Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method

1 Introduction.- 1.1 Introduction.- 1.2 Review of the Shape Optimization.- 1.2.1 Shape Optimization using the Finite Element Method.- 1.2.2 Shape Optimization using the Boundary Element Method.- 1.2.3 Shape Design Sensitivity Analysis.- 1.3 References.- 2 Basic Numerical Optimization Techniques.- 2.1 Introduction.- 2.2 Basic Concepts and Terminology.- 2.3 Mathematical Programming Method.- 2.4 References.- 3 The Boundary Element Method in Elastostatics.- 3.1 Introduction.- 3.2 Review of the Boundary Element Method in Elastostatics.- 3.3 The Boundary Element Method in Elastostatics.- 3.3.1 Basic Equations of Linear Elasticity.- 3.3.2 The Boundary Integral Formulation of Elasticity.- 3.3.3 Numerical Implementation of the Boundary Element Method.- 3.4 Conclusion Remarks.- 3.5 References.- 4 Shape Design Sensitivity Analysis using the Boundary Element Method.- 4.1 Introduction.- 4.2 Two Basic Approaches for Design Sensitivity Analysis.- 4.2.1 The Discretized Approach (DA).- 4.2.2 The Continuum Approach (CA).- 4.2.3 Comparisons of the Two Approaches.- 4.3 The Implementation of the Material Derivative of Displacements.- 4.4 Stress Sensitivity Analysis by CA.- 4.4.1 A Simple Case.- 4.4.2 The Stress Sensitivity Formulation for the General Case.- 4.5 The Modelling of the Adjoint Problem.- 4.5.1 Numerical Approaches for Problems with singular Loads.- 4.5.2 Mesh Refinement and Special Elements Methods.- 4.5.3 Local Singular Function Method.- 4.5.4 Smooth Loading Method.- 4.5.5 Singularity Subtraction Method.- 4.5.6 Concluding Remarks.- 4.6 Implementation of the Singularity Subtraction Method.- 4.7 A New Finite Difference Based Approach to Shape Design Sensitivity Analysis.- 4.7.1 A Simple Example.- 4.7.2 Derivation of the Finite Difference Load Method.- 4.7.3 Further Discussions of FDLM.- 4.7.4 Concluding Remarks.- 4.8 Numerical Examples.- 4.8.1 A Cantilever Beam.- 4.8.2 A Circular Plate Under Internal Pressure.- 4.8.3 A Fillet Example.- 4.8.4 An Elastic Ring under a Concentrated Load.- 4.9 Concluding Remarks.- 4.10 References.- 5 Shape Optimization Using the Boundary Element Method.- 5.1 Introduction.- 5.2 The Design Model and the Analysis Model.- 5.2.1 The Design Model.- 5.2.2 The Analysis Model - Remeshing Problem.- 5.3 Shape Optimization Implementation.- 5.3.1 SOP - A Shape Optimization Program.- 5.4 Numerical Examples.- 5.4.1 A Beam Example.- 5.4.2 A Fillet Example.- 5.4.3 A Plate With a Hole.- 5.4.4 A Connecting Rod.- 5.5 References.- A Fundamental Solutions of the Semi-infinite Plane.- B Derivatives of Boundary Stresses on the Normal Direction.