Preface. 1 Introduction to Optimization. 1.1 Introduction. 1.2 Historical Development. 1.3 Engineering Applications of Optimization. 1.4 Statement of an Optimization Problem. 1.5 Classification of Optimization Problems. 1.6 Optimization Techniques. 1.7 Engineering Optimization Literature. 1.8 Solution of Optimization Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 2 Classical Optimization Techniques. 2.1 Introduction. 2.2 Single-Variable Optimization. 2.3 Multivariable Optimization with No Constraints. 2.4 Multivariable Optimization with Equality Constraints. 2.5 Multivariable Optimization with Inequality Constraints. 2.6 Convex Programming Problem. References and Bibliography. Review Questions. Problems. 3 Linear Programming I: Simplex Method. 3.1 Introduction. 3.2 Applications of Linear Programming. 3.3 Standard Form of a Linear Programming Problem. 3.4 Geometry of Linear Programming Problems. 3.5 Definitions and Theorems. 3.6 Solution of a System of Linear Simultaneous Equations. 3.7 Pivotal Reduction of a General System of Equations. 3.8 Motivation of the Simplex Method. 3.9 Simplex Algorithm. 3.10 Two Phases of the Simplex Method. 3.11 MATLAB Solution of LP Problems. References and Bibliography. Review Questions. Problems. 4 Linear Programming II: Additional Topics and Extensions. 4.1 Introduction. 4.2 Revised Simplex Method. 4.3 Duality in Linear Programming. 4.4 Decomposition Principle. 4.5 Sensitivity or Postoptimality Analysis. 4.6 Transportation Problem. 4.7 Karmarkar's Interior Method. 4.8 Quadratic Programming. 4.9 MATLAB Solutions. References and Bibliography. Review Questions. Problems. 5 Nonlinear Programming I: One-Dimensional Minimization Methods. 5.1 Introduction. 5.2 Unimodal Function. ELIMINATION METHODS. 5.3 Unrestricted Search. 5.4 Exhaustive Search. 5.5 Dichotomous Search. 5.6 Interval Halving Method. 5.7 Fibonacci Method. 5.8 Golden Section Method. 5.9 Comparison of Elimination Methods. INTERPOLATION METHODS. 5.10 Quadratic Interpolation Method. 5.11 Cubic Interpolation Method. 5.12 Direct Root Methods. 5.13 Practical Considerations. 5.14 MATLAB Solution of One-Dimensional Minimization Problems. References and Bibliography. Review Questions. Problems. 6 Nonlinear Programming II: Unconstrained Optimization Techniques. 6.1 Introduction. DIRECT SEARCH METHODS. 6.2 Random Search Methods. 6.3 Grid Search Method. 6.4 Univariate Method. 6.5 Pattern Directions. 6.6 Powell's Method. 6.7 Simplex Method. INDIRECT SEARCH (DESCENT) METHODS. 6.8 Gradient of a Function. 6.9 Steepest Descent (Cauchy) Method. 6.10 Conjugate Gradient (Fletcher-Reeves) Method. 6.11 Newton's Method. 6.12 Marquardt Method. 6.13 Quasi-Newton Methods. 6.14 Davidon-Fletcher-Powell Method. 6.15 Broyden-Fletcher-Goldfarb-Shanno Method. 6.16 Test Functions. 6.17 MATLAB Solution of Unconstrained Optimization Problems. References and Bibliography. Review Questions. Problems. 7 Nonlinear Programming III: Constrained Optimization Techniques. 7.1 Introduction. 7.2 Characteristics of a Constrained Problem. DIRECT METHODS. 7.3 Random Search Methods. 7.4 Complex Method. 7.5 Sequential Linear Programming. 7.6 Basic Approach in the Methods of Feasible Directions. 7.7 Zoutendijk's Method of Feasible Directions. 7.8 Rosen's Gradient Projection Method. 7.9 Generalized Reduced Gradient Method. 7.10 Sequential Quadratic Programming. INDIRECT METHODS. 7.11 Transformation Techniques. 7.12 Basic Approach of the Penalty Function Method. 7.13 Interior Penalty Function Method. 7.14 Convex Programming Problem. 7.15 Exterior Penalty Function Method. 7.16 Extrapolation Techniques in the Interior Penalty Function Method. 7.17 Extended Interior Penalty Function Methods. 7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints. 7.19 Penalty Function Method for Parametric Constraints. 7.20 Augmented Lagrange Multiplier Method. 7.21 Checking the Convergence of Constrained Optimization Problems. 7.22 Test Problems. 7.23 MATLAB Solution of Constrained Optimization Problems. References and Bibliography. Review Questions. Problems. 8 Geometric Programming. 8.1 Introduction. 8.2 Posynomial. 8.3 Unconstrained Minimization Problem. 8.4 Solution of an Unconstrained Geometric Programming Program Using Differential Calculus. 8.5 Solution of an Unconstrained Geometric Programming Problem Using Arithmetic-Geometric Inequality. 8.6 Primal-Dual Relationship and Sufficiency Conditions in the Unconstrained Case. 8.7 Constrained Minimization. 8.8 Solution of a Constrained Geometric Programming Problem. 8.9 Primal and Dual Programs in the Case of Less-Than Inequalities. 8.10 Geometric Programming with Mixed Inequality Constraints. 8.11 Complementary Geometric Programming. 8.12 Applications of Geometric Programming. References and Bibliography. Review Questions. Problems. 9 Dynamic Programming. 9.1 Introduction. 9.2 Multistage Decision Processes. 9.3 Concept of Suboptimization and Principle of Optimality. 9.4 Computational Procedure in Dynamic Programming. 9.5 Example Illustrating the Calculus Method of Solution. 9.6 Example Illustrating the Tabular Method of Solution. 9.7 Conversion of a Final Value Problem into an Initial Value Problem. 9.8 Linear Programming as a Case of Dynamic Programming. 9.9 Continuous Dynamic Programming. 9.10 Additional Applications. References and Bibliography. Review Questions. Problems. 10 Integer Programming. 10.1 Introduction 588. INTEGER LINEAR PROGRAMMING. 10.2 Graphical Representation. 10.3 Gomory's Cutting Plane Method. 10.4 Balas' Algorithm for Zero-One Programming Problems. INTEGER NONLINEAR PROGRAMMING. 10.5 Integer Polynomial Programming. 10.6 Branch-and-Bound Method. 10.7 Sequential Linear Discrete Programming. 10.8 Generalized Penalty Function Method. 10.9 Solution of Binary Programming Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 11 Stochastic Programming. 11.1 Introduction. 11.2 Basic Concepts of Probability Theory. 11.3 Stochastic Linear Programming. 11.4 Stochastic Nonlinear Programming. 11.5 Stochastic Geometric Programming. References and Bibliography. Review Questions. Problems. 12 Optimal Control and Optimality Criteria Methods. 12.1 Introduction. 12.2 Calculus of Variations. 12.3 Optimal Control Theory. 12.4 Optimality Criteria Methods. References and Bibliography. Review Questions. Problems. 13 Modern Methods of Optimization. 13.1 Introduction. 13.2 Genetic Algorithms. 13.3 Simulated Annealing. 13.4 Particle Swarm Optimization. 13.5 Ant Colony Optimization. 13.6 Optimization of Fuzzy Systems. 13.7 Neural-Network-Based Optimization. References and Bibliography. Review Questions. Problems. 14 Practical Aspects of Optimization. 14.1 Introduction. 14.2 Reduction of Size of an Optimization Problem. 14.3 Fast Reanalysis Techniques. 14.4 Derivatives of Static Displacements and Stresses. 14.5 Derivatives of Eigenvalues and Eigenvectors. 14.6 Derivatives of Transient Response. 14.7 Sensitivity of Optimum Solution to Problem Parameters. 14.8 Multilevel Optimization. 14.9 Parallel Processing. 14.10 Multiobjective Optimization. 14.11 Solution of Multiobjective Problems Using MATLAB. References and Bibliography. Review Questions. Problems. A Convex and Concave Functions. B Some Computational Aspects of Optimization. B.1 Choice of Method. B.2 Comparison of Unconstrained Methods. B.3 Comparison of Constrained Methods. B.4 Availability of Computer Programs. B.5 Scaling of Design Variables and Constraints. B.6 Computer Programs for Modern Methods of Optimization. References and Bibliography. C Introduction to MATLAB(R) . C.1 Features and Special Characters. C.2 Defining Matrices in MATLAB. C.3 CREATING m-FILES. C.4 Optimization Toolbox. Answers to Selected Problems. Index .

Engineering Optimization : Theory and Practice

Introduction to linear and nonlinear programming

The use of statistical techniques to build approximations of expensive computer analysis codes pervades much of today’s engineering design. These statistical approximations, or metamodels, are used to replace the actual expensive computer analyses, facilitating multidisciplinary, multiobjective optimization and concept exploration. In this paper, we review several of these techniques, including design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning and kriging. We survey their existing application in engineering design, and then address the dangers of applying traditional statistical techniques to approximate deterministic computer analysis codes. We conclude with recommendations for the appropriate use of statistical approximation techniques in given situations, and how common pitfalls can be avoided.

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Metamodels for Computer-Based Engineering Design: Survey and Recommendations

Additive manufacturing (AM) is poised to bring about a revolution in the way products are designed, manufactured, and distributed to end users. This technology has gained significant academic as well as industry interest due to its ability to create complex geometries with customizable material properties. AM has also inspired the development of the maker movement by democratizing design and manufacturing. Due to the rapid proliferation of a wide variety of technologies associated with AM, there is a lack of a comprehensive set of design principles, manufacturing guidelines, and standardization of best practices. These challenges are compounded by the fact that advancements in multiple technologies (for example materials processing, topology optimization) generate a "positive feedback loop" effect in advancing AM. In order to advance research interest and investment in AM technologies, some fundamental questions and trends about the dependencies existing in these avenues need highlighting. The goal of our review paper is to organize this body of knowledge surrounding AM, and present current barriers, findings, and future trends significantly to the researchers. We also discuss fundamental attributes of AM processes, evolution of the AM industry, and the affordances enabled by the emergence of AM in a variety of areas such as geometry processing, material design, and education. We conclude our paper by pointing out future directions such as the "print-it-all" paradigm, that have the potential to re-imagine current research and spawn completely new avenues for exploration. The fundamental attributes and challenges/barriers of Additive Manufacturing (AM).The evolution of research on AM with a focus on engineering capabilities.The affordances enabled by AM such as geometry, material and tools design.The developments in industry, intellectual property, and education-related aspects.The important future trends of AM technologies.

The status, challenges, and future of additive manufacturing in engineering

The increasing complexity of engineering systems has sparked increasing interest in multidisciplinary optimization (MDO). This paper presents a survey of recent publications in the field of aerospace where interest in MDO has been particularly intense. The two main challenges of MDO are computational expense and organizational complexity. Accordingly the survey is focused on various ways different researchers use to deal with these challenges. The survey is organized by a breakdown of MDO into its conceptual components. Accordingly, the survey includes sections on Mathematical Modeling, Design- oriented Analysis, Approximation Concepts, Optimization Procedures, System Sensitivity, and Human Interface. With the authors'' main expertise being in the structures area, the bulk of the references focus on the interaction of the structures discipline with other disciplines. In particular, two sections at the end focus on two such interactions that have recently been pursued with a particular vigor: Simultaneous Optimization of Structures and Aerodynamics, and Simultaneous Optimization of Structures Combined With Active Control.

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Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments

Layout Optimization of Structures

Optimal Topologies of Structures

The topology of truss structures is optimized by assuming zero lower bounds on cross-sections. It is shown that the optimal topology might correspond to a singular solution even for simple structures. Assuming the force method analysis formulation, the problem can be stated in a linear programming (LP) form under certain assumptions. It is then possible to derive analytically some conditions related to optimal topologies. In addition, the difficulty of singular optimal solutions is eliminated. The effect of compatibility conditions on optimal topologies is studied. It is shown that for particular geometries or loading conditions, where some element forces change from tension to compression or vice versa, the optimal topology might represent an unstable structure. Analytical conditions are derived to obtain geometries of multiple optimal topologies. Part of the latter solutions usually represent statically determinate structures. It is shown that a transition in the set of active constraints at the optimum occurs at these particular geometries. The phenomena presented in this study might lead to a better understanding of some properties associated with optimum structural topologies, and to improved design procedures.

Optimal topologies of truss structures

A new solution procedure for improved approximate reanalyses of structures, using results of a single precise analysis, is presented. The proposed procedure is based on combining the computed terms of a series expansion, used as high-quality basis vectors, with coefficients of a reduced basis expression. The latter coefficients can readily be determined for each trial design by various criteria. The results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations. The proposed approach is suitable for various types of design variables and can be used with a general finite element model. A general reanalysis procedure is introduced and the physical significance of first-order approximations is demonstrated. Several numerical examples illustrate the effectiveness of the solution process. It is shown that high-quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables.

Reduced basis approximations of structural displacements for optimaldesign

This study presents a unified reanalysis approach for structural analysis, design, and optimization that is based on the Combined Approximations (CA) method. The method is suitable for various analysis models (linear, nonlinear, elastic, plastic, static, dynamic), different types of structures (trusses, frames, grillages, continuum structures), and all types of design variables (cross-sectional, material, geometrical, topological). The calculations are based on results of a single exact analysis. The computational effort is usually much smaller than that needed to carry out a complete analysis of modified designs. Accurate results are achieved by low-order approximations for significant changes in the design. It is possible to improve the accuracy by considering higher-order terms, and exact solutions can be achieved in certain cases. The solution steps are straightforward, and the computational procedures presented can readily be used with general finite element systems. Typical results are demonstrated by numerical examples.

Uri Kirsch

Papers

Layout Optimization of Structures

Optimal Topologies of Structures

Optimal topologies of truss structures

Reduced basis approximations of structural displacements for optimaldesign

A unified reanalysis approach for structural analysis, design, and optimization