: 1 (2.133) 2 Therefore, when the layer is not along a characteristic line, the thickness η of the layer is such that η = O(ε1/2 ) whatever the nature of the middle surface S is. α=

2.8.2

Case of a Layer along a Characteristic Line

Case of a parabolic shell If the line y 2 = constant is characteristic, we have b11 = 0. As the middle surface is parabolic, we have b11 b22 − b212 = 0, which implies b12 = 0. Equation (2.128) then becomes: 2 (4) ε 1 2 2 12 2 (2) 2 E a11 ∂1 + a22 ∂2 − 2a ∂1 ∂2 + (1 + ν) b22 ∂1 + O(ε ) uε3 = 12 a2 (2) (1 + ν) a11 ∂12 + a22 ∂22 − 2a12∂1 ∂2 + O(ε2 ) f 3 (2.134)

66


We perform the same scaling (2.130) in the direction perpendicular to the layer along y 2 = constant. The left-hand side of equation (2.134) becomes: ! 8 4 K ∂ ∂ b ε E ε2 + . . . + Km (2.135) 4 + . . . u3 ε8α ∂(z 2 )8 ∂(z 1 ) In the layer α has to satisfy: ε2 = O (1) ε8α

(2.136)

which gives: 1 (2.137) 4 Thus, when the shell is parabolic, the thickness of a layer along a characteristic line is of order ε1/4 . Equivalently, we have η = O(ε1/4 ). It is thicker than a layer along a non-characteristic when ε 0. α=

Case of a hyperbolic shell Finally, let us consider a hyperbolic shell. If the line y 2 = constant is characteristic, we have b11 = 0. Since the shell is hyperbolic, we have b11 b22 − b212 < 0, which implies that b12 = 0. Equation (2.128) then writes: E

(4) ε2 1 2 2 2 (2) 2 a ∂ + a ∂ − 2a ∂ ∂ + (1 + ν) b ∂ ∂ + b ∂ + O(ε ) uε3 = 11 22 12 1 2 12 1 2 22 1 2 1 12 a2

(2) (1 + ν) a11 ∂12 + a22 ∂22 − 2a12 ∂1 ∂2 + O(ε2 ) f 3 (2.138)

Performing the dilatation (2.130), the left-hand side of equation (2.138) becomes: ! 8 4 K ∂ 1 ∂ b E ε2 + . . . + Km 2α + . . . uε3 (2.139) ε8α ∂(z 2 )8 ε ∂(z 1 )2 (z 2 )2 which leads to:

ε2 =O ε8α

"

1 ε2α

# (2.140)

and finally: 1 (2.141) 3 Thus, when the shell is parabolic, the thickness of a layer along a characteristic line is of order ε1/3 (equivalently η = O(ε1/3 )). It is thicker than a layer along a non-characteristic line (when ε 0), but thinner than a layer along a characteristic line in the case of a parabolic shell. α=

2.9 Conclusion

67

We can summarize these last results about layer thicknesses η as follows: • η = O(ε1/2 ) when the layer is not along a characteristic line whatever the nature of the middle surface S is; • η = O(ε1/4 ) along the double characteristics, i.e. for the parabolic shells; • η = O(ε1/3 ) along the simple characteristics, i.e. for the hyperbolic shells. Equivalently, the thickness of the layers, which depends on the configuration encountered, can be summarized in the following table: Table 2.2. Layer thickness orders according to the considered case. Properties layer thickness

Non Characteristic O(ε1/2 )

Characteristic hyperbolic parabolic O(ε1/3 ) O(ε1/4 )

The numerical simulations which will be performed in the next chapters in various case of figures will enable to recover accurately these theoretical results on the layer thicknesses.

2.9

Conclusion

The brief recall on geometrical rigidity properties of surfaces (completing the considerations of chapter 1), and on the limit behavior of the Koiter and Nagdhi shell model when the thickness tends to zero, enabled us to well understand the singularity phenomenon appearing for very thin shells. These singularities are inherent to the geometric behavior of elastic very thin shells, including membrane and bending effects (and even transverse shear for the Nagdhi model) coupled at different orders. Starting from a reduced appropriate formulation of the limit membrane model, we then established some general results concerning the singularities (order, propagation) appearing in the layers for very small thicknesses: • for inhibited shells, boundary and internal layers appear for very small thicknesses when the loading (or its derivatives) is singular along a curve, • the thickness of these layers is directly linked to the geometrical nature of the middle surface (elliptic, parabolic or hyperbolic) and to the localization of the singularity (along an asymptotic curve or not), • in these layers, the displacements tend to be more and more singular when the thickness ε tends to zero, • the singularities may propagate in the layers according to the geometrical nature of the middle surface and to the localization of the singularity (along a characteristic or not), • the results obtained are general and would be similar when considering another shell model, like Nagdhi model including transverse shear for instance.

68


All these results will be completed in the next chapters, on one hand by theoretical developments in particular situations where analytical expressions of the singularities of the displacements can be established, and on the other hand by numerical simulations using adaptive anisotropic meshes which enable us to refine the mesh only around the layers containing the singularities, and to access very accurately to the singular displacements in the layers.

3 Anisotropic Error Estimates in the Layers

3.1

Introduction

As we saw in the last chapter, the normal displacement u3 , as a solution of the limit membrane model, is at best in L2 (Ω), whereas the solution uε3 of the Koiter model is in H 2 (Ω) (for ε > 0). Consequently, boundary and internal layers, leading to singularities of the displacements inside the layers, may appear during the asymptotic process when ε decreases toward zero. As the internal and boundary layers appear at elongated regions, the use of anisotropic elements to compute the solutions inside the layers should be an adequate solution to reduce considerably the number of elements. However, it is usually admitted in F. E. treatises that elements cannot be “too much elongated” to satisfy optimal interpolation estimates insuring the convergence of the approximate solution. Even if this is true in general, we can take advantage of the highly anisotropic variation of the displacements in the layers (they vary strongly in the direction normal to the layer) to get better estimates with anisotropic elements in the energy norm. Moreover, as most of the deformation energy is contained in the layers, improving the solution inside the layers amounts to improve significantly the solution in the whole domain. This kind of anisotropic elements have already been used in various mechanical problems including geometrical singularities or boundary layers (elliptic problems with edges or corners, diffusion problems), and the corresponding estimates were exhibited mainly for model problems (see [1][2][3][4]). In this chapter, we address error estimates for singular perturbation problems with layers, including isotropic and anisotropic elements. In particular, we prove that error estimates obtained with an anisotropic mesh are similar to those obtained with an isotropic one containing much more elements. The next chapter will present in detail the numerical procedure of resolution used in the sequel which is based on the F. E. method coupled with an adaptive and anisotropic mesh generator. E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 69–86. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

70


3.2

Estimate for Galerkin Approximation in Singular Perturbation and Penalty Problems

Let us come back to the Koiter shell model which is a singular perturbation problem P(ε) in V depending on the small parameter ε (the relative thickness). It writes (see 1.59): ⎧ ⎨ F ind uε in V such that (3.1) P(ε) ⎩ am (uε , v) + ε2 ab (uε , v) = f, v ∀v ∈ V where the bilinear membrane and bending forms are defined in (1.60) and (1.61). In the sequel we will denote a(·, ·) = am (·, ·) + ε2 ab (·, ·) As a classical result (see [14] for example), the bilinear form a(·, ·) is continuous and coercive on V such that: α v2V ≤ a(v, v) ≤ M v2V

(3.2) 1/2

for some α and M strictly positive real constants. Consequently, a(v, v) defines a norm (denoted ·en ) on V which is equivalent to the given norm of V . It is called classically the energy norm: ven = a(v, v)1/2

(3.3)

Let now Vh ∈ V be a subspace of V (in general, Vh is of finite dimension but this is not necessary). The Vh −Galerkin approximation of the Koiter model is then defined by: Find uεh ∈ Vh such that a(uεh , v) = f, v ∀v ∈ Vh Taking in (3.1) v ∈ Vh and subtracting the above equation from it, we get: a(uε − uεh , v) = 0 ∀v ∈ Vh

(3.4)

Equation (3.4) means that uεh is the orthogonal projection (in the scalar product defined in the energy space) of the exact solution uε on Vh (see Fig. 3.1). In other words, uεh is the best possible approximation of uε in Vh , when best is considered in the energy norm ·en . Before establishing error estimates, it is worthwhile to specify the difference between the “interpolation error” and the error of the Galerkin approximation. This we proceed to do. Once the space V is given, let us admit that the subspace Vh is such that we have an estimate of the error of approximation of any element uε ∈ V , satisfying to: F or any uε ∈ V, there exists u ˇεh ∈ Vh such that uε − uˇεh V ≤ e(uε , h) ε

(3.5) ε

The “function” e(u , h) is usually called “interpolation error of u in Vh ”.

3.2 Estimate for Galerkin Approximation

71

V uε

O uεh Vh

Fig. 3.1. Projection of u onto Vh

Remark 3.2.1. Clearly, (3.5) has nothing to do with the Galerkin problem. The norm is a fixed norm in V . Moreover, such an estimate always exists: for instance we shall consider e(uε , h) = uε V , with u ˇεh = 0. Finally, let us quote that even if h is often a parameter tending to zero, the properties in this section have nothing to do with that: Vh is merely the name of a subspace of V . Obviously, u ˇεh is the interpolation of the real solution in the subspace Vh , and in general it is different from the solution uεh of the approximate Galerkin problem. Indeed, uˇεh is concerned with uε , V and Vh , but not with a(uε , v), whereas uεh is the projection (in the energy norm) of the exact solution uε on Vh . If uε denotes the exact solution, (3.5) implies that there exists (at least) a ε u ˇh ∈ Vh satisfying (3.5). Otherwise, there exists a Vh −Galerkin approximation uεh of uε . But it is by no means evident that uεh enjoys an estimate analogous to (3.5). Nevertheless, the following new estimate may be obtained thanks to the projection property: uε − uεh en ≤ uε − u ˇεh en (3.6) as u ˇεh ∈ Vh . Using (3.2), (3.5) and (3.6) we have: 1 1 uε − uεh V ≤ √ uε − uεh en ≤ √ uε − u ˇεh en α √α √ M M ≤ √ uε − u ˇεh V ≤ √ e(uε , h) α α

(3.7)

Proposition 3.1. The interpolation estimate (3.5) with the error multiplied by the factor (M/α)1/2 (which is always ≥ 1) applies to the Galerkin approximation. It is important to quote that proposition 3.1 relies on the chain of inequalities (3.7). It does not mean that the Galerkin approximation is worse than the interpolation one. It may happen that uh = u ˇh . But (3.7) is an estimate of the Galerkin approximation which certainly holds true.

72


3.2.1

Degradation of the Estimate in a Singular Perturbation Problem

Now, we shall come back to the Koiter problem (3.1). When ε 0, we obviously have (3.2) with M = const (3.8) α = cε2

with c = const > 0

(3.9)

" # 1 e(uε , h) ε

(3.10)

Then estimate (3.7) leads to: uε − uεh V ≤ O

Therefore, as ε 0, the estimate becomes worse and with respect to worse the Vh interpolation approximation, the factor being O 1ε . In general, as ε decreases, in order to keep a constant accuracy of the Galerkin approximation, we must take h 0, as it is expected from heuristic considerations on boundary (an internal) layers. 3.2.2

Degradation of the Estimate in a Penalty Problem

Analogously, considering the following penalty problem in the case of noninhibited shells. As they are geometrically less rigid than inhibited shells, we shall consider the new scaling (ε2 f instead of f ) as in section 2.3.5. The rescaled Koiter shell model then writes: Find uε ∈ V such that 1 am (uε , v) + ab (uε , v) = f, v ∀v ∈ V ε2 with the standard hypotheses (it corresponds to the classical penalty problem leading to the pure bending model for non-inhibited shells, see section 2.3.5 or [88]). Then, as ε 0, we obviously have (3.2) with M =β

1 ε2

with β = const > 0 α = const

and (3.7) becomes

(3.11) (3.12)

" # 1 e(uε , h) (3.13) ε As ε 0, the estimate becomes once again worse and with respect to worse the Vh interpolation approximation, the factor being O 1ε . It is important to notice that estimate (3.13) constitutes a first (embryonary) version of the locking phenomenon. In general, as ε 0, in order to have a similar Galerkin approximation, we must take h 0. Clearly, this follows from the disparity between the V −norm and the energy norm which is only taken into account by the Galerkin process. But this is only concerned with inequalities, and a proof of effective locking needs a deeper study which will be done in chapter 4. uε − uεh V ≤ O

3.3 Interpolation Error for Isotropic Meshes in Layers

3.3 3.3.1

73

Interpolation Error for Isotropic Meshes in Layers The Basic F. E. Interpolation Error Estimate

Let us denote by · s the norm either in H s (Ω) or C s (Ω) as the general considerations here hold true in both cases. Obviously, we shall focus on H s (Ω) spaces where the results will be used in the sequel. Let m and k be positive integers and u ∈ H k+1 (or C k+1 ). We consider a “standard F. E.” approximation (this term is explained later). Then, if h denotes the diameter of the triangulation, there exists a F. E. approximation denoted by u ˇh satisfying the estimate of interpolation error: u − u ˇh m ≤ Chk+1−m uk+1

(3.14)

where k is the order of the polynomial approximation. The constant C only depends on the local nature of the F. E., not on the domain Ω which only appears in the expressions of the norms. For the proof of this classical estimate, refer to [29][53][85], where the reader may find general hypotheses accounting for “standard F. E.”. Remark 3.3.1. The usual approximation of H 1 (Ω) by polynomials of first order (k = 1) is possible, but the proof has nothing to do with (3.14). This estimate is only useful when u is sufficiently smooth for instance u ∈ H 2 (Ω). For instance, taking m = 0 in (3.14), we have for k = 1 u − u ˇh 0 ≤ Ch2 u2

(3.15)

which is only significant when u ∈ H 2 . To have an estimate of the convergence in H 1 (m = 1), we must once again have u ∈ H 2 (Ω), so that: u − u ˇh 1 ≤ Chu2

(3.16)

On the other hand, the effectiveness of the good approximation with element of high order only works for very smooth functions. Heuristic evidence of (3.14) follows from the Taylor formula written in a symbolic self-evident way: f (a + y) = f (a) + · · · +

yk k y (k+1) k+1 f (a) + f (a + θy) k! (k + 1)!

(3.17)

where θ ∈ [0, 1]; a + θy is an intermediate point of the interval. Then we have: k f (a + y) − f (a) + · · · + y f k (a) ≤ Cy k+1 f k+1 (3.18) k! Note that f k+1 is evident in C k+1 . In H k+1 , it also follows from other forms of the complementary term. This gives the estimate

74


f − fˇh ≤ Cy k+1 f k+1

(3.19)

in one element. When we have several contiguous F. E., (3.14) is valid for each one (and the norm in this element), and therefore in the whole domain. Estimate (3.19) is obviously valid in C 0 by taking the sup, and in L2 also by taking the square root of the sum of the integral of the square. Moreover, using (3.19) for the derivatives of u of order 1, . . . , m (with polynomials of orders k − 1, . . . , k − m) and adding, (3.14) is easily obtained. Remark 3.3.2. The above heuristic considerations which are (almost) a proof, show the necessity of having a “sufficiently rich set of polynomials” for constructing the F. E. approximation. We also see that h is the “diameter” of the element, or the sup of the length in any direction. The term “standard F. E.” is precisely in conformity with the requirements of remark 3.3.2. It is supposed that we have “sufficiently many” polynomials in all the variables. Furthermore, h is the size of the triangles, supposed to be asymptotically isotropic (the ratio of the inscribed and circumscribed balls should be bounded; the constant then included in C). The precise hypothesis are specified in classical treatises of F. E. Remark 3.3.3. The estimate (3.14) is classically considered as “optimal”. The above heuristic considerations show the precise meaning of this assertion. Clearly, for functions with non-zero derivatives of order k+1, the complementary term of the Taylor expression does not vanish, so that the estimate is effectively of that order. Otherwise, for a specific function u, it may happen that the approximation is better (for instance, for a piecewise polynomial of order k, the approximation is exact). This point is very important. For functions having a specific structure, there is no contradiction when having an estimate better than the “optimal one”. 3.3.2

Case of a Layer: Interpolation Error for Isotropic Meshes

Let us consider now the case of a layer whose thickness is η 0, a small parameter such as η < 1. We consider the domain Ωη =]0, 1[×]0, η[ in the plane (x1 , x2 ). We also consider the domain Ω1 =]0, 1[×]0, 1[ in the plane (y1 , y2 ) (see Fig. 3.2). We consider the family of functions uη (depending on η) defined on Ωη by uη (x1 , x2 ) = u(y1 , ηy2 )

(3.20)

The function u is supposed to be smooth (belonging to H k+1 for the values of k considered in the sequel) and obviously independent of η. This amounts to saying that uη is obtained from u by the change of variable: x1 = y1 ⇐⇒ x2 = ηy2

∂ ∂ = ∂x1 ∂y1 ∂ 1 ∂ = ∂x2 η ∂y2

(3.21)

3.3 Interpolation Error for Isotropic Meshes in Layers

x2

75

y2 1

Ω1

η Ωη 1

1

x1

y1

Fig. 3.2. Domains Ωη and Ω1

The basic interpolation estimate (3.14) in Ωη gives (with C independent of η) uη − u ˇηh (Ωη ,m) ≤ Chk+1−m uη (Ωη ,k+1)

(3.22)

According to (3.20) and (3.21), the computation of the norm in the right hand side of (3.22) leads to: 2

uη (Ωη ,k+1) =

Ωη

k+1 2 ∂ η u ∂xk+1 dx1 dx2 + . . .

(3.23)

2

where + . . . denotes terms not involving the highest order of differentiation with respect to x2 . We have: Ωη

k+1 2 ∂ η −2(k+1)+1 u dx dx = η 1 2 ∂xk+1 2

Ω1

k+1 2 ∂ ∂y k+1 u dy1 dy2

(3.24)

2

where the factor η 1 comes from the change of domain and η−2(k+1) from the differentiation, so that (3.23) leads to: uη (Ωη ,k+1) ≤ η −(k+1)+1/2 u(Ω1 ,k+1)

(3.25)

Moreover, as the terms + . . . involve lower order differentiation with respect to x2 , the asymptotic behavior as η 0 is: $ k+1 $ $ $ η −(k+1)+1/2 $ ∂ $ u (Ωη ,k+1) η (3.26) $ ∂y k+1 u$ 2 (Ω1 ,0) Remark 3.3.4. When u is not a polynomial with respect to y2 , the right hand side of (3.26) does not vanish and the asymptotic behavior is significant.

76


Coming back to estimate (3.22), we have: u − η

u ˇηh (Ωη ,m)

≤ Ch

k+1−m

η

−(k+1)+1/2

$ k+1 $ $∂ $ $ $ $ ∂y k+1 u$ 2

(3.27)

(Ω1 ,0)

The factor η −(k+1)+1/2 measures the deterioration of the estimate with respect to Ω1 . But the norm uη (Ωη ,m) is itself large, so that we may consider the relative error. According to (3.26) with k + 1 = m, we have: $ m $ $ $ η −m+1/2 $ ∂ $ u (Ωη ,m) η (3.28) $ ∂y m u$ 2 (Ω1 ,0) As the estimate is better described in terms of h/η than of h, we obtain finally: $ k+1 $ $∂ $ $ u$ η " # $ k+1 $ η k+1−m u − u ˇh (Ωη ,m) ∂y2 h $ m $ (Ω1 ,0) ≤C (3.29) η $ ∂ $ u (Ωη ,m) η $ $ $ ∂y m u$ 2

(Ω1 ,0)

Therefore, as u is independent of η, we finally obtain: uη − uˇηh (Ωη ,m) uη (Ωη ,m)

≤C

" #k+1−m h η

(3.30)

for some other C independent of η, which is exactly analogous to (3.22) but involving relative error and relative diameter. It is worthwhile noticing that estimate (3.27) is obviously concerned with isotropic triangulation (h × h, say) in the variables x1 , x2 . But it is clear that the dependence of u in y2 plays the leading role. We shall see in the next section that better estimates may be obtained using anisotropic meshes.

3.4

Interpolation Error for Anisotropic Meshes in Layers

We consider the same problem as in the previous section. We now start with a standard F. E. problem approximation of the function u in the domain Ω1 (isotropic in y) with triangles of diameter H. The basic estimate gives: u − u ˇH (Ω1 ,m) ≤ CH k+1−m u(Ω1 ,k+1)

(3.31)

Next, for each value of η, we construct: u ˇηH (x1 , x2 ) = u ˇH (y1 , ηy2 )

(3.32)

which is defined on Ωη and may be considered as an approximation of uη with F. E. involving standard polynomials of order k + 1 in anisotropic triangles H × ηH. As in (3.23), we have:

3.4 Interpolation Error for Anisotropic Meshes in Layers

u −

2 u ˇH (Ω1 ,m)

= Ω1

m 2 ∂ ∂y m (u − uˇH ) dy1 dy2 + . . .

77

(3.33)

2

∂m . Performing the change of variables ∂y2m (3.21) in the right hand side of (3.33), we get: where + . . . denotes terms not involving Ω1

m 2 ∂ 2m−1 (u − u ˇ ) dy dy = η H 1 2 ∂y m 2

Ωη

m 2 ∂ η η (u − u ˇ ) H dx1 dx2 ∂xm

(3.34)

2

and analogous expressions with lower powers of η for the other derivatives. As the norms are positive, we have: 2

u − u ˇH 2(Ω1 ,m) ≥ η 2m−1 uη − u ˇηH (Ωη ,m)

(3.35)

and from (3.31) and (3.35), we obtain: u − u ˇηH (Ωη ,m) ≤ C H k+1−m η−m+1/2 u(Ω1 ,k+1)

(3.36)

which is obviously concerned with a H × ηH mesh. Comparing (3.36) and (3.27), we observe that taking h H= (3.37) η both estimates coincide up to the values of the constants u(Ω1 ,k+1) and $ k+1 $ $∂ $ $ $ . $ ∂y k+1 u$ 2 (Ω1 ,0) Analogously, we may consider the relative error. From (3.36) and (3.26) with k + 1 = m, we get: uη − u ˇηH (Ωη ,m) uη (Ωη ,m)

u(Ω1 ,k+1) $ ≤ C H k+1−m $ $ ∂m $ $ $ $ ∂y m u$ 2

(3.38)

(Ω1 ,0)

As u is still independent of η, we finally obtain: uη − u ˇηH (Ωη ,m) uη (Ωη ,m)

≤ C H k+1−m

(3.39)

for some other constant C, which obviously coincides (up to the constants) with (3.30) for H = h/η. It is important to quote that the change of constants in the estimates obviously preserves the asymptotics for η 0. Moreover, we observe that the constant u(Ω1 ,k+1) involves the derivatives of u in all directions. Finally, we have the following proposition concerning the estimates obtained for isotropic and anisotropic elements (see in particular (3.30) and (3.39)):

78


Proposition 3.2. The estimates with an isotropic h×h mesh and an anisotropic H ×ηH mesh coincide (up to the values of the constant) with H = h/η, either for absolute or relative error. However, the numbers of triangles are in the asymptotic ratio 1/η. Equivalently, this proposition, based on the comparisons of estimates (3.30) and (3.39) for isotropic and anisotropic meshes, means that η −1 h × h anisotropic meshes are as efficient as h × h isotropic meshes (with a number of triangles 1/η times smaller). Moreover, when considering relatives errors, the anisotropic estimates of the interpolation relative error is analogous to those on the function on the reference domain Ω1 =]0, 1[×]0, 1[ with isotropic mesh with (asymptotically) the same number of triangles.

3.5

Galerkin Error Estimates in a Layer

The calculations and the error estimates which are obtained in this section will be established in the particular case of hyperbolic shells subjected to the singular normal loading (3.40). However, we will see that they are more general, as they are concerned with relative and not absolute errors. In this section, we consider a layer Ωη as in the previous sections, corresponding to a characteristic layer of hyperbolic type. To fix ideas, we consider the case of a layer produced by a normal loading f 3 = δ(C)

(3.40)

where C is the characteristic curve and δ denotes the Dirac mass on it. According to the theoretical analysis of singularities of section 2.6, a δ-singularity of the normal loading implies singularities of the displacements such that (see table 2.1): uεα ∼ δ (C), uε3 ∼ δ (C) (3.41) The corresponding layer thickness η is (see table 2.2): η = ε1/3

(3.42)

and consequently in the layer we have (see Fig. 5.16 and the considerations of chapter 5): 1 1 uεα 2 Uα (y) , uε3 3 U3 (y) (3.43) η η x2 with y1 = x1 , y2 = , Uα (y) and U3 (y) being independent of ε (i. e. we only η consider the leading asymptotic term).

3.5 Galerkin Error Estimates in a Layer

79

Let us admit1 that: • uε3 is approximated with polynomials of order k • uεα is approximated with polynomials of order k − 1 We shall denote in the sequel k−1 = β. Using (for uεα and uε3 ) the interpolation estimate (3.27) with m = 1 for uεα and m = 2 for uε3 , we get: uεα − u ˇεαh (Ωη ,1) ≤ Ch(k−1)+1−1 η −k+1/2 η −2 = Chβ η −β−5/2

(3.44)

uε3 − u ˇε3h (Ωη ,2) ≤ Ch(k−1)+1−2 η −k−1/2 η−3 = Chβ η −β−9/2

(3.45)

where the factors η −2 and η −3 come from (3.43), giving the corresponding norms on the right-hand side of (3.27). Remark 3.5.1. Obviously, estimates (3.44) and (3.45) hold for h × h isotropic elements. Moreover, according to previous considerations, they also hold true for η −1 h × h anisotropic elements. We shall also need in the sequel a L2 estimate of uε3 . As in (3.45) but with m = 0, we get: uε3 − u ˇε3h (Ωη ,0) ≤ Chβ+2 η −β−9/2 (3.46) From (3.44) and (3.45) we may immediately construct the interpolation error in H 1 × H 1 × H 2 . But, according to the considerations of section 3.2, only the estimate in the energy norm holds true for F. E. approximations. We obviously may use (3.2) after computing α and M , but this is not an easy task. We shall merely estimate the energy norm error (for either interpolation or F. E. approximation), and we shall comment directly on it. Note that this is often done, when considering locking phenomena for instance in [82]. Obviously, taking w = error (interpolation or F. E.), we have: 2 2 2 wen γαβ (w)(Ωη ,0) + ε2 ραβ (w)(Ωη ,0) (3.47) α,β

α,β

with ε2 = η 6 . We shall estimate γαβ (w)(Ωη ,0) using (3.44) and (3.46):

2 2 2 γαβ (w)(Ωη ,0) ≤ C hβ η −β−5/2 + hβ+2 η−β−9/2 ≤ C hβ η −β−5/2

α,β

(3.48) This last inequality was obtained using the fact that we shall always use h/η < 1. 1

Other choices are possible, but this one is recommended, and in any case, it is taken as an example.

80


Remark 3.5.2. According to the construction, estimate (3.48) in our context (see (3.43) and the general considerations at the end of section 3.3.2) is exactly analogous to the approximation of a standard (i. e. coercive, not depending on a parameter) problem in H 1 (Ω1 ) with polynomials of order β = k − 1 (for the relative error). In other words, the L2 (Ωη ) approximation of the γαβ with β polynomials in uεα and β + 1 polynomials in uε3 is analogous to a H 1 (Ω1 ) approximation in the reference domain Ω1 with β polynomials. Now, in order to take account for the ε2 term in (3.47), we should merely add ε2 times in the H 2 estimate of the norm of uε3 (as ε 1, the other terms are of the same order as the γαβ ones). This gives (with (3.45)): w2en ≤ Ch2β η −2β−5 (1 + ε2 η −4 ) = Ch2β η −2β−5 (1 + η 2 ) Ch2β η −2β−5 (3.49) with η < 1. The energy norm involved is (3.47), and w is the error (either interpolation or F. E.). Remark 3.5.3. Inequality (3.49) constitutes the basic rigorous estimate for the F. E. approximation. Nevertheless, it is very difficult to throw useful estimates for the errors wα , w3 in standard spaces. Indeed, in (3.47), the terms γαβ do not allow estimates in classical spaces (as the hyperbolic shell is not well-inhibited2 ). Of course, the standard coerciveness inequality γαβ 2(Ω1 ,0) + ε2 ραβ 2(Ω1 ,0) ≥ Cε ·2H 1 ×H 1 ×H 2 (3.50) α,β

α,β

holds true in the whole domain, but it is practically of no use, as according to (3.47), this gives a factor ε−2 in the estimates (note that the energy is concentrated in the layer so that we may consider Ω1 instead of the region of the layer).

3.6

First Remarks on Approximations in Layers

It is evident that the previous considerations show that H 1 × H 1 × H 2 estimates are practically fallacious. On the other hand, it is clear that the very strong anisotropy and the great difference between the two variables show that significant magnitudes are associated with the structure of the solutions in the layers. Moreover, as the various functions have different asymptotic magnitudes, significant errors are relative, not absolute errors. In order to compare approximations in different situations, we first define a certain (well known) type of approximation which will be taken as reference in the sequel. To this end, let us consider a standard second order problem (Dirichlet problem for the Laplacian for instance) in a domain Ω1 , for instance the square 2

Only the elliptic shells clamped along all their lateral boundaries are well inhibited (the Spapiro-Lopatinskii condition is verified, see chapter 9).

3.6 First Remarks on Approximations in Layers

81

]0, 1[×]0, 1[. We admit that the solution u is smooth. The standard interpolation error estimate for F. E. of diameter H and polynomials of order γ is (see (3.14)): δinter um ≤ CH γ+1−m uγ+1

(3.51)

where the symbol δinter denotes the “interpolation error of”. Moreover, the relative error is the quotient given by: uγ+1 δinter um ≤ CH γ+1−m um um

(3.52)

Moreover, taking m = 1, the interpolation error estimate holds also true for the Galerkin approximation as the problem is continuous and coercive on H 1 . The constants α and M of section 3.2 (see inequalities (3.2) and (3.7)) are independent of the parameters, so that they may be included in C. We have: uγ+1 δGal u1 ≤ CH γ u1 u1

(3.53)

As u is smooth, the norms in the right-hand side are constants and can also be included in C. Note that the same may be said of u1 in the denominator of the left-hand side, but we prefer to keep it. Finally, we have: δGal u1 ≤ CH γ u1

(3.54)

Moreover, for ulterior use, we prefer to write it in terms of the L2 norms of the gradient rather than of H 1 norm of the functions (the norms are equivalent). In the case of a Ωη layer, we have (with anisotropic elements): δGal grad u(Ωη ,0) grad u(Ωη ,0)

≤ CH γ

(3.55)

This type of estimate will be taken as reference. Definition 3.3. If we have an approximation of some function ϕ in Ωη satisfying δGal ϕ(Ωη ,0) ≤ CH γ (3.56) ϕ(Ωη ,0) we shall say that the approximation is “analogous to that of the first order derivatives in a standard elliptic second order problem with polynomials of order γ. It is important to notice that the previous definition is invariant by multiplication of ϕ and δϕ by a constant. Therefore, it is concerned with the relative error. Now, let us consider the functions in a Ωη layer (as in previous section) defined by:

82


⎧ ⎪ ⎨ y 1 = x1 ϕη (x) = ϕ (y(x))

with

x ⎪ ⎩ y2 = 2 η

(3.57)

Taking h = ηH in the context of previous sections (either h × h isotropic or η −1 h × h anisotropic), we have: δGal ϕ(Ωη ,0)

≤C

ϕ(Ωη ,0)

" #γ h η

(3.58)

(as both norms are multiplied by the same factor η1/2 ) Definition 3.4. In the context of the previous sections, if we consider functions ϕη of the form ϕη = λ(η)ϕ(y(x)) such that δGal ϕη (Ωη ,0) ϕη (Ωη ,0)

≤C

" #γ h η

(3.59)

we shall say that the approximation is analogous to that of the first order derivative in a standard elliptic problem of second order, with polynomials of order γ. We shall quote that the choice of L2 norms in this definition was motivated for ulterior use (when taking norms including derivatives, there are ambiguities concerning derivatives in y1 and y2 ). The counterpart is that it refers to the convergence of the derivatives, not of the function of the well-known problem of second order. Moreover, obviously definition 3.4 is independent of a factor λ(η), which has no influence on relative errors. Remark 3.6.1. Obviously, when an estimate is μ times larger than the reference one (3.59), we need to take h narrower in the proportion μ1/γ in order to have the same relative error.

3.7

Estimates for Significant Entities in the Layer: Local Locking in Layers

We consider again the general context of section 3.5, in particular (3.43) and the approximation order of the components of uε . More explicit explanations may be found in [19]. Keeping only the leading order term, we have: γ11 ∼ ∂1 uε1 ∼ η −2

∂ U1 (y) + . . . ∂y1

(3.60)

3.7 Estimates for Significant Entities in the Layer: Local Locking in Layers

γ12

1 ∂ ε ε −3 ∼ ∂2 u1 − b12 u3 + · · · ∼ η U1 (y) − b12 U3 + . . . 2 ∂y2

83

(3.61)

γ22 ∼ ∂2 uε2 ∼ η −3

∂ U2 (y) + . . . ∂y2

(3.62)

ρ22 ∼ ∂22 uε3 ∼ η−5

∂2 U3 (y) + . . . ∂y22

(3.63)

(the other ραβ are of lower orders and are not significant in the layer). Moreover, denoting β = k − 1 and β + 1 = k (the orders of the polynomials for uεα and uε3 respectively), (3.49) becomes (where w denotes the F.E or interpolation error): " #β h wen ≤ C η −5/2 (3.64) η From (3.64) and accounting of the expression (3.47) of the energy norm (we recall that the symbol δ denotes “interpolation error of”), we obtain : δγαλ 2(Ωη ,0) 2

ε

" #2β h ≤C η −5 η

2 δρ22 (Ωη ,0)

or with ε = η 3 2 δρ22 (Ωη ,0)

" #2β h ≤C η −5 η

" #2β h ≤C η −11 η

(3.65)

(3.66)

(3.67)

Moreover, from (3.60) to (3.63), we have: ⎧ " #2β h ⎪ 2 ⎪ ⎪ δγ11 (Ωη ,0) ≤ C η−2 γ11 2(Ωη ,0) ⎪ ⎪ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " #2β ⎪ ⎪ h ⎪ 2 2 ⎪ δγ ≤ C γ12 (Ωη ,0) ⎪ 12 (Ω ⎪ η ,0) ⎨ η

⎧ 2 γ11 (Ωη ,0) ∼ η −3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ 2 −5 ⎪ 12 (Ωη ,0) ∼ η ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ γ22 2(Ωη ,0) ∼ η −5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ρ22 (Ωη ,0) ∼ η −9 or equivalently:

=⇒

" #2β ⎪ ⎪ ⎪ h 2 2 ⎪ ⎪ δγ22 (Ωη ,0) ≤ C γ22 (Ωη ,0) ⎪ ⎪ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " #2β ⎪ ⎪ ⎪ h ⎪ ⎩ δρ22 2(Ω ,0) ≤ C η−2 ρ22 2(Ωη ,0) η η

(3.68)

84


⎧ " #β h ⎪ ε ⎪ ⎪ δ∂1 u1 (Ωη ,0) ≤ C η −1 ∂1 uε1 (Ωη ,0) ⎪ ⎪ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " #β ⎪ ⎪ $ 1 $ 1 $ $ h ⎪ ⎪ $δ ∂1 uε − b12 uε $ $ ∂1 uε − b12 uε $ ≤ C ⎪ 2 3 2 3 (Ωη ,0) 2 2 ⎪ (Ω ,0) η ⎨ η " #β ⎪ ⎪ ⎪ h ⎪ ε ⎪ δ∂2 u2 (Ωη ,0) ≤ C ∂2 uε2 (Ωη ,0) ⎪ ⎪ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " #β ⎪ ⎪ $ $ $ $ ⎪ h ⎪ ⎩ $δ∂22 uε3 $ ≤ C η−1 $∂22 uε3 $(Ωη ,0) (Ωη ,0) η

(3.69)

We then have the significant result: Result 3.7.1. The error estimates (3.69) of 12 ∂2 uε1 − b12 uε3 and ∂2 uε2 are analogous to those of the first order derivatives in a standard elliptic second order problem with polynomials of degree β, whereas ∂1 u1 and ∂22 uε3 are poorer in the proportion η −1 . This result has a transparent interpretation. When considering the structure of the solution in the layer (see [19]), on account of (3.43), we perform the change of functions ⎧ ε ⎨ uα (x) = η−2 Uα (y), (3.70) ⎩ ε u3 (x) = η−3 U3 (y). and of variables y1 = x1 , y2 = x2 /η. Moreover, as the singularity of the normal loading considered f3ε is defined by (3.40), we also perform the change of function: f 3,ε (x) = η −1 F3 (y) The variational Koiter problem, first written in the layer Ωη and then transformed in Ω1 , becomes: 0 1 0 1 Aαβλμ [η −1 γαβ (U ) + γαβ (U )][η −1 γλμ (V ) + γλμ (V )]dy+ Ω1

+ Ω1

∂22 U3 ∂22 V3 dy

(3.71)

3

+ ... =

F V3 dy Ω1

0 1 where +... denotes asymptotically small terms and the γαβ and γαβ are given by: 0 1 γ11 (V ) = 0 γ11 (V ) = ∂1 V1 0 γ12 (V ) = 12 ∂2 V1 − b12 V3

1 γ12 (V ) = 12 ∂1 V2

0 γ22 (V ) = ∂2 V2

1 γ22 (V ) = 0

3.8 Conclusion

85

This variational problem is obviously a penalty problem starting by η−2 terms. Writing that the penalty terms vanish, we obtain the constraints 12 ∂2 V1 −b12 V3 = 0 and ∂2 V2 = 0, and the limit-constrained problem 1 2 2 ∂1 U1 ∂1 V1 dy + ∂2 U3 ∂2 V3 dy = F 3 V3 dy (3.72) C1111 Ω1 This limit problem only involves the “significant entities” ∂1 U1 and ∂22 U3 . Coming back to (3.69), we see that the error estimates of these entities have a degradation of order η −1 which follows from the penalty terms (“local locking” in the layer, see (3.13)). Moreover, the functions 12 ∂2 uε1 − b12 uε3 and ∂2 uε2 which are better approximated are the functions in the penalty terms. The limits of the leading terms vanish; in other words, the effective values of these functions are asymptotically small with respect to the order coming from the scaling of uε . Consequently, the relative error is likely larger than the estimate given in result 3.7.1. Result 3.7.2. On the basis of the previous comments, it may be said that the significant functions ∂1 uε1 and ∂22 uε3 are computed with a degradation of order η −1 coming from the phenomenon of local locking (see also [90]) associated with the local structure of a penalty problem in the layer.

3.8

Conclusion

Before performing numerical simulations with adaptive and anisotropic mesh procedure in the next chapters, let us conclude on the error estimates established in this chapter. First, it is important to note that, even if the previous considerations were established in the case of the loading (3.40), they are more general as they are concerned with relative (and not absolute) errors. Moreover, the previous calculations were performed taking into account only the leading order terms of the asymptotic expansions in the layer. For instance, in (3.43), Uα (y) is supposed to be independent of ε (or η) which amounts to discard lower order terms. Obviously, errors coming from numerical integration and other were also neglected. The conclusion should be considered rather as “general trends” than as rigorously established results. On the other hand, the previous results were established on the domain Ωη of the layer itself, whereas the region outside the layer was disregarded. Likely, as the F. E. are “smaller than the regions”, the adaptivity of the mesh (either with anisotropy or not) leads to a separate management of the various regions, so that the above considerations should be valid in each region. Moreover, inspection of examples of adaptive meshes shows that in the region of intersection of layers, the anisotropy disappear, not the adaptivity (smallness of the triangles), whereas the overall trend remains out of the intersections.

86


Finally, let us consider the “ideal model” example of a layer of relative thickness O(η) in the domain Ω. The layer bearing most of the energy (in the asymptotic sense, i.e. the energy out of the layer is negligibly small with respect to that in the layer for small ε). Let us consider that h is the mesh step necessary to obtain a fairly good computation. According to the previous results, it should be equivalent to use a η −1 h×h anisotropic mesh or a h×h isotropic one. Neglecting the triangles out of the layer, the number of triangles should be: 1 • O h×h for uniform mesh η • O h×h for adaptive isotropic mesh 2 η • O h×h for adaptive anisotropic mesh so that the number of triangles is divided by O(η −1 ) by the adaptive process and again by O(η−1 ) by the anisotropic one (in the previous approximate context).

4 Numerical Simulation with Anisotropic Adaptive Mesh

4.1

Introduction

Chapters 1 and 2 revealed all the diversity of the limit problems and of the associated singularities existing in shell theory when the thickness ε tends to be zero. Obviously, this induces serious difficulties for numerical computations of Koiter shell model (or of any other shell model). The main difficulty is to describe accurately the singularities of the displacements in the layers, and to avoid (or at least to reduce) the locking present in most of the situations. On the first hand, in the layers the resulting displacements vary strongly and anisotropically, essentially in the direction perpendicular to the layers that appear along lines. On the other hand, we do not know a priori the position and propagation of the layers in complex situations. Thus, one way to obtain accurate numerical results would be to refine strongly and isotropically the mesh in the whole shell. However, in practice, that would lead to non-reasonable computation times. Another more appropriate solution to conserve a reasonable number of elements, is to refine strongly the mesh only in the layers, using anisotropic elements. However, this is only possible with an adaptive mesh generator, which re-meshes automatically in the necessary areas, as in complex situations, it is impossible to predict a priori the position of the layers. This way, the mesh will be refined only in the areas and in the directions where the displacements vary the most. This we propose to do in all the numerical simulations presented in the next chapters, using the adaptive and anisotropic mesh generator BAMG (Bidimentional Anisotropic Mesh Generator) developed by INRIA1 . This mesh generator will be coupled with the finite element software MODULEF developed by INRIA as well. The numerical simulations so performed, which enable us to access very accurately to all the singularities, will be compared all along this book, to the theoretical expressions of singularities (when they exist). 1

Institut de Recherche en Informatique et Automatique, France.


88


An important point to quote is that the use of adaptive anisotropic re-meshing procedure enables to reduce the locking phenomenon that appears when the thickness becomes very thin (see results of chapter 3). The first part of this chapter contains a review on the locking phenomena in elastic thin shell theory. We then focus on the presentation of the DKT shell element used with MODULEF for the finite element computations, and on the anisotropic adaptive remeshing procedure used by BAMG. Finally, we will present a procedure of computation of the membrane and bending energies implemented in MODULEF which gives precious information on the formation of boundary and internal layers.

4.2 4.2.1

Review on the Numerical Locking Introduction

Generally speaking, when dealing with problems depending on a little parameter ε, it is said that there is locking for ε = 0 (or when ε 0), if the finite element approximation is not uniform with respect to ε on intervals of the form ]0, ε0 [ (see [9]). In other words, this amounts to saying that, to have some desired approximation, the F. E. should be taken smaller and smaller as ε tends to zero. This is due to some kind of singular behavior of the limit problem denoted P (0) (the membrane model in our case) with respect to the generic one P (ε) (the Koiter shell model). Nevertheless, this singular behavior spreads out at ε = 0 and amounts to difficulties of convergence for small ε. Usually, the problem for ε > 0 does not exhibit peculiarities, and the convergence is uniformly good on intervals of the form ]ε1 , ε2 [ with ε1 > 0. It should be pointed out that locking is concerned with non-uniformity either on the open interval ]0, ε0 [ or the closed interval [0, ε0 ]. In other words, the limit ε = 0 may be included or not. The essential point is some kind of pathology of the convergence of the F. E. for small values of ε (not necessarily ε = 0), as a consequence of the peculiarities of the limit. This is a little subtle, as the problem “well-behaves” for ε > 0. However, it is “less and less good” as ε decreases. Roughly speaking, constants in estimates defining the F. E. convergence depend on ε and “explode” as ε 0”. In thin shell theory, locking for ε 0 arises in the two cases of inhibited and non-inhibited shells (geometrically rigid or not), but under different forms. We shall explain this in the sequel. 4.2.2

Locking in the Non-inhibited Case (Classical Locking Associated with a Limit Constraint)

As already mentioned in the last chapter, when the shell is not inhibited or non-geometrically rigid, via an appropriate re-scaling, the Koiter model can be written as the following penalty problem:

4.2 Review on the Numerical Locking

89

F ind uε ∈ V such that 1 am (uε , v) + ab (uε , v) = f, v ∀v ∈ V ε2

(4.1)

Moreover, we recall that in the non-inhibited case, the subspace G of V of inextensional displacements defined by: G = {v ∈ V ; am (v, w) = 0 ∀ w ∈ V }

(4.2)

does not reduce to the zero element. We will see in this case that the locking is due to a change of space for ε 0. Merely, the limit is the solution of a problem which “lives” in a subspace G of the space V when ε > 0, and this has unexpected consequences. Let us consider the classical limit pure bending model: F ind u0 ∈ G such that ab (u0 , v) = f, v ∀v ∈ G

(4.3)

It is known that both problems (4.1) and (4.3) are well-posed in V and G, respectively, and we have the classical convergence theorem (see theorem 2.3.3 of section 2.3.5): uε −→ u0 strongly in V (4.4) ε0

Let us prove for ulterior use a result stronger than (4.4). To do this, let us define the following energy norm: v2En(ε) =

1 am (v, v) + ab (v, v) ε2

(4.5)

which is equivalent to the norm (3.3) for a fixed value of ε (they are proportional to ε−2 ). Let us now consider, for each value of ε, the energy of the difference between the solution uε and its limit u0 : uε − u0 2En(ε) =

1 am (uε − u0 , uε − u0 ) + ab (uε − u0 , uε − u0 ) ε2

Using (4.1) and (4.3) with suited v, it follows easily: % & % & uε − u0 2En(ε) = f, uε − 2 f, u0 + f, u0 −→ 0 ε0

(4.6)

(4.7)

which clearly improves (4.4), as (4.4) is concerned with the norm in V , defined for instance by: 2 · V = am ( · , · ) + ab ( · , · ) (4.8) whereas the membrane energy appear in (4.5) multiplied by ε−2 . Let us now consider a F. E. approximation in V , i. e. a sequence Vh (h 0) of subspaces of V (with finite dimension, but this is not relevant) which “densifies on V ”, i. e. such that for any v ∈ V , there exists a sequence vh such that vh −→ v strongly in V h0

(4.9)

90


The Vh approximation of uε denoted uεh is defined by F ind uεh ∈ Vh such that 1 am (uεh , v) + ab (uεh , v) = f, v ∀v ∈ Vh ε2

(4.10)

We recall that a classical property of F. E. approximation (or Galerkin approximate) of a problem (with a fixed value of ε) is that the approximate solution uεh is the element of Vh which is the nearest to the exact solution in the energy norm .En(ε) defined by (4.5) (see section 3.2 if necessary). In other words, uεh is the orthogonal projection of uε on Vh in the metrics associated with the energy norm .En(ε) . Considering (4.9) with v = uε , there exists elements whε ∈ Vh tending to uε in V (and then in the norm of energy as ε is fixed). We have uε − whε En(ε) −→ 0

(4.11)

uε − uεh En(ε) −→ 0

(4.12)

uε − uεh V −→ 0

(4.13)

h0

and by the above property h0

Consequently, we obtain: h0

as well. This is the classical F. E. or Galerkin approximation (with fixed ε). The consideration of the norm of V or the energy norm is almost trivial as the corresponding norms are equivalent. Now, when considering the dependence of this convergence with respect to the parameter ε ≥ 0, it is obvious that the “equivalence” disappears because of the factor ε−2 in (4.5). Remark 4.2.1. The locking can be seen as the consequence of the discrepancy between the norms (4.5) and (4.8) when ε 0. Convergence is concerned with the norm (4.8), whereas the F. E. approximation takes into account the very different criterion of distance associated with (4.10). To go on with a geometric interpretation of the locking, let us consider the case when for each h, we have G ∩ Vh = {0} (4.14) This is often the case when the elements of G are solution of the system γαβ (v) = 0, i. e. very specific functions, whereas elements of Vh are usually piecewise polynomials. As a consequence, the only common element is 0. We are now considering the paths of the approximate solutions uεh as functions of ε in the two different metrics, with fixed h. Let us decompose the whole space V as product of G and its orthogonal G⊥ (in the fixed metric of V of course): V = G ⊗ G⊥

(4.15)


91

G⊥

Vh uεh uε O

G

u0

Fig. 4.1. Schematic decomposition of V = G ⊗ G⊥ and approximation of uε

In figure 4.1, G and G⊥ are represented by two orthogonal axes. The exact solution is plotted for small values of ε and as ε 0, its limit is u0 which is located in G. We also represented the subspace Vh by a straight line which according to (4.14) has only a common point with G, the origin. Now, the approximation uεh is the orthogonal projection of uε on Vh , but orthogonality is understood in the sense of the energy norm (4.5), not (4.8) of figure 4.1. To compare the two metrics, we have for any v: $ $2 2 vV = $v ⊥ + vG $V = am (v ⊥ , v ⊥ ) + ab (v⊥ , v ⊥ ) + ab (vG , v G )

(4.16)

and analogously 2

vEn(ε) =

1 am (v ⊥ , v ⊥ ) + ab (v⊥ , v ⊥ ) + ab (vG , v G ) ε2

(4.17)

In other words, passing from the norm V to the energy norm amounts to replace distances along G⊥ from [am (v, v) + ab (v, v)]1/2 to:

1 am (v, v) + ab (v, v) ε2

1/2 ∼

1 1/2 [am (v, v) + ab (v, v)] ε

(4.18)

1 i.e approximatively to multiply the distances along G⊥ by . That corresponds ε to the approximate picture in the energy norm in figure 4.2. When h and the subspace Vh are fixed, its representation is a straight line which tends to be vertical as ε 0. Otherwise, the points representing uε with ε ≥ 0 are not the same as in Fig. 4.1; they are also “dilated” in the vertical direction. Nevertheless, because of the property (4.7), the representative point tends to u0 ∈ G in this distance as well. It then appears from figure 4.2 with ε 0 that the orthogonal projection uεh tends to 0 in the energy norm, i. e.: 1 am (uεh , uεh ) + ab (uεh , uεh ) −→ 0 ε0 ε2

(4.19)

92


1 ⊥ G ε Vh

uεh uε

G

u0

O

Fig. 4.2. Schematic explanation of the locking phenomenon when ε 0

and consequently uεh −→ 0 strongly in V ε0

(4.20)

In other words, coming back to Fig. 4.2, the projection uεh of uε on Vh tends to the origin. The projection is orthogonal in the energy metrics, but in the V -metrics, it becomes “more and more tangential” as ε decreases. The phenomenon (4.20) is quite natural and has nothing to do with a poor approximation of V by Vh . The reason is rather that the F. E. approximation deals with problem (4.10) which involves large coefficients (because of the factor ε−2 ), whereas the solution is of order O(1) in the V -norm and inherits large errors from the computation. Remark 4.2.2. The property (4.20) may be proved directly from (4.10) letting ε 0 with fixed h. Our proof is longer and less rigorous (as it includes the approximation (4.18)), but it explains better the locking by the drastic difference between the norms (4.5) and (4.8). The “pathological” nature of the F. E. approximation (4.13) in the vincinity of ε = 0 is now evident: the two limits ε 0 with a fixed h and h 0 with a fixed ε do not commute: " # lim lim (uεh ) = lim (0) = 0 (4.21) h0

whereas

ε0

h0

" # lim lim (uεh ) = lim (uε ) = u0

ε0

h0

ε0

(4.22)


93

It is very easy to deduce from this that the convergence (4.13) cannot be uniform on ]0, ε0 [. Indeed, we consider uεh as a sequence (with index h) of functions of ε, defined on ]0, ε0 [. As the limit of each one for ε ≥ 0 exists and is 0, we may consider uεh as continuous functions of ε on ]0, ε0 [ (or obviously [0, ε0 ]) taking the value 0 for ε = 0. We then use the classical property that the limit of a sequence of continuous functions which converges uniformly is continuous. Here the limit of the sequence uεh indexed with h is uε for ε ≥ 0 and 0 for ε = 0. If the convergence was uniform, this limit should be a continuous function of ε, and this is false as # " lim lim (uεh ) = u0 = 0. (4.23) ε0

h0

This kind of locking can be encountered in various situations : the “shear locking” for bending beams including shear (Timoshenko beams [23]), the “membrane locking” for plane arches [24] or hyperbolic shells [27, 28, 88]. In every case, it is shown that some space containing the limit solution for ε = 0 (similar to Vh ∩ G) reduces to {0}. Thus, the F. E. are unable to take bending (or shear) effects into account for very small ε. The resulting displacements are then very underestimate. 4.2.3

Locking in the Inhibited Case (Singular Perturbations)

In this case, the proof of the locking is very easy and follows from the properties of the general result of convergence of uε → u0 , without taking into consideration the approximation Vh (see theorem 2.3.2). Nevertheless, the results are more subtle than in the non-inhibited case, as various topologies are involved and the definition of locking itself depends on the required topology. The general framework is the singular perturbation problem associated with the Koiter model in the inhibited case: F ind uε ∈ V such that am (uε , v) + ε2 ab (uε , v) = f, v ∀v ∈ V

(4.24)

which may equivalently be written as (see (2.21)): Am uε + ε2 Ab uε = f

(4.25)

where Am and Ab are continuous operators from V into V defined by (2.20). We then consider the space VA which is the completion of V with the norm .VA defined by (see 2.22): vVA = Am vV As we saw in the proof of theorem 2.3.2, the operator A can be extended by continuity to an operator A¯ which is continuous from VA into V and which defines an isomorphism (i. e. a one-to-one bi-continuous mapping) between VA

94


and V . Moreover, according to the theorem 2.3.2, there is a unique u0 ∈ VA such that ¯ 0=f Au (4.26) and which satisfies:

uε −→ u0 strongly in VA

(4.27)

It is obvious that VA is a space larger than V and accordingly its topology is weaker. It should be emphasized that u0 is an element of the enlarged space VA ; only exceptionally it is in smaller spaces (as V ). In other words, as VA is filled by the u0 coming from f ∈ V , the property u0 ∈ VA is optimal for any f ∈ V . Result 4.2.1. Let us consider the usual situation of f such that u0 is not in V . Then, the convergence of the F. E. approximation uεh −→ uε strongly in V

(4.28)

cannot be uniform with respect to ε ∈ ]0, ε0 [. Obviously, uεh denotes the F. E. approximation of uε with a sequence of subspaces Vh which “densify on V ” as in (4.9). The proof is by contradiction, as at the end of section 4.2.2. The classical property that the uniform limit of continuous functions is a continuous function is applied to functions with values in V . If the convergence was uniform, the limit, uε , considered as a function of ε with values in V should be continuous at ε = 0, i. e. uε should have a limit in V for ε 0. But this limit u0 is not in V and we have a contradiction. Remark 4.2.3. The same proof shows that, when u0 is out of some space V intermediate between V and VA , the convergence of the F.E. approximation cannot be uniform with values in V. The only possibility of uniform convergence in general is with values in VA , but the corresponding topology is so weak that such a result is irrelevant for practical purposes.

4.3

Shell Element and Associated Discrete Problem

We recall that the formulation of the Naghdi model without shear stress is similar to (1.59) where ραβ is replaced by (see 2.46): 1 Dα θβ + Dβ θα − bλα (Dβ uλ − bλβ u3 ) − bλβ (Dα uλ − bλβ u3 ) 2 (4.29) where θα denote the two rotations of the normal vector a3 during the deforma√ tion. Note that the element of surface dS can be written dS = ady 1 dy 2 where a is the determinant of the metric tensor. χαβ (u, θ) =

4.3 Shell Element and Associated Discrete Problem

95

Then, two different finite element spaces are used to discretize the problem: - P2-Lagrange elements (6 nodes, 1 unknown per node) for the tangent displacements u1 , u2 and for the rotations θ1 , θ2 . - P3’-Hermite elements (3 nodes, 3 unknowns per node) for the normal displacement u3 . 4.3.1

The Shell Element D.K.T.

The D.K.T (Discrete Kirchhoff Triangle) element is based on a non-conforming approximation method: the discrete approximation sub-space Vh is not contained in the space V . This allows us to use a simpler element which only needs a C 0 continuity, whereas a conforming approximation method such that Vh ⊂ V would need a C 1 continuity [15]. An example of a conforming element is the Ganev– Argyris one which is C 1 and has 51 DOF, whereas the D.K.T element has only 21 degrees. Thus the use of D.K.T elements allows us to limit the number of degrees of freedom during the mesh adaptation. The non-conformity of the D.K.T. element and the convergence of the discrete formulation to the continuous one is proved in [15]. The D.K.T element of MODULEF has been developed by M. Bernadou [15] from the D.K.T. element. The construction of the later starts from the Nagdhi model with some simplifications: the transverse shear is neglected and Kirchhoff–Love kinematic constrains are imposed in a finite number of points. Applying Kirchhoff–Love kinematic constrains (no transversal shear) θα = −u3,α − bλα uλ on the six nodes, we finally have 21 DOF (Fig. 4.3). This constitutes the DKT element corresponding to the Koiter shell model which is nothing else than the Naghdi model without transversal shear. In all this book, the numerical simulations will be performed with the finite element software MODULEF using the DKT element associated with the Koiter model without transverse shear. However, the use of more complex shell elements

a3 b2 a1

u1

u1 u2 u3 β1 β2

b1

u2 b3 a2

Fig. 4.3. Unknowns of the discrete problem

96


including transverse shear (associated with Nagdhi model) would lead to the same results (in terms of singularities) for very thin shells. Indeed, as explained in section 2.4 of chapter 2, Nagdhi and Koiter models have the same limits, when ε 0. Therefore, the limit displacements and the associated singularities given by both models are identical. 4.3.2

Discretization of Naghdi Model

The discretization of Naghdi model leads to a discrete problem of the form : aS (uh , β h , vh , δ h ) = aSK (uh , β h , vh , δ h ) = f S (vh , δ h ) (4.30) K

where aS (., .) is the discretized bilinear form of εam (., .)+ε3 ab (., .) and aSK (., .) the discretized bilinear form of the element K. The right-hand side f S (., .) represents the discretized linear form of the loadings. It can be written under the form: t t UK RK UK = FK UK (4.31) K

K

where UK is the unknown vector, RK the elementary rigidity matrix, and FK the elementary loading vector of the element K. More precisely, the discretized problem can be written as: ' L ( t DLDKT (uh , β h ) [B] ωl,K [LAM BD] [AIJ ] [LAM BD] (bl,K ) K t

l=1 t

[B] [DLDKT (vh , δ h )] =

' L K

ωl,K

t

P [LAM BD] (bl,K ) t

(t t

[B] [DLDKT (vh , δ h )]

l=1

(4.32) where K is the number of the element considered. The matrices [B] and [LAM BD] contain, respectively, the constant and the variable part of the interpolation. The vector [DLDKT (vh , δ h )] is the unknowns vector. Finally, Ee √ e2 t α β t α β t α β t α β [AIJ ] = a (1 − ν) Λβ Λα + ν Λα Λβ + (1 − ν) Nβ Nα +ν Nα Nβ 1 − ν2 12 (4.33) contains the membrane and bending terms Λβα and Nαβ defined by γαβ (uh ) = Λβα t [LAM BD]t [B]t DLDKT (uh , β h ) ρβα (uh ) = Nαβ t [LAM BD]t [B]t DLDKT (uh , β h )


97

When considering Λβα = 0, we obtain the elementary bending rigidity matrix. On the other hand, considering Nαβ = 0, we obtain the elementary membrane rigidity matrix. This enables us to compute separately the membrane and bending energy of each element. Taking into consideration, the area of each element (computed with the metrics), we obtain the energy surface density due to membrane and bending deformation. First, displacements and rotations are computed with the complete elementary rigidity matrix. The elementary bending and membrane rigidity matrixes are then built to compute both energy surface densities separately. In MODULEF, the mesh for this element is realized on the 2D domain Ω of the local mapping of the middle surface. That avoids geometrical approximations of planar facet decomposition. When computing F.E. shell problems, the main difficulties occur inside the layers where singularities appear when ε 0. To obtain accurate results, we need to refine the mesh essentially in these layers. As they appear along lines, it is obvious that an efficient mesh should be anisotropic: we need to refine more significantly in the directions perpendicular to the layers. Error estimates for finite elements using anisotropic meshes inside the layers were presented in chapter 3 (see also [89][90] in the case of parabolic shells). It is clear that anisotropic elements lead to a better description of the singularities with a reduced number of elements. We recall (see sect. 3.8) that if η denotes the thickness of the layer, the number of triangles necessary to get a fairly good approximation is divided by O(η−1 ) by the adaptive process, and again by O(η −1 ) by the use of anisotropic elements, in comparison with a classical (non-adapted) isotropic mesh. Therefore, the efficiency of an anisotropic adapted mesh, which needs O(η−2 ) less elements than a classical uniform one, increases drastically when the relative thickness ε (and so that the layer thickness) of the shell tends towards zero. There exists various methods of adaptive mesh. The main ones are presented below. • The h-methods enable us to adapt the mesh using a determined criterion, by adapting isotropically or anisotropically (in a privileged direction) the size of the elements. This can be done locally (only a part of the mesh is modified) or globally (the whole mesh is entirely reconstructed). During the mesh adaptation, the degree of interpolation remains constant. Moreover, when anisotropic elements are allowed, the angles cannot approach 180◦ [6]. • The p-methods consist in adapting the degree of interpolation of the polynomials of some elements, without changing the geometry of the mesh, to obtain more accurate results. These methods are only relevant for simple geometries [7]. • The hp-methods couple the two last strategies, combining the respective advantages of the two approaches [8]. In all these cases, and irrespective of the method used, it is necessary to have a criterion (gradient of the solution, error estimator, . . . ) to detect the areas, where the adaptation is necessary.

98


4.3.3

Adaptive Mesh Strategy: BAMG

We use the software BAMG (Bidimensional Anisotropic Mesh Generator), developed at INRIA2 [58]. It is an anisotropic adaptive mesh generator based on a global h-method of approximation, and on a classical mesh generator of Delaunay type. From an initial mesh and the numerical results given by the finite element simulation, BAMG generates an adapted mesh, with anisotropic elements when necessary (according to the problem). For shell problems, BAMG revealed to be particularly adapted to the study of boundary and internal layers, because of their locally anisotropic character. Therefore, an adapted mesh refined more particularly in the direction perpendicular to the layer, leads to more accurate results than a uniform or isotropic mesh, with much less elements. We shall see in the numerical simulations performed in the next chapters that with such an anisotropic adaptive mesh generator, we approach very accurately the singularities in the layers predicted by the theory. The Bidimensional Anisotropic Mesh Generator BAMG performs an anisotropic mesh adaptation using metric control technique. It was initially developed to compute supersonic aerodynamic flows that exhibit shock waves [16, 17, 22]. It has already been successfully used for shell and shell-like problems computations in [11, 36, 37]. More precisely, the length of elements are adapted to satisfy an imposed criterion based on the Hessian of the approximate solution in our concern (as we will see in the sequel). To understand the procedure of re-meshing of BAMG, we shall consider the following example. Let Ω be a domain of R2 . By definition (see [17]), a mesh of Ω adapted using metric control technique is a mesh having all edges of unit length with respect to the associated Riemannian structure (whose metrics is noted M). In other words, a unit mesh has a good size quality with respect to the associated control metrics. −− → Let us consider the mesh edge AB parametrized by AB = (A + tAB)t∈[0,1] . Assuming that M is a continuous metrics on Ω, noted M(t) at current point − −→ A + tAB, the length of a mesh edge AB (with respect to metrics M) is given by: lM (AB) =

1

)

−− →

− −→

T ABM(t)ABdt

(4.34)

0

−− → − −→ where T AB denotes the transposed of vector AB. In the isotropic case, the metrics is constant and obviously equal to: ⎛ ⎞ 1 ⎜ 2 0 ⎟ M(t) = ⎝ h 1 ⎠ (4.35) 0 h2 h being the length of the mesh edge AB. In the anisotropic case, the appropriate metrics to be considered is: 2

Institut national de Recherche en Informatique et Automatique, France.


⎛

1 2 (t) ⎜ h M(t) = T R(t) ⎝ 1 0

99

⎞ 0

⎟ 1 ⎠ R(t) h22 (t)

(4.36)

where R(t) denotes the rotation matrix that indicates the directions of the anisotropy, T the transposition operator, and hi (t) the length of the considered element in the direction i of anisotropy. The adaption of the mesh is based on an a posteriori error estimate between the discrete and the real solution. To this end, the solution at each node is considered as exact (which may not be the case). If we use a linear interpolation between the results at each node, the error can be estimated for a 2D problem, due to the Taylor development (see [5] or [47], p.283), by: |η − ηh |∞ ≤ c0 h2 H(η)∞ H being the Hessian of η defined by: ⎛

∂ 2η ∂ 2η ⎜ ∂(y 1 )2 ∂y 1 ∂y 2 ⎜ H(η) = ⎜ ⎜ ⎝ ∂ 2η ∂ 2η 2 1 ∂y ∂y ∂(y 2 )2

(4.37)

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(4.38)

and where c0 is a constant and h the size of the mesh. | |∞ denotes the L∞ norm in R2 and ∞ the matrix L∞ norm. This method is not based on an a priori error estimate, but on an estimate of the areas, where the solution vary the more. Then, BAMG generates a new mesh using a new metrics M(t) determined from the Hessian of η, and from the relative error calculated on the edge ai of the present mesh [47]: = c0 | t ai Hai | (4.39) Because H is not in general definite positive and does not define a metrics, we define |H| from the diagonal matrix composed of the eigenvalues λ1 and λ2 of H: ⎛ ⎞ |λ1 | 0 ⎠ R−1 |H| = R ⎝ (4.40) 0 |λ2 | Finally, BAMG generates a new mesh using the new metric M(t) calculated as follows: c0 M(t) = |H| (4.41) Thus, lengths are increased (respectively diminished) in the areas and the directions where the second-order derivatives are large (respectively small). Consequently, the mesh is refined (respectively enlarged) when the second order derivatives are large (respectively small).

100


That can be explained in a very simpler manner in the case of a definite positive Hessian H(η) which defines a metrics. In that case, the elementary length ds along the surface S can be calculated as follows: ds2 = (∂12 η)dy12 + (∂22 η)dy22 + 2(∂1 ∂2 η)dy1 dy2 where dy1 and dy2 are the elementary lengths in the plane of the parameters. Obviously, it is clear that the lengths dyi are diminished in the direction or in the areas, where the derivatives ∂i2 η are large, so as to conserve the “measured lengths”. This way, the mesh is refined only in the areas where the second-order derivatives of η are large. The generalization to |H| defined by (4.40) when the Hessian of η is not definite positive and does not define a metrics is left to the reader. 4.3.4

Coupling BAMG-MODULEF for Shell Computations

For the numerical simulations performed in that follows, we use the version v0.68 of April 2001 developed by Frédéric Hecht. It is a freeware that can be downloaded on the following site: http://www-rocq1.inria.fr/gamma/cdrom/www/bamg/fra.htm For our problems, we will use the normal displacement u3 for the remeshing procedure (it will plays the role of the parameter η). Indeed, on the one hand, as we saw previously, u3 is the most singular displacement for a given f 3 . Moreover, the second-order derivatives of u3 are the leading terms of the curvature variation ραβ (the leading terms of ραβ are ∂α ∂β u3 ). Thus, the results given by MODULEF for u3 at each iteration are used by BAMG to estimate the Hessian and then the new metrics. From it, an adapted mesh is generated using this new metrics. This way, the mesh will be refined in the areas where the second-order derivatives of u3 are most relevant (inside the layers). They correspond to the areas, where the bending is important.

Other Data

Initial Mesh

MODULEF

Results

Adapted Mesh

BAMG

Fig. 4.4. Scheme of the adaptation

4.4 Membrane and Bending Energies Computation with MODULEF

101

More precisely, a first computation is performed with MODULEF using a large mesh approximatively isotropic. Then, from the results of this first calculation, BAMG generate a new adapted mesh. The finite element computations are performed again, but with this new mesh, whose results are used again to adapt the mesh, and so on. Several adaptations are needed to get a better mesh. The scheme of the adaptation procedure, used in the numerical computations that follows, is summarized in Fig. 4.4. The adaptation process ends when the difference between the results of two successive iterations is small. To estimate this difference, the solution of the iteration i − 1 is projected onto the mesh of the iteration i, and the results of both iterations are compared due to a norm of the form: n 1 i u3 (j) − ui−1 3 (j) n j=1 × 100 Δui3 = sup(ui3 ) − inf (ui3 )

(4.42)

where n is the number of nodes at the iteration i, ui3 (j) the value of u3 at the node j of the mesh of iteration i, and ui−1 3 (j) the value of the solution of iteration i − 1 projected onto the mesh of iteration i at node j. Thus, Δui3 is expressed in percentage and the stop criterion that we choose is Δui3 < 0.10% except when another specific criterion is given.

4.4

Membrane and Bending Energies Computation with MODULEF

In all the shell problems that will be considered (inhibited, well inhibited, or sensitive problems), the distribution of the energy surface densities constitutes an important feature for the interpretation of the results. Indeed for inhibited shells, the limit problem contains only membrane energy (except in the layers). Oppositely, for ill-inhibited problems, when instabilities appear, a more important part of energy due to the bending is contained in the oscillations. So that the comparison of membrane and bending energies gives supplementary information concerning the limit problem. On the other hand, the numerical computations of energies can be used also to measure the internal layer thickness. We implemented in MODULEF a new procedure which evaluates the bending and membrane surface energy densities in each element, and then in the whole shell domain. This we proceed to describe. 4.4.1

Implementation Procedure in MODULEF

In the numerical computations performed with MODULEF using the DKT shell element, the discrete problem contains both membrane and bending energies (coming from the discretization of am (., .) and ab (., .) respectively). So that from the elementary rigidity matrix RK associated with the discrete problem (4.31)

102


in MODULEF, the elementary energy (per element K) can be written as EK = t UK RK UK where UK denotes the unknown vector of element K. To compute separately membrane and bending energies, we created two “new elements” which are very close to the DKT one. The first one contains only the membrane part of the energy, and the second one only the bending part. This way, we access separately to the elementary membrane and bending rigidity matrices (RmK and RbK , respectively), their sum being equal to RK . The elementary membrane and bending energies per element are then given, respectively, by EmK = t UK RmK UK and EbK = t UK RbK UK . Because the mesh size is not constant, it is more consistent to compare the energy surface densities, given by dividing EmK and EbK by the surface of the element K. Finally, the total membrane and bending energies on the whole shell can be computed by summing the elementary energies on each element. 4.4.2

Validation on Simple Examples

We propose to validate the computation of membrane and bending energies implemented in MODULEF before performing more complex simulations for shell problems. To this end, we perform simple numerical simulations for classical plate and shell problems, where an analytical solution is known. The first example of validation for the bending energy is a simple problem of bending of a circular plate. The second example is a half-sphere subjected to a normal pressure. It enables us to validate the computation of both energies in a case where they are coupled (since the curvatures are not null). Circular plate under normal pressure We consider a circular plate with a radius R = 100 mm, a constant thickness ε = 3.28 mm and the following material properties: E = 210 000 M P a and ν = 0.33. The plate is clamped at r = R. We consider a normal loading p3 = −0.025 M P a constant on the whole plate. We can find u3 from the Kirchhoff–Love bending equations. In the case of a circular plate, using polar coordinates, the bending displacement is given by: u3 =

3 p (1 − ν 2 )(R2 − r2 )2 16 Eε3

In the pure bending problem considered, we have Em = 0 and Eb = Etot . Moreover, we easily obtain: Π Eb = p3 u3 dS = α p 3 R6 (4.43) 3 ω 3 p where we have set α = (1 − ν 2 ). The numerical application for the 16 Eε3 particular values considered here gives Eb = 44.77 mJ.

4.4 Membrane and Bending Energies Computation with MODULEF

103

Numerical results Now let us perform several numerical computations with MODULEF and with an increasing number of elements. With 964 elements, we already find that the bending displacement at the center of the plate is u3 (0) = −1.707 10−5 , which corresponds to an error of 0.2% in comparison with the theoretical value. The next table gives the values of both membrane and bending energies: Nel Eb (mJ) Em (mJ) Etot (mJ) 964 45.22 0 45.22 1602 44.92 0 44.92 2762 44.86 0 44.86 We notice that the bending energy approaches the theoretical value when the number of elements increases. The membrane energy vanishes exactly. Moreover, the distribution of the energy surface density (Fig 4.5) agrees with the theory: near the inflexion point for u3 (at r = 0.577), we have ρrr = 0 and the bending energy surface density is much weaker.

Fig. 4.5. Bending energy surface density Ef s

Half-sphere under uniform normal pressure Let us consider the domain Ω = (y 1 , y 2 ) [0, 2π] × [0, π2 ] and the half-sphere defined by the mapping (Ω, ψ) where : ψ(ξ 1 , ξ 2 ) = (Rcos(ξ 1 )sin(ξ 2 ), Rsin(ξ 1 )sin(ξ 2 ), Rcosξ 2 )

(4.44)

We consider a radius R = 100 mm, a thickness ε = 3.14 10−2 mm and the material constants: ν = 0.3 and E = 210 000 M P a. The shell is assumed to be subjected to an internal pressure p = 1 M P a on the whole surface. The imposed

104


k

ξ2

11m 00 00 11 j

i

ξ1

Fig. 4.6. Half-sphere and variables considered

boundary conditions on the boundary ξ = π2 are u1 = u2 = 0 and u3 is free. The solution of the membrane problem is classically given by [42]: u1 = u2 = 0 and u3 =

(1 − ν)pR2 2εE

(4.45)

The numerical application for the data considered leads to u3 = 5.307 10−1 mm. Moreover, as p and u3 are constant, the total energy reduces to Etot = p u3 dS = pu3 S ω

where S denotes the surface of the half sphere. For the given data of the problem, we obtain Etot = 5.307 10−1 S mJ . Even if most of this energy is membrane like, u3 the bending energy does not vanish completely because: ρ11 = ρ22 = − R 2 = 0. So that we obtain: ε3 Eb = T r(Mt ρt )dS (4.46) 8 ω As ρ11 = ρ22 , using the constitutive law, we obtain : " 2# Eε3 u3 Eb = S 6(1 − ν) R4

(4.47)

whose numerical application leads to Eb = 4.361 10−9 S mJ . Therefore, the bending energy is negligible with respect to the membrane one. We finally obtain the energy surface densities Ems = 5.307 10−1 mJ/mm2 and Ebs = 4.361 10−9 mJ/mm2 which are constant on the whole half sphere. Numerical results As the energy surface densities are constant on the whole domain, we can now compare the numerical results of both energy surface densities to the theoretical ones. These results are obtained with a mesh adaptation.

4.5 Conclusion

105

Nel Ems (mJ/mm2 ) Ebs (mJ/mm2 ) 1631 5.325 10−1 4.375 10−9 −1 2795 5.317 10 4.369 10−9 −1 4725 5.313 10 4.366 10−9 The energy surface densities on the whole half sphere are almost constant, and very close to the theory with only 1 613 elements. They become more accurate as the number of elements increases. Thus, the numerical results are in good agreement with the theory, and the numerical procedure of computation of energies may be considered as validated.

4.5

Conclusion

In this chapter, after a review on the numerical locking in shell theory, we presented the numerical simulation softwares that will be used in the sequel to study the singularities of the displacements present in the boundary and internal layers. The numerical simulations will be performed with the finite element software MODULEF coupled with the adaptive anisotropic mesh generator BAMG. This way, the mesh will be refined anisotropically only in the areas where the bending displacement vary drastically: the boundary and internal layers. We will see in the next chapters that the simulations performed in various configurations enable us to approach very accurately the singularities predicted by the theory, palliating considerably the locking phenomena. Finally, in the last part, we presented a new procedure of computation of membrane and bending surfacic energies, implemented in MODULEF and validated on examples where analytical solutions are known.

5 Singularities of Parabolic Inhibited Shells

5.1

Introduction

In chapter 2, we developed a study of the singularities in linear thin elastic shell theory, based on the reduction of the coupled membrane system to a partial differential equation involving only one component of the displacement. We obtained general results on the orders of the singularities of the displacements and on their propagation, which depend strongly on the nature of the middle surface of the shell (parabolic, hyperbolic or elliptic). In the case of parabolic and hyperbolic shells (but not for elliptic shells), an alternative accurate calculation of the singularities is possible. It consists in integrating directly the membrane system, that is possible if we choose the coordinate system such as the coordinate curves correspond to the asymptotic ones. With such a direct integration of the membrane system, we first determine the singularity orders of the membrane stress tensor with respect to the singularities of the loading (only a normal loading leading to the highest order singularities is considered). Then, using the elastic constitutive law and the boundary conditions, we determine explicitly the highest order singularities for the three components of the displacements. Obviously, the obtained results coincide with those of chapter 2. The two alternative ways are possible for parabolic and hyperbolic shells (hyperbolic shells will be considered in the next chapter). The first part of this chapter is devoted to the direct integration of the membrane system for parabolic shells, using a particular system of coordinates to decouple the membrane equations. We determine explicitly at the leading order the more singular terms for the membrane stresses and the displacements. In section 5.3, in the particular case of a loading applied on a circular zone, we describe completely the singularities of the loading and of the corresponding displacements. This example will exhibit a non-classical family of singularities which have a fractional order compared to the family of Dirac function singularities. The results obtained will be used to test the reliability of the anisotropic adaptive mesh procedure. E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 107–145. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

108


The second part of the chapter is devoted to numerical computations of the singularities performed with the adaptive anisotropic procedure described in chapter 4. It enables us to refine automatically the mesh in the direction perpendicular to the layer where the variations of the solution are important. This procedure is particularly well adapted to obtain an accurate description of the various kinds of singularities appearing inside the layers. The results obtained for the singularities of the displacements (u1 , u2 and mainly u3 ), for the particular circular loading considered, are compared to the theoretical ones obtained in section 5.3. In particular, the efficiency and the accuracy of an anisotropic adaptive mesh are put in a prominent position by comparison with results obtained with classical uniform meshes. In section 5.6, we study numerically the case of singularities along non-characteristic lines in various problems. Finally, in sections 5.7 and 5.8, we consider more complex problems where no analytical solutions are known. We will focus in particular on the case of a loading singular along the boundary of the shell, and on singularities due to a complex shape of the middle surface of the shell.

5.2

Study of the Singularities and of Their Propagation

In this section, from the integration of the membrane system, we will prove that the singularities of the displacements are of very different nature either the loading is singular along a characteristic line or along a non-characteristic one. Let us consider a parabolic shell, or equivalently a developable surface, and the coordinate system associated to the main curvature, such as y 2 = const corresponds to the asymptotic line of the middle surface1 . In this coordinate system, we have b11 = b12 = 0 and b22 = b22 (y 1 , y 2 ) = 0. As in chapter 2, as the normal loading leads to the higher singularities for the displacements, the tangential components f 1 and f 2 of the loading are not considered. Therefore, the membrane system (1.64) reduces to : ⎧ D1 T 11 + D2 T 12 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ D2 T 22 + D1 T 12 = 0 (5.1) ⎪ ⎪ ⎪ ⎪ ⎩ −b22 (y 1 , y 2 )T 22 = f 3 Moreover, to simplify the problem, the analytical calculations will be performed in this chapter in the particular case of a cylindrical shell. In that case, considering the appropriate system of coordinates, the equations reduce considerably and a complete analytical calculation of the singularities of the displacements (resulting from a singularity of the loading) is possible. Of course, the results obtained for the singularity orders are still valid for general parabolic (developable) surface, only the expressions of the coefficients of the singularities would differ. 1

We recall that for a parabolic surface, one of the two main curvatures is equal to zero.

5.2 Study of the Singularities and of Their Propagation

109

Let us consider in that follows, for the calculation of the singularities and their propagation, a normal loading: f 3 = ψ(y 1 ) ϕ(y 2 )

(5.2)

where ψ(y 1 ) or ϕ(y 2 ) will be singular, according to the case studied. Let us now deduce, from the singular displacement f 3 , the higher order singularities for the displacements u1 , u2 and u3 . To do this, we develop a local study of the singularities inspired by microlocal analysis techniques, as presented in chapter 2. In the calculations, we will only keep the higher order singular terms in the integration of the membrane system (5.1). With the appropriate system of coordinates chosen, all the Christoffel symbols vanish and the membrane system reduces to: ⎧ ∂1 T 11 + ∂2 T 12 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ∂2 T 22 + ∂1 T 12 = 0 (5.3) ⎪ ⎪ ⎪ ⎪ ⎩ −b22 (y 1 , y 2 )T 22 = f 3 The microlocal analysis of the singularities will now be performed near the lines y 1 = const or y 2 = const, where the loading is assumed to be singular. The latter corresponds to a characteristic line. 5.2.1

Singularity along a Characteristic Line

We first consider here the case of a normal loading f 3 singular on a characteristic line y 2 = k of the middle surface S, or equivalently of the membrane system. We will study the system locally near by the characteristic line y 2 = k. More precisely, we consider a normal loading f 3 under the form: f 3 = ψ(y 1 ) ϕ(y 2 − k)

(5.4)

where ϕ(y 2 − k) plays the role of the singularity S0 (x) in (2.73). In figures 5.1(a) and 5.1(b), we present two examples of singular loadings along the characteristic line y 2 = k. In both cases, we have f 3 = ψ(y 1 ) ϕ(y 2 − k) with ψ(y 1 ) = constant. In the first case, we have ϕ(y 2 − k) = H(y 2 − k) and ϕ(y 2 − k) = δ(y 2 − k) in the second one. Now, we deduce the higher order singularity for the displacements. From the third equation of (5.3), near the line y 2 = k, it follows: T 22 = τ 22 (y 1 ) ϕ(y 2 ) with τ 22 (y 1 ) = −

1 ψ(y 1 ) b22 (y 1 , k)

(5.5)

(5.6)

110

5 Singularities of Parabolic Inhibited Shells y1

y1

y2

y2

y2 = k

y2 = k

(a)

(b)

Fig. 5.1. Examples of singular loadings along the characteristic line y 2 = k

Replacing T 22 in the second equation of (5.3), we obtain: ∂1 T 12 = −τ 22 (y 1 )ϕ (y 2 − k)

(5.7)

from which we deduce the leading term of T 12 :

with

T 12 = τ 12 (y 1 ) ϕ (y 2 − k) + . . .

(5.8)

d 12 1 τ (y ) = −τ 22 (y 1 ) dy 1

(5.9)

and where . . . denotes lower order terms. Finally, replacing T 12 in the first equation of (5.1), we obtain the expression of T 11 : T 11 = τ 11 (y 1 ) ϕ (y 2 − k) + . . . (5.10) with

d 11 1 τ (y ) = −τ 12 (y 1 ) dy 1

(5.11)

Remark 5.2.1. According to (5.9) and (5.11), τ 11 , τ 12 and τ 22 are only determined up to two integration constants. They can be determined with the boundary conditions on the stresses themselves (and also those on displacements if necessary), as we shall see on two different examples in section 5.3. We can see that the most singular tension is T 11 which is two orders more singular than ϕ(y 2 − k). To determine the displacements from the components of the stress tensor, we use the inverse constitutive law of (1.53) which writes: γλμ = Bαβλμ T αβ

(5.12)

where Bαβλμ are the membrane compliance coefficients. Then, we get from (5.12) and (1.69):


⎧ 11 1 2 ⎪ ⎪ ∂1 u1 = B1111 τ (y ) ϕ (y ) + . . . ⎪ ⎪ ⎪ ⎨ ∂2 u2 − b22 (y 1 , k)u3 = B2211 τ 11 (y 1 ) ϕ (y 2 ) + . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (∂1 u2 + ∂2 u1 ) = 2B1211 τ 11 (y 1 ) ϕ (y 2 ) + . . . 2

111

(5.13)

Remark 5.2.2. The coefficients B1211 and B1222 of the inverse constitutive law frequently vanish (they depend directly on the chosen coordinate system). In that case, we must consider the second more singular stress T 12 in the third equation of (5.13). The reader can refer to [13] where the calculations are performed in that particular case. However, we will see that this has no consequences on the higher order singularities (when they are along a characteristic line). Indeed, in the third equation of (5.13), ∂2 u1 is always more singular than the right hand side, whether B1211 is null or not. Now, we can deduce u1 from the first equation of (5.13). We then deduce u2 from the third equation where we notice that ∂2 u1 is more singular than the righthand side. Finally, we get u3 from the second equation of (5.13). We obtain the general form of the displacements: ⎧ dU1 ⎪ ⎪ = B1111 τ 11 (y 1 ) ⎧ ⎪ 1 ⎪ 1 2 1 2 dy ⎪ u (y , y ) = U (y ) ϕ (y − k) + ... ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ dU2 u2 (y 1 , y 2 ) = U2 (y 1 ) ϕ(3) (y 2 − k) + ... with = −U1 (y 1 ) ⎪ ⎪ dy 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ u3 (y 1 , y 2 ) = U3 (y 1 ) ϕ(4) (y 2 − k) + ... ⎪ ⎪ 1 ⎪ ⎩ U3 = U2 (y 1 ) b22 (5.14) We obtain ordinary differential equations for Ui which can be uniquely solved on account of the boundary conditions on the displacements. Two aspects of the results must be underlined. On one hand, u1 , u2 and u3 are, respectively, 2, 3 and 4 orders more singular than f 3 (they are proportional to ϕ , ϕ(3) and ϕ(4) ). On the other hand, the singularities propagate along the corresponding generator y 2 = k. Indeed, U1 , U2 and U3 do not vanish when ψ(y 1 ) vanishes. As they are primitives of ψ(y 1 ) with respect to y 1 , they contain smooth terms generally different from 0. This implies that the singular terms in y 2 exist all along the concerned generator y 2 = k even if ψ(y 1 ) vanishes on a part of the generator (see Fig. 5.2). An explicit computation exhibiting the propagation phenomena will be presented in section 5.3. 5.2.2

Singularity along a Non-characteristic Line

We consider now the case when ψ(y 1 ) is somewhat singular along the line y 1 = q. This case corresponds to a singularity of the loading f 3 along a non-characteristic

112


y1

y2 y2 = k propagation of singularities

Fig. 5.2. Propagation of des singularities along the characteristic line y 2 = k

line2 . Note that the results would be similar on every non-characteristic line y 1 = m(y 2 ). Now let us write f 3 under the form: f 3 = ϕ(y 2 )ψ(y 1 − q)

(5.15)

As in the previous section, we shall only consider the more singular terms corresponding to the highest order derivatives of ψ(y 1 − q) with respect to y 1 . From the third equation of (5.1), we get: T 22 = τ 22 (y 2 ) ψ(y 1 − q) with τ 22 (y 2 ) = −

1 ϕ(y 2 ) b22

(5.16)

(5.17)

The tension T 22 is the most singular one because when solving the system (5.1), from the third to the first equation, ψ(y 1 − q) is always integrated but never differentiated. Remark 5.2.3. In general, there is no need to compute the other tensions to determine the most singular term of the displacements, except when B1211 = B1222 = 0. In that case, we need to compute T 12 to get the singularity of u2 (see remark 5.2.2). For a singularity along a non-characteristic line, u2 will be one order less singular (see [13] where the calculations are performed in that particular case). Considering the inverse elastic linear constitutive law γαβ = Bαβλδ T λδ , we obtain the following system: 2

We will see that in this case, the results are the same as those obtained for an elliptic shell which has no real asymptotic line (see chapter 2).


⎧ B1122 2 1 ⎪ ⎪ ⎪ ∂1 u1 = − b (q, y 2 ) ϕ(y ) ψ(y − q) + . . . ⎪ 22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ B2222 ∂2 u2 − b22 u3 = − ϕ(y 2 ) ψ(y 1 − q) + . . . 2 ⎪ b 22 (q, y ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1 B1222 ⎪ ⎩ (∂1 u2 + ∂2 u1 ) = − ϕ(y 2 ) ψ(y 1 − q) + . . . 2 b22 (q, y 2 )

113

(5.18)

where . . . denotes terms bearing a lower singularity in y 1 . The solution of system (5.18) leads to the singularity orders for the displacements, when B1222 is not equal to zero. Considering the first equation of (5.18), the singularity of u1 will be one order lower than the singularity of ψ(y 1 − q): u1 = U1 (y 2 )ψ (−1) (y 1 − q) + . . .

(5.19)

where + . . . denotes lower order terms and where we denote ψ (−1) a primitive dψ (−1) of ψ, i.e. = ψ(y 1 ). Using the third equation of (5.18), we get a similar dy 1 expression for u2 : u2 = U2 (y 2 )ψ (−1) (y 1 − q) + . . . (5.20) where ψ (−1) (y 1 −q) is a first order primitive of ψ(y 1 −q) and where + . . . denotes lower order terms. Finally, using the second equation of (5.18), we obtain: u3 = U3 (y 2 )ψ(y 1 − q) + . . . The factors Ui are respectively: ⎧ B1122 ⎪ ⎪ U1 (y 2 ) = − ϕ(y 2 ) ⎪ ⎪ b22 (q, y 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2B1222 U2 (y 2 ) = − ϕ(y 2 ) ⎪ b22 (q, y 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B2222 ⎪ ⎩ U3 (y 2 ) = ϕ(y 2 ) (b22 (q, y 2 ))2

(5.21)

(5.22)

In the case of a singularity of the loading along a non-characteristic line, we observe that u3 has the same kind of singularity as f 3 , whereas u1 and u2 are one order less singular than f 3 . However, u3 = 0 (and so are u1 and u2 ) only when ϕ(y 2 ) = 0, i.e. when f 3 = 0. In other words, the higher order terms of the singularities of the displacements u1 , u2 and u3 are obtained without integration along the singular loading line (here y 1 = q) and without using the boundary conditions. Therefore, they vanish when ϕ(y 2 ) vanishes, that is the case outside of the loading domain. That proves

114

5 Singularities of Parabolic Inhibited Shells no propagation of singularities

y1

y2 y1 = q

propagation of singularities

Fig. 5.3. No propagation of singularities along the non-characteristic line y 1 = q

that, in the case of a singularity of the loading across a non-characteristic line, there is no propagation of singularity (see Fig. 5.3). Note that in the case of a parabolic shell, with an appropriate choice of the coordinates system3 , we succeeded in integrating directly the membrane system. One can verify that the singularity orders of the displacements correspond to those obtained differently in chapter 2.

5.3

Example of a Half-Cylinder

In this section, we will particularize the results obtained for the singularities of the displacements in the particular case of a half-cylinder of revolution. In that case, according to the loading considered, we will see that the singularities existing on non-characteristic lines are negligible with respect to those along characteristic lines. The singularities on non-characteristic lines will be addressed separately in section 5.6. 5.3.1

Geometric Description of the Cylinder

Let us consider the half-cylinder of figure * 5.4 whose middle surface + S is defined by the local mapping (Ω, Ψ ) with Ω = (y 1 , y 2 ) [0, L] × [0, Rπ] and: Ψ (y 1 , y 2 ) =

"

" R cos

y2 R

#

, y 1 , R sin

"

y2 R

## (5.23)

The constants R and L denote, respectively, the radius and the length of the cylinder. We shall take in the sequel L = 4R with R = 25mm. With the mapping (5.23) considered, the tangent plane to the surface S is defined at each point p = Ψ (y 1 , y 2 ) by the two tangent unitary vectors: 3

Where the coordinates curves correspond to the asymptotic curves of the middle surface.

5.3 Example of a Half-Cylinder

" " 2# " 2 ## y y a1 = 0, 1, 0 and a2 = − sin , 0, cos R R a1 ∧ a2 or equivalently: | a1 ∧ a2 | " " 2# " 2 ## y y N = − cos , 0, − sin R R

115

(5.24)

The normal vector is N =

(5.25)

In the considered coordinates, the local covariant basis (a1 , a2 , N ) is orthonormal and the metric tensor is given by aαβ = δαβ where δαβ denotes the Kronecker symbol. Therefore, the contravariant basis and the covariant one coincide. Using (1.12) and (1.14), the components of the curvature tensor reduce to: ⎛ ⎞ 0 0 ⎠ bαβ = bβα = ⎝ (5.26) 1 0 R One can notice that det(bαβ ) = 0 which proves that the surface S is parabolic at each point. Moreover, as the asymptotic directions cancel the second fundamental form, which reduces here to b22 (dy 2 )2 , they are given by: dy 2 = 0 ⇐⇒ y 2 = const

(5.27)

Thus, the asymptotic lines of the middle surface are the lines y 2 = const which correspond to the generators of the cylinder. Finally, according to the parametrization chosen, all the Christoffel symbols λ vanish (Γαβ = 0). Therefore, we have trivially: ∂α aβ = ∂α aβ = bαβ a3 = bβα a3 and the covariant derivatives become the classical ones. N

a1 a2 e3 O

p

e2

L

e1

2R Fig. 5.4. Surface S

(5.28)

116

5.3.2


Constitutive Law

As the local covariant basis is orthonormal, the constitutive law (1.52)-(1.53) reduces to: ⎛ ⎞ 1 ν 0 ⎛ ⎛ 11 ⎞ ⎞ ⎜ 1 − ν 1 − ν ⎟ γ11 T ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ 22 ⎟ ⎜ ν ⎟⎜ ⎟ E 1 ⎜T ⎟ = ⎜ ⎟ ⎜ ⎟ (5.29) ⎜ ⎟ (1 + ν) ⎜ 1 − ν 1 − ν 0 ⎟ ⎜ γ22 ⎟ ⎜ ⎟⎝ ⎝ ⎠ ⎠ ⎜ ⎟ ⎝ T 12 1 ⎠ γ12 0 0 2 The inverse constitutive law involving the compliance tensor Bαβλμ writes: ⎛ ⎞ ⎛ ⎞ ⎛ 11 ⎞ γ11 1 −ν 0 T ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 22 ⎟ ⎜ γ22 ⎟ = 1 ⎜ −ν 1 ⎟ ⎜ ⎟ 0 (5.30) ⎜ ⎟ E ⎜ ⎟⎜T ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ γ12 0 0 2(1 + ν) T 12 For the local coordinates and mapping considered in this example, we have B1211 = B1222 = 0. Therefore, according to remark 5.2.3, u2 will be one order less singular than u1 , itself one order less singular than f 3 , when the singularity of the loading is along a non-characteristic line. 5.3.3

Loading and Boundary Conditions

Let us define the loading and the boundary conditions in the plane of parameters (y 1 , y 2 ) ∈ R2 (Fig. 5.5). The shell is clamped on the parts of the boundaries OA and CD, subjected to a constant normal loading f 3 = −1 applied in the hatched circular zone. Thus, the loading domain corresponds to the disk centered at (a, 2l ), with radius 4l (see Fig. 5.5). This example, although simple, is interesting because the loading induces singularities in all the directions. However, we will observe, as predicted by the theory, that only the singularities in the direction tangent to the characteristic propagate. Expression of the normal loading f 3 around the characteristic lines y 2 = 4l and y 2 = 3l 4 To deduce the singularity orders of the displacements from the singularities of the loading, we must first express f 3 as f 3 = ψ(y 1 )ϕ(y 2 ) in a neighborhood of the lines y 2 = 4l and y 2 = 3l 4 , i.e. the characteristics tangent to the support of the loading. To do this, let us consider f 3 as a distribution with respect to the

5.3 Example of a Half-Cylinder

117

y2 l

A

3l 4

B

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

l 2 l 4 O

D a−s

0

C

a

a+s

L

y1

Fig. 5.5. System of coordinates and boundary conditions

variable y 2 written: ⎧ ⎪ ⎪ F or ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ F or ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ F or

∈ [0, l] whose values belong to D (0, L)y1 . This distribution can be 0 < y2
0 3l 4 4 − 4 and

3l 4 + 3l 4 −

#−1/2 " # 1 " 3l 3l 2 2 −y H − y dy 2 < ∞, ∀ > 0 − 2 4 4

(5.55)

124


However, this does not stand for the next derivatives of ϕ. Indeed, the functions " #− 32 " # 1 3l 3l 2 2 2 ϕ (y ) = − −y H −y 4 4 4 and

" #− 52 " # 3 3l 3l 2 ϕ (y ) = − −y H − y2 8 4 4 are not locally integrable and cannot be associated to distributions. Therefore, the derivatives of ϕ must be considered in the sense of distributions as follows: " " ## − 12 d 1 3l 3l ϕ (y 2 ) = 2 − − y2 H − y2 (5.56) dy 2 4 4 " # 3l 3l such that, for any test function θ D − , + : 4 4 / 0 / 0 " # − 12 − 12 dd 1 3l 3l 1 3l 3l 2 2 2 2 − −y H( − y ) , θ = −y H( − y ), θ dy 1 2 4 4 2 4 4 3l4 − 12 1 3l = − y2 θ dy 2 (5.57) 3l 2 4 4 − (3)

2

and so on for the next derivatives. The problem is similar for the study of the singularity along the characteristic line y 2 = 4l . In this case, we have ϕ(y 2 ) = , " # l l 2 2 y − H y − . 4 4

5.4

Numerical Simulations with Anisotropic Adaptive Mesh

In this section, we perform a numerical resolution of Koiter model for small thicknesses, using the software BAMG and MODULEF coupled together as explained in chapter 4. The anisotropic and adaptive mesh procedure will be performed with respect to the more singular displacement uε3 . In the numerical simulation that follows, we consider that the loading is centered at the middle of the plane of the parameters (see Fig. 5.15), so that we L have a = . 2 Moreover, the values of the physical parameters are the following: • • • • • •

L = 100 mm l = 25π mm a = 50 mm E = 210, 000 M P a ν = 0.3 fˆ3 = 10ε M P a so that f 3 = 10 M P a

In the sequel, all the results for the displacements will be given in mm.

5.4 Numerical Simulations with Anisotropic Adaptive Mesh

125

y2

l

A

B

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

3l 4 l 2 l 4

O 0

C y 2 = cste

D a−s

a

a+s

L

y1

Fig. 5.15. Centered loading

5.4.1

Remark for the Interpretation of the Numerical Results in Terms of Singularities

In order to interpret the numerical results of simulations that will be performed in the sequel, we recall that the Dirac distribution δ corresponds to the limit of a function having a support length equal to η and an amplitude equal to 1/η when η tends towards zero. The distribution δ being the derivative of δ, it has the same support of length η, an amplitude 1/η 2 and one more oscillation. That characterizes the family of singularities δ, δ , δ . . . (see Fig. 5.16 )

1 η2

1 η

1 η3

η

η

δ

δ Fig. 5.16. Heuristic patterns of δ singularity family

η

δ

126

5.4.2


Convergence of the Adaptive Mesh Procedure

The convergence of the iteration process for the mesh and the computed uε3 is represented in figures 5.17 to 5.21. In figures 5.17 to 5.20, we can see the evolution of the mesh during the adaptation for a fixed ε = 10−5 . During the adaptation, the mesh is clearly refined in zones around both internal layers. We observe that the two singularities of the loading tangent to a characteristic line propagated all along this line. The effect of the boundary conditions on the mesh refinement is clearly revealed. The refined zone is larger 1 for y 2 = 3l 4 because of the free boundary at y = L which induces larger displacements near the free edge. The number of degrees of freedom is multiplied by 6 during the refining process. However, the final number of elements or equivalently of degrees of freedom is small for the very small thickness considered (ε = 10−5 ) and the accuracy of the results. Figure 5.21 shows the evolution of uε3 on the line y 1 = L4 during the mesh adaptation. We observe that uε3 evolves during the adaptation and stays almost

Fig. 5.17. Initial mesh (11275 DOF)

Fig. 5.18. Mesh at the 2nd iteration (23226 DOF)

Fig. 5.19. Mesh at the 5th iteration Fig. 5.20. Mesh at the 7th iteration (66778 DOF) (66283 DOF)


127

0.6

0.4

0 1 2 3 4 5 6 7

u3

0.2

0 0

10

20

30

40

50

60

70

80

90

-0.2

-0.4

-0.6

y2

Fig. 5.21. Convergence of uε3 during the mesh adaptation

constant from the 6th iteration. The adaptation process is quite fast. We notice that the maximum of uε3 at the 7th iteration is twice larger than at the 2nd iteration. That proves the necessity of an adaptive mesh which prevents from an important error. 5.4.3

Computing the Displacements

Let us first observe the deformed shape of the shell (Fig. 5.22). The results are not symmetrical because of the non-symmetrical boundary conditions: the displacements near the free edge are much larger than those near the fixed edge so that the last ones are nearly invisible. Figure 5.23 shows the displacement uε3 in the plane of parameters of the mapping. We clearly observe oscillations which propagate all along the internal layers. These oscillations are also visible in figures 5.24 to 5.26, where the three displacements are plotted on the line y 1 = L4 (with L = 100 mm) for a fixed relative thickness ε = 10−5 . We can remark that the singularities only propagate when they are tangent to a characteristic line. We will not study the singularities 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5

0 20 40 60 80 100 0

10

20

30

40

50

60

70

80

Fig. 5.22. Deformed shape of the shell for Fig. 5.23. Displacement uε3 in the plane of parameters for ε = 10−5 ε = 10−5

128

5 Singularities of Parabolic Inhibited Shells 0.06

0.006

0.05 0.004

0.04 0.002

0.03 0

0.02 10

20

30

40

50

60

70

80

u2

u1

0 -0.002

0.01 0

-0.004

0

10

20

30

40

50

60

70

80

-0.01 -0.006

-0.02 -0.008

-0.03 -0.04

-0.01

2

y2

y

Fig. 5.24. Displacement y 1 = L4 for ε = 10−5

uε1

on the line Fig. 5.25. Displacement uε2 on the line y 1 = L4 for ε = 10−5

0.6

0.4

u3

0.2

0 0

10

20

30

40

50

60

70

80

-0.2

-0.4

-0.6

y2

Fig. 5.26. Displacement uε3 on the line y 1 =

L 4

for ε = 10−5

along non-characteristic lines in this section because they are hardly visible (they are hidden by the much larger propagated singularities along both lines y 2 = 4l and y 2 = 3l 4 with l = 25π). They will be addressed separately in section 5.6. The oscillations visible around y 2 = l/4 and y 2 = 3l/4 correspond to the singularities that appear at the limit for ε = 0. However, as the thickness layer η = O(ε1/4 ) is rather large for parabolic problems4 , even for small values of ε, we can only observe the natural trend of the convergence process without attempting the convergence, which enables to interpret the global pattern of the deformation and the order of the singularities. These oscillations are larger for y 2 = 3l/4 than 1 for y 2 = l/4, because of the free edge boundary BC corresponding to y =εL and l 2 y ∈ 2 , l (see Fig. 5.15). Moreover, we observe more oscillations for u3 than for uε2 and uε1 in Figs. 5.24 to 5.26. As the number of oscillations increases with the order of the singularity, the results obtained are in good agreement with the theoretical developments of section 5.3. Using the properties of the Dirac family (see Fig. 5.16 of section 5.4.1) and especially the relations with η (the η of Fig. 5.16 and the layer thickness are confused), we can deduce some specificities of the three displacements. Indeed, we saw in section 5.3 that for the loading considered, the singularities of f 3 are between xH(x) and H(x). Therefore, the singularities in y 2 of the limit displacements u1 , u2 , u3 are, respectively, between δ and δ , δ and δ and δ 4

For layers along characteristic lines.


129

and δ . Therefore, uε1 should have between one and two oscillations5 , uε2 between two and three oscillations, and so on for uε3 (see Fig. 5.16 for comparison). On the other hand, even if it is not easy to determine accurately which oscillations are significant, we can observe that the relations between the various displacements exhibited in sections 5.2 and 5.3 are satisfied. In particular, we can see that uε3 has more oscillations than uε2 and uε2 more than uε1 . In fact, uε2 appears to be the derivative of uε1 up to multiplicative factor: uε1 vanishes wherever uε2 reaches an extremum. We observe the same relation between uε2 and uε3 . This is in good agreement with (5.45)-(5.47). Influence of the Relative Thickness ε

5.4.4

In this section, we focus on the asymptotic process of the Koiter model when ε tends to zero. We perform several numerical computations for decreasing values of ε, and plot the corresponding displacements and energies. We recall that for the inhibited shell considered (parabolic and clamped on the whole determination domain of the generators), the Koiter model converges to the membrane problem when ε tends to zero. This result should be observed numerically. Bending displacements and internal layer thickness Let us first consider the variations of the normal displacement uε3 with respect to ε inside the layers. The amplitude of uε3 will enable us to determine the thickness of the internal layers corresponding to y 2 = 4l and y 2 = 3l 4 . Figures 5.27 to 5.30 present the bending displacement uε3 for various values of ε. 0.1

0.02

0.08 0.015

0.06 0.01

0.04

0.005

u3

u3

0.02

0

0 0

10

20

30

40

50

60

70

0

80

10

20

30

40

50

60

70

80

-0.02 -0.005

-0.04 -0.01

-0.06

-0.08

-0.015

2

2

y

y

Fig. 5.27. uε3 on the line y 1 = 10−3

L 4

for ε = Fig. 5.28. uε3 on the line y 1 = 10−4

L 4

for ε =

We observe that the maximum of uε3 increases when ε decreases. Moreover, the two zones affected by the larger normal displacements (i.e. the two internal layers) decrease, from nearly the whole domain for ε = 10−3 , to only two small zones around the two generators tangent to the loading domain for ε = 10−6 . Thus, the bending effects concentrate in the layers which become themselves 5

Inside the layer and especially for y2 = visible.

3l 4

= 58, 9 where the singularities are more

130


0.6

4

3

0.4

2 0.2

u3

u3

1 0 0

10

20

30

40

50

60

70

80

0 0

10

20

30

40

50

60

70

80

-0.2 -1

-0.4

-2

-0.6

-3 2

2

y

y

Fig. 5.29. uε3 on the line y 1 = 10−5

L 4

for ε = Fig. 5.30. uε3 on the line y 1 = 10−6

L 4

for ε =

1.2

0.6

1

0.4

0.8

y = 0.2492x + 1.9571 R2 = 0.9997

log(η)

u3

0.2

0 0

10

20

30

40

50

60

70

0.6

80

0.4 -0.2

0.2 -0.4

0

η

-0.6

-6.5

Fig. 5.31. Measure of η on uε3 for y 1 =

-6

-5.5

-5

-4.5

-4

-3.5

-3

log(ε)

y2

L 4

Fig. 5.32. Determination of the layer thickness η for y 1 = L4

thinner when ε 0. All these observations are in good agreement with the theoretical analysis developed previously. Now, let us determine the internal layer thickness from the numerical results obtained for uε3 . To do this, we define the distance η between the two highest extrema of uε3 around y 2 = 3l , as represented in figure 5.31. Other distances 4 between different extrema would give similar results. For each value of ε, we measure a corresponding value of η and we plot log(η) with respect to log(ε) in figure 5.32. This way we find that η = O(ε0.2492 ). This result is very close to the classical theoretical result η = O(ε1/4 ) (see section 2.8 of chapter 2). Singularity orders of the displacements From the results of section 5.4.3 and knowing the relation between η and ε, we can deduce the order of magnitude of the amplitude of the three displacements with respect to ε. Indeed, according to the theoretical analysis of section 5.3, the three displacements have the following singularities with respect to y 2 : • u1 is between δ and δ . Its amplitude varies like η−α1 , with 1 < α1 < 2 • u2 is between δ and δ . Its amplitude varies like η−α2 , with 2 < α2 < 3 • u3 is between δ and δ . Its amplitude varies like η −α3 , with 3 < α3 < 4. Considering that η = O(ε1/4 ), which has been verified numerically in section 5.4.4, we finally have:


131

• u1 amplitude varies like ε−λ1 , with 1/4 < λ1 < 1/2 • u2 amplitude varies like ε−λ2 , with 1/2 < λ2 < 3/4 • u3 amplitude varies like ε−λ3 , with 3/4 < λ3 < 1. Now, to compare the numerical results to the theoretical predictions, we measure the maximum of uε3 for y 1 = L4 obtained from numerical computations for several values of ε (we do the same for the other displacements). We obtain the following results (all with a R-squared6 larger than 0.9996): • uε1 amplitude varies like ε−0.3758 • uε2 amplitude varies like ε−0.5584 • uε3 amplitude varies like ε−0.8021 . All the results are in good agreement with the orders of singularities predicted by the theory for the displacements. 5.4.5

Localization of Membrane and Bending Energies

Let us now observe the repartition of membrane and bending energy surface densities (respectively denoted Ems and Ebs ) in Figs. 5.33 and 5.34. These computations have been performed for the relative thickness ε = 10−5 and with a new routine implemented in MODULEF [11, 12].

Fig. 5.33. Repartition of Ems for ε = 10−5 Fig. 5.34. Repartition of Ebs for ε = 10−5

The main part of both energies is located along the two internal layers around y 2 = 4l and y 2 = 3l with l = 25π = 78.53 mm for the considered example. 4 Moreover, this repartition is influenced by the boundary conditions. Indeed, for y 2 = 3l , there is more bending energy near the free edge in y 1 = L than near the 4 fixed edge in y 1 = 0 where the membrane energy is predominant. On the other hand, in y 2 = 4l , the repartition of both energies is symmetrical with respect to the line y 1 = L2 , with L = 100 mm. This is due to the symmetrical boundary 6

The (R-squared) coefficient R2 equals the square of the correlation coefficient between the observed data values (obtained from numerical simulations) and the modelled (predicted) ones (obtained by linear regression).

132


conditions on the generator y 2 = 4l . Finally, for y 1 = L4 we observe precisely three zones with a higher bending energy surface density around y 2 = 4l . They correspond to the three main oscillations observed for uε3 around y 2 = 4l in figure 5.26. Between each of these zones, we observe a weaker bending density energy which corresponds to an inflexion point of uε3 . The same phenomenon is observed around y 2 = 3l 4. Evolution of the part of bending energy during the singular perturbation process To finish, let us observe the layer thicknesses from the energy surface densities (Figs. 5.35 and 5.36). We can observe that both internal layers (at y 2 = 4l and y 2 = 3l ) become thinner when ε tends to zero, as predicted by the 4 theory. However, as there are several layers in this example, it is not easy to consider them precisely and separately, especially for rather large ε (10−3 or 10−4 ). When ε 0, we can observe that the bending energy concentrates not only in the two internal layers around y 2 = l/4 and y 2 = 3l/4 but also in the two boundary layers tangent to the characteristic lines at y 2 = 0 and y 2 = l.

Fig. 5.35. Evolution of the percentage of Ebs for ε = 10−3 and 10−4

Fig. 5.36. Evolution of the percentage of Ebs for ε = 10−5 and 10−6

5.5 Comparison between Uniform and Adapted Meshes

5.5

133

Comparison between Uniform and Adapted Meshes

In this section, we present some comparisons between uniform and adapted meshes to illustrate the necessity (and the efficiency) of adaptive meshes in such configurations. The ratio of the maximum displacement uε3max =

L sup uε3 ( , y 2 ) 4 y 2 ∈[0,L]

to uε3ref is plotted in figures 5.37 to 5.39 for various thicknesses versus the numbers of elements. The reference displacement uε3ref is the maximum one obtained at the last iteration of the adaptive process thought it may not be the exact solution. We can see that an anisotropic adapted mesh is more efficient than a uniform one, especially for small values of the relative thickness ε. The results converge to the reference solution with less elements. This trend becomes all the more significant when the thickness ε tends towards zero. For ε = 10−3 , the performances of an adapted mesh and of a uniform mesh are similar. Indeed, even if an adapted mesh requires several iterations for the adaptation that increases the 1.12 1.1 1.08 1.06

uniforme adapté

u3max/u3ref

1.04 1.02 1

0.98 0.96 0.94 0.92 0

2000

4000

6000

8000

10000

12000

14000

16000

nombre d'éléments

Fig. 5.37. Scaled displacement uε3 versus the number of elements for ε = 10−3 1.2

1

u3max/u3ref

0.8

uniforme adapté

0.6

0.4

0.2

0 0

2000

4000

6000

8000

10000

12000

14000

16000

nombre d'éléments

Fig. 5.38. Scaled displacement uε3 versus the number of elements for ε = 10−4

134


1

uniform adapted (isotropic) adapted (anisotropic)

u3max/u3ref

0.8

0.6

0.4

0.2

0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

Number of elements

Fig. 5.39. Scaled displacement uε3 versus the number of elements for ε = 10−5

time of computation, it requires less elements than a uniform mesh for the same accuracy. Oppositely, for ε = 10−4 , an anisotropic adapted mesh gives much better results than a uniform one. For ε = 10−4 , in figure 5.38, we can see that an adapted mesh of 6000 elements gives the same results as a uniform mesh of 14000 elements. Thus, the time of computation with an adapted mesh is strongly reduced and the convergence is much better (the uniform mesh did not converge with 14000 elements). For ε = 10−5 , we compare three different meshes : the classical uniformed mesh (non-adapted), the adapted isotropic mesh, and the adapted anisotropic mesh. We observe that the solution uε3 given by a uniform mesh does not converge at all, whereas the convergence is quite fast with an anisotropic adapted mesh. This is due to an important locking [82] for this value of ε, which is considerably reduced with the use of an adapted anisotropic mesh. Finally, the results obtained with an adapted isotropic mesh are intermediate between a uniform mesh and an adapted anisotropic mesh. However, even with 14000 elements, an adapted isotropic mesh does not converge, because of the locking which is present in the layers for very thin shells. The refinement of the mesh isotropically, even only inside the layer (the mesh is adapted), is not sufficient to insure the convergence. We must use an adapted anisotropic mesh to insure the convergence for very small thicknesses. Let us notice that we find again the theoretical results of chapter 3 based on the error estimates. With an adaptive isotropic mesh, the number of element is divided by O(η −1 ) compared to a classical uniform mesh, and once again by O(η −1 ) with an adaptive and anisotropic mesh, η being the layer thickness (η = O(ε1/4 ) for the parabolic shell considered). Finally, for rather small values of ε, even if we consider the total time of computation7 (including the time of each iteration), it is clear that the anisotropic adapted mesh is much more efficient than a uniform one (see Fig. 5.40 for ε = 10−4 ). Therefore, for small values of ε (smaller than 10−4 ), the use of an adaptive anisotropic mesh is necessary to ensure the convergence with a 7

The computations have been performed on a Pentium IV 3GHz with 1Gb of RAM.

5.6 Numerical Study of Singularities on Non-characteristic Lines

135

1.2

1

u3max/u3ref

0.8

uniform adapted

0.6

0.4

0.2

0 0

100

200

300

400

500

600

time of computation (seconds)

Fig. 5.40. Maximum of the scaled displacement uε3 versus the time of computation for ε = 10−4

reasonable time of computation (the uniform and adapted isotropic meshes do not converge for ε = 10−5 ).

5.6

Numerical Study of Singularities on Non-characteristic Lines

In the last example, the singularities existing on non-characteristic lines are negligible with respect to the other ones. In this section, we will consider another simple example which enables to observe them precisely. To avoid singularities along characteristic lines, we consider a full cylinder8 clamped all along its boundary and subjected to a normal loading f 3 on the hatched area (Fig. 5.41). Due to axisymmetric geometry and loading, the results are independent of y 2 : there are only two layers on two non-characteristic lines y 1 = a and y 1 = b, with a = 30 and b = 70 in the considered example. As f 3 = H(y 1 − a) − H(y 1 − b), considering the theoretical results of section 5.2.2, we can deduce the singularity orders of the limit displacements (solution of the membrane model): • u1 should have singularities of the form (y 1 − a)H(y 1 − a) − (y 1 − b)H(y 1 − b) • u3 should have singularities of the form H(y 1 − a) − H(y 1 − b) Because of the symmetry of the problem (it only depends on y 1 ), the displacement u2 vanishes everywhere, that is observed numerically. Indeed, for the cylinder and the parametrization considered, we have B1222 = 0 and we are in the particular case of remark 5.2.3. In that case, we need to compute T 12 to get the singularity of u2 . The result obtained for U2 (y 2 ) is then a little different from (5.22): it involves ϕ (y 2 ) instead of ϕ(y 2 ) (see expression (41) of 8

If we consider the same half-cylinder, there are two boundary layers at y 2 = 0 and y 2 = l which contain the largest displacements when ε is small enough (the displacements would have the same order as in an internal layer, see [61]), then hiding the layers along non-characteristic lines.

136

5 Singularities of Parabolic Inhibited Shells y2 l

A

B

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

O 0

a

D

b

y1

L

Fig. 5.41. 2D domain of the middle surface definition

[13]). According to the symmetry of the problem, ϕ is independent of y 2 and u2 vanishes at the highest order singularity. If we computed the lower order singularities, we would find that they are proportional to ϕ(n) (y 2 ), with n ≥ 1, and vanish as well. That explained why u2 is exactly equal to zero. The displacements uε1 and uε3 on a line y 2 = const, given by a numerical resolution of Koiter model for ε = 10−4 , are represented in Figs. 5.42 and 5.43. The results obtained numerically are those expected by the theory. In particular, the displacement uε3 , and the derivative of uε1 with respect to y 1 , have two discontinuities in y 1 = a and y 1 = b. 5.E-05

5.0E-05

4.E-05

0.0E+00 0

3.E-05

10

20

30

40

50

60

70

80

90

100

-5.0E-05 2.E-05 -1.0E-04

u3

u1

1.E-05 0.E+00 0

10

20

30

40

50

60

70

80

90

-1.5E-04

100

-1.E-05

-2.0E-04

-2.E-05 -2.5E-04 -3.E-05 -3.0E-04

-4.E-05 -5.E-05

-3.5E-04

y1

Fig. 5.42.

5.7

uε1

2

y1

−4

for y fixed for ε = 10

Fig. 5.43.

uε3

2

for y fixed for ε = 10−4

Singularity along a Boundary

We address in this section a more complex case where the singularities of the loading are tangent to the boundary of the shell. Very few theoretical results are known in that case, so that the numerical simulations performed with adaptive mesh give precious information on the structure of the singularity behavior near the boundary.

5.7 Singularity along a Boundary

5.7.1

137

Theoretical Considerations

Let us consider the same problem as previously in section 5.4. Only the plane l 3l of the parameters in R2 is reduced to Ω = [0, L] × [ , ] such that the edges 4 4 OD and AB are tangent to the hatched loading domain (see Fig. 5.44). We will then consider two different boundary conditions leading to different behaviors: • the shell is clamped all along its lateral boundary • the shell is clamped on the boundary [OA] ∪ [BD] and free on the other part [OD] ∪ [AB] y2 l 3l 4

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

A

l 2 l 4

O

0

L 4

a

B

C

D

L

y1

Fig. 5.44. Loading tangent to the boundary

Let us extend the solution (T ij , uk ) of the membrane model in Ω = [0, L] × l 3l [ , ] to zero outside of Ω. So that we will have: 4 4 ⎧ l 3l ⎪ ⎪ ui f or y2 [ , ] ⎪ ⎨ 4 4 u ˜i = (5.58) ⎪ ⎪ l 3l ⎪ 2 ⎩ 0 f or y 2 ≤ and y ≥ 4 4 According to the theoretical results of [95], the solution u˜i of the membrane problem in [0, L]×[0, l] should have the same singularities as previously in section l 3l l 5.4 for y2 ∈ [ , ], and should be equal to zero for y 2 ≤ and y 2 ≥ 3l . 4 4 4 4 The numerical simulations that follow will confirm these qualitative theoretical considerations. 5.7.2

Numerical Simulations

First case: the shell is clamped all along its boundary Let us consider, for the numerical resolution of the Koiter model with BAMG and MODULEF coupled together, the same numerical data as in section 5.4. In

138


0.02

0.015 "Internal" singularity

u2

0.01

Singularity along a fixed boundary

0.005

0 0

10

20

30

40

50

60

70

80

-0.005

-0.01

-0.015 2

y

Fig. 5.45. Displacement uε1 for a loading singularity inside the domain or along the clamped boundary for y 1 = L4 and ε = 10−5 0.004

0.003

0.002

0.001

0

u1

0

10

20

30

40

50

60

70

80

-0.001

-0.002 "Internal" singularity -0.003

Singularity along a fixed boundary

-0.004

-0.005

y2

Fig. 5.46. Displacement uε2 for a loading singularity inside the domain or along the clamped boundary for y 1 = L4 and ε = 10−5

figures 5.45 to 5.47 are represented the displacements on the line y 1 = L4 . The numerical results for the displacements, given in mm for a relative thickness ε = 10−5 , are compared to those obtained in section 5.4 were the loading singularities where inside the domain Ω. l 3l These results agree with the theory in [95]. The singularities for y 2 ∈ [ , ] 4 4 are qualitatively analogous to those of section 5.4, though truncated near the clamped edges. Second case: the shell is free along OD and AB • Numerical computation of the displacements We still consider the same problem as previously, with the edges OD and AB which are now free. The displacements computed numerically for a relative thickness ε = 10−5 are plotted on the line y 1 = L/4 in figures 5.48 to 5.50.


139

0.2

0.15

"Internal" singularity Singularity along a fixed boundary

0.1

u3

0.05

0 0

10

20

30

40

50

60

70

80

-0.05

-0.1

-0.15 2

y

Fig. 5.47. Displacement uε3 for a loading singularity inside the domain or along the clamped boundary for y 1 = L4 and ε = 10−5

0.02

1

0.01

0.8

0.6 0 0

10

20

30

40

50

60

70 0.4

-0.01 0.2 -0.02 0 0

-0.03

10

20

30

40

50

60

70

-0.2 -0.04

-0.4

-0.05

-0.6

-0.8

-0.06

Fig. 5.48. Displacement for a loading Fig. 5.49. Displacement uε2 for a loading singularity tangent to the free boundary singularity tangent to the free boundary for y 1 = L4 and ε = 10−5 for y 1 = L4 and ε = 10−5 uε1

0.1

2

0.05 0 0

10

20

30

40

50

60

0

70

10

20

30

40

50

60

70

-0.05

-2

-6

u3

-0.1 -4

-0.15 -0.2 -0.25

-8

-0.3 -0.35

-10

-0.4 -12

y2

Fig. 5.50. Displacement uε3 for a loading Fig. 5.51. Displacement uε3 for y 1 = L 4 singularity tangent to the free boundary and ε = 5 · 10−4 for y 1 = L4 and ε = 10−5

We do not observe the same result as previously: the displacements are l 3l much larger near the two free edges for y 2 = and y 2 = . Elsewhere, 4 4 ε the respective order of the singularities is respected: u3max is about 10 times

140


larger than uε2max , itself 10 times larger than uε1max . The number of oscillations which characterizes the singularity order is the same as previously, even if the oscillations are flattened because of the large maximum value of uε3max which imposes the scale of the figure. • Determination of the singularity order for uε3 It is possible to determine the singularity order of the displacement uε3 (the more singular one), as in section 5.4.4. 0.4

2

0.2 0 0

10 10

20

30

40

50

60

20

30

40

50

60

70

70

-0.2

-2

-0.4 -4

u3

u3

-0.6 -0.8

-6

-1 -8

-1.2 -1.4

-10 -1.6 -1.8

-12

y2

y2

Fig. 5.52. Displacement uε3 for y 1 = and ε = 10−4

L L Fig. 5.53. Displacement uε3 for y 1 = 4 4 and ε = 10−5

L We plot in figures 5.52 and 5.53 the displacement uε3 on the line y 2 = , for 4 various values of the relative thickness ε. Then, using also the results of Figs. 5.50 and 5.51, we plot the variation of the amplitude of the first oscillation with respect to ε in Fig. 5.54. We find that it varies like ε−0.8388 , that is between the values ε−3/4 and ε−1 , as predicted by the theory for internal layers of section 5.4.4. 0.5

0

-12

-11

-10

-9

-8 -0.5

lnu3

y = -0.8388x - 9.8611 R2 = 0.9971

-1

-1.5

-2

-2.5

-3

-3.5

lnε

Fig. 5.54. Amplitude of ln uε3 versus ln ε


141

• Comparison with singularities inside the domain To finish, let us compare the results for the displacements uε1 , uε2 and uε3 obtained here in the case of a singularity of the loading tangent to the free edges OD and AB, and in section 5.4 for the same loading singularity inside the domain with the free edges y 2 = 0 and y 2 = l. The singularities of the displacements are plotted for ε = 10−5 , in both cases with the same scale (the displacements are in mm) in the same figure (Figs. 5.55 to 5.57). In both cases, the singularities of the displacements are of the same order: we observe the same number of oscillations, even if they are narrower in the case when the loading singularity is tangent to the free edge. In the latter case, the amplitude of the displacements is obviously much larger. 0.01

0.005

0

u1

0

10

20

30

40

50

60

70

80

-0.005

-0.01

"Internal" singularity -0.015

Singularity along a free boundary

-0.02

y2

Fig. 5.55. Displacement uε1 for y 1 = L/4. Cases of a loading singularity inside the l 3l domain and along the free edges y 2 = and y 2 = 4 4 0.1 0.08 0.06 0.04

u2

0.02 0 0

10

20

30

40

50

60

70

80

-0.02 -0.04 -0.06

"Internal" singularity -0.08


-0.1

y

2

Fig. 5.56. Displacement uε2 for y 1 = L/4. Cases of a loading singularity inside the l 3l domain and along the free edges y 2 = and y 2 = 4 4

142

5 Singularities of Parabolic Inhibited Shells 1 0.8 0.6 0.4

u3

0.2 0 0

10

20

30

40

50

60

70

80

-0.2 -0.4

"Internal" singularity -0.6 -0.8 -1


y2

Fig. 5.57. Displacement uε3 for y 1 = L/4. Case of a loading singularity inside the l 3l domain and along the free edges y 2 = and y 2 = 4 4

5.8

Singularities due to the Shape of the Domain

We propose a last example to finish the study of the singularities in the case of parabolic shells, which reveals that even with a very simple uniform loading, the complexity of the resulting singularities is directly linked to the shape of the domain. We will see that in that case, the singularities resulting from the boundary layer CD will propagate inside the domain along the characteristic CF (see Fig. 5.58). Indeed, let us consider a uniform loading f 3 = −1 applied on the whole domain Ω of the mapping represented in figure 5.58. This time, the shape of the mapping domain Ω is more complex. The middle surface of the shell is still defined by the same mapping (5.23): " " 2# " 2 ## y y Ψ (y 1 , y 2 ) = R cos , y 1 , R sin R R The shell is clamped all along its lateral boundary, and then inhibited. Let us study the displacements on the two lines y 1 = a and y 1 = b, with the particular values a = 25 mm and b = 75 mm considered here. We recall that we have L = 100mm and l = 25π 78.5mm. The resulting displacements are plotted in mm for ε = 10−5 in figures 5.59 to 5.61. For y 1 = b with b = 75 mm, we observe a singularity near the two clamped boundaries y 2 = 0 and y 2 = 2l , due to a classical boundary layer studied in section 5.7.2 (but with a different loading). Moreover, we observe two different singularities on the line y 1 = a, with a = 25 mm. At the two extremities y 2 = 0 and y 2 = l, we have boundary layer singularities as on the line y 1 = b. Conversely, the singularity appearing on the middle line y 2 = l/2 39 mm is not classical. It is due indirectly to the boundary layer existing on the clamped boundary CD. Indeed, the singularity first appear along the clamped boundary CD and then propagated along the characteristic line CF . It is important to notice that the shape of this non-classical singularity is different from that due to the boundary layer near CD. This problem is complex

5.8 Singularities due to the Shape of the Domain y2

l

A B

F

C

D

O

E a

0

L

b

y1

Fig. 5.58. Mapping domain Ω considered

6.E-07

4.E-07

y1=25 y1=75

u1

2.E-07

0.E+00 0

10

20

30

40

50

60

70

80

-2.E-07

-4.E-07

-6.E-07

2

y

Fig. 5.59. Displacement uε1 on the lines y 1 = a and y 1 = b for ε = 10−5

1.E-05

8.E-06

6.E-06

y1=25 y1=75

4.E-06

u2

2.E-06

0.E+00 0

10

20

30

40

50

60

70

80

-2.E-06

-4.E-06

-6.E-06

-8.E-06

-1.E-05

y2


143

144

5 Singularities of Parabolic Inhibited Shells 0.E+00 0

10

20

30

40

50

60

70

80

-5.E-05

y1=25 y2=25

-1.E-04

-2.E-04

u3

-2.E-04

-3.E-04

-3.E-04

-4.E-04

-4.E-04

-5.E-04

-5.E-04

y2


(even if the loading is very simple): we have both a boundary layer singularity and a singularity propagation along a characteristic line CF . To our knowledge, there exists no theoretical result for such problems. Therefore, this last example reveals all the complexity of the interaction between the different kinds of singularities. It put in a prominent position all the interest of the adaptive (and anisotropic) remeshing procedure performed to access very accurately to all the kinds of the existing singularities. 5.8.1

Conclusion

In this chapter, we first calculated analytically the singularity orders of the displacements resulting from a singular loading, by a direct integration of the membrane system, in the general case of parabolic shells. This integration is possible for parabolic shells (and for hyperbolic ones also as we will see in chapter 6) if the coordinate lines corresponds to the asymptotic curves. We found again the theoretical results on the singularities obtained in chapter 2 from a reduction of the membrane system to a PDE involving only one component of the displacement. We then focused on the particular case of a half-cylinder of revolution. First we studied accurately, analytically and numerically, a singular loading on a circular domain. We observed that the singularities propagate only along the characteristic lines tangent to the singular loading, leading to two internal layers. The mesh has been refined automatically and anisotropically inside these layers, leading to very accurate numerical results for the displacements even for very small thicknesses, with a reduced number of elements. We recovered the theoretical results obtained in the first part of this chapter and in chapter 2, concerning the singularity orders and the layer thicknesses. Moreover, some comparisons with uniform mesh revealed the necessity to use an adapted mesh to ensure the convergence of the iterative numerical procedure, to get accurate results with a reduced number of elements, and to reduce the locking phenomenon for very small thicknesses.

5.8 Singularities due to the Shape of the Domain

145

The study of the singularities along a non-characteristic line necessitated to consider another singular loading, because the singularities along the characteristic lines hid all the other ones on the first considered singular loading (singular loading on a circular domain). We recovered as well the theoretical results obtained previously. We next focused on the particular case of a loading domain tangent to the boundary of the shell. We observed numerically the qualitative behavior predicted by the theory: the singularities along a boundary which is a characteristic of the middle surface propagate and have the same order than in the case of an internal layer. The oscillations characterizing the singularity order are truncated near the clamped boundary and narrowed near the free boundary. Finally, we performed numerical computations of the singular displacements in the case of a more complex shell geometry, subjected to a very simple uniform loading. We put in a prominent position the existence of non-classical singularities due to boundary layers (near the clamped boundary) which propagate inside the domain along the characteristic lines. This last example, where no analytical solutions are known, revealed the interest and the necessity of an adaptive anisotropic remeshing procedure to access accurately to all the complex singularities existing and not predictable theoretically.

6 Singularities of Hyperbolic Inhibited Shells

6.1

Introduction

This chapter is devoted to hyperbolic shells, whose principal curvatures are of opposite sign. Consequently, at each point of the middle surface, there are two asymptotic directions. Concerning the singularities emerging when ε 0, some aspects are very similar to the case of parabolic shells. For instance, singularities along characteristics are more singular than the loading f 3 (at least for the normal displacement u3 ) and propagate. However, the existence of two distinct families of characteristics implies important differences in the structures of these singularities with respect to parabolic shells. When the loading f 3 is singular along a characteristic, the resulting displacement u3 is “only” two orders more singular than f 3 , whereas it is four order more singular for parabolic shells (see section 2.6.2 of chapter 2). Moreover, some singular loadings (for instance a point force) propagate along both asymptotic directions (which is also different from the parabolic case). An important consequence of the existence of two distinct families of characteristics is the apparition of a pseudo-reflection phenomenon which is described in section 2.7. When a singularity reaches a point of a boundary which is not a characteristic, it “reflects” along the two characteristics starting from this point of the boundary. When considering a real problem, we may have propagation and reflection in the same time. It is then all more difficult to create “manually” an efficient mesh to describe these singularities, as in most of cases, there is no analytical solution predicting a priori the positions of the layers where the mesh needs to be refined. These complex situations, considered in section 6.4, will reveal once again all the interest and the necessity of an anisotropic adaptive mesh procedure to describe accurately the singularities.

6.2

The Limit Problem for a Hyperbolic Inhibited Shell

Taking a special parametrization (y 1 , y 2 ), where the coordinate lines correspond to the asymptotic curves of the middle surface, the covariant components bαβ E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 147–170. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

148


of the second fundamental form reduce to b11 = b22 = 0 and b12 = 0. In what follows, we still consider only a normal loading f 3 (with f 1 = f 2 = 0), giving the most singular displacements. Thus, the membrane system (1.65) reduces to: ⎧ D1 T 11 + D2 T 12 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ D2 T 22 + D1 T 12 = 0 (6.1) ⎪ ⎪ ⎪ ⎪ ⎩ −2b12 T 12 = f 3 In the sequel, this system will be used to determine the higher order term of singularities of the three displacements (solutions of the limit problem) for various loadings. Numerical simulations using FE method and an adaptive anisotropic mesh procedure will confirm this result when ε 0 (see also [63, 40]). 6.2.1

Example of a Hyperbolic Paraboloid

In what follows, we consider the case of a shell whose middle surface S is a hyperbolic paraboloid (see Fig. 6.1). The surface S is defined by the mapping (Ω, Ψ ) with * + Ω = (y 1 , y 2 ) ∈ R2 , (y 1 , y 2 ) ∈ [−L, L]2 (6.2) and

" # 1 2 1 2 y y Ψ (y , y ) = y , y , c 1

2

(6.3)

We will consider the values L = 50 mm and c = 250 mm for the numerical computations. Moreover, the material considered is isotropic and homogeneous, with a Young modulus E = 28, 500 MPa and a Poisson ratio ν = 0.4. The specific parametrization (6.3) corresponds to that of the asymptotic lines. Indeed, the mapping (Ω, Ψ ) implies the following covariant coefficients of the first and second fundamental forms: ⎛ ⎞ (y 2 )2 y1 y 2 1 + ⎜ ⎟ c2 c2 ⎜ ⎟ aαβ = ⎜ (6.4) ⎟ ⎝ 1 2 1 2 ⎠ y y (y ) 1+ 2 c2 c b11 = b22 = 0

1 b12 = b21 = . 2 1 c + (y )2 + (y 2 )2

(6.5)

The only non-vanishing Christoffel symbols reduce to: 1 Γ12 =

y2 c2

+

(y 1 )2

+ (y 2 )2

(6.6)

6.2 The Limit Problem for a Hyperbolic Inhibited Shell

149

y3 10

0

-10

-50

50 y1

0

0 50

y2

-50

Fig. 6.1. The hyperbolic paraboloid

The considered middle surface is uniformly hyperbolic and the asymptotic directions satisfy 2b12 (dy 1 )(dy 2 ) = 0 (6.7) which gives, as b12 > 0, two distinct families of asymptotic curves: y 1 = const

and

y 2 = const

(6.8)

The asymptotic curves play an important role as they correspond to the characteristic lines of the membrane and rigidity systems (see section 1.7.2 of chapter 1). We will see that if the loading is somewhat singular along an asymptotic line, the resulting displacements will be singular on the whole asymptotic line. In what follows, we shall call these lines indifferently “asymptotic” or “characteristic” lines. We now consider the case of a normal loading f 3 singular along a characteristic line, and we shall exhibit the leading order singularities and their propagative character. The case of singularity along non-characteristic lines is similar to that of parabolic shells which were considered in chapter 5. 6.2.2

Singularities of the Displacements due to a Loading Singular on the Line y 1 = 0

Let f 3 be singular on a characteristic line y 1 = 0. We shall rewrite it under the form: f 3 = ψ(y 1 ) ϕ(y 2 ) (6.9) where ψ(y 1 ) plays the role of S0 (see (2.73)) and ϕ(y 2 ) denotes its weight along y 1 = 0. Let us study the system (6.1) near y 1 = 0. It becomes: ⎧ 11 12 ⎪ ⎪ D1 T + D2 T = 0 ⎪ ⎪ ⎨ D2 T 22 + D1 T 12 = 0 (6.10) ⎪ ⎪ ⎪ ⎪ ⎩ −2b12 (0, y 2 )T 12 = f 3

150


From the third equation of (6.10), we obtain T 12 = τ 12 (y 2 ) ψ(y 1 ) with τ 12 (y 2 ) = −

1 ϕ(y 2 ) 2b12 (0, y 2 )

(6.11)

(6.12)

Replacing T 12 in the second equation of (6.10), we obtain: 1 ∂2 T 22 + Γ12 (0, y 2 )T 22 = −τ 12 (y 2 )ψ (y 1 ) + . . .

(6.13)

γ where Γαβ are the Christoffel symbols and . . . denotes lower order terms. We then deduce the leading term of T 22 :

with

T 22 = τ 22 (y 2 ) ψ (y 1 ) + . . .

(6.14)

1 22 2 ∂2 τ 22 (y 2 ) + Γ12 τ (y ) = −τ 12 (y 2 )

(6.15)

and where . . . denotes lower order terms. Finally, replacing T 12 in the first equation of (6.10), we obtain the expression of T 11 : T 11 = τ 11 (y 2 ) ψ (−1) (y 1 ) + . . . (6.16) with

τ 11 (y 2 ) = −∂2 τ 12 (y 2 ) (−1)

(−1)

1

(6.17)

1

where ψ is such that ∂1 ψ (y ) = ψ(y ). The most singular tension is T 22 which is one order more singular than ψ(y 1 ). Using the inverse constitutive law (5.12), we obtain the system ⎧ D1 u1 = B1122 τ 22 (y 2 ) ψ (y 1 ) + . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ D2 u2 = B2222 τ 22 (y 2 ) ψ (y 1 ) + . . . (6.18) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (D1 u2 + D2 u1 ) − 2b12 u3 = 2B1222 τ 22 (y 2 ) ψ (y 1 ) + . . . 2 We deduce u1 and u2 , respectively, from the first and the second equations of (6.18). Finally, we get u3 from the third equation of (6.18) where ∂1 u2 is more singular that the right hand side. We obtain the general form of the displacements (at the leading order of singularity) in the vicinity of the line y 1 = 0: ⎧ u1 (y 1 , y 2 ) = U1 (y 2 ) ψ(y 1 ) + ... ⎪ ⎪ ⎪ ⎪ ⎨ u2 (y 1 , y 2 ) = U2 (y 2 ) ψ (y 1 ) + ... ⎪ ⎪ ⎪ ⎪ ⎩ u3 (y 1 , y 2 ) = U3 (y 2 ) ψ (y 1 ) + ...

⎧ U1 (y 2 ) = B1122 τ 22 (y 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂2 U2 (y 2 ) = B2222 τ 22 (y 2 ) (6.19) with ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ U3 (y 2 ) = 1 U2 (y 2 ) 4b12

6.2 The Limit Problem for a Hyperbolic Inhibited Shell

151

Therefore, the leading order terms of the singularities are defined up to two arbitrary constants (coming from the integration of the first order equations (6.15) and (6.19)). They can be easily specified using the boundary conditions at the first order. Two aspects of the results must be underlined. On one hand, u1 , u2 and u3 are, respectively, 0, 1 and 2 order(s) more singular than f 3 (their highest order singularity are respectively ψ, ψ and ψ ). On the other hand, the singularities propagate along the corresponding characteristic line (the generator y 1 = 0). Indeed, as we saw for the parabolic case, U1 , U2 and U3 do not vanish when ϕ(y 2 ) vanishes as they involve primitives of ϕ(y 2 ) with respect to y 2 . 6.2.3

Three Cases of Loading

In the sequel, we consider three kinds of loading. A uniform normal pressure, proportional to the thickness is applied, respectively, in the regions A, B, and C (see Figs. 6.2 to 6.4). These three loadings induce different types of singularities on characteristic or non-characteristic lines. They will produce different kinds of layers: boundary layers (along characteristic lines AD, DG, GJ and JA) and internal layers (along characteristic or non-characteristic lines). These examples will also enable us to observe the propagation of the singularities. • Loading A I H J 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 K 000000 111111 000000 111111 y2 000000 111111 A 000000 111111 O 000000 111111 L 000000 111111 y1 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 A

B

C

G F E

A = (−50, −50) B = (0, −50) C = (25, −50) D = (50, −50) E = (50, 0) F = (50, 25) G = (50, 50) H = (25, 50) I = (0, 50) J = (−50, 50) K = (−50, 25) L = (−50, 0) O = (0, 0)

D

Fig. 6.2. Loading A

A normal pressure is applied in the rectangle ABIJ. The expression of the loading A is1 f 3 = H(−y 1 ). The singularity near y 1 = 0 is then: ψ(y 1 ) = H(−y 1 ) and ϕ(y 2 ) = 1

1

H(.) denotes the classical Heaviside jump function.

(6.20)

152


• Loading B J

I

111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 K 000000 111111 000000 y2 111111 000000 111111 B 000000 111111 O 000000 111111 L 1 000000 111111 000000 y 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 A

B

H

G

F E

C

A = (−50, −50) B = (0, −50) C = (25, −50) D = (50, −50) E = (50, 0) F = (50, 25) G = (50, 50) H = (25, 50) I = (0, 50) J = (−50, 50) K = (−50, 25) L = (−50, 0) O = (0, 0)

D

Fig. 6.3. Loading B

A normal pressure is applied on the triangle AOJ . The expression of the singularity of loading B near y 1 = 0 is not direct. For a fixed y 1 < 0, we have: f 3 = H(y 2 − y 1 ) − H(y 2 + y 1 ) = (−2y 1 )

H(y 2 − y 1 ) − H(y 2 + y 1 ) −2y 1

(6.21)

and f 3 vanishes for y 1 > 0. It then appears that the expression of the singularity of the loading B near y 1 = 0 is finally: f 3 ≈ −2y 1 H(−y 1 )δ(y 2 )

(6.22)

where δ denotes the Dirac mass. A rigorous proof of (6.22) may be obtained using distributions acting on test functions. This gives: ψ(y 1 ) = y 1 H(−y 1 ) and ϕ(y 2 ) = −2δ(y 2 )

(6.23)

A similar singularity (i.e. with the same order) exists along y 2 = 0. • Loading C Finally, the last loading domain considered is a disc centered in O with a radius equal to L/4 with L = 50. The expression of the singularity of the loading C nearby the line y 1 = L/4 = 12.5 is obtained in the same way. Let us exhibit the singularity of the loading along the line y 1 = 12.5. For a fixed −12.5 < y 1 < 12.5, we have f 3 = H(y 2 + s) − H(y 2 − s) (6.24) where s has to be expressed using the equation of the circle (see section 5.3.3). This finally gives √ . f 3 ≈ 2 L L/4 − y 1 H(L/4 − y 1 )δ(y 2 ) (6.25)

6.2 The Limit Problem for a Hyperbolic Inhibited Shell I

J K

H

G A = (−50, −50) B = (0, −50) = (25, −50) F C D = (50, −50) E = (50, 0) F = (50, 25) E G = (50, 50) H = (25, 50) I = (0, 50) J = (−50, 50) K = (−50, 25) L = (−50, 0) O = (0, 0)

y2

L

1111 0000 0000 1111 0000 1111 C O y1 0000 1111 0000 1111 0000 1111

A

B

153

D

C

Fig. 6.4. Loading C

so that we have: . √ ψ(y 1 ) = L/4 − y 1 H(L/4 − y 1 ) and ϕ(y 2 ) = 2 Lδ(y 2 )

(6.26)

Obviously, the singularity of ψ(y 1 ) in (6.26) is not very classical and it is not an element of the classical chain (2.74). However, it may be taken as S0 for a chain of the type (2.73) describing the singularities of the solutions. Remark 6.2.1. It is worth noticing that despite the presence of a corner at (0, 0) in the domain B, the loading B is the less singular along the characteristic y 1 = 0. The loading A is the most singular, whereas the loading C is intermediate between the loadings A and B. However, the loading B is the only one singular Table 6.1. Orders of the singularities of the three displacements at y 1 = 0 for the three loadings Loading A

Loading B

u1 = U1 (y 2 )H(−y 1 )

u1 = U1 (y 2 )y 1 H(−y 1)

u2 = U2 (y 2 )δ(−y 1 )

u2 = U2 (y 2 )H(−y 1)

u3 = U3 (y 2 )δ (−y 1 )

u3 = U3 (y 2 )δ(−y 1 )

Loading C u1 = U1 (y 2 )

.

L/4 − y 1 H(L/4 − y 1 ) 1

u2 = U2 (y 2 )(L/4 − y 1 )− 2 H(L/4 − y 1 ) u3 = U3 (y 2 )

1 d (L/4 − y 1 )− 2 H(L/4 − y 1 ) 1 dy

154


along the line y 2 = 0 (in y 2 H(−y 2 )), which is due to the corner. We shall see later that two internal layers (in both directions) appear when ε 0. 6.2.4

The Singularities of the Resulting Displacements

Using (6.19), we deduce the singularities of the three displacements at y 1 = 0 for each loading. The corresponding results are summarized in table 6.1, where d is the derivative in the sense of distributions. dy 1

6.3

Numerical Computations Using Adaptive Meshes

6.3.1

Numerical Results for Loading A

First, let us observe the adaptation process at some iterations.

(a) Initial mesh

(b) Iteration 2

(c) Iteration 5

(d) Iteration 7

Fig. 6.5. Evolution of the mesh during the adaptation process for ε = 10−4

6.3 Numerical Computations Using Adaptive Meshes

155

20 15 10

u3

5 0 -5 ite0 ite1 ite2 ite4 ite7

-10 -15 -20 -25

-20

-15

-10

-5

0

5

10

15

20

25

y1

(a) Displacement uε3 for ε = 10−4 90

60

u3

30

0

-30 ite0 ite1 ite2 ite4 ite7

-60

-90 -15

-10

-5

0

5

10

15

y1

(b) Displacement uε3 for ε = 10−5 Fig. 6.6. Displacement uε3 at y 2 = 0 for y 1 ∈ [−25, 25] and at various iterations for loading A

The initial mesh is quite uniform (Fig. 6.5(a)). At the first iteration, the refinement is nearly isotropic and uniform (the initial mesh was too coarse). From the second iteration, the elements start to become clearly anisotropic and we can observe a refinement mainly inside the layers which appear progressively. The final mesh (Fig. 6.5(d)) is highly anisotropic and non-uniform: the elements are small and lengthened in the layers, especially along y 1 = 0 corresponding to an internal layer. The displacements at various iterations and for two relative thicknesses are plotted in Figs. 6.6(a) and 6.6(b). In Fig. 6.6(a), we can see that the results are nearly the same at the 4th and the 7th iteration for ε = 10−4 as the process has nearly converged. The results obtained with the initial mesh and at the first and second iterations are quite far from those of the 7th. Subsequent iterations do not improve the solution. The efficiency of the adaptive process is more visible for ε = 10−5 (Fig. 6.6(b)). For this thickness, the process is longer to converge, and a refinement is

156

6 Singularities of Hyperbolic Inhibited Shells 0.03

0.3 eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

0.025 0.02 0.015

0 -0.3 -0.6 u2

u1

0.01 0.005 0

-0.9

-0.005 eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

-1.2

-0.01 -0.015

-1.5 -0.02 -0.025 -50

-40

-30

-20

-10

0 y1

10

20

30

40

50

-1.8 -50

-40

-30

-20

-10

0 y1

10

20

30

40

50

Fig. 6.7. Displacement uε1 at y 2 = 25 for Fig. 6.8. Displacement uε2 at y 2 = 25 for various ε various ε 60 90 eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

75 60 45 30

50

40

u3

15

30

0 -15

20

-30 -45

10

-60

numerical theoretical

-75 -90 -50

0 -40

-30

-20

-10

0 y1

10

20

30

40

50

Fig. 6.9. Displacement uε3 at y 2 = 25 for various ε

0

0.02

0.04

0.06

0.08

0.1

Fig. 6.10. Layer thickness η vs. ε1/3

essential to obtain accurate results. The displacements at the 7th iteration are about twice larger than those at the first iteration. Convergence toward the limit problem The solutions (in fact sections of them at y 2 = 25) of the three displacements for various values of the relative thickness ε are represented in figures 6.7 to 6.9. These three figures show that the displacements converge to the predicted distributions: uε1 converge to a Heaviside jump in y 1 = 0, uε2 to a δ and uε3 to a δ (see Fig. 5.16 and the interpretation of the numerical results in terms of distributions). When ε 0, the amplitudes of uε2 and uε3 keep increasing, tending to infinity. A way to confirm the nature of these singularities is to study how the amplitudes vary with ε. As we did in the parabolic case, we measure the internal layer thickness on the displacement uε3 . We take the thickness η as being the distance between the two main extrema visible in Fig. 6.9. It is known (see [63, 84] and section 2.8) that the thickness vary like ε1/3 when an internal layer is along a characteristic


157

90 1.6 75

1.4 1.2

60

1 45 0.8 30

0.6 0.4

15 0.2



0

0 0

10

20

30

40

0

50

Fig. 6.11. Value of uε2max vs. ε−1/3

500

1000

1500

2000

Fig. 6.12. Value of uε3max vs. ε−2/3

line (here y 1 = 0). The numerical and theoretical results are compared in Fig. 6.10. A very good concordance is observed. We now analyze the variation of the amplitude of uε2 and uε3 for the different values of ε to define the singularities in presence. To this end, we shall consider the maximum of these two displacements. We recall that, according to the analysis developed in section 6.2, uε2 and uε3 are supposed to tend, respectively, to be a Dirac singularity and its derivative. In section 5.4.1, we recalled that the Dirac distribution δ can be seen as the limit of a distribution whose support width is η and amplitude is 1/η when η tends toward zero. Taking η as the thickness of the internal layer, the amplitude of uε2 should vary like 1/η i.e. ε−1/3 . With the same reasoning on δ , we conclude that uε2 should vary like 1/η 2 = ε−2/3 . Figs 6.11 and 6.12 are in good agreement with that. On the other hand, all along the clamped boundary, we have a boundary layer. The most important one (in amplitude) is located along the edge AJ because AJ is also a boundary of the loading domain. As AJ is a characteristic line, the layer thickness should also be proportional to ε1/3 . Fig. 6.13 shows the normal displacement uε3 in this layer for various thicknesses. 16 0 eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

-0.5

14 12 10

-1 u3

8 6

-1.5

4 -2

2 -2.5 -50


0 -45

-40 y1

-35

-30

0

0.02

0.04

0.06

0.08

0.1

Fig. 6.13. Displacement uε3 at y 2 = 0 for Fig. 6.14. Boundary layer thickness η vs. various ε ε1/3

158


In Fig. 6.13, we see that the description of the boundary layer is not so good for ε = 10−5 . In fact, most elements concentrate in the internal layer where singularities are more important (in amplitude but not in order) and there are few elements outside. Without taking into consideration, the value obtained for ε = 10−5, we see that the layer thickness varies like ε1/3 (see Fig. 6.14). 6.3.2

Results for the Loading B

Mesh adaptation and displacements We now consider loading B. The meshes obtained at various iterations are represented in Fig. 6.15. The evolution of the mesh is different from that of loading A, because there is a visible propagation of singularities. In the previous case, we had propagation of singularities, but it was not visible because it was along the line y 1 = 0 which is also a boundary of the loading domain. With loading B, the mesh is refined along the loading domain boundary AOJ (corresponding to a singularity along a non-characteristic line), but also along the characteristic lines tangent to the point O where the loading is locally singular in y 1 (see (6.22)), and also in y 2 (with the same orders as the singularity near y 1 = 0). Thus, we clearly have propagated singularities along the lines y 1 = 0 and y 2 = 0. Let us now consider the singularities of the displacements along the line y 2 = 0 (to have a better visualization of them). In this case, the order of the singularities of u1 and u2 are exchanged as compared to the results presented in Tab. 6.1 (where the singularity was considered along the line y 1 = 0). The results concerning the two most singular displacements along the line y 2 = 0, i.e. uε1 and uε3 , are presented in figures 6.16(a) and 6.16(b) at y 1 = 25 i.e. on CH (see Fig. 6.3).

(a) Iteration 2

(b) Iteration 5

Fig. 6.15. Evolution of the mesh during the adaptation process ε = 10−4


159

0.06 eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

0.04

u1

0.02

0

-0.02

-0.04

-0.06 -20

-15

-10

-5

0

5

10

15

20

y2

(a) Displacement uε1 0.5 0 -0.5

u3

-1 -1.5 -2 -2.5

eps 0.00001 eps 0.0001 eps 0.0002 eps 0.0003 eps 0.0005 eps 0.001

-3 -3.5 -4 -20

-15

-10

-5

0

5

10

15

20

y2

(b) Displacement uε3 Fig. 6.16. Displacements at y 1 = 25 for y 2 ∈ [−25, 25] and for various thicknesses

In particular, we can notice that the singularities propagate along the characteristic line y 2 = 0 (as it is provoked by a singularity of the data at the origin). As predicted, the displacements are less singular than for loading A, uε1 tending to a Heaviside and uε3 to a Dirac. The displacement uε2 is not presented here. Comparison with structured meshes Let us exhibit some numerical computations using uniform structured meshes. Fig. 6.17 displays the normal displacement for various meshes and for ε = 10−5 . We observe the efficiency of the adaptive process: adapted meshes of 34 171 degrees of freedom (iteration 3) and 48 437DOF (iteration 7) give more accurate results than a structured uniform mesh with 71 525 DOF (called S4). The structured uniform mesh S4 is composed of equal squares, each being divided in two triangles.

160

6 Singularities of Hyperbolic Inhibited Shells 0.5 0 -0.5 -1

u3

-1.5 -2 -2.5 -3 -3.5 -4 -15

ite7 - DOF = 48437 ite3 - DOF = 34171 S4 - DOF = 71525 -10

-5

0 y2

5

10

15

Fig. 6.17. Displacement uε3 on the line y 1 = 25 for several meshes and for ε = 10−5

1

0.9

0.8

0.7 adapted structured 0.6 0

10000

20000

30000

40000 DOF

50000

60000

70000

Fig. 6.18. Comparison between the convergence of adapted and structured meshes for ε = 10−5

Fig. 6.18 shows the convergence of the results for ε = 10−5 versus the number of degrees of freedom. The reference result uε3ref is that of the last iteration of the adaptive process although it may not be the exact result. We can see that the computations using adapted meshes converge much faster that those using structured meshes2 . In fact, the latter does not converge at all, because of the locking phenomenon encountered for very small thicknesses. This is due to the fact that the layers become very thin as ε tend to zero (like ε1/2 or ε1/3 ). Thus, 2

The difference is not so important for thicker shells.


161

the elements need to be very thin and very elongated in the layers to get accurate results with a reasonable number of elements, and to reduce the locking. 6.3.3

Results for the Loading C

In Figs. 6.19(a) and 6.19(b), we see that during the adaptation process, the mesh is refined along the loading domain boundary and along the characteristic lines tangent to the loading domain. A main difference with the loading B is that with loading C the singularity at (12.5, 0) only propagate along y 1 = 12.5 and no propagation (or refinement) is seen along y 2 = 0.

(a) Iteration 2

(b) Iteration 5

Fig. 6.19. Evolution of the mesh during the adaptation process ε = 10−4

10 5 0 -5 -10 -15

60 40 20 0 -20 -40 -60

-60

-40

-20

0

20

Fig. 6.20. Deformed shape

40

60

162


Fig. 6.21. Displacement uε2 on the line y 1 = 37.5

Fig. 6.22. Displacement uε3 on the line y 1 = 37.5

The mesh is refined along the four lines y 1 = −12.5, y 1 = 12.5, y 2 = −12.5 and y 2 = 12.5 corresponding to the four characteristics tangent to the loading domain. Around the intersections between these layers, the mesh is isotropic. The deformed shape is plotted in Fig. 6.20. The shapes of the two displacements uε2 and uε3 plotted in Figs. 6.21 and 6.22 do not enable to determine accurately the type of singularity. We do not have one distinct oscillation as in figure 6.8, or two distinct oscillations as in figure 6.9. This is due to the fact that these singularities are intermediate with respect to the Dirac family. Moreover, the four layers interact. According to the results of table 6.1 and to the singularity order of the loading3 , the singularity of the displacement uε3 is between δ and δ , and that of the displacement uε2 between a Heaviside and a Dirac singularity. That could be verified by considering, respectively, the variations of |uε2 |max and |uε3 |max when ε tends to zero. For instance, |uε3 |max should vary between ε1/3 and ε2/3 . A similar study was carried out in the parabolic case (see section 5.4.4). 3

Which is similar to that obtained in chapter 5 for the same loading.

6.4 Some Examples Including Pseudo-reflections

6.4

163

Some Examples Including Pseudo-reflections

In this section, we shall illustrate the pseudo-reflection phenomena described in section 2.7. It occurs as soon as a singularity (propagated or not) reaches a boundary which is not (locally) tangent to a characteristic curve. To this end, we consider the hyperbolic paraboloid defined by the same mapping (see (6.3)) with c = 1, but on the triangular domain of figures 6.23 and 6.24. This way, a part of the boundary (i.e. BC) is not along a characteristic line, which allows pseudo-reflection phenomena. Numerical computations are done with thickness 10−4 .

y2 C δ

F E

δ δ

δ

δ δ or δ

G

δ

y1

δ

B

Fig. 6.23. First problem considered

6.4.1

A = (0, 0) B = (4, 0) C = (0, 4) G = (2, 2)

C

D δ

A

y2

A = (0, 0) B = (4, 0) C = (0, 4) D = (3, 1) E = (2, 2) F = (1, 3)

δ

A

y1

δ or δ δ

B

Fig. 6.24. Second problem considered

Reflection of a Characteristic Layer

The first problem considered (corresponding to Fig. 6.23) is similar to that of [62], but the computations are now performed with the adaptive mesh procedure. The shell is clamped along the part BC of the boundary and is consequently inhibited. A δ normal force with unit weight is applied on the segment y 2 = 1 with 1 ≤ y 1 ≤ 2, whose expression specifically writes f 3 = [H(y 1 −1)−H(y 1 −2)]δ(y 2 −1). As we saw in section 6.2.2, there is propagation of singularities for u3 (see Fig. 6.23): δ -like singularities propagate along the lines y 1 = 1 and y 1 = 2 whereas a δ -like singularity propagate along the line y 2 = 1. Moreover, we have pseudo-reflection of the propagated singularities as the boundary BC is not along a characteristic line (Rule 1 stated in section 2.7). In the case considered here (reflection of a singularity along a characteristic line), the reflected singularity is one order lower than that of the incident one. Then, the singularity in δ (along y 2 = 1) reflects at D and gives a δ -like singularity

164


4

4 3.5

2

3

-2

0 -4 -6

2.5

-8 2

0

1.5

0.5 1

1

1.5 2 2.5

0.5

3 3.5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.25. Mesh at the 5th iteration

4 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.26. Displacement uε3 on the whole domain

along y 2 = 3. In the same way, the reflections of the two other propagated singularities in δ (along y 1 = 1 and y 1 = 2) at E and F give δ-like singularities along the lines y 2 = 2 and y 2 = 3. Note that there would be no reflection if the shell was free along BC. We handled this first problem numerically, using the adaptive and anisotropic remeshing procedure. The last mesh and results for uε3 are presented in Figs. 6.25 to 6.28. In Fig. 6.25, we clearly observe that the mesh is mainly refined along the characteristic line y 2 = 1, corresponding to the discontinuity of the loading, and leading to a propagated δ − like singularity as seen previously (it is the higher order singularity). Moreover, the mesh is also refined along the characteristic lines y 1 = 1 and 1 y = 2. As quoted above, the singularities also propagate along these characteristic lines, but are of a lower order (only in δ ) than the previous one. So that the mesh is less refined than in the vicinity of the characteristic y 2 = 1. The mesh is also slightly refined along the characteristic line y 1 = 3, corresponding to the pseudo-reflection of the singularity of uε3 at the fixed edge. Finally, there is no refinement along the lines y 2 = 2 and y 2 = 3 where the existing singularities are negligible (“only” in δ). Fig 6.27 displays uε3 along the line y2 = 0.5 exhibiting three δ -like singularities (“double oscillations”). The two first ones, at y 1 = 1 and y 1 = 2 are those propagated from the extremities of the applied loading, whereas the third one at y 1 = 3 is the pseudoreflected from D (see Fig. 6.23). It has to be noted that this last one is weaker that the original singularity, in order (δ against δ ), but also in amplitude (0.05 against 2.5). Finally, figure 6.28 displays uε3 along the line y1 = 0.5, exhibiting the singularity of the highest order (δ -like or “triple oscillation”).


165

0.5

0.4

0.3

0.2

0.1

0 0

0.5

1

1.5

2

2.5

3

3.5

-0.1

-0.2

-0.3

-0.4

-0.5

Fig. 6.27. Displacement uε3 along the line y 2 = 0.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

-0.5 -1 -1.5 -2

Fig. 6.28. Displacement uε3 along the line y 1 = 0.5

6.4.2

Reflection of a Non-characteristic Layer

We now consider the second problem of Fig. 6.24. The shell is subjected to a normal loading bearing a δ-singularity with unit weight along the line y 1 = y 2 for 1 ≤ y 1 ≤ 2. This singular loading can be rewritten as f 3 = δ(y 1 − y 2 )H(y 1 − 1)H(y 2 − 1). The loading is singular along a non-characteristic line that has an intersection with the boundary BC (which is a non-characteristic line). The singularity of the displacement u3 is then of the same order as that of the loading f 3 (but there is no propagation). Consequently, u3 has a δ-like singularity along the line y 1 = y 2 (for 1 ≤ y 1 ≤ 2). Moreover, there are some classical propagated singularities of u3 along the lines y 1 = 1 and y 2 = 1. On the other hand, according to the Rule 2 of section 2.7, some reflected singularities also appear on the two characteristic lines issued from the point G = (2, 2) which is the intersection between the loading domain and the boundary BC. The order of the singularity of u3 in these two reflected layers depends on the boundary condition along the boundary BC. According to the Rule 2, u3 has a δ-like singularity (respectively, a δ -like singularity) when BC is clamped (respectively free).

166


Let us observe the numerical results for two types of boundary conditions: (a) the shell is clamped along the BC and free elsewhere (b) the shell is free along the BC and clamped elsewhere. With any of these two boundary conditions, the shell is inhibited. Case a (BC is clamped) According to the Rule 2 of section 2.7, the singularity of u3 arriving at the boundary BC reflects. Because the shell is clamped along BC, the reflected singularity is of the same order as the initial one. Figure 6.29 shows that the mesh is mainly refined along five layers: - three classical layers along the loading domain and the characteristic lines y 1 = 1 and y 2 = 1 issued from the discontinuity of the loading in (1, 1), - two reflected layers along the lines y 1 = 2 and y 2 = 2. The other reflected singularities are negligible. Figs. 6.30 clearly shows a δ like singularity (two oscillations) along y 1 = 1 and a δ-like singularity (one main oscillation) along the line y 1 = 2 that was predicted by the theory. 3

4

2

3.5

1

3

0

2.5

-1

2

-2

1.5

1

-3

0.5

-4

-5

0 0

0.5

1

1.5

2

2.5

3

3.5

0

4

0.5

1

1.5

2

2.5

3

3.5

Fig. 6.29. Refined mesh for Fig. 6.30. Displacement uε3 on the line y 2 = 0.5 for the second problem - Case a the second problem - Case a 40 35 30 25 20 15 10 5 0 -5 -10 -15

0 0.5 1 1.5 2 2.5 3 3.5 4 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.31. Displacement uε3 on the whole domain for the second problem - Case a


167

Case b (BC is free) In this case, as the shell is free along BC, the reflected singularities are one order more singular than the initial one (Rule 2). Figure 6.32 displays the mesh of the last iteration: it is here clearly refined along the reflected layers which are the most important ones (δ -like). The classical propagated singularities (which are δ -like too) are less important because of the boundary conditions. In Fig. 6.33, we see two δ -like singularities: in y 1 = 1 there is a small singularity (in amplitude), whereas the reflected singularity in y 1 = 2 is much more important. The previous examples show that despite the various types of layers (propagated or reflected), the automatic and adaptive numerical procedure allow us to get accurate results for the normal displacement by refining automatically the mesh where it needs to be. Doing a manual refinement should need a priori knowledge of all the existing singularities, their order and amplitude everywhere on the shell. It would have been much longer in the two considered examples, and would be nearly impossible in general for complex shell geometries and loadings. 8 4

6 3.5

4 3

2 2.5

0 2

-2 1.5

-4

1

-6

0.5

-8

-10

0 0

0.5

1

1.5

2

2.5

3

3.5

0

4

0.5

1

1.5

2

Fig. 6.32. Refined mesh for Fig. 6.33. Displacement the second problem - Case b case b

uε3

2.5

0 0.5 1.5 2 2.5 3 3.5 4 0

0.5

1

1.5

2

2.5

3

3.5

3.5

on the line y = 0.5 in

140 120 100 80 60 40 20 0 -20 -40 -60

1

3

2

4

Fig. 6.34. Displacement uε3 on the whole domain - Case b

168


6.4.3

Reflection of a Characteristic Layer when the Loading “Touches” the Non-characteristic Boundary

Finally, we shall consider a more complex case which is not concerned with the two rules of section 2.7: a singular loading (here a δ-singularity) along a characteristic line “touches” the non-characteristic boundary (see Fig. 6.35). Once again, we consider the two different boundary conditions a and b leading to different results. Case a (BC is clamped) When the loading domain “touches” a fixed non-characteristic boundary, the singularity reflects and seems to be a δ -singularity: it has two main oscillations

y2 C F E

A = (0, 0) B = (4, 0) C = (0, 4) D = (3, 1) E = (2, 2) F = (1, 3)

D y1 A

B

Fig. 6.35. Loading along a characteristic line touching the boundary

0.6

4 0.4

3.5

0.2

3

2.5

0

2

-0.2

1.5

-0.4

0

1

0.5

1

1.5

2

2.5

3

3.5

-0.6

0.5

-0.8

0 0

0.5

1

1.5

2

2.5

3

3.5

4

-1

Fig. 6.36. Refined mesh for Fig. 6.37. Displacement uε3 on the line y 2 = 0.5 in case a case a


169

30 25 20 15 10 5 0 -5 -10 -15

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.38. Displacement uε3 on the whole domain

visible in Fig. 6.37. Note that this behavior is similar to that in section 6.4.1 predicted by Rule 1 of section 2.7. Case b (BC is free) Now, the non-characteristic boundary is free and the loading domain still “touches” the boundary. We are in the case when a singular loading along a characteristic line “touches” the free non-characteristic boundary. This case is not concerned by Rule 1 of section 2.7, only dealing with propagated singularities which arrives to the free boundary. In the case considered here, it seems to have a δ -reflected singularity when the loading domain “touches” the boundary (see Fig. 6.40). Thus, the reflected singularity seems to be of the same order as the original one (u3 bears a singularity in δ along the characteristic). 0.6 4

0.4 3.5

0.2 3

0 2.5

u3

0

0.5

1

1.5

2

2.5

3

3.5

-0.2

2

-0.4 1.5

-0.6 1

-0.8 0.5

-1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

y1

Fig. 6.39. Refined mesh in Fig. 6.40. Displacement uε3 on the line y 2 = 0.5 in case b case b

170

6 Singularities of Hyperbolic Inhibited Shells 100 80 60 40 20 0 -20 -40

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.41. Displacement uε3 on the whole domain in case b

This example shows that as we cannot predict theoretically all the possible cases, accurate numerical simulations enable us to explore other (more complex) situations, and to enunciate the associated rules of reflection of singularities, which are still to prove theoretically. On the other hand, such numerical simulations allow a very accurate access to the reflected singularity that is, only possible using an anisotropic adaptive mesh procedure which detects automatically all the singularities (reflected or not) and refine in consequence.

6.5

Conclusion

In the first part of this chapter, we addressed the singularities of the displacements by solving the membrane problem for the most singular terms. When the loading is singular, we deduced the most singular term of the resulting displacements. In particular, we proved that the normal displacement u3 is two orders more singular than the normal loading f 3 when f 3 is singular along a characteristic line, and only as singular as f 3 when f 3 is singular on a non-characteristic line4 . A second part was devoted to the numerical illustration of pseudo-reflection phenomena in various situations. The examples presented proved once again the pertinence and the efficiency of the automatic anisotropic adaptive re-meshing used. This is all more true when considering real and complex situations, when the theoretical description of the asymptotic behavior is practically impossible, because of the simultaneous presence of propagation and pseudo-reflection of singularities in the shell.

4

Moreover, the propagation is exactly described by exhibiting the ODEs satisfied by the coefficients Ui .

7 Singularities of Elliptic Well-Inhibited Shells

7.1

Introduction

The theoretical analysis developed in Chapters 2 and 5 revealed the singular displacements which can appear in the internal and boundary layers when the loading is singular. These singularities and internal layers are linked to the loss of regularity of the bending displacement uε3 , solution of the Koiter model, when the thickness ε tends to zero. However, for elliptic shells, and only for elliptic shells, other kinds of singularities can appear when the boundary of the loading domain has corners. These singularities which appear at the corners of the singular loading domain are logarithmic, of a completely different nature than those described in the previous chapters, and have an amplitude much more important. To our knowledge, there exists no work on these logarithmic singularities in the case of elastic shells. In this chapter, starting from the reduced membrane formulation (2.60), we propose to establish theoretically the existence of such logarithmic singularities in the case of well-inhibited shells. In a second part, we will perform numerical simulations in the particular case of an elliptic paraboloid, using the anisotropic adaptive mesh procedure already presented in the last chapters. We will confirm numerically the existence of logarithmic singularities appearing at the corners of the loading domain. We will also consider the more classical singularities existing in the internal layers.

7.2

Existence of Logarithmic Point Singularities at the Corners of the Loading Domain

Let us consider an elliptic shell clamped along all its lateral boundary. According to the theoretical results recalled in Chapter 2, it is well-inhibited and the Koiter model tends to the membrane model when the thickness ε tends to zero. We propose in this section to study theoretically why (and when) such logarithmic point singularities may appear at the corners of the loading domain. E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 171–194. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

172


To do this, let us start again from reduced formulation (2.60) of the membrane model obtained in Chapter 2. We recall that this later only involved the normal displacement u3 and writes: (2) (2) 3 E b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 u3 = a2 a11 ∂12 + a22 ∂22 + 2a12 ∂1 ∂2 f (7.1) For an elliptic shell, this equation can be simplified if we consider the system of coordinates linked to the lines of principal curvature of the middle surface S. As the local basis associated is orthogonal, we have a12 = 0 and b21 = b12 = 0. So that b12 = 0 also (b11 and b22 are strictly different from zero and of the same sign because the shell is elliptic). Equation (7.1) then becomes: (2) (2) 3 E b22 ∂12 + b11 ∂22 u3 = a2 a11 ∂12 + a22 ∂22 f (7.2) It is important to notice that (as the shell is elliptic), the operators involved in both sides of the two members of equation (7.2) are elliptic. For such equations, it is known (see for instance [44][67][68]) that for specific characteristic exponents λk , singularity of the kind1 : rλk ϕk (θ) (7.3) or rλk [ϕk (θ) + log(r) ψk (θ)]

(7.4)

or even involving higher powers of log(r) may appear. The specific values of λk , which give the complete description of the singularities, depend on the coefficients of the equation. In general, these values (which are the eigenvalues of certain operator) may only be obtained by numerical computation (see [68]). In certain specific examples (involving the Laplacian or other analogous operators), a Fourier expansion with respect to the azimuthal angle θ enables us to have a complete description of the structure of the solution. To simplify the theoretical study of singularities, (7.2) can be reduced to an equation involving only the bi-Laplacian operator. Indeed, by the change of variables: ⎧ −1/2 ⎪ ⎨ z 1 = b22 y 1 1 2 1 2 (y , y ) ⇒ (z , z ) with (7.5) ⎪ ⎩ z 2 = b−1/2 y 2 11 equation (7.2) reduces to:

Δ2 u3 = C4 f 3

(7.6) 2

where C4 is a fourth-order operator, which can be written C4 = (D2 ) with: ! E −1/2 a11 ∂ 2 a22 ∂ 2 1 ∂2 1 ∂2 −1/2 D2 = 11 22 + =E + 1 a a b22 ∂(z 1 )2 b11 ∂(z 2 )2 b22 ∂(z 1 )2 b1 ∂(z 2 )2 (7.7) The proof of the existence of logarithmic singularity point is now easier from reduced equation (7.6), using Fourier expansions. It is the subject of the next sections. 1

For a polar coordinate system (r,θ).

7.2 Existence of Logarithmic Point Singularities at the Corners

7.2.1

173

Model Problem of Second Order

In this section, for a better understanding of the problem, we begin with studying a model problem for elliptic second-order operators instead of fourth-order. Let us consider a current point P of the domain Ω and use a polar coordinate system with P as origin (see Fig. 7.1).

x2 r

1111111111 0000000000 00θ 11 0000000000 1111111111 00 11 0000000000 1111111111 00 11 0000000000 1111111111 P 000000 111111 x 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 1

Fig. 7.1. Considered polar coordinate system

We will consider the problem: Δu = E2 f 3 (θ) in a vicinity of r = 0

(7.8)

where E2 is a second-order operator analogous to Δ (but not proportional to Δ) and where f 3 (θ) is independent of r. Remark 7.2.1. The case where the operator E2 is proportional to Δ is not interesting. Indeed, if E2 = KΔ, then equation (7.8) writes: Δ(u − Kf 3 ) = 0

(7.9)

As a consequence, u − Kf 3 is a harmonic function and then smooth (analytic). Therefore, u and f 3 have the same singularities. This remark also stands for equation Δ2 u3 = C4 f 3 where we consider that Δ2 and C4 are proportional. The aim of that follows is to prove that the solution u of (7.8) has log(r)-like generic singularity, and not only a singularity corresponding to that of f 3 (θ). For the analysis of singularities of problem (7.8), we just need to consider the singularity of one solution, because all the solutions have the same singularities. Indeed, let us take two solutions u and u˜, we have: Δu = E2 f 3 (θ) 3

Δ˜ u = E2 f (θ)

(7.10) (7.11)

which gives by linearity Δ(u − u ˜) = 0. So that, as u − u ˜ is a harmonic function (a smooth function), u and u ˜ will have the same singularities.

174


• The problem Δv = f 3 (θ) Let us first consider the problem Δv = f 3 (θ)

(7.12)

without the operator E2 . In polar coordinate system, the problem writes: " #(2) 1 ∂ ∂2 r + 2 v = f 3 (θ)r0 (7.13) r2 ∂r ∂θ The functions v and f 3 being 2π-periodic in θ, we can perform a Fourier expansion: ∞ v = v0 (r) + (vn (r) cos(nθ) + wn (r)sin(nθ)) (7.14) n=1

and f 3 (θ) = a0 +

∞

(an cos(nθ) + bn sin(nθ))

(7.15)

n=1

Problem (7.13) then becomes: " #(2) d r v0 (r) = a0 r2 dr " #(2) d 2 r − n vn (r) = an r2 dr " #(2) d 2 r − n wn (r) = bn r2 dr

(7.16) (7.17) (7.18)

After a new change of variable log(r) = t, we obtain:

"

2

d2 v˜ (t) = a0 e2t 2 0 dt#

d − n2 vñ (t) = an e2t dt2 " 2 # d 2 − n w ñ (t) = bn e2t dt2

(7.19) (7.20) (7.21)

which can be solved easily. The solution of the non-homogeneous equation will be of the form vñ = An e2t except for n = 2. For n = 2, the solutions will be A2 t e2t and B2 t e2t with A2 = a42 and B2 = b42 . So we can find v˜: v˜ =

a2 2t b2 t e cos(2θ) + t e2t sin(2θ) + terms in ent with n = 2 (7.22) 4 4


175

Coming back to the variable r, we get: a2 b2 log(r).r2 cos(2θ) + log(r).r2 sin(2θ) 4 4 + terms in rn with n = 2

v=

(7.23)

• Back to problem Δu = E2 f 3 (θ) Let us now go back to the problem (7.8). Let us apply operator E2 to the both sides of the previous problem. We get: E2 Δv = E2 f 3 (θ). As both operators have constant coefficients, they commute. Then, we have: ΔE2 v = E2 f 3 (θ)

(7.24)

Thus, u = E2 v is a solution of (7.8). As E2 is a second-order operator, the most singular terms will be those differentiated twice with respect to y 1 and/or y 2 . In polar coordinates, the power compared to r loses two orders either because ∂ of two differentiation(s) and/or division(s) by r (see the expressions of ∂y 1 and ∂ in polar coordinates). So, finally, we obtain: ∂y 2 u = G(θ) log(r) + more regular terms

(7.25)

where G(θ) is a function depending only on θ. 7.2.2

The Membrane Problem Δ2 u3 = C4 f 3 (θ)

Let us go back to the initial problem (7.6) which can be studied due to the previous one. As in the second-order case, all the solutions have the same singularity, we only need to find a solution. Let us first consider the problem without the operator C4 , which is equivalent to: Δ2 u = f 3 (θ) ⇔ ΔΔu = f 3 (θ) ⇔

Δv = f 3 (θ) v = Δu

(7.26)

(7.27)

The first equation of (7.27) has been studied in the previous section. Its general solution is given by (7.23). Then replacing v in the second equation of (7.27), we obtain: Δu =

a2 b2 log(r).r2 cos(2θ) + log(r).r2 sin(2θ) 4 4 + terms of the f orm rn with n = 2

(7.28)

Let us only keep terms which are not regular. We then consider the problem: Δu =

a2 b2 log(r).r2 cos(2θ) + log(r).r2 sin(2θ) 4 4

(7.29)

176


After a Fourier expansion of u: u = u0 (r) +

∞

(un(r) cos(nθ) + u∗n (r)sin(nθ))

(7.30)

1

we obtain the system: " #(2) 1 ∂ r − 4 u2 (r) = a2 r2 log(r) r2 ∂r " #(2) 1 ∂ r − 4 u∗2 (r) = b2 r2 log(r) r2 ∂r ⇔

#(2) ∂ r − 4 u2 (r) = a2 r4 log(r) ∂r " #(2) ∂ r − 4 u∗2 (r) = b2 r4 log(r) ∂r

(7.31)

(7.32)

"

With the same change of variable log(r) = t as previously, we get: " 2 # ∂ −4 u ˜2 (t) = a2 t e4t ∂ t2 " 2 # ∂ −4 u ˜∗2 (t) = b2 t e4t ∂ t2

(7.33) (7.34)

(7.35) (7.36)

As e4t is not a solution of the homogeneous equation, u ˜2 (and similarly u ˜∗2 ) has the form: 2 d 4t 4t u ˜2 = A t e + B e ⇒ − 4 u˜2 = A(8 + 16t)e4t + 16Be4t (7.37) dt2 So that u ˜2 is solution of (7.35) only if: ⎧ ⎨ 8A + 16B = 0 ⎩

16A = a2

⎧ a2 ⎪ ⎪A = ⎨ 16 ⇔ ⎪ ⎪ ⎩ B = − a2 32

(7.38)

Now we can determine the most singular terms of u which write: u=

a2 4 b2 r log(r) cos(θ) + r4 log(r) sin(θ) + more regular terms 16 16

(7.39)

Finally, to solve problem (7.6), let us apply the operator C4 to equation (7.26) as we did in the previous section. If we set u3 = C4 u, with C4 a fourth-order operator, we obtain:


Δ2 u3 = C4 f 3 (θ)

177

(7.40)

and using (7.39): u3 (r, θ) = H(θ) log(r) + more regular terms

(7.41)

where H(θ) is a function depending only on θ. In conclusion, we obtain a logarithmic point singularity for the elliptic membrane problem considered (which is of the fourth-order). However, this singularity only exists if at least one of the coefficients a2 or b2 (of Fourier expansion of the loading f 3 (θ)) is not equal to zero. In other words if: 1 2π 3 a2 = f (θ)cos(2θ)dθ = 0 (7.42) π 0 and/or 2π 1 b2 = f 3 (θ)sin(2θ)dθ = 0 (7.43) π 0 7.2.3

Particular Case when the Logarithmic Point Singularity Vanishes

The reduced equation (7.6) was established in the general case in the system of coordinates corresponding to the principal curvatures (for which b21 = b12 = 0). During the resolution, we assumed that the two operators of equation (7.6) were not proportional (see Remark 7.2.1). However, the later case can occur when b11 = b22 = b at the studying point, which is then an umbilic point of the middle surface S. In that case, (7.6) becomes : (2) E b2 ∂12 + ∂22 u 3 = Δ2 f 3 2

2

2

3

⇐⇒ E b Δ u3 = Δ f ⇐⇒ Δ2 E b2 u3 − f 3 = 0

(7.44) (7.45) (7.46)

So E b2 uε3 − f 3 is a smooth function. So as a consequence, u3 and f 3 have the same singularities. This means that even if, as seen previously, a2 or b2 is not equal to zero, there cannot exist any logarithmic singularity in a point where the principal curvatures are equal. 7.2.4

Existence Condition of a Logarithmic Singularity

Let us now examine concrete examples illustrating various situations (shape of the loading domain, position of the point P considered) and consequently various functions f 3 (θ). We will see in which cases a2 or b2 do not vanish and when a logarithmic singularity exists at the point P . In the following examples, we consider an elliptic surface S with any local mapping (Ω, Ψ ). The loading f 3 is purely normal, constant and applied on the hatched area.

178


Case 1 Let us consider a point P inside the loading domain where the normal loding is equal to 1 (Fig. 7.2).

x2

y2 B

1

C

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 O 000000000 111111111 P 000000000 111111111 000000000 111111111 000000000 111111111

−1

−1

A

r

1 y1

1

D

Fig. 7.2. Position of the point P

We then have:

1111111111 0000000000 00θ 11 0000000000 1111111111 00 11 0000000000 1111111111 00 11 0000000000 1111111111 P 000000 111111 0000000000x 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

Fig. 7.3. Polar coordinate system

f 3 (θ) = 1 f or 0 ≤ θ ≤ 2π

(7.47)

And consequently: a2 =

1 π

2π

f 3 (θ) cos(2θ)dθ = 0

0

and

b2 =

1 π

2π

f 3 (θ) sin(2θ)dθ = 0

0

Thus, if the point P is inside the loading domain, we have a2 = b2 = 0 and no logarithmic point singularity may appear. This result was clearly previsible. Case 2 Let us now consider a point P at the boundary of the loading domain, where the tangent to the boundary is smooth. Two possible situations are represented in Figures 7.4 and 7.6. In both cases, f 3 (θ) can be written in the local coordinate system: 1 f or 0 ≤ θ ≤ π f 3 (θ) = (7.48) 0 f or π ≤ θ ≤ 2π We then have: 1 2π 3 1 π f (θ) cos(2θ)dθ = cos(2θ)dθ = 0 π 0 π 0 1 2π 3 1 π b2 = f (θ) sin(2θ)dθ = sin(2θ)dθ = 0 π 0 π 0 a2 =

7.2 Existence of Logarithmic Point Singularities at the Corners y2 B

1

C

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 O P 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

−1

−1

A

1 y1

D

111111 000000 000000 111111 000000 111111 000000 x 111111 P 111111 000000 10 000000 111111 1010 000000 111111 0000 1111 000000 111111 0000 1111 1010θ r 000000 111111 000000 111111 10 x

179

2

1



y2 B

1

C

11111111111 00000000000 P 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 O 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111

−1

A

−1

1 y1

11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 P 00000 11111 00000 11111 00000 11111 0000 1111 00000 11111 00000 11111 00000 11111 0000 1111 00000 11111 00000 11111 00000x 11111 0000 1111 00000 11111 00000 11111 0000 1111 θ 11111 00000 00000 11111 0000 1111 00000 11111 r x 00000 11111 00000 11111

2

D

Fig. 7.6. Another possible position of the point P

1


We find again that a2 = b2 = 0. Thus, if the loading domain boundary at P is a straight line, a circle or any other smooth curve (with a regular tangent), there is no logarithmic point singularity. We recover the classical result: when f 3 is singular along a curve (which is not characteristic in the elliptic case), u3 has the same singularity as f 3 . Case 3 Finally, let us consider the following case: the point P is on the boundary of the loading domain where the tangent is not continuous. In other words, there is a “corner” at the point P (Figs. 7.8 and 7.9). In the case of a 90◦ corner, the function f 3 (θ) writes locally: ⎧ ⎨ 1 f or 0 ≤ θ ≤ π 3 2 f (θ) = (7.49) ⎩ 0 f or π ≤ θ ≤ 2π 2

180

7 Singularities of Elliptic Well-Inhibited Shells y2

B

1

P

−1

C

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 O 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 −1

A

x 0000000 1111111 00000 11111 000000 P111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 00000 11111 000000 111111 r θ 000000 111111 2

1 y1

D


x1 Fig. 7.9. Polar coordinate system

We have then: a2 =

1 π

1 b2 = π

2π

0

0

2π

f 3 (θ) cos(2θ)dθ =

1 π

1 f 3 (θ) sin(2θ)dθ = π

π/2

cos(2θ)dθ = 0 0

0

π/2

(7.50) 2 sin(2θ)dθ = π

Therefore, one of the coefficients of the Fourier expansion of f 3 (here b2 ) does not vanish. Consequently, we have a logarithmic singularity at this point provided the principal curvatures are different at this point. The results obtained in Section 7.2 can be summarized as follows: Proposition 7.1. Let us consider a well-inhibited elliptic shell subjected to a normal loading f 3 constant on its support, and a point P of the boundary of the support. Let us consider a local polar coordinate system whose origin is the point P (corresponding to r = 0). When the two following conditions are satisfied at the point P : • the Fourier expansion of the loading f 3 with respect to θ contains at least a second range term (a2 or b2 ) different from zero. This is the case if the loading zone has a “corner”. • the principal curvatures b11 and b22 are different at this point (the point P is not an umbilic point of the surface S) then a logarithmic singularity exists at the point P . Such singularities will be observed in the next section thank to numerical computations.

7.3 Example of an Elliptic Paraboloid

7.3

181

Example of an Elliptic Paraboloid

In this section, we propose to visualize concretely the singularities existing in the elliptic well-inhibited case. Numerical computations using an adaptive mesh strategy will allow to observe the singularities of displacements due to a singular loading. In particular, we will confirm the existence of logarithmic point singularities (only for u3 ) when thehypothesis of proposition 7.1 is satisfied. 1 2 Let us consider the domain Ω = (y , y ) ∈ [−1, 1] × [−1, 1] and the elliptic surface S defined by the mapping (Ω, ψ) where : ψ(y 1 , y 2 ) = (y 1 , y 2 , (y 1 )2 + (y 2 )2 )

(7.51)

The considered shell is defined by S and the thickness ε which is constant in Ω. The shell is clamped on its whole boundary so that it is wellinhibited. We then apply a loading on the hatched domain F defined by * + F = (y 1 , y 2 ) ∈ [− 12 ; 12 ] × [− 12 ; 12 ] (see Fig. 7.10). According to (1.62), we consider a normal loading f˜3 = 10 ε M P a proportional to the thickness ε, so that f 3 is constant. We also take the following material constants which correspond to a standard steel : E = 210, 000 M P a and ν = 0.3. y2

B

1

C (y1)2+(y2)2

Ω

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 O 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 F 0000000000 1111111111 0000000000 1111111111

−1

2 1.5

1

y1

1 0.5 1

F

0 0.5 0 -1

A

−1

D

Fig. 7.10. Domains Ω and F

y2

-0.5

-0.5

0 y1

0.5

1 -1

Fig. 7.11. Middle surface S

As the problem is symmetric with respect to the lines y 1 = 0 and y 2 = 0, we only consider a quarter of the domain and the following symmetries on the displacements: uε1 = 0 and uε3,1 = 0 on the line y 1 = 0

(7.52)

uε2 = 0 and uε3,2 = 0 on the line y 2 = 0

(7.53)

182


7.3.1

Geometric Properties

First, let us determine the geometric properties of the shell (local basis, curvature tensors) of the elliptic paraboloid defined by (7.51). • Covariant and contravariant basis The tangent plane at S is defined at each point p = ψ(y 1 , y 2 ) of S by the two tangent vectors aα = ∂α ψ: a1 = 1, 0, 2y 1 and a2 = 0, 1, 2y 2 (7.54) a1 ∧ a2 whose components write: | a1 ∧ a2 | 1 N=. − 2y 1 , −2y 2 , 1 1 + 4(y 1 )2 + 4(y 2 )2

The normal vector is N =

(7.55)

The covariant and contravariant components of the metric tensor are, respectively: ⎛ ⎞ 1 + 4(y 1 )2 4y 1 y 2 ⎠ aαβ = ⎝ (7.56) 4y 1 y 2 1 + 4(y 2 )2 and aαβ = (aαβ )−1

⎛ ⎞ 1 + 4(y 2 )2 −4y 1 y 2 1 ⎝ ⎠ = 1 + 4(y 1 )2 + 4(y 2 )2 −4y 1 y 2 1 + 4(y 1 )2

(7.57)

1 . The contravariant basis is then 1 + 4(y 1 )2 + 4(y 2 )2 a1 = C(y 1 , y 2 ) 1 + 4(y 2 )2 , −4y 1 y 2 , 2y 1 (7.58) a2 = C(y 1 , y 2 ) − 4y 1 y 2 , 1 + 4(y 1 )2 , 2y 2

Let us set C(y 1 , y 2 ) = defined by:

Finally, the only non vanishing Christoffel symbols are: 1 2 Γαα = 4y 1 C(y 1 , y 2 ) and Γαα = 4y 2 C(y 1 , y 2 ), α = 1, 2

(7.59)

• Curvature tensor The covariant components of the curvature tensor are given by bαβ = N · aα,β (see (1.12)). They write here: ⎛ ⎞ 10 . ⎠ bαβ = 2 C(y 1 , y 2 ) ⎝ (7.60) 01


183

2 1.5 1 0.5 0 -0.5

1 0.5 0 -0.5 -1 -1

-0.5

0

0.5

1

Fig. 7.12. Deformed shape for ε = 10−4

The fact that det(bαβ ) = 4C > 0 proves that the surface S is elliptic at any point of S. The asymptotic directions are not real. The mixed components are then: ⎛ ⎞ 1 + 4(y 2 )2 −4y 1 y 2 1 2 3/2 ⎝ ⎠ bα (7.61) β = 2C(y , y ) −4y 1 y 2 1 + 4(y 1 )2 We notice that the surface S has only one umbilic point located in (0, 0). The principal curvatures are equal at this point, and all the bases are principal bases. Thus, according to the results of paragraph 7.2, no logarithmic singularity can appear at this point. A similar example is addressed in [11]. 7.3.2

Numerical Results

For every ε, we start the computation with a uniform mesh made of 236 elements (Fig. 7.13). Then, the mesh is refined automatically with the anisotropic mesh generator BAMG as in the previous chapters (see Figs. 7.15 and 7.16). Figure 7.14 displays the normal displacement uε3 on a quarter of the domain after seven iterations. We can see the two phenomena that happen: • a singularity along the loading domain boundary • a peak or a logarithmic singularity at the corner of the loading domain F . The next sections concern the analysis of these two singularities. The internal layer due to the loading Using the Result 2.6.1, we can deduce the singularity of the displacement u3 from that of the loading f 3 . Indeed, the loading f 3 writes as the product of two crenel functions:

184


1

2e-005 0 -2e-005 -4e-005 -6e-005 -8e-005 -0.0001 -0.00012 -0.00014 -0.00016

0.8

0.6

0.4

1 0.8 0.6

0.2

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Fig. 7.13. Initial mesh (1/4 of the domain) (236 elts)

0 0

0.2

0.4

0.6

0.8

1

Fig. 7.14. Displacement uε3 for ε = 10−4

f 3 = −10 H(y 1 + 0.5) − H(y 1 − 0.5) H(y 2 + 0.5) − H(y 2 − 0.5) (7.62) 0 when y 1 < a where H(.) is the Heaviside step function: H(y 1 − a) = . 1 when y 1 > a Since u3 has the same singularity as f 3 , u3 has Heaviside-like singularities except near the corners, where we have a logarithmic point singularity. So that the numerical simulations performed should put in a prominent position a jump (a Heaviside-like singularity) at y 1 = 0.5 on the line y 2 = 0 (also on the lines y 2 = c with c < 0.5 but not too close to 0.5 (we will see why in the next section). 7.3.3

Mesh Adaptation

We consider the case of a small thickness ε = 10−4 fixed. First, let us observe the evolution of the mesh on Figures 7.15 and 7.16. We can see that the mesh is mainly refined along the boundary of the loading domain and in particular near the corner [ 12 ; 12 ]. We saw in the previous sections that the discontinuity of the loading induces an internal layer along the border of F and a logarithmic singularity at the corner. With the anisotropic mesh generator used (BAMG), the areas where the mesh is more refined correspond to those where the displacement uε3 varies drastically and tends to be singular2 (inside the layers). The mesh aspect is different according to the part of the mesh. In the layer, the elements are anisotropic: the length of the edge in the direction of the layer is more important than the length of the edge in the perpendicular direction (Fig. 7.17). This is due to the singularity which only exists in the direction perpendicular to the loading boundary. On the other hand, the mesh around the 2

We recall that uε3 denotes the solution of the Koiter model for a thickness ε, whereas u3 is the solution of the limit membrane model (corresponding to ε = 0), whose singularities are determined theoretically.

7.3 Example of an Elliptic Paraboloid 1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

185

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 7.15. Mesh at the third iteration Fig. 7.16. Mesh at the seventh iteration (3076 éléments) (5351 éléments)

0.6

0.28 0.26 0.24

0.55

0.22 0.2 0.5 0.18 0.16 0.45

0.14 0.12 0.1

0.4 0.46

0.48

0.5

0.52

0.54

0.4

0.45

0.5

0.55

0.6

Fig. 7.17. Mesh inside the layer along Fig. 7.18. Mesh around the point singuy 1 = 12 larity

point singularity is isotropic as the logarithmic singularity exists all around the point (0.5, 0.5) (Fig. 7.18). Let us now focus on uε3 which is the most singular displacement of this problem. Figures 7.19 and 7.20 show the efficiency of the remeshing procedure on the results for the computation of uε3 , whereas the first mesh gives very bad results around the internal layer (the shape of the singularity is not correctly described), the results of the third iteration are close to the theory. Indeed, we can really notice the jump of uε3 near the line y 1 = 0.5, which corresponds to a Heaviside-like singularity as predicted by the theory. The next iterations give even a better approximation of the singularity shape.

186

7 Singularities of Elliptic Well-Inhibited Shells 1.5E-05

1.0E-05

5.0E-06

0.0E+00

u3

0

0.25

0.5

0.75

-5.0E-06

1

Iteration 0 Iteration 3

-1.0E-05

Iteration 7 -1.5E-05

Uniform mesh

-2.0E-05

-2.5E-05

y1

Fig. 7.19. Convergence of uε3 during the adaptation for y 2 = 0 and y 1 ∈ [0, 1] 1.5E-05

1.0E-05

5.0E-06

0.0E+00

u3

0.4

0.45

0.5

-5.0E-06

0.55

0.6

Iteration 0 Iteration 3

-1.0E-05

Iteration 7 -1.5E-05

Uniform mesh

-2.0E-05

-2.5E-05

y1

Fig. 7.20. Convergence of uε3 during the adaptation for y 2 = 0 and y 1 ∈ [0.4, 0.6]

Another important fact to quote is that the number of nodes (or of elements) increases inside the internal layer, whereas it is nearly constant outside. Moreover, between the lines y 1 = 0 and y 1 = 0.25, and around y 1 = 0.75, the results are quite the same for each iteration as for the initial mesh. This is due to the absence of internal layer in these zones where u3 (solution of the membrane model) is smooth. Finally, let us compare the results of the different iterations to the results obtained with a uniform mesh containing the same number of elements as the seventh iteration (5 582 elements). It is clear in figures 7.19 and 7.20 that a uniform mesh gives very bad results in the layer, even with a great number


187

of elements. The approximation of uε3 obtained at the third iteration of the adapted mesh with about twice less elements (3 076 exactly), gives better results than the uniform mesh with 5 582 elements. This enforces the interest of an anisotropic automatic (adaptive) re-meshing inside the internal layers or near the singularities. 7.3.4

Thickness of the Internal Layer along y 1 = 0.5

Let us now determine the internal layer thickness by numerical simulations performed on the same example. The variations of uε3 for various values of ε are represented in Figure 7.21. We recall that we apply a loading proportional to ε in the loading domain F . Thus the displacement uε3 stays quite the same out of the internal layer (in the smooth regions) and it is easy to compare the results on the same figure. 1.5E-05

1.0E-05

5.0E-06

0.0E+00

u3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-5.0E-06

10-2

-1.0E-05

10-3 10-4

-1.5E-05

10-5 -2.0E-05

-2.5E-05 1

y

Fig. 7.21. uε3 for different ε from 10−2 to 10−5

When ε tends to zero which corresponds to the membrane model, the theoretical analysis developed in the previous section proves that uε3 tends to a Heaviside singularity. For thicker shells (ε = 10−2 for instance), uε3 is a smooth function and it is difficult to measure the internal layer thickness. A first possibility would consist in measuring the distance between the two extrema near y 1 = 0.5, but this is not always an easy task. So that it is preferable to measure this distance η (see Fig. 7.22) for 10−6 ≤ ε ≤ 10−3 and we search for a relation of the form η = εα . To do this, in Figure 7.23, we plot log(η) with respect to log(ε) for different values of ε. The result is once again very close to the theory3 : we find η = O(ε0.5163 ). 3

We recall that for such problems, the thickness of the internal layers or of the 1 boundary layers are of the order of ε 2 (see [96] or Section 2.8.1).

188

7 Singularities of Elliptic Well-Inhibited Shells 2.E-05

u3

1.E-05

0.E+00 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

-1.E-05

η -2.E-05

y1

Fig. 7.22. Measure of η on the displacement uε3 0

-6

-5

-4

-3

-2

-1

0

-0.5

log(η)

y = 0.5163x + 0.61 R2 = 0.9959

-1

-1.5

-2

-2.5

log(ε)

Fig. 7.23. Plot of log(η) vs. log(ε) measured on uε3

The results obtained for the displacements uε1 and uε2 are similar. The symmetries of the problem imply uε1 (y 1 , y 2 ) = uε2 (y 2 , y 1 ). That is why we plot only uε1 along the line y 2 = 0 (due to the symmetries of the problem, uε2 vanishes along this line). According to the Result 2.6.1 of Chapter 2, in the case of an inhibited elliptic shell, the displacements u1 and u2 are one order (at least) less singular than f 3 . In other words, they have a singularity in (y 1 − 0.5)H(y 1 − 0.5). That is what we can observe: uε1 is close to a function with a jump of the derivative at y 1 = 0.5 (see Fig. 7.24). During the asymptotic process (when ε 0), the displacement uε1 along the line y 2 = 0 does not vary significatively: we can just observe on Fig. 7.25 that the derivative at y 1 = 0.5 tends to have a jump.


189

0.0E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2.0E-06 -4.0E-06 -6.0E-06

u1

-8.0E-06 -1.0E-05 -1.2E-05

10-3

-1.4E-05

10-4 -1.6E-05

10-5

-1.8E-05 -2.0E-05 1

y

Fig. 7.24. Displacement uε1 along y 2 = 0 for ε = 10−4

-1.5E-05 0.45

0.47

0.49

0.51

0.53

0.55

10-3 -1.6E-05

10-4

u1

10-5

-1.7E-05

-1.8E-05

y1

Fig. 7.25. Zoom of uε1 along y 2 = 0 for 0.45 < y 1 < 0.55 and for various ε

7.3.5

The Logarithmic Singularity at the Corner

As we saw in Section 7.2, there may exist logarithmic singularities in some particular points for an elliptic shell problem. At the point P (0.5, 0.5), corresponding to a corner of the loading domain, we are in Case 3 of Section 7.2.4. Therefore, the first condition of proposition 7.1 is satisfied: a2 = 0 but b2 = 0 (see (7.49) and (7.50)). Moreover, at P the two principal curvatures are different, the only umbilic point being at the origin (0, 0). That ensures the existence of a logarithmic singularity. Let us now compare the numerical simulations performed to the theory which predicts the existence of a logarithmic singularity at the corner P . Figure 7.26

190

7 Singularities of Elliptic Well-Inhibited Shells 2.0E-05 0.0E+00 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2.0E-05 -4.0E-05

u3

-6.0E-05 -8.0E-05 -1.0E-04 -1.2E-04 -1.4E-04 -1.6E-04

y1=y2

Fig. 7.26. Displacement uε3 on the line y 2 = y 1 pour ε = 10−4

0.0E+00 0

0.02

0.04

0.06

0.08

0.1

-2.0E-05 -4.0E-05

y = 3E-05Ln(x) + 6E-05 2 R = 0.9944

u3

-6.0E-05 -8.0E-05

node results

-1.0E-04 -1.2E-04 -1.4E-04

r

Fig. 7.27. Comparison between uε3 and a logarithmic function at the neighborhood of P

represents the normal displacement uε3 along the line y 2 = y 1 . We note that the maximum of uε3 is at about y 1 = 0.48. As we are not exactly at the limit problem (ε = 10−4 = 0), uε3 is not exactly a logarithmic function near by P . There is a smooth link between the two logarithmic-like functions which exist at each side of P . To specify the form of uε3 at the corner, let us zoom on a zone near the peak for y 1 > 0.5. In Figure 7.27, we remove the points near the maximum (that make the smooth link between the two sides of the singularity). We take as origin the point where uε3 reaches its maximum, and we denote by r the distance between the origin and the considered point (we take polar coordinates). The graph of


191

uε3 for 0 < r < 0.1 (for the side y 1 > 0.5) shows that uε3 is very close to a logarithmic function. Therefore, the numerical results obtained with the anisotropic adaptive re-meshing enable to approach precisely the logarithmic point singularity at the corner predicted by the theory. Evolution of the logarithmic singularity Figure 7.28 shows the evolution of the logarithmic singularity with respect to ε. It is clear again that when ε tends to zero, the bending displacement uε3 tends to the solution of the membrane model which bears a logarithmic singularity near the corner for y 1 = y 2 = 0.5. For a thick shell ε = 10−2, the solution uε3 is much smoother and does not present any singularity. This one appears more clearly for small values of ε (for ε < 10−4 ). The maximum of uε3 becomes larger: 3.5 × 10−5 for ε = 10−2 and 2.1 × 10−4 for ε = 10−5. It is easy to verify that it is proportional to log(ε). A second point to quote is that the smooth link between the two sides of the singularity is more and more narrow. These two phenomena are linked together. Indeed uε3 is like a logarithmic function (uε3 ≈ c log(r)), but not in a zone around its maximum. The thickness of this zone decreases as ε decreases. If the thickness of this zone is of the order O(εα ) with a given α, the logarithmic part starts from r = a εα . As the maximum of uε3 is near the maximum of the logarithmic part of uε3 reached for r = a εα , we have: uε3max ≈ c log(a εα ) ∼ c α log(ε)

(7.63)

0

Finally, let us quote that the position of the maximum progressively tends to the point P = (0.5, 0.5) as ε tends to zero. 5.0E-05

0.0E+00 0.25

0.5

0.75

-5.0E-05

u3

10-2 -1.0E-04

10-3 10-4 10-5

-1.5E-04

-2.0E-04

-2.5E-04 y1=y2

Fig. 7.28. Evolution of the logarithmic singularity for various values of ε from 10−2 to 10−5

192


7.3.6

Membrane and Bending Energies

In this section, we consider again the elliptic paraboloid presented at paragraph 7.3. The repartition of membrane and bending energy surface densities are represented in Figs. 7.29 and 7.30.

Fig. 7.29. Membrane energy surface density for ε = 10−4

Fig. 7.30. Bending energy surface density for ε = 10−4

For ε = 10−4 , both energies have a similar repartition: along the boundary of the loading domain and especially at the corner (where is located the logarithmic point singularity). In paragraph 7.3.4, we determined the thickness of the internal layer from the shape of the displacement uε3 . We can also determine this thickness from the energy surface densities. To this end, let us observe the evolution of the percentage of bending energy surface density compared to the total energy surface density (Figs. 7.31 and 7.32). We clearly observe in Figs. 7.31 and 7.32 that the layer thickness diminishes when ε tends to zero. Inside layers, bending effects remain important, whereas 1

1

0.8 0.7

y2

0.6 0.5

0.8 0.7 0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.25

0.5

y1

0.75

1

50 45 40 35 30 25 20 15 10 5 1

0.9

y2

55 50 45 40 35 30 25 20 15 10 5 1

0.9

0

0

0.25

0.5

0.75

1

y1

Fig. 7.31. Percentage of bending energy surface density for ε = 10−2 and 10−3

7.4 Conclusion 1

1

0.8 0.7

y2

0.6 0.5

0.8 0.7 0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.25

0.5

y1

0.75

1

35 30 25 20 15 10 5 1

0.9

y2

50 45 40 35 30 25 20 15 10 5 1

0.9

0

193

0

0

0.25

0.5

0.75

1

y1

Fig. 7.32. Percentage of bending energy surface density for 10−4 and 10−5

Fig. 7.33. log(η) vs. log(ε) measured on Figures 7.31 and 7.32

they become negligible outside it. Both energies are of the same order in the layers. Thus, we may define the thickness as the zone where bending energy represents more than 10% of the total energy4 . This way, we can measure for each value of ε, the thickness η where Eb is of the same order as Em . Plotting log η versus log ε in Figure 7.33, we find that the thickness η varies like ε0.5012 , whereas the theoretical value is ε1/2 (see Section 2.6.1). This constitutes an alternative way to determine the thickness of layers.

7.4

Conclusion

In this chapter, we addressed the singularities existing in the case of wellinhibited elliptic shells. First, the general theoretical results obtained in 4

Another choice of criterion is possible. For instance, we may consider that the limit is 20% or 30% instead of 10%. In that cases, the results obtained are similar.

194


Chapter 2 on singularities in internal layers were verified with the adaptive mesh procedure. In the case of elliptic shells, we have only layers along noncharacteristic lines and we recovered the classical results: • the normal displacement uε3 tends to have the same singularities as f 3 when ε 0, whereas uε1 and uε2 are of a lower order, • membrane and bending energies concentrate inside the internal layers, • the internal layer thickness is of the order of ε1/2 . Moreover, we put in a prominent position the existence of logarithmic point singularities for u3 (to our knowledge unknown until now), which appear when the loading domain has corners. The theoretical existence of these singularities were established in Section 7.2 using Fourier expansions. Numerical simulations using an adaptive anisotropic mesh procedure confirmed that uε3 tends to a logarithmic function, and that both membrane and bending energies reach their maximum at this point singularity. The use of a numerical adaptive procedure of remeshing enabled us to approach precisely these two kinds of singularities, by refining the mesh in an anisotropic way inside the layer, and in a isotropic way around the logarithmic point singularity. It should be noticed that other numerical computations and comparisons with theoretical results in more complex and various situations can be found in [11]. In particular, the influence of the loading domain on the resulting singularities, and situations where the logarithmic point singularities vanish, were examined.

8 Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems

8.1

Introduction

This chapter constitutes a general heuristic study of sensitive problems, and in particular of sensitive elliptic shell problems, i.e. elliptic shells clamped (or fixed) by a part Γ0 of the boundary and free by the rest Γ1 . Note that such sensitive problems or “ill-posed problems”, have already been considered in general in [70], and in some very particular case of shells in [10][23][83]. The first part of this chapter is devoted to a recall on boundary conditions for equations and systems (hyperbolic, parabolic and elliptic), and constitutes an introduction to elliptic sensitive problems which will be considered in the second part. We will see that when the Shapiro–Lopatinskii condition is not satisfied on a part of the boundary for the limit problem, the associated elliptic problem is ill-posed and a pathological behavior progressively emerges as ε tends to zero. In that case, the solution is not anymore contained in the distribution space, in particular in any Sobolev space, leading to a complexification phenomenon: large oscillations appear in the direction tangential to the free boundary, exponentially decreasing toward the interior of the domain. It should be emphasized that very little is known on sensitive problems, and in particular on the limit membrane problem for elliptic shells, when a part of the boundary is free. In particular, the “solutions” out of the distribution space are far from admitting a qualitative description as the “classical” ones of the clamped case. They basically consist in a “global singularity” all over the domain of the shell, mainly localized on the free boundary where the “pathological boundary condition” is prescribed, but spreading out all over the domain. The exponential decay inwards the domain implies some kind of “boundary layer”, but in any case it is asymptotically “very large”. Indeed, we will see that the thickness of this layer converges to zero very slowly as 1/log(1/ε). As far as we know, the explicit descriptions of these situations available in the literature [39][71] are only concerned with special simplified models, namely model problems with one equation (instead of a system) with constant coefficients and posed on a strip. We propose in the last section of this chapter, a E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 195–217. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

196

8 Generalities on Boundary Conditions for Equations and Systems

“more general” theoretical justification of these heuristic properties for sensitive shells. An extended version of this work is in [45] for a model problem and in [46] for the shell case.

8.2 8.2.1

The Cauchy Problem for Equations and Systems Generalities

When dealing with ordinary differential equations (ODEs) of order n in one variable t, in order to define a solution in a unique way, the values of the unknown and its derivatives up to order n − 1 (or some equivalent data) must be given. Obviously, in PDE with two variables x1 , x2 , some functions must be given on certain curves. We recall in what follows the main general results. Let us consider the very simple equation: ∂2 u(x1 , x2 ) = f (x1 , x2 )

(8.1)

which is in fact an ODE in x2 with a parameter x1 . Obviously, when u(x1 , 0) is known, the solution is completely defined, but this should be specified, on account of the domain where u is defined. More generally, the “initial point” x2 = 0 may depend on the parameter x1 . For instance, when u(x1 , x2 ) is defined on the domain Ω of Figure 8.1, the value of u should be given either on AD or BC, or any curve crossing Ω from left to right, without coming back. x2

1111111111 B 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 C 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 Ω 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 D A

x1

Fig. 8.1. Domain Ω

Otherwise, the data on DC are of no use and generally inconsistent with the equation. For instance, let us consider the boundary condition for (8.1): u(0, x2 ) = ϕ(x2 ) on DC

(8.2)

8.2 The Cauchy Problem for Equations and Systems

197

It is only consistent when the data ϕ and f satisfy the compatibility condition: ∂2 ϕ(x2 ) = f (0, x2 ) on DC

(8.3)

Let us now search for the characteristics of equation (8.1). As already seen in Chapter 1 (Section 1.5), they are the curves orthogonal to the vectors (ζ1 , ζ2 ) satisfying ζ2 = 0 (8.4) i.e. the lines x1 = const. This is the simplest case of a Cauchy problem where the equation is completed by the values on a curve of the unknown u(x1 , x2 ) and of its derivatives up to order n − 1 (n being the order of the equation). In the case of a system, the data and the value of each unknown and its derivatives must be given up to the order minus one (of this unknown). Remark 8.2.1. For other examples, the reader is invited to modify Ω in Fig. 8.1 to “become” non-convex (for instance replacing the segment CB by a curve oscillating up and down) as well as the previous comments on unique determination of u. 8.2.2

Role of the Characteristics

The role of the characteristics on the determinacy of the solution of a Cauchy problem is obvious, as well as the fact that Cauchy data on a characteristic are subjected to compatibility conditions and do not determine a solution (see (8.3) in the example of Section 8.2.1). This fact may be compared to Remark 1.5.1 on the existence of sinusoidal solutions in eiζ1 x1 +iζ2 x2 , as in (1.49). Hyperbolic equations or systems (i.e. having all characteristics real) may be analyzed in terms of Cauchy problems (see the sequel). This gives a good insight on the structure of the solutions, even if certain problems for hyperbolic equations are not of the Cauchy type. As an example, let us consider the system: ∂1 u1 = f1 (8.5) ∂2 u 2 = f 2 which is immediate to solve as it is done of two equations of the form (8.1) in the direction x1 and x2 . The characteristics are the curves orthogonal to the vectors (ζ1 , ζ2 ) which satisfy: ζ1 0 (8.6) 0 ζ2 = 0 Thus the characteristics are the two families of real curves x1 = const and x2 = const, and the system is hyperbolic. The Cauchy data consist in giving u1 and u2 on a curve. If u1 , u2 should be defined on Ω (Fig. 8.2), the Cauchy data on the arc AB determine uniquely u1 , u2 on the curvilinear triangle ABC which is the

198

8 Generalities on Boundary Conditions for Equations and Systems E D

Ω

C

F

A B

Fig. 8.2. Example of Cauchy data

domain of determinacy of the arc AB. It is also said that ABC is the domain of dependence of the point C, as the values of the unknowns at C depend on the values of the data (including f1 , f2 ) on ABC. Otherwise, the Cauchy data on AB have some influence on the regions ACED and CBF , without determining completely the solutions. Finally, they have no influence at all on the regions ECF . The domain swept by the characteristics issued from AB is called its domain of influence. Remark 8.2.2. The reader is invited to consider variants of the previous situation. Note, as an example, that in general Cauchy data given on the arc BADE are subjected to compatibility conditions. In the particular case when the data vanish (that appears when the interest is only on uniqueness properties) the compatibility conditions are automatically satisfied. The above properties of Cauchy problems are very general (but the proofs are more complicated) and hold true for general hyperbolic systems. For instance, the second-order hyperbolic equation: (∂12 − ∂22 )u = f

(8.7)

which is called the wave equation, has the characteristic polynomial: ζ12 − ζ22 = (ζ1 + ζ2 )(ζ1 − ζ2 ) = 0

(8.8)

The corresponding characteristics are x1 − x2 = const and x1 + x2 = const. A change of variables taking the characteristics as axes changes (8.8) into: ∂ 1 ∂2 u = F

(8.9)

Taking as new unknowns u1 = ∂1 u ,

u2 = ∂ 2 u

(8.10)


199

we have automatically (in order for u to exist): ∂1 u2 = ∂2 u1

(8.11)

and (8.9) is equivalent (there is some additive constant to control) to the system: ∂1 u2 = F (8.12) ∂2 u1 = F which is of the class (8.5) (with a different notation). The same holds true for the Cauchy data on a curve. For (8.7) they are the values of u and of a non-tangent derivative. For (8.12) they are the values of u1 and u2 . The verification of the equivalence is left as an exercise. Obviously, the fact that the characteristics are in the direction of the axes is not essential. Moreover, all the previous considerations hold true when lower order terms (i.e. the unknowns themselves) appear in (8.5). Also coefficients may depend on (x1 , x2 ) so that the characteristics are curves. The proofs are (for instance) done using a fixed point theorem for successive iterations. At each iteration, the lower order terms receive values from the previous iteration then sent to the right-hand side, allowing to compute the next iteration. 8.2.3

Normal Form of a Hyperbolic System: Riemann Invariants

System (8.5) (even with the above generalizations) is very particular, as each equation only contains the derivative of an unknown in the direction of a characteristic. But a hyperbolic system of first order admits a change of unknowns such that each one of the new unknowns is only differentiated in the direction of only one characteristic. This is the very important process of diagonalization, the new unknowns being called Riemann invariants. Let us explain this a little in the case where the system only contains the principal terms (of first-order) and the coefficients are constant. We consider a first-order hyperbolic system of n equations and unknowns written in matrix form: A∂1 u + B∂2 u = f

(8.13)

where u = (u1 , . . . , un )T and A and B are n × n matrices. If a certain direction is characteristic, we first make a change of variables such that the considered characteristics is x2 = const. Let ˜ 2 u = f˜ ˜ 1 u + B∂ A∂

(8.14)

be the new form of the system. As x2 = const is characteristic, (ζ1 , ζ2 ) = (0, ζ2 ) is a solution of the characteristic equation: ˜ 2 ) = 0 ⇔ detB ˜=0 ˜ 1 + Bζ det(Aζ

(8.15)

˜ T also vanishes and there Obviously, the determinant of transposed matrix B exists a non-vanishing vector β = (β1 , . . . , βn ) such that:

200


˜T β = 0 B

(8.16)

Let us take the scalar product of the system (8.14) with β. We get: ˜ β) + ∂2 (Bu, ˜ β) = F˜ ∂1 (Au,

(8.17)

˜ ∂1 β)+ (Bu, ˜ ∂2 β). with F˜ = (f˜, β). Otherwise, we have trivially F˜ = (f˜, β)+ (Au, Then, using the transposed matrices, we observe that by virtue of (8.16): ˜ β) = (u, B ˜ T β) = 0 (Bu,

(8.18)

˜ β = F˜ ∂1 Au,

(8.19)

so that (8.17) becomes:

˜ β) is only differentiated which amounts to saying that the (scalar) unknown (Au, in the characteristic direction x2 = const. It is the Riemann invariant associated with the characteristic x2 = const. Coming back to the initial variables, it is only differentiated in the direction of the initially chosen characteristic. Moreover, as the system is hyperbolic, there are n real characteristic directions and we may find n Riemann invariants, which are taken as new unknowns. Result 8.2.1. As a general result for hyperbolic equations or systems, the Cauchy problem defines a solution and only one in the determination domain swept by all the characteristics issued from the arc of curve AB bearing the Cauchy data (see Fig. 8.3).

A

B Fig. 8.3. Example of determination domain

Obviously, the fact that all the characteristics are real plays an essential role in the above considerations. The case of equations or system of total order 2 with one double characteristic direction at each point (parabolic) is somewhat analogous, the determination domain becoming a strip. Finally, the case of an elliptic equation or system is very different. It is the subject of that follows. Interesting examples of these issues are in Sect. 2.2 and the figures herein.


8.2.4

201

Elliptic Equations or Systems

The case of elliptic equations or systems (when all the characteristic directions are imaginary, i.e. no one is real) presents very different features. Roughly speaking, the Cauchy problem in that case is not well-posed (i.e. it does not enjoy existence uniqueness and stability of the solutions), but its main properties allow to understand certain pathological situations. As a model example, let us consider the Laplace equation on the strip R×[0, 1] with Cauchy data on the boundary x2 = 0: (∂12 + ∂22 )u = 0 on Ω

(8.20)

u(x1 , 0) = ϕ(x1 )

(8.21)

∂2 u(x1 , 0) = 0

(8.22)

x2 1

Ω

0

Γ0

x1

Fig. 8.4. Strip R × [0, 1]

The right-hand sides of (8.20) and (8.22) were taken equal to zero in order to get simpler expressions, but this is not essential. Taking for instance ϕ(x1 ) = cosλx1 , the solution is: u(x1 , x2 ) = cos λx1 cosh λx2

(8.23)

Letting λ → ∞, we see that this solution is unstable: the data are O(1) whereas the solution is exponentially large (for any x2 > 0). Moreover, this is the basic ingredient for studying the problem by Fourier transform, which leads in most cases (this depends on ϕ) to the non-existence of the solution and instability in any usual space or topology.

202


Direct and inverse Fourier transform in the variable x are defined by +∞ fˆ(ξ) = f (x)e−iξx dx (8.24) −∞

1 f (x) = 2π

+∞

fˆ(ξ)eiξx dξ

(8.25)

−∞

for functions, but there are generalizations for certain classes of distributions. The images of ∂x and ∂ξ are multiplied by iξ and −ix, respectively. Accordingly, (direct and inverse) Fourier transform exchanges “smoothness” by “decay at infinity” (or “singularity” by “growing at infinity”). In particular, the images of

are

δ(x), δ (x), δ (x) . . .

(8.26)

1, iξ, ξ 2 . . .

(8.27)

By Fourier transform from x1 to ξ, problem (8.20)–(8.22) becomes an ordinary differential equation in x2 with initial conditions and parameter ξ. The (formal) Fourier transform of the solution is u ˆ(ξ, x2 ) = ϕ(ξ) ˆ cosh ξx2

(8.28)

The (formal) solution u(x1 , x2 ) is obtained using (8.25) (or a generalization of it to distributions), while x2 is a parameter. We observe that fixing x2 > 0, cosh ξx2 grows exponentially as |ξ| → ∞, so that u(x1 , x2 ) is “very very singular” with respect to x1 , even if ϕ(x1 ) is not. To fix ideas, let us take ϕ(x1 ) = δ(x1 )

(8.29)

i.e. the Dirac mass at the origin. Using (8.26), (8.27) we have at x2 = 1 (for instance): ∞ ξ 2n u ˆ(ξ, 1) = cosh ξ = (8.30) (2n)! n=0 and using again (8.26),(8.27), the “solution” at x2 = 1 is: ∞ (−i)2n (2n) u(x1 , 1) = δ (x1 ) (2n)! n=0

(8.31)

In other words, a δ-singularity of the data implies “a singularity of order infinity” of the solution. But, expressions involving derivatives of any order of δ(x) (as (8.31)) are not distributions. Distributions are “locally of finite order”, i.e. on any bounded region, they are derivatives (of some finite order) of a continuous function. Fourier transform within distribution theory only works for “temperate distributions” not allowing exponential growth for x or ξ tending to infinity. The space of


203

(direct or inverse) Fourier transforms of general (not necessarily temperate) dis tributions is usually called Z (see for instance [43]) which is a space of analytic functionals (i.e. acting upon analytic test functions). Specifically, Z is the dual of the space called Z of analytic functions. Obviously, the same kind of results are obtained for other values of x2 > 0 and other data ϕ(x1 ), unless for very special ϕ, having in particular a Fourier transform decaying at infinity exponentially (with sufficiently large exponent). Generally speaking, elliptic Cauchy problems have no solution. If exceptionally they have, it is unstable (under a very small modification of the data, the solution disappears or is highly modified). Otherwise, if the solution exists, it is unique (theorem of Holmgren and Calderon, see for instance [35] or [88]). Remark 8.2.3. When the previous problem has a “solution”, it generally involves a generalization of the concept of distribution (analytical functionals). But such “solutions” have no much practical utility. They are functionals on spaces of analytic functions, so that their effect can only be tested on the support of the analytic functions which is the whole line! Such abstract solutions can not be “localized”, they have no sense at a point or a small neighborhood of it. As we shall see later, “solutions” of the kind (8.31) often appear not directly, but as limit of sequences of genuine functions. It is then useful to see an example of a sequence converging (in the sense of analytic functionals or specifically in Z ) to (8.31). Let us consider the following example presented in [38]: ⎧ ⎨ cosh(ξ) f or |ξ| < λ fˆλ (ξ) = (8.32) ⎩ 0 f or |ξ| > λ When λ ∞, the function fˆλ (ξ) tends to cosh(ξ) in the sense of distributions. But cosh(ξ) is not a tempered distribution. Then the inverse expansion of f λ (x) converges in the sense of Z only, but not in D . The inverse Fourier expansion f λ (x) of fˆλ (ξ) then writes: * λ + 1 f λ (x) = e [cos(λx) + x sin(λx)] − e−λ [cos(λx) − x sin(λx)] 2 2π(x + 1) (8.33) Roughly speaking, f λ (x) is a distribution oscillating with frequency proportional to λ, and amplitude proportional to eλ . At the limit, when λ ∞, f λ (x) is a “function” whose frequency and amplitude are infinite, which is definitely not a distribution. Indeed, we have: f λ (x)

eλ λ F (x) 2π

with F λ (x) =

1 [cos(λx) + x sin(λx)] x2 + 1

(8.34)

The function F λ (x) for some values of λ is plotted on Figure 8.5. We observe that when λ → +∞, the graph of F λ (x) occupies all the space comprised inside the “envelop1” ± x21+1 . 1

And in reality more than that because of the factor x in front of sin(λx).

204

8 Generalities on Boundary Conditions for Equations and Systems 1

λ=2 λ=5

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -10

-5

0

5

10

Fig. 8.5. Graph of F λ (x) for some values of λ.

Finally, as f λ (x) is F λ (x) multiplied by whole R2 plane when λ → +∞.

8.3 8.3.1

eλ , 2π

the graph of f λ (x) occupies the

Boundary Value Problems for Elliptic Equations and Systems Regularity of the Solution

In the previous section, we saw that the Cauchy problem consists in giving on a suited part of the boundary a number of boundary conditions equal to the order of the equation (or the total order of the system). Oppositely, elliptic problems are often associated with boundary conditions all over the boundary, this number being half the total order (note that, when the coefficients are real, in order to have all roots imaginary, the characteristic equation is of even degree: elliptic systems with real coefficients have even total order). As we saw in the last section, we have the following property of solutions of elliptic equations or systems: Property 8.3.1. Solutions of the elliptic equations (or systems) are “smooth” inside the domain apart from singularities locally provoked by the singularity of the right-hand side. Here, “smooth” means of class C ∞ if the coefficients are of class C ∞ , or analytic if the coefficients are analytic. In particular, there is no propagation of singularities (see the analysis of singularities performed in Chapter 2). The heuristic reasons of this property are easy to understand. Let us consider (to simplify) an elliptic equation with constant coefficients having only the terms of higher order (with vanishing right-hand side) in a domain Ω. Let O be an interior point, and x, y two arbitrary orthogonal axes (see Fig. 8.6). The elliptic equation considered writes:

8.3 Boundary Value Problems for Elliptic Equations and Systems

E(∂x , ∂y )u(x, y) = 0

205

(8.35)

Let us perform “locally” a Fourier transform of the solution in the direction x only, as for problems (8.20)–(8.22). Here, “locally” means “considering the solution only on a neighborhood of O”; this makes rigorously sense with the microlocal analysis techniques. The Fourier transform satisfies: E(iξ, ∂y )ˆ u(ξ, y) = 0

(8.36)

which is an ODE with constant coefficient in y with parameter ξ. The solution is obviously expressed in terms of eλy with λ solution of E(iξ, λ) = 0. By homogeneity, λ is proportional to ξ, then of the form λ = μξ. Moreover, μ has real part different from zero. Indeed, if μ were of the form iζ with real ζ, the characteristic equation should have the solution (ξ, ζξ) real and this is impossible as the equation is elliptic. It is then easily seen that the local structure of the Fourier transform writes: u ˆ(ξ, y) = · · · + u ˆ(ξ, 0)eμξy + . . .

(8.37)

with Re(μ) = 0. This situation is somewhat analogous to that of the Cauchy elliptic problems (8.20)–(8.22), whose solution is (8.28). The factor eμξy grows exponentially in ξ for fixed y = 0 (either positive or negative). Thus, in order for u ˆ(ξ, y) to be the Fourier transform of a function (or even a distribution), u ˆ(ξ, 0) decays necessarily exponentially for |ξ| → ∞ which implies that u(x, 0) is (locally) smooth. As the point O and the direction x were chosen arbitrarily, we see that u(x, y) in (8.35) is smooth everywhere. Obviously, the expression (8.37) is not very precise but the reader may complete it using the “Cauchy conditions” on y = 0 (for instance for a second-order equation, u and ∂y u on y = 0) which define uniquely the solution on both sides y < 0 and y > 0. Considering the boundary conditions, the treatment is somewhat analogous (see Fig. 8.7). Up to a diffeomorphism which (locally) transforms the boundary into the x axis, we locally deal with the half-plane.

x

y O

Ω

Fig. 8.6. A point O of the domain Ω

206


x 1 0 00000 11111 0 1 00000 11111 0 1 00000 11111 0 1 00000 11111 D 0 1 00000 11111 0 1 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 ex

ey y

Fig. 8.7. Treatment of the boundary conditions

The lower order and locally small terms induced by the diffeomorphism are taken into account by microlocal analysis techniques. With respect to (8.37), we only deal presently with y > 0 and we have the boundary conditions to be satisfied. To have analogous results, the role of the boundary condition is to “replace the part y < 0”. This amounts precisely to satisfy the Shapiro– Lopatinskii condition, explained in the next section. 8.3.2

The Shapiro–Lopatinskii Condition

The Shapiro–Lopatinskii condition for a given equation (or a system of equations) and the associated boundary conditions amounts to some kind of well-behaved relation between the boundary conditions and the equation. Roughly speaking, we consider a point of the boundary (Fig. 8.7) and “the principal part” of the equation (or of the system of equations) and of the associated boundary conditions, i.e. discarding terms of lower order of differentiation (which is justified according to the microlocal analysis technique). Then, we search for solutions in the half-plane (for instance, the half-plane y > 0 on Fig. 8.7) with the associated homogeneous boundary conditions. We search for the existence of solutions which are sinusoidal along the boundary and exponentially decreasing toward infinity in the direction normal to the boundary inside the domain. If such solutions exist, different from zero, it means that the Shapiro–Lopatinskii condition is not satisfied at this point. Oppositely, it is satisfied if zero is the only solution of such kind. Obviously, as the data of the problem are zero (as the equation and the boundary conditions are homogeneous), the fact that this condition is not satisfied amounts to some kind of non-uniqueness: the simplified problem in the half-space has non zero solutions with zero data. It should be emphasized that these solutions with zero data “live in the half-plane”. When considering solutions in some domain Ω, they appear as some kind of “local solution” in the vicinity of points of the boundary where the Shapiro–Lopatinskii condition is not satisfied. They may be considered as local trend to the presence of such kind of solutions, whereas the real structure of the solutions (of the full problem, provided they exist) also involves non-local considerations coming from other

8.4 The Shapiro–Lopatinskii Condition and the Membrane Problem

207

points of the boundary (either with the Shapiro–Lopatinskii condition satisfied or not). Coming back to problem (8.35), the “fictitious y < 0 region” should imply the impossibility of exponentially growing solutions, i.e. with Re(μξ) < 0. This amounts to “exponentially decaying toward y > 0”. The boundary conditions must avoid this kind of solutions: this is the Shapiro–Lopatinskii condition. The number of conditions concerned is half the total order of the equation or system (as we consider equations or systems with real coefficients so that the roots iμ are conjugate complex, non-real). Result 8.3.1. In order to have well-posed problems for elliptic equations or systems, boundary conditions satisfying the Shapiro–Lopatinskii condition must be prescribed at any point of the boundary. Their number is half the total order of the system. Obviously, the specific boundary conditions may differ from a point to another of the boundary. In particular, each connected component of the boundary may have its own set of boundary conditions. Otherwise, local changes of boundary conditions (as well as local non-smoothness of the boundary) induces local singularities. Remark 8.3.1. Let us consider a boundary value problem formed by an elliptic equation (or system) with boundary conditions satisfying the Shapiro–Lopatinskii condition. In what sense is it “well-behaved”? Obviously, existence and uniqueness of the solutions are far from ensured, as follows from the example: −Δu − λu = f

(8.38)

with Dirichlet boundary conditions when λ is an eigenvalue. Then, existence of u only occurs when f satisfies the compatibility conditions (orthogonality with the kernel of the adjoint). When these conditions are satisfied, uniqueness only holds up to an arbitrary eigenvector (which is in a space of finite dimension). This is exactly the situation in general: as we do not know a priori if 0 is an eigenvalue of the considered problem, existence and uniqueness of the solution only hold up to a finite number of compatibility conditions for f , and existence up to a finite dimension kernel. More precise properties need specific properties (positivity or others) of the equation. Remark 8.3.2. In general problems, the Shapiro–Lopatinskii condition is generally satisfied and it is not explicitly checked. Construction of well-behaved solutions follows often from coerciveness properties on Sobolev spaces and others. The interest of this condition arises in very special situations when usual methods fail.

8.4

The Shapiro–Lopatinskii Condition and the Membrane Problem

We are considering the membrane system for elliptic shells with a part of the boundary fixed (or clamped). As we said above (in Section 2.2.2) it is inhibited,

208


so that the asymptotic process when ε 0 leads to the membrane problem. We are checking if the Shapiro–Lopatinskii condition is satisfied or not, and we shall find that it is satisfied on a fixed boundary, not on a free one. As in our problem the shell is certainly fixed by a part, we must expect pathological phenomena when another part of the boundary is free. Let us consider an elliptic shell with a boundary represented in Fig. 8.7. The direction ex corresponds to the tangent to the boundary and ey to the interior normal. Then, we consider two possible boundary conditions on this boundary: F ixed edge : ux = uy = 0 (Dirichlet condition)

(8.39)

F ree edge : T αβ nβ = 0 (N eumann condition)

(8.40)

with T = A γλμ (u) and n = −ey in our case. It should be noticed that the Dirichlet condition (8.39) only involves the components ux and uy , not u3 (normal to the shell). Indeed, the Dirichlet condition is a “principal” boundary condition for the variational problem, to be prescribed on all the functions of the configuration space. Then, as u3 is at most in L2 (Ω), its trace does not make sense. Oppositely, the Neumann condition (8.40) is a “natural” boundary condition for the variational problem, only satisfied by the solution itself (not by all the functions in the space), and it may involve traces not making sense for any function in the space. According to Section 8.3.2, the Shapiro–Lopatinskii condition is not verified (see [88]) if there exists a non-zero solution of the limit problem in the half-plane y > 0 with f i = 0, satisfying the homogeneous boundary conditions, which has the form: u0 = U (y)eiμx (8.41) αβ

αβλμ

with μ ∈ R∗ and U (y) → 0 when y → +∞. Obviously, as we pointed out above, we only consider the “principal terms”. This amounts to neglecting all λ the Christoffel symbols Γαβ . Moreover, the function U (y) = (ux (y), uy (y), u3 (y)) denotes a function exponentially decreasing into the interior of the domain. We can notice that the function (8.41) results to perform a Fourier transform x → μ. It is periodic with respect to x with period 2π μ . So that we limit the study to a periodicity strip B. Let us consider a solution u0 of the membrane system with f i = 0, on the form (8.41). We multiply each line of the membrane system, respectively, by u0x , u0y and u03 , where u0i denotes the complex conjugate of u0i . Summing the three lines of the system and after an integration by parts, we obtain: αβ 0 0 T (u )γαβ (u ) = Aαβλμ γλμ (u0 )γαβ (u0 ) = 0 (8.42) B

B

We note that the integrated terms which have the form nα T αβ (u)uβ vanish for several possible reasons for both boundary conditions: - on the line y = 0 because of the boundary condition (8.39) or (8.40). - for y → ∞ because of the exponential decreasing of U (y).

8.4 The Shapiro–Lopatinskii Condition and the Membrane Problem

209

- on the lines x = 0 and x = 2π μ because of the periodicity (n in x = 0 is 2π opposed to that in x = μ ) so that the sum of the two terms vanishes. Finally, using the property of positivity of the coefficients Aαβλμ , (8.42) is equivalent to: γαβ (u0 ) = 0 , (α, β) ∈ (1.2)2 (8.43) with u0 given by (8.41). This leads to (as the Christoffel symbols were neglected): ⎧ ⎧ ∂x u0x − b11 u03 = 0 iμux − b11 u3 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ∂y u0y − b22 u03 = 0 ⇐⇒ ∂y uy − b22 u3 = 0 (8.44) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ∂x u0y + ∂y u0x − 2b12 u03 = 0 iμuy + ∂y ux − 2b12 u3 = 0 where we recall that u0 = u0x , u0y , u03 and U (y) = (ux (y), uy (y), u3 (y)). The system (8.44) is a linear system with constant coefficients. The general solutions are of the form: U (y) = U 0 eηy (8.45) with U 0 ∈ C3 . The total order of the system is 2 and non-zero solutions exist if: iμ 0 −b11 0 = 0 η −b22 = b11 η2 − 2iμb12η − b22 μ2 (8.46) η iμ −2b12 This gives two solutions η1 and η2 for each μ. The discriminant 4(b11 b22 − b212 )μ2 > 0 because μ = 0 and the surface is elliptic (b11 b22 − b12 2 > 0). So that we have two distinct solutions η1 and η2 , both being constants of C, for μ fixed: ! ! . . ib12 − b11 b22 − b212 ib12 + b11 b22 − b212 η1 = μ and η2 = μ b11 b11 (8.47) The solution of (8.43) is then of the form: u0 = U 0 eηy eiμx and the system (8.44) becomes the algebraic system: ⎧ iμUx0 − b11 U30 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ηUy0 − b22 U30 = 0 ⎪ ⎪ ⎪ ⎪ ⎩ 0 ηUx + iμUy0 − 2b12 U30 = 0

(8.48)

(8.49)

210


From system (8.49), we obtain: # iαb11 αb22 U = = − , ,α (8.50) μ η Then, we only consider the solution exponentially decreasing with respect to y corresponding to the solution η = η1 given by (8.47) (provided μ > 0; the case μ < 0 uses η2 and leads to an analogous conclusion). From here, we will consider separately the two boundary conditions (8.39) and (8.40). 0

(Ux0 , Uy0 , U30 )

"

Case of a free edge Let us prove that the limit problem in a point where the boundary is free (Neumann boundary condition (8.40)) does not satisfy the Shapiro– Lopatinskii condition. Indeed, we have γαβ (u0 ) = 0 (according to (8.43)), and so that T αβ = 0 for all (α, β). Then, the Neumann condition (8.40) is automatically satisfied. We easily exhibit a non trivial solution (taking α = 1) of the form 8.48): ! . " # ib12 − b11 b22 − b212 ib11 b22 0 μ f ixed, η = μ , U = − , ,1 (8.51) b11 μ η with μ > 0 (which gives oscillations on the free edge) and Re(η) < 0 (which gives an exponential decay into the interior of the domain) as searched. This proves that the Shapiro–Lopatinskii condition is not satisfied at a point of the boundary where the shell is free (Neuwmann condition (8.40)). Case of a fixed edge Oppositely when the edge is fixed, u0 must satisfy the boundary condition (8.39) which writes: u0x (x, 0) = u0y (x, 0) = 0 for all x of the boundary. Considering their form (8.48) and (8.50) and the fact that b11 > 0 and b22 > 0, we get α = 0 and u0x and u0y vanish everywhere. Consequently, we have u03 = 0 and finally: u0 = 0 (8.52) Therefore, the only solution of the form (8.41) is zero; the Shapiro–Lopatinskii condition is satisfied for the limit membrane problem for an elliptic shell fixed on all its boundary.

8.5 8.5.1

Sensitive Problems Elliptic Shell Clamped by a Part Γ0 of the Boundary and Free by the Rest Γ1

Very few is known concerning elliptic problems with boundary conditions not satisfying the Shapiro–Lopatinskii condition, and there is no general theory concerning them. Shell theory is, up to our knowledge, the only physical theory

8.5 Sensitive Problems

211

where they are naturally involved. We saw in the last section that the Shapiro– Lopatinskii condition is not satisfied on a free boundary. Specifically, the membrane problem is of total order 4. When it is elliptic (i.e. for elliptic surfaces) the number of boundary conditions should be 2. On a fixed boundary Γ0 , they are: u1 = u 2 = 0

(8.53)

(note that the trace of u3 does not make sense in the membrane framework), which satisfy the Shapiro–Lopatinskii condition. On a free boundary Γ1 , the conditions are T αβ nβ = 0 (8.54) which does not satisfy the Shapiro–Lopatinskii condition. Specifically, considering the problem (principal terms) on a half-plane, there are solutions (sinusoidal in the directions of the boundary and exponentially decreasing inwards the domain) with u = 0 and γαβ = 0 (and then T αβ nβ = 0, so that (8.54) is automatically satisfied). On the other hand, (8.53) constitutes Cauchy conditions for the rigidity system γαβ (u) = 0 (of total order 2). According to the uniqueness theorem for elliptic Cauchy problems (Holmgren, Calderon, see Section 8.2.4), an elliptic shell is inhibited (or geometrically rigid) provided that it is fixed (or clamped) on a part of (or the whole) boundary. When the boundary is everywhere free, the shell is not inhibited. Coming back to the inhibited shells, we see that when the whole boundary is fixed, the membrane problem is classical (the boundary conditions satisfy the Shapiro–Lopatinskii condition). But when a part of the boundary Γ0 is fixed whereas another one (Γ1 ) is free, the boundary conditions satisfy the Shapiro–Lopatinskii condition on Γ0 , not on Γ1 . The membrane problem is out of the classical theory of elliptic boundary problems and is called sensitive for reasons which will be later self-evident. To fix ideas (but other dispositions may be considered as well), we shall consider an elliptic shell defined in the plane of the parameters on Ω, the fixed and the free boundary being disposed as in Fig. 8.8.

Ω

Γ0

Γ1 Fig. 8.8. Disposition of Γ0 and Γ1

212


Let us consider formally the variational formulation of the membrane problem: Find u ∈ Vm such that am (u, v) =< f, v > ∀ v ∈ Vm

(8.55)

where Vm is the completion of the “Koiter space” V with the energy norm of (2.6). It is not very hard to see that this space is “very very large” (its topology is “very very weak”). Indeed, as Shapiro–Lopatinskii condition is not satisfied on Γ1 , we may construct corresponding solutions with u = 0 and γαβ (u) = 0 which are rapidly oscillating along Γ1 and exponentially decaying inwards Ω. This is only concerned with the higher order terms. When taking into account, lower order terms (which are “small” for rapidly oscillating functions), we see that we may have “large u” with “small γαβ ” and then small membrane energy. Accordingly, the dual space Vm where f must be taken for (8.55) to make sense is “very small”, and in general f is out of Vm . Another way to be convinced of the pathological character of the problem (8.55) is to write it in the local formulation of the membrane system: ⎧ ⎨ −Dα T αβ = f β (8.56) ⎩ −bαβ T αβ = f 3 where T αβ are taken as unknowns (this is the tension system, Sect. 1.7.2). Note that according to (8.55) they are elements of L2 (Ω). As we saw in Section 1.7.2, (8.56) is an elliptic system of total order 2, so that the two conditions (8.54) constitute Cauchy data on Γ1 . As we know, this problem is not well-posed and the solution does not exist in general. It only exists for very restricted classes of functions f , not containing the space D of test functions of distributions. This amounts to saying that the space Vm dual of Vm does not contain D . Accordingly, the energy space Vm is not a space of distributions. The above property originates the term “sensitive” and can be summarized in the following result. Result 8.5.1. A sensitive problem is unstable, and very small and smooth variations of f (even in D(Ω)) induce modifications of the solution which are large and singular (out of the space of distributions2 ). 8.5.2

Qualitative Description of the Solution of Sensitive Problems

Coming back to the Koiter problem for ε > 0 in the sensitive case, our aim is not really to describe the limit problem (which in general, has no solution in distribution spaces3 ) but rather to give a good description of the solution uε 2

3

A classical property of distributions in bounded domains Ω is that the space D(Ω) of all distributions (generalized functions in the sense of Schwartz) on Ω coincides with the union of all the Sobolev spaces H s (Ω) with real s (including negative s). In particular, the space Vm where there is always a limit is not a space of distributions.


213

for very small values of ε. This is what we shall try to do. We shall see that heuristic considerations allow to construct a simplified model accounting for the main features of the problem. A more explicit version of this work may be seen in [45] and [46]. A first remark in this context is that sensitive problems may be considered as intermediate between “inhibited” and “non-inhibited”. Indeed, “inhibited” means that v ∈ V and γαβ (v) = 0 implies v = 0 (see Definition 2.2.2), whereas “non-inhibited” means that there exists at least one v = 0 such that γαβ (v) = 0. Sensitive problems enter in the class “inhibited”, but there exist some v = 0 in V with γαβ (v) very small. This remark is the starting point of our heuristic procedure. To minimize the energy: am (v, v) + ε2 af (v, v) − 2 < f, v >

(8.57)

it is clear that we may reason as in non-inhibited problems: the solution with small ε “avoids” the (large) membrane energy am , so that roughly speaking, solutions for small ε satisfy: γαβ (v) = 0 or at least γαβ (v) 0 is very small with respect to v ∈ V (8.58) Obviously, it is impossible to impose the condition (8.53) on Γ0 with the strict system (8.58) as the equations imply v = 0. We shall see that it is possible to construct functions satisfying (8.53) and (8.58) in the non-strict sense of “very small γαβ involving a boundary layer in the vicinity of Γ0 ”. To this end, we shall first construct a set of functions v with one component vanishing on Γ0 . Choosing (for instance) the normal component, we define G0 = {v; vn = 0 on Γ0 }

(8.59)

The regularity is not specified as we shall later take the completion: we may consider C ∞ functions for instance. The local structure of a function w of G0 is easily obtained taking the Fourier transform in the tangential direction x, in local coordinates (x, y) as indicated on Fig. 8.9. According to general features of microlocal analysis, taking only the principal orders and localizing in a neighborhood of a point O, we have (see (8.37)): vˆ(ξ, y) = vˆx (ξ, 0)c eα|ξ|y

(8.60)

where α > 0 and c is some triplet (note that v and vˆ have three components, whereas the tangential component vˆx is scalar). The expression (8.60) is easily obtained by solving locally the Cauchy problem with the data vx and vy , on account of the fact that vy = 0; moreover, it constitutes a condition satisfying the Shapiro–Lopatinskii condition for the system γαβ (v) = 0 so that the term involving e−α|ξ|y vanishes. Obviously (8.60) as well as the forthcoming developments are concerned with |ξ| ! 1, disregarding smooth parts; this is consistent with the very singular foreseen asymptotic behavior. When taking into account the system associated with am (u, v) instead of the subspace (8.59) we deal with the membrane system of total order 4, which is

214


y O

Ω

x

Γ0

Γ1 Fig. 8.9. Local coordinate system

composed, as we know, of the membrane tensions (which is the adjoint of the rigidity system) and the tensions expressed in terms of displacements. This gives double characteristics as we saw in Chapter 1 (Section 1.7). Consequently, in addition of the local solutions in e±α|ξ|y there are in ye±α|ξ|y . The two with the sign “−” are exponentially decreasing toward the domain, then accounting for boundary layer terms (for |ξ| ! 1). It is then easily seen that they may be added to (8.60) in order to satisfy both conditions (8.53). As a result, each v ∈ G0 may be modified into another function denoted by v a which differs from v only on a boundary layer of thickness |ξ|−1 (then disregarding smooth terms) and such that v a is locally a solution (up to lower order terms for large |ξ| and small ε) of the whole equation issued from (8.57). The membrane energies of v and va are, respectively, zero and a very small quantity (which may be asymptotically evaluated) concentrated inside the layer. 8.5.3

Heuristic Treatment of the Problem

The idea of our heuristic treatment of the problem consists in solving the minimization of (8.57) in the class of function v a for v ∈ G. For more details, the reader can refer to [45] and [46]. To evaluate the bending energy, we also need asymptotic structure of the v3 component in a neighborhood of the free boundary Γ1 . Considering the leading order terms in γαβ (v) = 0 we see that v ∈ G0 implies that v3 satisfies a secondorder elliptic equation. Taking the trace on Γ1 as a (Dirichlet non-homogeneous) boundary condition, the asymptotic behavior of the “local Fourier transform” is vˆ3 (ξ, y) = vˆ3 (ξ, 0) eβ|ξ|y

(8.61)

in the vicinity of Γ1 where y denote the local inner normal component. It appears that the bending energy is concentrated in the vicinity of Γ1 . It then appears that, within our approximation, the expression to be minimized (8.57) becomes: |P (vt |Γ0 )|2 dS + ε2 |Q (v3 |Γ1 )|2 dS − 2 f vdx (8.62) Γ0

Γ1

Ω


215

in the space G0 . Here, P and Q are expressions acting upon the traces of vt on Γ0 and of v3 on Γ1 accounting for the membrane and bending energies, respectively. They are elliptic pseudo-differential operators of order 1/2 and 3/2, respectively. At the present state, (8.62) exhibits the non-local character of the asymptotic behavior: the solution is a function v defined all over Ω, whereas the energy forms only deal with the traces of Γ0 and Γ1 . Moreover, the traces vt |Γ0 and v3 |Γ1 are not independent: according to the structure of G0 (see 8.59), a function v ∈ G0 is defined by the trace of v3 on Γ1 (denoted by w) by solving the problem ⎧ γαβ (v) = 0 on Ω ⎪ ⎪ ⎪ ⎪ ⎨ vn = 0 on Γ0 (8.63) ⎪ ⎪ ⎪ ⎪ ⎩ v3 = w on Γ1 where w (defined on Γ1 ) is the data. Indeed, γαβ (v) = 0 is an elliptic system of total order 2 and the boundary condition vn (as well as v3 ) satisfies the Shapiro– Lopatinskii condition. Problem (8.63) is “well-posed” (perhaps the existence holds true up to a finite number of compatibility conditions, and up to a kernel of finite dimension but it is easily seen that this has no consequence in the sequel). Moreover, and this point is essential, it follows from standard regularity theory for elliptic systems that v is of class C ∞ on Ω ∪ Γ0 for any w (smooth or not). It follows that for v ∈ G0 , we have: vt |Γ0 = Sv3 |Γ1

(8.64)

where S is a smoothing operator (sending any distribution on Γ1 into C ∞ (Γ0 )). It follows that (8.62) may be written only in terms of the trace w of v3 on Γ1 : 2 2 |P Sw| dS + ε2 |Qw| dS − 2 F, w (8.65) Γ0

Γ1

where F is a functional on w defined in a obvious way from f . Under the form (8.65) for ε > 0, the problem is a classical variational problem continuous and coercive on H 3/2 (Γ1 ). The corresponding equation (there is no boundary conditions as Γ1 is a manifold without boundary) is: (A + ε2 B)wε = F with (the symbol

∗

(8.66)

denotes the adjoint): A = S ∗ P ∗ P S,

B = Q∗ Q

(8.67)

As the adjoint of a smoothing operator is also smoothing, we see that A is smoothing, whereas B is pseudo-differential elliptic of order 3. Essentially, they act on the fourier transforms by multiplication by e−|ξ| (or by any other function decaying at infinity faster than algebraically) and |ξ|3 , respectively, so that (8.67) amouts essentially to:

216


e−|ξ| + ε2 |ξ|3 w ˆ ε (ξ) = Fˆ

(8.68)

Taking for instance F = δ(x), i.e. Fˆ (ξ) = 1, we get: w ˆ ε (ξ) =

1 e−|ξ| + ε2 |ξ|3

(8.69)

where it appears that w ˆε (ξ) is mainly formed by components |ξ| O(log(1/ε)), and accordingly w(x) consists mainly of oscillations of wave length O((log(1/ε))−1 ).

w(ξ) ˆ e|ξ|

ε−2|ξ|−3

O log ε−1

|ξ|

Fig. 8.10. Heuristic pattern of w(ξ) ˆ

Coming back to uε (solving (8.63) and on account of (8.61)) the main features of uε with small ε are: • uε consists essentially of oscillations of wave length O((log(1/ε))−1 ) along Γ1 , • the oscillations decay as eβ|ξ|y inwards Ω, • the whole structure of the solution is highly non-local; it recalls the “local non-uniqueness” allowed by the fact that the Shapiro–Lopatinskii condition is not satisfied on Γ1 .

8.6

Conclusion

In this chapter, after recalls and generalities on boundary conditions for equations and systems (Cauchy’s problem and boundary value problems), we considered sensitive problems which are less known are more complex. They correspond

8.6 Conclusion

217

to elliptic shells with a non-satisfied Shapiro–Lopatinskii condition on the free edge. In that case, a pathological behavior emerges: there appear oscillations along the free edges, in the direction tangent to the free boundary, and exponentially decreasing toward the interior of the domain. For these sensitive problems, the limit solutions are out of the distributions space and are analytical functionals , having “infinite frequency and amplitude”. Then, we considered the case of elliptic membrane problems, which are illposed or sensitive, as soon as a part of the boundary is free. Indeed, we proved that the membrane limit problem with free boundary conditions does not satisfy the Shapiro–Lopatinskii condition, and so that enters in the framework of sensitive problems. Then, when ε 0, a heuristic description of the instability appearing proved that the characteristic frequency (i.e., the inverse of the wave length) of the oscillations which appear on the free boundary is of O(log(1/ε)).

9 Numerical Simulations for Sensitive Shells

9.1

Introduction

Chapter 7 was devoted to the study of singularities for elliptic well-inhibited shells, or equivalently elliptic shells clamped all along their lateral boundary. In chapter 8, we considered elliptic shells having a part of their boundary which is free. In that case, they are ill inhibited (and even “sensitive”, i.e. Vm is not a space of distribution), and the problem is more complex. A pathological behavior emerges progressively when ε tends toward zero. A complexification phenomenon (large oscillations corresponding to a new kind of instability) appears on the free boundary. As already quoted in chapter 8, very little is known on the limit problem for sensitive shell problems. The “solutions” out of the distribution space are far from admitting a qualitative description as the “classical” ones of the clamped case. We only established, using heuristic considerations, that the wave length of the oscillations appearing on the free boundary is proportional to [log(1/ε)]−1 . But these properties were never proved accurately for an arbitrary domain, and particularly for shells, so that they can only be considered as reasonable conjectures needing explicit description in more general situations. It is the goal of this chapter, which is devoted to numerical computations of various sensitive shell problems, and to comparisons with the (few) existing theoretical results. As very little is known on sensitive shells, we have increased the numerical examples to understand the influence of the parameters, in particular the geometric ones (the numerical simulations are still performed with adaptive anisotropic meshes). The numerical results obtained will be compared, on one hand to the qualitative results of chapter 8 for sensitive problems, and on the other hand to the numerical results obtained in chapter 7 for well-inhibited shells. Finally, in the last section, we study the influence on the resulting singularities of the localization of the loading domain near a clamped or a free boundary. E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 219–233. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

220


9.2

First Examples of Numerical Computations for Sensitive Problems (Ill-Inhibited Shells)

Let us consider the same shell as in chapter 7 subjected to the same loading f 3 applied on the domain (see Fig. 9.1) 1 1 1 1 F = (y 1 , y 2 ) ∈ − ; × − ; 2 2 2 2 We recall that the middle surface S of the shell is defined by the mapping ψ(y 1 , y 2 ) = (y 1 , y 2 , (y 1 )2 + (y 2 )2 )

(9.1)

with the same domain of definition of the local variables: Ω = (y 1 , y 2 ) ∈ [−1, 1] × [−1, 1] . The shell considered is defined by its middle surface S and its thickness ε. y2

B

1

C

Ω

1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 O 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 F 0000000000 1111111111 0000000000 1111111111

−1

F

−1

A

y1

1

D

Fig. 9.1. Considered problem

2 1.5 1 0.5 0 -0.5

1 0.5 y2

0 -0.5 -1

-1

-0.5

0

0.5

1

1.5

y1

Fig. 9.2. Deformed shape of the shell for ε = 10−4

9.2 First Examples of Numerical Computations for Sensitive Problems

221

The only difference with the well-inhibited shells considered in chapter 7 is that the part CD of the boundary is now free (see Fig. 9.1). In that case, we will see that the solutions are very different from those obtained in the well-inhibited case. The deformed shape of the shell is represented on Fig. 9.2 for a fixed relative thickness ε = 10−4. Some oscillations along the free boundary CD are visible. The corresponding displacement uε3 may be seen in figure 9.3. 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008

-1 -0.5 0 0.5 0

-0.5

1 -1

1

0.5

Fig. 9.3. uε3 on the whole domain for ε = 10−4

0.05

0

-0.05 0.9

Fig. 9.4. Final mesh for ε = 10

−4

0.95

1

Fig. 9.5. Zoom of the final mesh for ε = 10−4 in the box [0.9, 1] × [−0.05, 0.05]

The final mesh of the adaptive numerical procedure is plotted on Fig. 9.4. We observe that the mesh is mainly refined along the free boundary. In this zone, the mesh is isotropic as represented on Fig. 9.5. The general trends for uε3 described in chapter 7 may be recognized here. The oscillations of uε3 on the free edge y 1 = 1 appear clearly in figure 9.6. This

222


0.008

0 -0.8

-0.6

-0.4

-0.2

0 -0.001

0.004

-0.002

0.002

-0.003

u3

u3

-1 0.006

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1

-0.004

1

-0.002

-0.005

-0.004

-0.006

-0.006

-0.007

-0.008

-0.008 2

y

y1

Fig. 9.6. uε3 on the line y 1 = 1 for ε = Fig. 9.7. uε3 on the line y 2 = 0 for ε = 10−4 10−4

Fig. 9.8. Interpolation of uε3 by an exponential function on the line y 2 = 0 for y 1 ∈ [0.43; 1]. Case of a free boundary CD.

instability is so important that the two other singularities present in the wellinhibited case (internal layer and logarithmic point singularity) are invisible in comparison. This is put in a prominent position in figure (9.7) where uε3 is plotted on the line y2 = 0. The Heaviside-like singularity disappeared completely. Figure 9.8 shows that uε3 is exponentially decreasing toward the interior of the domain from y 1 = 0.43 to y 1 = 1. All these simulations are in good agreement with the theory developed in section 8.4 when the Shapiro-Lopatinskii condition is not satisfied. Finally, let us notice that all the numerical simulations have been performed in this section with the same thickness ε = 10−4. However, it is clear that the results directly depend on ε. This is the subject of next section.

9.3

Asymptotic Process when ε Tends to Zero

First, let us study the behavior of uε3 on the whole domain for different values of ε (Figs. 9.9 to 9.12).

9.3 Asymptotic Process when ε Tends to Zero

223

0.001 0.0008 0.0006 0.0004 0.0002 0 -0.0002 -0.0004 -0.0006 -0.0008 -0.001

6e-005 5e-005 4e-005 3e-005 2e-005 1e-005 0 -1e-005 -2e-005 -3e-005 -4e-005

1

1

0.5

0.5 -1

-1

0

-0.5 0

0

-0.5 0

-0.5 0.5

-0.5 0.5

1 -1

1 -1

Fig. 9.9. uε3 on the whole domain for ε = Fig. 9.10. uε3 on the whole domain for ε = 10−2 10−3

0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008

0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 1

1

0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 9.11. uε3 on the whole domain for ε = Fig. 9.12. uε3 on the whole domain for ε = 10−4 5 10−5

For rather large ε, from 10−2 to 10−3 , we can notice that the singularities around the loading domain are still present as in the well-inhibited problem (Figs. 9.9 and 9.10). For ε = 10−2 , these singularities are as large as the oscillations due to the free edge. When ε decreases, they progressively “disappear”. In fact, they are still present, but they are negligible compared to the instability arising from the free edge. For ε = 10−4, the are about 40 times smaller than the oscillations, so that they are visible neither in figure 9.11, nor in figure 9.12 for ε = 5 10−5. In other words, for the same applied forces, the corresponding displacements will be much larger in the sensitive case than in the classical well-inhibited one. Thus, free boundary conditions (even only on a small part of the boundary) leading to a sensitive problem, imply an important weakening of the structure. In figures 9.13 to 9.15, we compare the oscillations on the free edge for different values of ε from 10−2 to 10−4 . We observe that the number of oscillations increases when ε tends toward zero, but the phenomenon is quite slow. We recall that the number of oscillations behaves like log(ε−1 ), which corresponds to a balance between the bending terms and the exponential behavior (see section 8.5 of chapter 8 for more details). In order to put in a prominent position such a similar behavior, we plot on figure 9.16 the number of oscillations with respect to log(ε−1 ). As the number of oscillations is a discrete function, we can only see the steps corresponding to the formation of a new oscillation, which is a slow phenomenon. However the general trend is still like log(ε−1 ), or more precisely

224


8.E-05

0.001 0.0008

6.E-05 0.0006

4.E-05

0.0004 0.0002

u3

u3

2.E-05

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.E+00 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.0002

1

-0.0004

-2.E-05

-0.0006

-4.E-05 -0.0008

-6.E-05

-0.001

2

2

y

y

Fig. 9.13. uε3 on the line y 1 = 1 for ε = Fig. 9.14. uε3 on the line y 1 = 1 for ε = 10−3 10−2 10

0.008

9

0.006 8

u3

0.002

0 -1

-0.8

-0.6

-0.4

-0.2

0

-0.002

0.2

0.4

0.6

0.8

1

number of oscillations

0.004 7

6

5

y = 0.8607x - 0.8819 2 R = 0.9349

4

3

-0.004 2

-0.006

1

0

-0.008

0

2

y

2

4

6

8

10

12

ln(1/ε)

Fig. 9.15. uε3 on the line y 1 = 1 for ε = Fig. 9.16. Number of oscillations vs. 10−4 log(ε−1)

like log(Kε−1 ) for some K positive (Fig. 9.16). However this is equivalent1 as we are concerned with the asymptotic behavior when ε tends to zero. On the other hand, in figure 9.17 are represented the variations of the normal displacement uε3 on the line y 2 = 0 for different values of ε. We observe that the magnitude of uε3 increases very quickly when ε decreases and tends toward zero. Moreover the zone in large bending affected by this singularity (near the free edge) increases as well. Indeed, for ε = 10−2 only the vicinity of the free edge is in bending state. Oppositely, for ε = 10−4 , the exponential behavior has propagated inside the domain and the Heaviside-like singularity has totally disappeared at y 1 = 0.5 (in comparison with the large bending due to the free edge). The numerical results are once again in good agreement with the general trend predicted when the Shapiro-Lopatinskii condition is not satisfied (see chapter 8). For large ε, the instability on the free edge is as important as the singularities around the loading domain boundary. But when ε tends toward zero, uε3 tends to something hard to imagine: a function having oscillations on the free edge with a frequency and with an amplitude growing to infinity. However, as the phenomenon is very slow (depending on log(ε−1 )), it is hard to reach small wave lengths. This would mean to perform simulations for very small values 1

Because of the classical property of log-function, the additive term logK has no influence when ε tends to zero.

9.4 Influence of the Free Edge Length

225

0.0002

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u3

-0.0002

-0.0004

10-2 10-3 10-4

-0.0006

-0.0008

-0.001 1

y

Fig. 9.17. uε3 on the line y 2 = 0 for different ε

of ε (smaller than 10−7 ) which cannot be attempted because of numerical limitations (especially locking).

9.4

Influence of the Free Edge Length

Let us consider in this section the same elliptic problem as previously. We just change the length of the free edge: a part of the boundary CD with length λ is free (Fig. 9.18). Figures 9.19 to 9.22 show the evolution of the normal displacement uε3 when λ increases2 for ε = 10−4 fixed. When λ is small (see Fig. 9.19), the problem is close to the well-inhibited one addressed in chapter 7. We still observe the existence of internal layers due to the discontinuity of the loading, and of logarithmic point singularities at the corners. On the other hand, a new type of singularity appears on the free edge: oscillations near the middle of the side. When λ increases, these oscillations become so large that the other singularities around the loading domain become negligible in comparison. They correspond to oscillations in the y 2 direction, which decrease exponentially in the y 1 direction. It is important to note that the phenomenon is not local: it is present in a large part of the shell, and propagates when λ increases. The scaled normal displacement uε3 is presented in Figure 9.23. It clearly shows that uε3 behaves like an exponential function in a part of the domain, which becomes larger when λ increases: it starts near from y 2 = 0.8 for λ = 0.2, and near from y 1 = 0.4 for λ = 2. For λ = 0.5, we can still observe the singularity around the boundary of the loading domain (at y 1 = −0.5 and y 1 = 0.5) due to the discontinuity of f 3 . Oppositely, for λ = 1.5, the exponential part of uε3 propagated, and it is clearly impossible to see the singularity at y 1 = −0.5 or y 1 = 0.5. 2

The case λ = 2 corresponds to a boundary CD completely free, as considered previously.

226

9 Numerical Simulations for Sensitive Shells y2 B

1

C

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 O 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111

−1

y1

−1

A

λ

1

D

Fig. 9.18. Considered problem

0.0001

0.001

5e-005

0.0005

0

0

-5e-005

-0.0005

-0.0001

-0.001

-0.00015

-0.0015 1

1

0.5

0.5 -1

-1

0

-0.5 0

0

-0.5 -0.5

0

-0.5 0.5

0.5

1 -1

1 -1

Fig. 9.20. uε3 for λ = 0.5

Fig. 9.19. uε3 for λ = 0.2

0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008

0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 1

1 0.5

0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 9.21. uε3 for λ = 1.5

-1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 9.22. uε3 for λ = 2

On the other hand, we can also observe that, on the free edge, the number of oscillations increases with λ (Figs. 9.24 and 9.25). This result was predictable according to the theoretical development of chapter 8. To finish, we shall see what happens when more boundaries are free. Figures (9.27) and (9.28) correspond respectively to two and three free edges. In both cases, as for the previous problem, we can observe some oscillations on the free edges. Their amplitudes reach also their maximum at the middle of the free edge.

9.4 Influence of the Free Edge Length

227

0.2

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u3/u3max

-0.2

-0.4

λ=0.2 λ=0.5 λ=1.5 λ=2

-0.6

-0.8

-1

y1

Fig. 9.23. Scaled displacement uε3 /uε3max on the line y 2 = 0 for various values of λ 0.0015

0.008

0.006 0.001 0.004 0.0005

u3

u3

0.002

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.002 -0.0005 -0.004 -0.001 -0.006

-0.0015

-0.008

y2

y2

Fig. 9.24. Three extrema for uε3 on the Fig. 9.25. Five extrema for uε3 on the line line y 1 = 1 for λ = 0.5 y 1 = 1 for λ = 1.5 0.008

0.006

0.004

u3

0.002

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.002

-0.004

-0.006

-0.008

y2

Fig. 9.26. Seven extrema for uε3 on the line y 1 = 1 for λ = 2

228


150 100 50 0 -50 -100 -150 -200 -250

500 400 300 200 100 0 -100 -200 -300 -400 -500 1

1

0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 9.27. uε3 for two free edges

0.5 -1

0

-0.5 0

-0.5 0.5

1 -1

Fig. 9.28. uε3 for three free edges

For three free edges, we can observe that there are few oscillations on the middle free edge, due to the symmetry of the problem.

9.5

Energy Repartition in Sensitive Problems

In this section, we propose first to compute separately the two energy surface densities for the sensitive problem considered in section 9.2, with the edge CD completely free and for a fixed thickness ε = 10−4 . The repartition of the membrane energy surface density Ems (Fig. 9.29) is quite similar to the one obtained for the well-inhibited problem (see Fig. 7.29 of chapter 7). The membrane energy concentrates at the 4 corners near the logarithmic point singularities (even if we cannot see them in the graph of uε3 because of the more important instability on the free edge). The membrane energy is also high near the boundary of the loading domain corresponding to the internal layers. The main difference with the well-inhibited case is a higher membrane energy around the two edges perpendicular to the free edge. The repartition of the bending energy surface density Ebs is very different compared to the well-inhibited case (see Fig. 7.30). It concentrates only near the

Fig. 9.29. Ems for ε = 10−4

Fig. 9.30. Ebs for ε = 10−4

9.6 Influence of the Loading Domain

229

free edge (Fig 9.30). We observe seven zones where Ebs is larger than on the rest of the shell. They correspond to the seven extrema of the oscillations of uε3 on the free edge (see Fig. 9.15). Between each extremum, the bending energy surface density is lower and corresponds to an inflexion point, where the second derivative of uε3 with respect to y 2 vanishes. On the rest of the shell, the bending energy surface density is much lower. To finish, let us study the evolution of the percentage of bending energy surface density Ebs compared to the total energy surface density (Ems + Ebs ) when ε decreases (Fig. 9.31). For ε = 10−2, the bending energy Ebs is predominant (more than 70%) at the center of the free edge and near the internal layers parallel to the free edge. It is also present (between 20% and 50%) at the corners of the loading domain and near the internal layers perpendicular to the free edge. The rest of the shell is dominated by membrane energy.

Fig. 9.31. Evolution of the percentage of Ebs for various ε from 10−2 to 10−4

As long as ε decreases, the proportion of Ebs increases along the free edge and inside the shell, close to the free edge. Oppositely, it decreases in the internal layer which becomes thinner. There is a main difference between the singularities existing in the well-inhibited case (see Figs. 7.31 and 7.32) and the instabilities which appear near the free edge in the sensitive problem (Fig. 9.31). Whereas the internal layers become thinner when ε tends toward zero, the complexification phenomenon propagates inside the rest of the domain.

9.6

Influence of the Loading Domain

In this section, we will consider the same problem as previously but with a different loading. With this new loading, we shall illustrate again the qualitative and quantitative difference existing between well-inhibited and ill-inhibited elliptic shells. Let us consider again the shell defined by the mapping (9.1) but with a different loading domain F = [0.25, 1] × [0.25, 1] (see Fig. 9.32). The surface loading is still normal and constant f 3 = 10ε M P a. As previously, the shell is assumed to be clamped along the edges AB, BC and AD. The last edge CD will be either clamped that corresponds to the

230

9 Numerical Simulations for Sensitive Shells y2

1111111 0000000 0000000 1111111 0000000 1111111 F 0000000 1111111 0000000 1111111 0000000y 1111111 0000000 1111111

B

C

2

= 0.25

O

y1

Ω

D

y 1 = 0.25

A

Fig. 9.32. Loading domain F

well-inhibited case or free that corresponds to the ill inhibited or sensitive case. These two different boundary conditions will be addressed separately. Figures 9.33 and 9.34 represent the displacement uε3 on the whole domain (in meters) for the two boundary conditions on CD (clamped or free). On Fig. 9.33, we observe a logarithmic singularity at the four corners of the loading domain F , even if that at point (0.25,0.25) is very weak because the principal curvatures are nearly equal at this point (and the logarithmic point singularity must vanish according to section 7.2.3). In the sensitive case (Fig. 9.34), the displacements are much larger along the free edge where tangential oscillations appear. 0.08

0.0004

0.06

0.0003

0.04

0.0002

0.02

0.0001 0

0

-0.0001

-0.02

-0.0002

-0.04 -0.06

-0.0003

1

1 0.5

-1 0

-0.5 0

-0.5

0.5 1 -1

0.5

-1 0

-0.5 0

-0.5

0.5 1 -1

Fig. 9.33. Displacement uε3 on the domain Fig. 9.34. Displacement uε3 on the domain Ω for ε = 10−4 when the boundary CD is Ω for ε = 10−4 when the boundary CD is free clamped

Figure 9.35 represents the displacement uε3 in both cases (well-inhibited and sensitive). We observe a large difference of amplitude: the displacements in the sensitive case are about 50 times larger than in the well-inhibited case. The shapes of the displacements are also very different. This is visible more clearly in figures 9.36 to 9.38, where the normalized displacements (in the well inhibited and in the sensitive case) are plotted on the same figure.

9.6 Influence of the Loading Domain

231

0.01

0.008

Well-inhibited case Sensitive case

0.006

0.004

u3

0.002

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.002

-0.004

-0.006

-0.008

2

y

Fig. 9.35. Displacement uε3 on the line y 1 = 0.75 for ε = 10−4 1.25

Well-inhibited case

1

Sensitive case 0.75 0.5

u3

0.25 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.25 -0.5 -0.75 -1 -1.25

2

y

Fig. 9.36. Normalized displacement uε3 /uε3max on the line y 1 = 0.75 for ε = 10−4 1.2


1

0.8

0.6

u1

0.4

0.2

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.2

-0.4

-0.6

y2

Fig. 9.37. Normalized displacement uε1 /uε1max on the line y 1 = 0.75 for ε = 10−4

232

9 Numerical Simulations for Sensitive Shells 1.5


1

u2

0.5

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5

-1

-1.5 2

y

Fig. 9.38. Normalized displacement uε2 /uε2max on the line y 1 = 0.75 for ε = 10−4

On figure 9.36, we can see clearly in the well-inhibited case, the Heaviside jump at y 2 = 0.25 which corresponds to the singularity due to the singular loading. In the sensitive case, we observe oscillations which have nothing to do with the wellinhibited case, even if the Heaviside jump at y 2 = 0.25 is still present but totally hidden. This result is very different from that obtained for parabolic shells in section 5.7, for which having a free or clamped edge does not modify drastically the shape of the displacements. For the elliptic shells considered here, the shape of the displacements are completely different (in both directions), regardless of whether we have a free edge or not. The same phenomenon can also be observed on Figs 9.37 and 9.38 for uε1 and uε2 .

9.7

Conclusion

Numerical simulations performed with the adaptive mesh procedure allowed to visualize qualitatively et quantitatively the particular behavior of sensitive shells when a part of the boundary is free. As predicted theoretically in chapter 8, when ε tends to zero, instabilities appear along the free edges, leading to displacements which are oscillating in the direction tangent to the free boundary, and exponentially decreasing toward the interior of the domain. We confirm that the frequency3 and the amplitude of these oscillations tend to infinity, with a characteristic frequency of O(log(1/ε)), as predicted theoretically with heuristic consideration on simplified problems. Moreover, they progressively spread in all the shell domain. On the other hand, we observe that the singularities described in chapter 7 for well-inhibited shells still exist, but are negligible in comparison. When ε tends to zero, bending energy concentrates at the neighborhood of the free boundary, whereas membrane energy is localized in the internal layers and at the point singularities around the corners of the loading domain. 3

Corresponding to the inverse of the wave length.

9.7 Conclusion

233

Finally, during the adaptive mesh procedure, we observe that the mesh is refined mainly along the free boundary in a quite isotropic way, in order to describe accurately both the oscillations in the direction tangent to the free boundary and the exponential decrease toward the interior of the domain.

10 Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems)

Introduction In this chapter, we shall see how the numerical anisotropic adaptive procedure of remeshing reacts when the shell is not inhibited. In that case, the middle surface naturally deforms with inextensional displacements, and the Koiter model tends to a pure bending problem when ε tends to zero (see section 2.3.5 of chapter 2). Indeed, in order to minimize elastic energy, the natural trend of the shell (for small ε) is to deform by “pure bendings”: it avoids the (large) membrane energy, and only uses the (small) bending energy. In these very particular deformations, involving inextensional displacements, the asymptotic lines of the surface play a peculiar role, leading to an anisotropic behavior. Indeed, the elements of the subspace G of inextensional displacements (see section 2.2) are solutions of the “rigidity system” (1.44). As we know (see section 1.5.2), this system is of the type of the middle surface. We choose in the sequel examples of hyperbolic surfaces. Obviously, after eliminating one of the unknowns, u3 for instance from the third equation, the rigidity system (1.44) is a hyperbolic system of first order with two equations and two unknowns. The characteristics of the rigidity system are the asymptotic curves of the middle surface. Thus, the characteristics play in some sense the role of “local infinitesimal hinges” which allow infinitesimal rotations along them. This is particularly obvious in the case of a ruled surface, made a of family of straight lines (the generators): if the shell is not inhibited, the natural trend is to deform with infinitesimal rotations around the generators, which preserve the intrinsic metrics of the surface. Clearly, such kind of inextensional displacements is highly anisotropic along the generators. The simplest and clearest example is that of the “ruled quadrics” (the hyperbolic hyperboloid and the hyperbolic paraboloid) which have two families of generators (corresponding to the asymptotic lines). All the isometric deformations are made of combination of the above-mentioned rotations around the two families of generators. According to boundary conditions, one or both rotations may be impeached, giving various specific examples of subspace G (the case when the boundary conditions impeach both rotations everywhere is the E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 235–245. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

236

10 Examples of Non-inhibited Shell Problems

“inhibited” case). The general case (when the characteristics are not straight) is analogous “at the leading order” of deformation, whereas other smoother terms are also present. Thus, we shall encounter several possible situations: • rotations around both families of characteristics are allowed: the shell is “totally non-inhibited” • rotations around only one family of characteristics are allowed: the shell is “partially non-inhibited” • both rotation are impeached: the shell is inhibited. The partially non-inhibited shells, which deform by rotations around one family of characteristics are intermediates between the inhibited and the noninhibited shells. Computing the solution of thin shell problems (inhibited or not) with F. E. methods, it is known that locking occurs when the relative thickness ε decreases, leading to an important underestimate of the displacements. The reader may refer to section 4.2.2 for a general review on the different cases of locking existing and its consequences on shell computation. The numerical examples presented in this chapter will contain in the same time inhibited, partially non-inhibited and totally non-inhibited areas in complex mixed situations (where solutions are hardly predictable theoretically). They will illustrate once again the efficiency and the accuracy of the adaptive anisotropic mesh procedure used for the numerical computations.

10.1

Examples of Partially Non-inhibited Shells

We shall consider the hyperbolic paraboloid of chapter 6 defined by the mapping (6.3) with c = 1: Ψ (y 1 , y 2 ) = (y 1 , y 2 , y 1 y 2 ) The reference domain of the local variables (y 1 , y 2 ) is that of Fig. 10.1. The shell is clamped along the edges BG and BE. The distances EC and GA are, respectively, equal to β and α which are taken as parameters. A constant normal loading is applied in the hatched area. When β and α vary, the (partially or totally) non-inhibited areas vary as well, leading to different results, as explained in the introduction. All the numerical simulations that follow are performed with a thickness ε = 10−4 . 10.1.1

First Case: α = 0 and β = 0.25

In that case, the shell is only non-inhibited in the rectangle [0.75, 1] × [−1, 1] (it is inhibited elsewhere). In fact, referring to the considerations on rotations explained in the introduction, this zone is “partially non-inhibited” because the whole family of characteristics y 2 = const is clamped at y 1 = −1, so that rotations around them are prevented. Only rotations around y 1 = const for 0.75 < y 1 < 1 are allowed.

10.1 Examples of Partially Non-inhibited Shells

237

y2 β

00000C 11111 B 1 10E 10 11111111111111 00000000000000 Ω y1

1

−1 O

α

01 10 10 1010 10

0 G1 000000000000000000y 111111111111111111 000000000000000000 111111111111111111 F 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 −1 D A

2

= − 34

Fig. 10.1. Domain in R2 1 0.5

0.4

0.5

0.3

0

0.2

-0.5 0.1

-1

0

-1

-0.5

0

0.5

1

0.5

0.55

0.6

0.65

0.7

Fig. 10.2. Mesh at the 5th iteration for Fig. 10.3. Zoom of the mesh for ε = 10−4 ε = 10−4

The mesh after convergence of the adaptive remeshing process is represented in Fig. 10.2. We observe that the mesh is mainly refined along the line y 1 = 0.75, which is the border of the non-inhibited zone. In the non-inhibited zone, the mesh is strongly anisotropic (anisotropy factor of about 10, see Fig. 10.3) in order to describe accurately the bendings which occur around the directions y 1 = const, for y 1 ∈ [0.75, 1], and mainly near y 1 = 0.75. Moreover, the mesh is also refined anisotropically along the lines y 2 = −0.75 and y 2 = −1 corresponding to an

238


0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 -0.002

1 0.5 0 -0.5 -1

1 0.5

1

0 -0.5 -1-1

-0.5

0

0.5

1

Fig. 10.4. Displacement uε3 in the domain Ω for α = 0, β = 0.25 and ε = 10−4

0.5 0 -0.5 -1 -1

-0.5

0

0.5

1

1.5

Fig. 10.5. Deformed shape (zoom ×10) for α = 0, β = 0.25 and ε = 10−4

internal and a boundary layer due to the singular loading applied in the inhibited domain. The displacement uε3 represented in the domain Ω, and the resulting deformed shape are plotted in Figs. 10.4 and 10.5. The displacements uε3 are much more important in the non-inhibited region, and also significant in the two layers. This put in a prominent position the capacity of the remeshing procedure to describe automatically phenomena qualitatively and quantitatively very different. 10.1.2

Second Case: α = 0.25 and β = 0.25

The shell is now non-inhibited in a more important area composed of three “subareas”: - in the rectangle [0.75, 1] × [−0.75, 1], only bendings around the directions y 1 = const are allowed, - in the rectangle [−1, 0.75] × [−1, −0.75], only bendings around the directions y 2 = const are allowed, - the square [0.75, 1] × [−1, −0.75] admits bendings in the two directions, and is “totally non-inhibited”. The shell is obviously inhibited in the square [−1, 0.75] × [−0.75, 1]. The converged adapted mesh (at the 5th iteration) is plotted on Fig. 10.6. It is automatically refined along the lines y 1 = 0.75 and y 2 = −0.75, and mainly around the latter which corresponds to the boundary of the loading domain. In these areas, the anisotropy factor of the elements (in the directions y 1 and y 2 ) reaches 20. In the totally “non-inhibited domain”, as bending is allowed in both directions, the mesh is nearly isotropic (see Fig. 10.7). The displacement uε3 is plotted on Fig. 10.8: it is important in the three noninhibited areas, but reaches its maximum in the totally “non-inhibited domain”

10.1 Examples of Partially Non-inhibited Shells

239

1 -0.7

-0.72

0.5

-0.74

0

-0.76

-0.5 -0.78

-1 -1

-0.5

0

0.5

1

Fig. 10.6. Mesh at the 5th iteration for ε = 10−4

-0.8

Fig. 10.7. Zoom of the mesh around the point (0.75, −0.75) for ε = 10−4

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01

1 0.5 1

0 0.5 -0.5

0 -0.5 -1-1

Fig. 10.8. Displacement uε3 on the domain Ω for α = 0.25, β = 0.25 and ε = 10−4

corresponding to the square [0.75, 1] × [−1, −0.75] (which is also subjected to the normal loading). Remark 10.1.1. In figures 10.4 and 10.8, the displacements are very different in the area [−1, 0.75] × [−1, −0.75], which is inhibited in the first case and partially non-inhibited in the second case considered. Obviously, they are much larger in the second case in the non-inhibited area. Finally, elsewhere and in particular in the inhibited area, the resulting displacements are rather similar. On the other hand, it is interesting to observe the repartition of membrane and bending energies as this problem contains inhibited and non-inhibited areas. The percentage of bending energy surface density with respect to the total energy surface density is displayed on Fig. 10.9.

240

10 Examples of Non-inhibited Shell Problems 1

95 90 80 70 60 50 40 30 20 10 1

0.75 0.5

y2

0.25 0

-0.25 -0.5 -0.75 -1 -1

-0.5

0

0.5

1

y1

Fig. 10.9. Percentage of bending energy surface density with respect to the total energy surface density for ε = 10−4

In the inhibited region [−1, 0.75] × [−0.75, 1], whose limit behavior (when ε = 0) is membrane-like, the membrane energy is dominating. Oppositely, bending energy concentrates along the characteristics y 2 = −0.75 and y 1 = 0.75 (separating the inhibited from the non-inhibited areas), which behave like “hinges” and contains about 95% of the total energy surface density. Bending energy is also important in the rest of the “partially non-inhibited” domain (between 70 and 80%), and in particular in that containing the loading. Finally, bending energy is dominating in the “totally non-inhibited region” (between 80 and 95%).

10.2

Propagation of Singularities in the Partially Non-inhibited Regions

In this section, we focus on the propagation of singularities in the partially noninhibited regions. We shall consider the same mapping (Ω, ψ) and boundary condition as previously and only different domains of loading (applied loading in the inhibited area or not). We shall see that the propagated singularities are of very different natures. 10.2.1

Loading Applied in the Inhibited Area

Let us consider the same problem as in section 10.1, with the normal loading f 3 which is now applied on the domain F = [−0.5, 0.25] × [−0.25, 0.5] situated in the inhibited region (see Fig. 10.10). We consider here the case α = β = 0.25. The final mesh and the associated normal displacement uε3 are plotted on Figs. 10.11 and 10.12, respectively.

10.2 Propagation of Singularities in the Partially Non-inhibited Regions

241

y2 β

00000C 11111 0 1 0 1 0 1 1111111111111 0000000000000 0 1 0 1 0 1 0000000 1111111 0 1 0000000 1111111 0 1 F 0000000 1111111 0 1 1 0000000 1111111 −1 1 0 0000000 1111111 0 1 O 0000000 1111111 0 1 0000000 1111111 0 1 B

y1

G

α

1 0 0 1 0 1 0 1 0 1 0 1

E

1

Ω

A

−1

D

Fig. 10.10. Domain Ω of R2 and loading domain F

0.0001 8e-005 6e-005 4e-005 2e-005 0 -2e-005 -4e-005 -6e-005 -8e-005 -0.0001

1 0.5 0 -1 -0.5

-0.5 0 0.5 1

th

-1

uε3

Fig. 10.11. Mesh at the 5 iteration Fig. 10.12. Displacement for a loading infor a loading inside the inhibited area side the inhibited area for ε = 10−4 for ε = 10−4

We observe that the mesh is refined along the characteristics tangent to the loading domain F , like for classical inhibited shell problems. The singularities of uε3 (two orders more singular than f 3 , i.e. in δ ) propagated along the lines y 1 = −0.5, y 1 = 0.25, y 2 = −0.25 and y 2 = 0.5, even in the parts where the shell is partially non-inhibited. However, no pure bending like in section 10.1.2 can be seen (we have only foldings in the four internal layers). Let us observe more precisely the singularity along the line y 1 = −0.5. In the associated layer, the displacements are mainly bendings1 around the characteristic y 1 = −0.5. According to the clamped boundary condition on BE, the

1

They are not pure bending displacements, so that they are not inextensional.

242


0.0001 8e-005 6e-005 4e-005 2e-005 0 -2e-005 -4e-005 -6e-005 -8e-005 -0.0001

1 0.5 0 -1

-0.5

-0.5 0 0.5 1

-1

Fig. 10.13. Displacement uε3 for the inhibited problem with α = β = 0 and for ε = 10−4

Fig. 10.14. Displacement uε3 on the line y 2 = −1. Comparison between “the inhibited problem” (α = β = 0) and the non-inhibited one (α = β = 0.25) for ε = 10−4

rotation around y 1 = −0.5 is impeached, so that for −1 < y 2 < −0.75, the shell reveals as rigid as in the inhibited zone, and the singularity propagates as if the shell was inhibited. On the other hand, the rotations around y 2 = const are possible for −1 < y 2 < −0.75 in that partially non-inhibited area, but are not activated by the loading which creates only bendings around the lines y 1 = −0.5 and y 1 = 0.25. The comparison with “the analogous inhibited problem” (with α = β = 0) confirms that behavior (see Figs. 10.13 and 10.14). The loss of rigidity for α = β = 0.25, leading to non-inhibited areas, is very weak compared to “the inhibited problem” for α = β = 0 (see Fig. 10.14).

10.2 Propagation of Singularities in the Partially Non-inhibited Regions

10.2.2

243

Loading Domain Tangent to the Non-inhibited Area

Let us now move the loading domain to F = [0, 0.75] × [−0.25, 0.5], with the same boundary conditions corresponding to α = β = 0.25. The loading domain is now tangent to the characteristic y 1 = 0.75 which separates the inhibited zone [−1, 0.75]× [−0.75, 1] from the “partially non-inhibited” one [0.75, 1]× [−0.75, 1]. Thus, a part of the internal layer along y 1 = 0.75 will be in the “partially noninhibited” area. Fig. 10.15 represents the last adapted mesh, whereas Fig. 10.16 displays the associated displacement uε3 on the domain Ω.

0.00015 0.0001 5e-005 0 -5e-005 -0.0001 -0.00015 -0.0002 -0.00025 -0.0003 -0.00035

1 0.5 0

-1 -0.5

-0.5

0 0.5

1 -1

uε3

Fig. 10.15. Mesh at the last itera- Fig. 10.16. Displacement on the whole tion when F is tangent to y 1 = 0.75 domain when F is tangent to y 1 = 0.75 for for ε = 10−4 ε = 10−4

In this example, a part of the layer (for y 2 ≤ −0.75) along the line y 1 = 0.75 is in the partially non-inhibited area where the bending rigidity around y 1 = const is very weak. Consequently, the displacements in this part of the layer are much more important for the same loading. To enforce the comparison with the case considered in section 10.2.1, we shall plot on the same figure 10.17, the displacement uε3 on the lines y 2 = −1 and y 1 = 1: - when the loading domain F is inside the inhibited zone (called “internal” loading) - when the loading domain F is tangent to the partially non-inhibited zone (called “tangent” loading). On Fig. 10.17(a), we observe that the normal displacement “explodes” in the partially non-inhibited area, whereas it was nearly zero in the previous case. The displacement uε3 on the line y 1 = 1 is plotted on Fig. 10.17(b): the shapes of the normal displacement are very similar in both cases; only the order of magnitude of uε3 differs according to the weaker stiffness in the non-inhibited area.

244

10 Examples of Non-inhibited Shell Problems 0.0001

0.00005

0 -0.8

-0.6

-0.4

u3

-1

-0.2

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

-0.00005

-0.0001

"Internal" loading "Tangent" loading

-0.00015

-0.0002

y1

(a) On the line y 2 = −1 0.0001

0.00005

0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

u3

-0.00005

-0.0001

-0.00015

"Internal" loading "Tangent" loading

-0.0002

-0.00025

y1

(b) On the line y 1 = 1 Fig. 10.17. Comparison between the displacement uε3 when F is inside the inhibited zone or tangent to the non-inhibited zone for ε = 10−4

10.2.3

Loading Partially Applied in the Non-inhibited Area

Finally, we shall consider a last case, when the loading domain F = [0, 0.8] × [−0.25, 0.5] has a very little part (for y 1 ∈ [0.75, 0.8]) contained inside the partially non-inhibited zone. We will see that the results are drastically different. Figures 10.18 and 10.19 display the adapted mesh at the last iteration and the associated normal displacement uε3 on the domain Ω. We observe that the displacements are larger in the partially non-inhibited area [0.75, 1] × [−1, 1], where rotations around y 1 = const are allowed. Pure bendings are now activated directly by the loading. The layers, due to a discontinuity of the loading, are still present (they are visible on Fig. 10.18), but are of a lower order. Note that in the rest of the non-inhibited domain, the order of the displacements are like in the previous examples, and no pure bendings occur.

10.3 Conclusion

245

0.001 0 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 -0.009

1 0.5 0

-1 -0.5

-0.5

0 0.5

Fig. 10.18. Mesh at the last iteration when F is partially inside the non-inhibited domain for ε = 10−4

10.3

1 -1

Fig. 10.19. Displacement uε3 on the whole domain when F is partially inside the noninhibited domain for ε = 10−4

Conclusion

In this chapter, we addressed various shell problems containing both inhibited and non-inhibited areas. We saw in particular that the results are very different whether the loading domain is inside the inhibited zone, or inside the (partially) non-inhibited zone. When the loading is entirely inside the inhibited zone, the shell behaves as a totally inhibited one, and the mesh is refined uniquely along the internal and boundary layers due to the singular loading. Oppositely, as soon as the loading is applied (even very partially) in the non-inhibited area, the displacements are much larger: they are of a totally different nature mainly consisting in pure bendings. In that case, the mesh is refined anisotropically along the “free” characteristics (those that are not fixed anywhere), which behave like “hinges” where important rotations occur. The pertinence and the efficiency of the automatic anisotropic adaptive mesh procedure used for the numerical computations is once again revealed on the more complex examples considered in this chapter, where several kinds of limit behaviors (inhibited or non-inhibited) exist in different parts of the shell.

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A Characteristics of the Membrane System

In order to prove that the characteristic equation of the membrane system (1.68) expressed in terms of the displacements is equivalent to equation (1.71), we shall procced as follows. First, let us replace system (1.68) by another equivalent one, by considering as unknowns (u1 , u2 , u3 ), and the supplementary unknowns (T 11 , T 22 , T 12 ). Inverting the matrix of stiffness Aαβλμ involved in (1.69), system (1.68) is clearly equivalent to: ⎧ −D1 T 11 − D2 T 12 = f 1 ⎪ ⎪ ⎪ ⎪ ⎨ −D1 T 12 − D2 T 22 = f 2 (A.1) ⎪ ⎪ ⎪ ⎪ ⎩ −b11 T 11 − b22 T 22 − 2b12 T 12 = f 3 with

⎧ D1 u1 − b11 u3 − B11αβ T αβ = 0 ⎪ ⎪ ⎪ ⎪ ⎨ D2 u2 − b22 u3 − B22αβ T αβ = 0 ⎪ ⎪ ⎪ ⎪ ⎩1 (D1 u2 + D2 u1 ) − b12 u3 − B12αβ T αβ = 0 2

(A.2)

where Bαβλμ are the coefficients of the compliance matrix (inverse of the stiffness matrix of Aαβλμ , see (1.57). With that order of the unknowns and equations, we obtain a system of 6 equations with the 6 unknowns (u1 , u2 , u3 , T 11 , T 22 , T 12). We recognize (see (1.65) and (1.44) for comparison) the membrane tension system in (A.1) and the rigidity system in (A.2)), unless concerning the membrane tensions T αβ . By analogy with the results of sections 1.5.2 and 1.7.2, we should define the indices (1, 1, 0, 0, 0, 0) both for unknowns and equations. Then replacing again the derivatives ∂α with dzα , and taking the determinant of the system obtained, we have a determinant of order 6 with the structure: 0 C12 =0 (A.3) C21 C22

254

A Characteristics of the Membrane System

where the Cαβ are 3x3 matrices, and where 0 denotes the zero 3x3 matrix. Moreover, C12 and C21 are precisely those of the membrane tension (1.65) and of the rigidity system (1.44), respectively. Obviously, C22 comes from the terms in T αβ of (A.2). But it follows immediately from the definition and elementary properties of determinants that the determinant of (A.3) is given by the product of the determinants of C11 and C22 . The conclusion follows immediately.

B Reduced Membrane and Koiter Equations

This appendix contains the detailed calculations which lead to the reduced formulation of the problem (1.64) comprising three PDEs respectively for u1 , u2 and u3 . A similar development is then carried out for the full Koiter problem (section B.2) but only for u3 .

B.1

Membrane Problem

First, we start from membrane system (obtained after integration by parts of the variational formulation of the membrane problem): ⎧ ⎨ −Dα T αβ = f β (B.1) ⎩ −bαβ T αβ = f 3 Using the constitutive law, we get: ⎧ ⎨ −Dα Aαβλμ γλμ = f β ⎩

−bαβ Aαβλμ γλμ = f 3

(B.2)

As our aim is to study the singularities and their propagations, according to the microlocal analysis [43], it is sufficient to keep only the higher order terms for the displacements u1 , u2 and u3 and to consider the geometrical coefficients aαβ and bαβ as constants (at least locally at the point considered). Thus, we obtain the following system which only involves the displacements u1 , u2 and u3 : ⎧ 1βγ1 ∂β ∂γ u1 − A1βγ2 ∂β ∂γ u2 + A1βγδ bγδ ∂β u3 + · · · = f 1 ⎪ ⎪ −A ⎪ ⎪ ⎨ −A2βγ1 ∂β ∂γ u1 − A2βγ2 ∂β ∂γ u2 + A2βγδ bγδ ∂β u3 + · · · = f 2 (B.3) ⎪ ⎪ ⎪ ⎪ ⎩ −A1βγδ bγδ ∂β u1 − A2βγδ bγδ ∂β u2 + Aαβγδ bαβ bγδ u3 + · · · = f 3

256


ξ η ξ where + . . . denotes lower orders terms of the form Γμδ ∂γ uβ and Γμδ Γβη uβ or

ξ ξ bημ u3 in the two first lines of the system, and of the form Γμδ bημ uβ in the Γμδ third one. In the sequel, we only keep the highest order derivatives.

Now, let us write the simplified system (B.3) as follows: Au = f with

⎛

−A1βγ1 ∂β ∂γ

⎜ ⎜ 2βγ1 ∂β ∂γ A=⎜ ⎜ −A ⎝ −A1βγδ bγδ ∂β

(B.4)

−A1βγ2 ∂β ∂γ −A

2βγ2

∂β ∂γ

−A2βγδ bγδ ∂β

A1βγδ bγδ ∂β

⎞

⎟ ⎟ A bγδ ∂β ⎟ ⎟ ⎠ αβγδ A bαβ bγδ 2βγδ

Developing the terms of the matrix A, we get: ⎞ ⎛ B∂1 −A1111 ∂12 − A1212 ∂22 −A1112 ∂12 − A1222 ∂22 ⎟ ⎜ −2A1112 ∂1 ∂2 −(A1122 + A1212 )∂1 ∂2 +C∂2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2222 2 1212 2 ⎟ ⎜ −A1112 ∂12 − A1222 ∂22 −A ∂2 − A ∂1 C∂1 ⎟ A=⎜ 1222 ⎟ ⎜ −(A1122 + A1212 )∂1 ∂2 −2A ∂1 ∂2 +D∂2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ −B∂1 −C∂1 Bb11 + Db22 ⎠ −C∂2 −D∂2 +2Cb12

(B.5)

(B.6)

with B = A11αβ bαβ , C = A12αβ bαβ and D = A22αβ bαβ , where the Einstein summation convention for the indexes α and β is used. B.1.1

Case of the Normal Displacement u3

In order to obtain a reduced equation for u3 , we need to compute the cofacC tor AC 33 (see section 2.5 of chapter 2). By definition, A33 = A11 A22 − A12 A21 , so that: 1111 1212 A − (A1112 )2 ∂14 + A2222 A1212 − (A1222 )2 ∂24 AC 33 = A + A1111 A2222 + (A1212 )2 +4A1112 A1222 −2A1112 A1222 −(A1122 + A1212 )2 ∂12 ∂22

2A1111 A1222 + 2A1112 A1212 − 2A1112 (A1122 + A1212 ) ∂13 ∂2

+ 2A2222 A1112 + 2A1222 A1212 − 2A1222 (A1122 + A1212 ) ∂1 ∂23

B.1 Membrane Problem

257

It then follows that: 1111 1212 AC A − (A1112 )2 ∂14 + A2222 A1212 − (A1222 )2 ∂24 33 = A + A1111 A2222 + 2A1112 A1222 − (A1122 )2 − 2A1122 A1212 ∂12 ∂22 1111 1222 2A A − 2A1112 A1122 ∂13 ∂2 + 2A2222 A1112 − 2A1222 A1122 + ∂1 ∂23 Let us now recall the expression of the coefficients of the linear elastic isotropic constitutive law: Aαβλδ =

αλ βδ E a a + aαδ aβλ + Jaαβ aλδ 2(1 + ν)

with J =

2ν 1−ν

(B.7)

Taking the symmetries into account, we have: A1111 =

E (2 + J)(a11 )2 2(1 + ν)

(B.8)

A1112 =

E (2 + J)a11 a12 2(1 + ν)

(B.9)

A2222 =

E (2 + J)(a22 )2 2(1 + ν)

(B.10)

A1222 =

E (2 + J)a22 a12 2(1 + ν)

(B.11)

E (1 + J)(a12 )2 + a11 a22 2(1 + ν)

(B.12)

E (2)(a12 )2 + Ja11 a22 2(1 + ν)

(B.13)

A1212 =

A1122 =

In the expression of AC 33 , we shall compute separately the different terms: • Terms in ∂14 : E2 (2 + J)(a11 )2 a11 a22 + (1 + J)a12 a12 − (2 + J)2 (a11 a12 )2 4(1 + ν)2 =

E2 11 2 11 22 12 2 (2 + J)(a ) a a − (a ) 4(1 + ν)2

• Terms in ∂24 (obtained symmetrically): E2 22 2 11 22 12 2 (2 + J)(a ) a a − (a ) 4(1 + ν)2

(B.14)

(B.15)

258


• Terms in ∂12 ∂22 E2 2 11 22 2 2 12 2 11 22 12 12 11 22 2 (2 + J) (a a ) + 2(2 + J) (a ) a a − 2a a + Ja a 4(1 + ν)2 −2 2a12 a12 + Ja11 a22 a11 a22 + (1 + J)a12 a12 E2 2 11 22 11 22 12 2 12 4 2 11 22 2 11 22 12 2 (2 + J) (a a )(a a + 2(a ) ) − 4(a ) + J (a a ) + 4Ja a (a ) 4(1 + ν)2

=

! −2 2a11 a22 (a12 )2 + 2(1 + J)(a12 )4 + J(a11 a22 )2 + J(1 + J)a11 a22 (a12 )2 E2 11 22 2 12 2 2(2 + J)(a a ) − 4(2 + J)(a ) 2 4(1 + ν)

=

+2(2 + J)a11 a22 (a12 )2

E2 2(2 + J) a11 a22 − (a12 )2 (a11 a22 + 2(a12 )2 ) 4(1 + ν)2

=

(B.16)

• Terms in ∂13 ∂2 E2 2 11 2 22 12 11 12 12 2 11 22 2(2 + J) (a ) (a a ) − 2(2 + J)a a (2(a ) + Ja a ) 4(1 + ν)2 =

E2 11 12 11 22 12 2 4(2 + J)a a a a − (a ) 4(1 + ν)2

• Terms in ∂1 ∂23 (obtained symmetrically) E2 4(2 + J)a22 a12 a11 a22 − (a12 )2 2 4(1 + ν)

(B.17)

(B.18)

After simplification, we obtain: AC 33 =

E2 11 22 12 2 (2 + J) a a −(a ) (a11 )2 ∂14 +(a22 )2 ∂24 + 2(a11 a22 + 2(a12 )2 )∂12 ∂22 4(1+ν)2

+4a11 a12 ∂13 ∂2

22 12

+4a a

∂1 ∂23 (B.19)

Replacing J by its expression, we get finally: AC 33

(2) E2 11 22 12 2 11 2 22 2 12 = a a − (a ) a ∂1 + a ∂2 + 2a ∂1 ∂2 2(1 + ν)2 (1 − ν) (B.20)

B.1 Membrane Problem

259

1 , expression (B.20) reduces to: a (2) (B.21) a11 ∂12 + a22 ∂22 + 2a12∂1 ∂2

As a11 a22 − (a12 )2 = (a11 a22 − (a12 )2 )−1 = AC 33 =

E2 2(1 + ν)2 (1 − ν)a

On the other hand, the term det(A) may be obtained in a similar way: Det(A) =

(2) E3 b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 2 3 2(1 + ν) (1 − ν)a

(B.22)

Replacing the expressions (B.21) and (B.22) of AC 33 and Det(A) in equation (2.57), we obtain the reduced membrane PDE (2.60) accounting for the displacement u3 : (2) (2) 3 E b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 u3 = a2 a11 ∂12 + a22 ∂22 + 2a12 ∂1 ∂2 f (B.23) B.1.2

Reduced Equation for the Tangential Displacements u1 and u2

In the same way, we can exhibit two PDEs for the displacements u1 and u2 . We present here the details for the case of u1 , that of u2 being obtained symmetrically. Let us start from the equation (2.62) characterizing u1 : 3 Det(A)u1 = AC 31 f

(B.24)

We only have to compute AC 31 whose expression is given by: AC 31 = A12 A23 − A22 A13 = −A1112 C∂13 −A1222 C∂1 ∂22 −(A1122 +A1212 )C∂12 ∂2 −A1112 D∂12 ∂2 −A1222 D∂23 −(A1122 + A1212 )D∂1 ∂22 + A1212 B∂13 + A2222 C∂23 + A2222 B∂1 ∂22 + A1212 C∂12 ∂2 +2A1222 B∂12 ∂2 + 2A1222 C∂1 ∂22 = (−A1112 C + A1212 B)∂13 + (A2222 C − A1222 D)∂23 +(A1212 C + 2A1222 B − (A1212 + A1122 )C − A1112 D)∂12 ∂2 +(A2222 B + 2A1222 C − (A1212 + A1122 )D − A1222 C)∂1 ∂22 (B.25)

260


After some long technical calculations, we obtain: 2 11 22 1 E 11 2 12 2 C A31 = (2+J)b11(a ) + a a −(2 + 2J)(a ) b22 ∂13 + a 4(1 + ν)2 2(2+J)(a22)2 b12 +2(2 + J)a22 a12 b11 ∂23 + 4(2 + J)a11 a12 b11 +(4(2 + J)(a12 )2 − 2a11 a22 )b12 − 2(2 + J)a22 a12 b22 ∂12 ∂2 ((3J + 4)a11 a22 + (6 + 2J)(a12 )2 )b11 −(2 + J)(a22 )2 b22 + (8 + 4J)a22 a12 b12 ∂1 ∂22

B.2

Koiter Problem

In order to obtained a reduced PDE of the Koiter model for u3 , we use the local formulation (1.63) of the Koiter model: ⎧ β αβ 2 αγ γ αβ β ⎪ ⎨ −Dα T − ε bγ Dα M + Dγ (bα M ) = f (B.26) ⎪ ⎩ −bαβ T αβ + ε2 Dα Dβ M αβ − bα bβδ M αβ = f 3 δ 1 αβλδ where T αβ = Aαβλδ γλδ and M αβ = A ρλδ denote, respectively, the mem12 brane stresses and the bending moments. In the equation involving u3 , we only consider the most important bending terms, i.e. those with the lowest power in ε, and the highest order derivatives. We can write system (B.26) under the form (B.4) with: ⎛ ⎞ −A1111 ∂12 − A1212 ∂22 −A1112 ∂12 − A1222 ∂22 B∂1 ⎜ ⎟ −2A1112 ∂1 ∂2 −(A1122 + A1212 )∂1 ∂2 +C∂2 ⎜ ⎟ 2 2 2 2 2 3 ⎜ ⎟ +ε (∂ + . . . ) +ε (∂ + . . . ) +ε (∂ + . . . ) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2222 2 1212 2 ⎜ −A1112 ∂12 − A1222 ∂22 ⎟ −A ∂ − A ∂ C∂ 1 2 1 ⎜ ⎟ 1122 1212 1222 ⎜ ⎟ −(A + A )∂ ∂ −2A ∂ ∂ +D∂ A=⎜ 1 2 1 2 2 ⎟ 2 2 2 2 2 3 ⎜ ⎟ +ε (∂ + . . . ) +ε (∂ + . . . ) +ε (∂ + . . . ) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −B∂ −C∂ +Bb + Db + 2Cb 1 1 11 22 12 ⎟ ⎜ ⎟ 2 ε ⎝ ⎠ −C∂2 −D∂2 + 12 F 2 3 2 3 2 3 +ε (∂ + . . . ) +ε (∂ + . . . ) +ε (∂ + . . . ) (B.27) with F = A1111 ∂14 +A2222 ∂24 +(2A1122 +4A1212 )∂12 ∂22 +4A1112 ∂13 ∂2 +4A1222 ∂1 ∂23 and where +ε2 (∂ n + . . . ) denote the ignored bending terms, n being the highest

B.2 Koiter Problem

261

order of differentiation contained in these terms. Let us quote that F is a fourthorder operator. Thus, the most important bending term of det(A) comes from: (A11 A22 − A12 A12 )

ε2 F = 12

ε2 C 1111 4 A33 A ∂1 + A2222 ∂24 +(2A1122 +4A1212 )∂12 ∂22 +4A1112 ∂13 ∂2 +4A1222 ∂1 ∂23 12 (B.28) This term is in ε2 and comprises 8th order derivatives. The other terms in ε2 involve lower order derivatives. The full expression of Det(A) then writes: 11 2 (4) ε2 1 a ∂1 + a22 ∂22 + 2a12∂1 ∂2 3 2 24 (1 + ν) (1 − ν) a ! 1 (2) + b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 + O(ε2 ) (B.29) 2(1 + ν)2 (1 − ν)a3

Det(A) = E 3

where O(ε2 ) denotes terms containing lower order derivatives or being factors of terms in εn with n > 2. On the other hand, we always have: 2 AC 33 = E

11 2 (2) 1 a ∂1 + a22 ∂22 + 2a12 ∂1 ∂2 + O(ε2 ) 2 2(1 + ν) (1 − ν)a

(B.30)

so that, when ε 0, the reduced PDE of the Koiter model involving only u3 writes: 2 (4) (2) ε 2 11 2 E a a ∂1 + a22 ∂22 + 2a12 ∂1 ∂2 + (1 + ν) b22 ∂12 + b11 ∂22 − 2b12 ∂1 ∂2 12 (2) +O(ε2 ) u3 = a2 (1 + ν) a11 ∂12 + a22 ∂22 + 2a12 ∂1 ∂2 + O(ε2 ) f 3 (B.31)

Index

a posteriori error estimate, 99 adapted mesh, 126, 133, 154, 158, 184 adaptive mesh, 97, 124, 154, 184, 221, 236 algebraic equation, 23 system, 23, 209 analytic functionals, 203, 217 analytic functions, 203 anisotropic mesh, 76, 126, 154, 237 asymptotic behavior, 33 curves, 24, 30, 32, 56, 61 directions, 16 lines, 17, 115, 149 BAMG, 98, 137, 184, 221, 236 bending energy, 101, 131, 192, 228, 239 energy bilinear form, 28 moments tensor, 27 bi-Laplacian operator, 172 boundary, 27 conditions, 27, 35, 116, 208, 211 edge layers, 136 layers, 157, 187 singularity along, 136 boundary conditions kinematical, 29 boundary layer, 38, 132 Cauchy problem, 196 characteristic case, 56, 65, 109, 119, 149

curves, 22, 23 equation, 24 lines, 24, 30 Christoffel symbol, 19, 109, 148, 182 classification Finite Element, 88 of surface, 16 of systems, 24, 30 complexification phenomenon, 203, 222 compliance coefficients, 28 constitutive law, 27, 34, 116 contravariant basis, 15, 115, 182 component, 16, 19 convergence asymptotic process, 38, 88, 129, 188, 191, 208 Finite Element, 88, 94, 95 strong, 42, 43, 45 weak, 39, 44, 45 coordinate vectors, 14 corners of the loading domain, 171, 181 covariant basis, 13, 14, 115, 182 component, 15, 19 derivative, 115 differentiation, 18 curvature tensor, 15, 115, 148, 182 curvature variation tensor, 21, 27 D.K.T. shell element, 94 deformation patterns, 5 degrees of freedom (DOF), 95

264

Index

determination domain, 36, 200 Dirac function, 52, 202 Dirichlet condition, 208 displacement, 21 eigenvalues, 172, 207 elastic material, 27 elliptic Cauchy problem, 203, 211 paraboloid, 181 point, 17 shell, 35 surface, 181 energy norm, 70 energy repartition, 131, 192, 228, 239 error estimates, 69, 73, 78, 97, 99 finite element, 94 P2-Lagrange, 95 P3’-Hermite, 95 first fundamental form, 15 Fourier expansion, 172, 174, 176, 177, 180, 203 transform, 201–203, 205, 213, 214 Galerkin approximation, 70 Galerkin error estimates, 78 Gauss formula, 18 geometrically rigid, see inhibited half-cylinder, 114 Heaviside step function, 117 Heaviside step function, 52 Hessian, 99 Holmgren and Calderon theorem, 203, 211 Hooke’s law, 27 hyperbolic paraboloid, 149 point, 17 shell, 6, 36, 147 surface, 149, 235 ill-inhibited shell, 36 inextensional displacement, 22, 34 infinitesimal area element, 15 infinitesimal lenght element, 15 inhibited shell, 3 surface, 34

internal layer, 38, 126 interpolation error, 73 anisotropic meshes, 76 isotropic meshes, 73, 74 interpolation error estimate, 99 isotropic mesh, 73, 74 refinement, 221 isotropic material, 27 iterated meshes, 126, 154, 158, 159, 184 Kirchhoff-Love hypothesis, 47 kinematical hypothesis, 27 kinematics, 95 Koiter hypothesis, 47 theorem, 22 Koiter model, 26 asymptotic behavior of, 33 asymptotic limit, 29 coerciveness, 28 convergence, 29, 129, 222 ellipticity, 37 limit behavior of, 33, 37 Korn’s inequality, 28 Kronecker symbol, 15, 115 Laplacian operator, 175, 201 Lax-Milgram theorem, 28 layer thickness, 63, 78, 130, 132, 156, 187, 192 linear elasticity, 26 linear theory, 22 loading, 26 locking, 88, 236 inhibited case, 93 local, 82, 93 non-inhibited case, 88 logarithmic singularity, 171, 183, 189 mapping, 13, 114, 148, 181 membrane bilinear form, 28 energy, 101, 131, 192, 228, 239 energy bilinear form, 28 locking, 93 model, 29, 38 problem, 175, 207 strain tensor, 27

Index stress tensor, 27 system, 30, 31, 108 tension system, 30 metric tensor, 15, 21, 115, 148, 182 metric variation tensor, 21 metrics, 22, 34, 91, 98 mixed component, 16, 115 Nagdhi model, 27, 45 Neumann condition, 208 non-characteristic case, 53, 64, 111, 135 non-characteristic layer, 187 non-conforming element, 95 non-inhibited, 34 non-inhibited shell, 3, 5, 235 normal vector, 14, 115, 182 parabolic point, 17 shell, 7, 36, 107 surface, 114 partially non-inhibited shell, 236 PDE, 47, 49, 172, 196, 198, 201, 207 system, 29, 199 penalty problem, 43, 72, 85 percentage of bending energy, 132, 192, 229, 239 Poisson’s ratio, 27 positivity property, 28, 39 principal curvatures, 17, 54, 177 propagation of singularities, 52, 61, 63, 108, 111, 127, 142, 151, 158, 161, 163, 240 pseudo-reflections, 63, 163 pure bending displacement, 34 model, 43 Reissner–Mindlin hypothesis, 27 remeshing h-methods, 97 hp-methods, 97 p-methods, 97 Riemann invariants, 199 rigidity system, 22

265

scaling, 66 second fundamental form, 15 sensitive problem, 195, 210 shell, 7, 212, 219 shape of the domain, 142 Shapiro-Lopatinskii condition, 206 shell, 1, 13 singular perturbation problem, 72 singularity chain, 52 singularity order, 61 structured mesh, 159 surface, 13, 16 deformation, 21 geometrically rigid, 35 rigidity, 34 symmetry condition, 181 symmetry property, 28 tangent plane, 14, 16, 114, 182 vectors, 182 Taylor formula, 73 tension system, 40 thickness of the layer, see layer thickness thickness of the shell, 26 totally non-inhibited shell, 236 transversal shear, 27 triangle P2-Lagrange, 95 P3’-Hermite, 95 umbilic point, 177, 180, 183 uniform convergence, 88, 93, 94 mesh, 133 Weingarten formula, 18 well-inhibited shell, 7, 171 surface, 34 Young’s modulus, 27 Z space, 203 Z space, 203, 217

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