Shape Design Sensitivity Analysis and Optimization Using the Boundary Element Method

ISBN-13: 9783540535188 / Angielski / Miękka / 1991 / 192 str.

Z. Zhao; Zhiye Zhao

Opis

This book investigates the various aspects of shape optimization of two dimensional continuum structures, including shape design sensitivity analysis, structural analysis using the boundary element method (BEM), and shape optimization implementation. The book begins by reviewing the developments of shape optimization, followed by the presentation of the mathematical programming methods for solving optimization problems. The basic theory of the BEM is presented which will be employed later on as the numerical tool to provide the structural responses and the shape design sensitivities. The key issue of shape optimization, the shape design sensitivity analy sis, is fully investigated. A general formulation of stress sensitivity using the continuum approach is presented. The difficulty of the modelling of the ad joint problem is studied, and two approaches are presented for the modelling of the adjoint problem. The first approach uses distributed loads to smooth the concentrated adjoint loads, and the second approach employs the singu larity subtraction method to remove the singular boundary displacements and tractions from the BEM equation. A novel finite difference based approach to shape design sensitivity is pre sented, which overcomes the two drawbacks of the conventional finite difference method. This approach has the advantage of being simple in concept, and eas ier implementation. A shape optimization program for two-dimensional continuum structures is developed, including structural analysis using the BEM, shape design sensitiv ity analysis, mathematical programming, and the design boundary modelling."

Spis treści

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.