Mechanical Engineering Series. Frederic F. Ling Series Editor. Springer. New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo

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1 Mechanical Engineering Series Frederic F. Ling Series Editor Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo

2 Kyung K. Choi Nam H. Kim Structural Sensitivity Analysis and Optimization 1 Linear Systems Springer

3 Kyung K. Choi Department of Mechanical and Industrial Engineering The University of Iowa Iowa City, IA USA Nam H. Kim Department of Mechanical and Aerospace Engineering The University of Florida Gainesville, FL USA Series Editor Frederick F. Ling Ernest F. Gloyna Regents Chair in Engineering, Emiritus Department of Mechanical Engineering The University of Texas at Austin Austin, TX , USA and William Howard Hart Professor Emeritus Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY , USA ISBN x 2005 Springer Science+Bussiness Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Bussiness Media, Inc., 233 Springer Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use of this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (EB) SPIN springeronline.com

4 To our wives Ho-Youn Jee-Hyun

5 Preface Structural design sensitivity analysis concerns the relationship between design variables available to the design engineer and structural responses determined by the laws of mechanics. The dependence of response measures such as displacement, stress, strain, natural frequency, buckling load, acoustic response, frequency response, noise-vibrationharshness (NVH), thermoelastic response, and fatigue life on the material property, sizing, component shape, and configuration design variables is implicitly defined through the governing equations of structural mechanics. In this text, first- and second-order design sensitivity analyses are presented for static and dynamics responses of both linear and nonlinear structural systems, including elastoplastic and frictional contact problems. Prospective readers or users of the text are seniors and graduate students in mechanical, civil, biomedical, industrial, and engineering mechanics, aerospace, and mechatronics; graduate students in mathematics; researchers in these same fields; and structural design engineers in industry. A substantial literature exists on the technical aspects of structural design sensitivity analysis. While some studies directly address the topic of design sensitivity, the vast majority of research is imbedded within texts and papers devoted to structural optimization. The premise of this text is that a comprehensive theory of structural design sensitivity analysis for linear and nonlinear structures can be treated in a unified way. The objective is therefore to provide a complete treatment of the theory and practical numerical methods of structural design sensitivity analysis. Design sensitivity supports optimality criteria methods of structural optimization and serves as the foundation for iterative methods of structural optimization. One of the most common structural design methods involves decisions made by the design engineer based on experience and intuition. This conventional mode of structural design can be substantially enhanced if the design engineer is provided with design sensitivity information that explains design change influences, without requiring a trial and error process. Such advanced, state-of-the-art analysis methods as finite element analysis, boundary element analysis, and meshfree analysis provide reliable tools for the evaluation of the structural design. However, they give the design engineer little help in identifying ways to modify the design to either avoid problems or improve desired qualities. Using design sensitivity information generated by methods that exploit finite element, boundary element, or meshfree formulations, the design engineer can carry out systematic trade-off analysis and improve the design. This text presents design sensitivity analysis (DSA) theory and numerical implementation to create advanced design methodologies for mechanical systems and structural components, which will permit economical designs that are strong, stable, reliable, and have long service life. The design methodologies can be used by design engineers in the academia, industry, and government to obtain optimal structural designs for ground vehicles, aircraft, space systems, ships, heavy equipment, machinery, biomedical devices, etc. Extensive numerical methods for computing design sensitivity are included in this text for practical application and software development. More importantly, the numerical method allows seamless integration of CAD-FEA-DSA software tools, so that design optimization can be carried out using CAD geometric

6 viii Preface models instead of FEA models. This capability allows integration of CAD-CAE-CAM so that optimized designs can be manufactured effectively. The book is organized into two volumes, four parts, and fourteen chapters. Parts I and II are in Volume 1: Linear Systems, and Parts III and IV are in Volume 2: Nonlinear Systems and Applications. Part I introduces structural design concepts that include the CAD-based design model, design parameterization, performance measures, costs, and constraints. Based on the design model, an analysis model is introduced using finite element analysis. A broad overview of design sensitivity analysis methods is provided. By relying on energy principles to develop design sensitivity analysis theory, the design sensitivity method is developed without requiring highly sophisticated mathematics. The energy method is introduced in order to develop the variational equation and its relationship to the finite element method. Chapters 2 and 3 are essentially a review for students who have already learned energy methods in structural mechanics. The finite element method is explained as a technique based on a piecewise polynomial approximation of the displacement field and as an application of the variational method for approximating a solution to the governing boundary-value problem. In Part II, this relationship is successfully used in the development of discrete and continuum design sensitivity analysis methods and their relationships. Part II treats design sensitivity analysis of linear structural systems. Both discrete and continuum design sensitivity analysis methods are explained. Chapter 4 describes finitedimensional problems in which the structural response is a finite-dimensional vector of structural displacements, and the design variable is a finite-dimensional vector of design parameters. Governing structural equations are matrix equations. Direct design differentiation and adjoint variable methods of design sensitivity analysis are presented, along with the design derivatives of eigenvalues and eigenvectors. The computational aspects of implementing these methods are treated in some detail in conjunction with finite-element analysis codes. Chapters 5 through 7 treat continuum problems in which response and design variables are functions (displacement field and material distribution) and governing structural equations are the variational equations introduced in Chapters 2 and 3. Sizing, shape, and configuration design variables are treated separately in Chapters 5 through 7, respectively. Both the direct differentiation and adjoint variable method of design sensitivity analysis are developed, and design derivatives of eigenvalues are derived. Analytical solutions to simple examples and numerical solutions to more complex examples are presented. For both shape and configuration design variables, the material derivative concept is taken from continuum mechanics to predict the effect of design changes on the structural response. For a structural component with curvature, a more general configuration design theory is presented in Section 7.5 of Chapter 7. For shape design sensitivity, the adjoint variable method is used to derive expressions for differentials of the structural response, either as boundary integrals (the boundary method) or domain integrals (the domain method). A similar method is used for the shape design sensitivity of eigenvalues. Part III treats design sensitivity analysis of nonlinear structural systems using continuum design sensitivity analysis methods. As with Chapters 2 and 3, the equilibrium equations for nonlinear structural systems are derived using the principles of virtual work from Chapter 8. Both geometric and material nonlinearities are treated. Nonlinear elasticity, buckling, hyperelasticity, elastoplasticity, nonlinear transient dynamics, and frictional contact problems are included. In nonlinear structural analysis, total and updated Lagrangian approaches have been introduced. The equilibrium equations are

7 Preface ix then linearized at the previously known configuration to yield incremental formulations for nonlinear analysis. The linearized equilibrium equation plays a key role in design sensitivity analysis in subsequent chapters, since the first-order variation with respect to the design parameter includes linearization of the energy form. The linearized form that appears during design sensitivity analysis is the same as the linearized form for nonlinear analysis. Sizing, shape, and configuration design variables are treated separately in Chapters 9 through 11, respectively. Both adjoint variable and direct differentiation methods are given for the nonlinear elastic problem. However, for nonlinear elastoplastic problems, only the direct differentiation method is used, since the design sensitivity is path-dependent. Analytical derivations of design sensitivity expressions for structural components are presented, along with numerical examples of sensitivity computations. Part IV is devoted to practical design tools and applications: sizing and shape design parameterization, design velocity field computation, numerical implementation of the sensitivity for general-purpose code development, and various other practical design applications. In Chapter 12, sizing design parameterization for line and surface design components is introduced. For shape design parameterization, a three-step process is developed. One important aspect of shape design parameterization is the connection between the design parameterization and the computation of the design velocity field, as explained in Chapter 13. In Chapter 13, the computational aspects of design sensitivity analysis are considered, using the finite element method to solve the original governing and adjoint equations. The numerical method allows seamless integration of CAD-FEA- DSA, so that design optimization can be carried out using CAD models instead of FEA ones. Chapter 14 includes a number of practical design applications of linear and nonlinear structural systems with additional applications in thermoelastic analysis and fatigue design optimization to aid application-oriented readers. A final comment on the notation used in this text. The structural design engineer may find that the notation conventionally used in structural mechanics has not always been adhered to. The field of design sensitivity analysis presents a dilemma regarding notation since it draws from fields as diverse as structural mechanics, differential calculus, calculus of variations, control theory, differential operator theory, and functional analysis. Unfortunately, the literature in each of these fields assigns a different meaning to the same symbol. Consequently, it is at times necessary to use symbols that look identical in an equation, but that come from very different notational systems. As a result, some notational compromise is required. The authors have adhered to standard notation except where ambiguity would arise, in which case the notation being used is indicated. This book has been made possible due to contributions from the authors former students, namely, Drs. R.-J. Yang, H.G. Lee, H.G. Seong, B. Dopker, T.-M. Yao, J.L.T. Santos, J.-S. Park, J. Lee, S.-L. Twu, K.-H. Chang, M. Godse, S.M. Wang, I. Shim, C.-J. Chen, Y.-H. Park, H.-Y. Hwang, X. Yu, W. Duan, S.-H. Cho, B.S. Choi, I. Grindeanu, J. Tu, B.-D. Youn, and Y. Yuan. Special appreciation is given to Professor K.-H. Chang at University of Oklahoma for his contributions to numerical methods and his examples in shape design sensitivity analysis and optimization. In addition, the authors value the contributions of colleagues Drs. J. Cea, B. Roussellet, J.P. Zolesio, R. Haftka, B.M. Kwak, G.W. Hou, H.L. Lam, and Y.M. Yoo. Finally, special thanks to Mr. R. Watkins for his outstanding work editing the manuscript. Kyung K. Choi Iowa City, Iowa September 2004

8 CONTENTS 1: Linear Systems Preface... vii PART I Structural Design and Analysis 1 Introduction to Structural Design Elements of Structural Design Structural Modeling and Design Parameterization Structural Modeling Design Parameterization Three-Bar Truss Example Structural Analysis Finite Element Analysis Structural Design Sensitivity Analysis Methods of Structural Design Sensitivity Analysis Finite Difference Method Discrete Method Continuum Method Summary of Design Sensitivity Analysis Approaches Second-Order Design Sensitivity Analysis Design Optimization Linear Programming Method Unconstrained Optimization Problems Constrained Optimization Problems Variational Methods of Structural Systems Introduction Energy Method Variational Formulation and the Principle of Virtual Work Hamilton's Principle Eigenvalue Problem Frequency Response Problem Structural Response Acoustic Response Thermoelastic Problem Thermal Analysis Elastic Analysis Variational Equations and Finite Element Methods Energy Bilinear and Load Linear Forms of Static Problems Truss Component Beam Component Plate Component Elastic Solid Deflection of a Membrane Torsion of an Elastic Shaft General Form of Static Variational Equations Vibration and Buckling of Elastic Systems... 85

9 xii Contents 3.3 Finite Element Structural Equations Truss Element Beam Element Plate Element Three-Dimensional Elastic Solid Global Matrix Equations for the Finite Element Method Construction of Global Matrices Variational Principles for Discrete Structural Systems Reduced Matrix Equation of Structural Mechanics Variational Equations for Discrete Structural Systems Numerical Integration PART II Design Sensitivity Analysis of Linear Structural Systems 4 Discrete Design Sensitivity Analysis Static Response Design Sensitivity Statement of the Problem Design Sensitivity Analysis with Reduced Stiffness Matrix Design Sensitivity Analysis with Generalized Stiffness Matrix Computational Considerations Second-Order Design Sensitivity Analysis Examples Design Sensitivity of the Eigenvalue Problem Eigenvalue Design Sensitivity Analysis Design Sensitivity Analysis of Eigenvectors Second-Order Design Sensitivity of a Simple Eigenvalue Systematic Occurrence of Repeated Eigenvalues in Structural Optimization Directional Derivatives of Repeated Eigenvalues Examples Transient Dynamic Response Design Sensitivity Design Sensitivity Analysis of Damped Elastic Structures Design Sensitivity Analysis of Undamped Structures Modal Reduction Method Using Ritz Vectors Functionals in a Structural Dynamic Design Continuum Sizing Design Sensitivity Analysis Design Sensitivity Analysis of Static Response Differentiability of Energy Bilinear Forms and Static Response Adjoint Variable Design Sensitivity Analysis Analytical Examples of Static Design Sensitivity Numerical Considerations Numerical Examples Eigenvalue Design Sensitivity Differentiability of Energy Bilinear Forms and Eigenvalues Analytical Examples of Eigenvalue Design Sensitivity Numerical Considerations Numerical Examples of Eigenvalue Design Sensitivity Transient Dynamic Response Design Sensitivity Design Sensitivity of Structural Dynamics Performance Analytical Examples Frequency Response Design Sensitivity Design Sensitivity Analysis of Frequency Response Analytical Examples

10 Contents xiii Numerical Examples Structural-Acoustic Design Sensitivity Analysis Design Sensitivity Analysis of Structural-Acoustic Response Analytical Example Numerical Examples Continuum Shape Design Sensitivity Analysis Material Derivatives for Shape Design Sensitivity Analysis Material Derivative Basic Material Derivative Formulas Static Response Design Sensitivity Analysis Differentiability of Bilinear Forms and Static Response Adjoint Variable Design Sensitivity Analysis Boundary Method for Static Design Sensitivity Shape Design Sensitivity Analysis of Local Performance Measures Domain Shape Design Sensitivity Method Parameterization of Design Boundary Regularity of Design Velocity Field Numerical Examples Eigenvalue Shape Design Sensitivity Analysis Differentiability of Bilinear Forms and Eigenvalues Boundary and Domain Methods of Eigenvalue Design Sensitivity Frequency Response Problem Design Sensitivity of Frequency Response Numerical Examples Thermoelastic Problem Design Sensitivity Analysis of Thermal Systems Design Sensitivity Analysis of Structural Systems Adjoint Variable Method in the Thermoelastic Problem Second-Order Shape Design Sensitivity Analysis Second-Order Material Derivative Formulas Direct Differentiation Method Hybrid Method Numerical Examples Configuration Design Sensitivity Analysis Material Derivatives for Configuration Design Sensitivity Analysis Line Design Component Surface Design Component Material Derivative of a General Functional Configuration Design Sensitivity Analysis Variation of the Static Response Eigenvalue Problems Analytical Examples Numerical Methods in Configuration Design Sensitivity Analysis Linear Approximation between Design Parameterization and Design Velocity Field Regularity of Design Velocity Fields Numerical Examples Structural-Acoustic Problem Variations for the Configuration Design Design Sensitivity Analysis Design Components Analytical Example

11 xiv Contents Numerical Example Configuration Design Theory for Curved Structure Geometric Mapping Degenerated Shell Formulation Material Derivative Formulas Direct Differentiation Method Adjoint Variable Method Numerical Example Appendix A.1 Matrix Calculus Notation A.2 Basic Function Spaces A.2.1 R k ; k-dimensional Euclidean Space A.2.2 C m ( ); m-times Continuously Differentiable Functions on A.2.3 L 2 ( ); The Space of Lebesgue Square Integrable Functions A.2.4 L ( ); Space of Essentially Bounded, Lebesgue-Measurable Functions A.2.5 H m ( ); Sobolev Space of Order m A.2.6 m H ( ) 0 ; Sobolev m-space with Compact Support A.2.7 The Sobolev Imbedding Theorem A.2.8 Trace Operator A.2.9 Product Spaces A.3 Differentials and Derivatives in Normed Space A.3.1 Mappings in Normed Spaces A.3.2 Variations and Directional Derivatives A.3.3 Fréchet Differential and Derivative A.3.4 Partial Derivatives and the Chain Rule of Differentiation References Index

12 Kyung K. Choi Nam H. Kim Structural Sensitivity Analysis and Optimization 2 Nonlinear Systems and Applications Springer

13 Kyung K. Choi Department of Mechanical and Industrial Engineering The University of Iowa Iowa City, IA USA Nam H. Kim Department of Mechanical and Aerospace Engineering The University of Florida Gainesville, FL USA Series Editor Frederick F. Ling Ernest F. Gloyna Regents Chair in Engineering, Emiritus Department of Mechanical Engineering The University of Texas at Austin Austin, TX , USA and William Howard Hart Professor Emeritus Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY , USA ISBN Springer Science+Bussiness Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Bussiness Media, Inc., 233 Springer Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use of this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (EB) SPIN springeronline.com

14 Contents 2: Nonlinear Systems and Applications Preface... vii PART III Design Sensitivity Analysis of Nonlinear Structural Systems 8 Nonlinear Structural Analysis Nonlinear Elastic Problems Static Problem Critical Load Hyperelastic Material Elastoplastic Problems Small Deformation Finite Rotation with Objective Integration Finite Deformation with Hyperelasticity Contact Problems Contact Condition and Variational Inequality Frictionless Contact Formulation Frictional Contact Formulation Nonlinear Dynamic Problems Implicit Method Explicit Time Integration Method Mathematical Formulas for Finite Deformation Elastoplasticity Principle of Maximum Dissipation Algorithmic Tangent Operator for Principal Stress Linearization of Principal Logarithmic Stretches Linearization of the Eigenvector of the Elastic Trial Left Cauchy-Green Tensor Nonlinear Sizing Design Sensitivity Analysis Design Sensitivity Formulation for Nonlinear Elastic Problems Static Problems Critical Load Hyperelastic Material Numerical Examples Design Sensitivity Analysis for Elastoplastic Problems Static problems Transient Problems with Explicit Time Integration Numerical Examples Nonlinear Shape Design Sensitivity Analysis Nonlinear Elastic Problems Static Problems Critical Load Hyperelastic Material Numerical Examples Elastoplastic Problems Small Deformation Finite Deformation

15 xvi Contents Numerical Examples Contact Problems Frictionless Contact Frictional Contact Die Shape Design Numerical Examples Dynamic Problems Implicit Method Explicit Method Numerical Examples Nonlinear Configuration Design Sensitivity Analysis Nonlinear Elastic Problems Configuration Design Sensitivity for Static Response Configuration Design Sensitivity for Critical Load Analytical Examples Numerical Examples Elastoplastic Problems Configuration Design Sensitivity for Elastoplastic Truss Numerical Examples PART IV Numerical Implementation and Applications 12 Design Parameterization Sizing Design Parameterization Line Design Components Surface Design Components Shape Design Parameterization Representation of Geometry in Parametric Space Shape Design Parameterization Method Curve Design Parameterization Surface Design Parameterization Design Variable Linking Across Geometric Entities Numerical Implementation of Sensitivity Analysis Sizing Design Sensitivity Computation Finite Element Approximation Structural Components Shape Design Sensitivity Computation Finite Element Approximation Structural Components Design Velocity Field Computation Boundary Velocity Field Computation Domain Velocity Field Computation Using Finite Difference Method Boundary Displacement Method Velocity Field Computation Using Isoparametric Mapping Method Combination of Isoparametric Mapping and Boundary Displacement Methods Design Applications Sizing Design Applications Design of Wheel Structure

16 Contents xvii Design of Wing Structure Shape Design Applications Design of a Three-Dimensional Clevis Design of Turbine Blade Configuration Design Applications Design Crane Structure Nonlinear Design Applications Design of Windshield Wiper Configuration Design of Vehicle A-Pillar Stability Design of Vehicle Passenger Compartment Frame Fatigue and Durability Design Applications Design of Engine Exhaust Manifold References Index