Singular Problems in Shell Theory - Computing and Asymptotics

Title

Contents

Notations

Introduction

Non-inhibited (or non-geometrically rigid) middle surface

Inhibited (or geometrically rigid) hyperbolic shells

Inhibited (or geometrically rigid) parabolic shells

Well-inhibited elliptic shells (fixed or clamped all along the boundary)

Ill-inhibited elliptic shells (fixed or clamped along a part of the boundary, free by the rest)

Geometric Formalism of Shell Theory

Introduction

Recall on Surface Theory

Mapping - Covariant Basis

First Fundamental Form of the Surface S - Contravariant Basis

Second Fundamental Form

Classification of Surfaces

Differentiation on the Surface S

Surface Rigidity

Deformation of a Surface

The Rigidity System and its Characteristic Curves

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

The Koiter Shell Model

The Limit Membrane Model

The Membrane Model

The System of Membrane Tension

Back to the Membrane System

Singularities and Boundary Layers in Thin Elastic Shell Theory

Introduction

Geometrically Rigid Surfaces

Inextensional Displacements

Examples of Geometrically Rigid Surface

Limit Behavior of Koiter Model

The Limit Membrane Problem

Boundary Layers and Singularities

Convergence to the Membrane Model in the Inhibited Case

A More General Result of Convergence

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

Case of the Normal Displacement u3

Tangential Displacements u1 and u2

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

Singularity along a Non-characteristic Line

Singularity along a Characteristic Line

Summary of the Results

Pseudo-reflections for Hyperbolic Shells

Thickness of the Layers

Case of a Layer along a Non-characteristic Line

Case of a Layer along a Characteristic Line

Conclusion

Anisotropic Error Estimates in the Layers

Introduction

Estimate for Galerkin Approximation in Singular Perturbation and Penalty Problems

Degradation of the Estimate in a Singular Perturbation Problem

Degradation of the Estimate in a Penalty Problem

Interpolation Error for Isotropic Meshes in Layers

The Basic F. E. Interpolation Error Estimate

Case of a Layer: Interpolation Error for Isotropic Meshes

Interpolation Error for Anisotropic Meshes in Layers

Galerkin Error Estimates in a Layer

First Remarks on Approximations in Layers

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

Conclusion

Numerical Simulation with Anisotropic Adaptive Mesh

Introduction

Review on the Numerical Locking

Introduction

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

Locking in the Inhibited Case (Singular Perturbations)

Shell Element and Associated Discrete Problem

The Shell Element D.K.T.

Discretization of Naghdi Model

Adaptive Mesh Strategy: BAMG

Coupling BAMG-MODULEF for Shell Computations

Membrane and Bending Energies Computation with MODULEF

Implementation Procedure in MODULEF

Validation on Simple Examples

Conclusion

Singularities of Parabolic Inhibited Shells

Introduction

Study of the Singularities and of Their Propagation

Singularity along a Characteristic Line

Singularity along a Non-characteristic Line

Example of a Half-Cylinder

Geometric Description of the Cylinder

Constitutive Law

Loading and Boundary Conditions

Numerical Simulations with Anisotropic Adaptive Mesh

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

Convergence of the Adaptive Mesh Procedure

Computing the Displacements

Influence of the Relative Thickness e

Localization of Membrane and Bending Energies

Comparison between Uniform and Adapted Meshes

Numerical Study of Singularities on Non-characteristic Lines

Singularity along a Boundary

Theoretical Considerations

Numerical Simulations

Singularities due to the Shape of the Domain

Conclusion

Singularities of Hyperbolic Inhibited Shells

Introduction

The Limit Problem for a Hyperbolic Inhibited Shell

Example of a Hyperbolic Paraboloid

Singularities of the Displacements due to a Loading Singular on the Line y1 = 0

Three Cases of Loading

The Singularities of the Resulting Displacements

Numerical Computations Using Adaptive Meshes

Numerical Results for Loading A

Results for the Loading B

Results for the Loading C

Some Examples Including Pseudo-reflections

Reflection of a Characteristic Layer

Reflection of a Non-characteristic Layer

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

Conclusion

Singularities of Elliptic Well-Inhibited Shells

Introduction

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

Model Problem of Second Order

The Membrane Problem .2u3 = C4 f3(.)

Particular Case when the Logarithmic Point Singularity Vanishes

Existence Condition of a Logarithmic Singularity

Example of an Elliptic Paraboloid

Geometric Properties

Numerical Results

Mesh Adaptation

Thickness of the Internal Layer along y1 = 0.5

The Logarithmic Singularity at the Corner

Membrane and Bending Energies

Conclusion

Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems

Introduction

The Cauchy Problem for Equations and Systems

Generalities

Role of the Characteristics

Normal Form of a Hyperbolic System: Riemann Invariants

Elliptic Equations or Systems

Boundary Value Problems for Elliptic Equations and Systems

Regularity of the Solution

The Shapiro–Lopatinskii Condition

The Shapiro–Lopatinskii Condition and the Membrane Problem

Sensitive Problems

Elliptic Shell Clamped by a Part G0 of the Boundary and Free by the Rest G1

Qualitative Description of the Solution of Sensitive Problems

Heuristic Treatment of the Problem

Conclusion

Numerical Simulations for Sensitive Shells

Introduction

First Examples of Numerical Computations for Sensitive Problems (Ill-Inhibited Shells)

Asymptotic Process when e Tends to Zero

Influence of the Free Edge Length

Energy Repartition in Sensitive Problems

Influence of the Loading Domain

Conclusion

Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems)

Examples of Partially Non-inhibited Shells

First Case: a = 0 and ß = 0.25

Second Case: a = 0.25 and ß = 0.25

Propagation of Singularities in the Partially Non-inhibited Regions

Loading Applied in the Inhibited Area

Loading Domain Tangent to the Non-inhibited Area

Loading Partially Applied in the Non-inhibited Area

Conclusion

References

Characteristics of the Membrane System

Reduced Membrane and Koiter Equations

Membrane Problem

Case of the Normal Displacement u3

Reduced Equation for the Tangential Displacements u1 and u2

Koiter Problem

Index