Pré-visualização parcial do texto
Baixe Elements of Structural Optimization - Haftka e outras Exercícios em PDF para Engenharia Aeronáutica, somente na Docsity!
SOLID MECHANICS AND ITS APPLICATIONS Volume 11 Series Editor: G.M.L. GLADWELL, Solid Mechanics Division, Faculty of Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3G4 Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative research- ers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics, Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynarmical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture, tribology; experimental mechanics; biomeçhanics and machine design, The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates, but essentially the emmphasis is on readability and clarity. For a list of related mechanics titles, see final pages. Elements of Structural Optimization Third revised and expanded edition by RAPHAEL T. HAFTKA Depariment of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA. and ZAFER GÚRDAL Depariment of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA. VA Al KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON This book is dedicated to Rose Pinar and Erin Contents Optimal Design of Euler-Bernoulki Columns Optimum Vibrating Euler-Bernoulli Beams .. 2.6 Use of Series Solutions in Structural Optimization . 2.7 Exercises 2.8 References . .52 57 .61 .64 -66 Chapter 3. Linear Programming........cecccccsessiieesiiiiiios n 3.1 Limit Analysis and Design of Structures Formulated as LP Problems72 3.2 Prestressed Concrete Design by Linear Programming ... 3.3 Minimum Weight Design of Statically Determinate Trusse 3.4 Graphical Solutions of Simple LP Problems . 3.5 A Lincar Program in a Standard Form . Basic Solution. .89 3.6 The Simplex Metho: +90 Changing the Basis. “9 Improving the Objective Function . 3.7 Duality in Lincar Programming... 3.8 An Interior Method—Karmarkar's Alg Direction of Move .... Transformation of Coordinates Move Distance... 3.9 Integer Linear Programming Branch-and-Bound Algorithm. 3.10 Exercises. 3.11 References, . Chapter 4. Unconstrained Optimization........ccisesiio 4.1 Minimization of Functions of One Variable Zeroth Order Methods First Order Methods Second Order Method. Safeguarded Polynomial Interpolation... 4.2 Minimization of Functions of Several Variables Zeroth Order Methods First Order Methods. . Second Order Methods Applications to Analysi: 4.3 Specialized Quasi-Newton Methods . Esploiting Sparsity...... Coercion of Hessians for Suitability with Quasi-Newton Mothods 144 Making Quasi-Newton Methods Globally Convergent .145 4,4 Probabilistic Search Algorithms Simulated Anneeling.. Genetic Algorithms. 4.5 Exercises............ 4.6 References viii Contents Chapter 5. Constrained Optimization.......,..... 5.1 The Kuhn-Tucker Conditions General Case... Convex Problems . 5.2 Quadratic Programming Problems . 5,3 Computing the Lagrange Multiplicrs . 5.4 Sensitivity of Optimum Solution to Problem Parameters . 5.5 Gradient Projection and Reduced Gradient Methods 5.6 The Feasible Directions Method 5.7 Penalty Function Methods. Exterior Penalty Function Interior and Extended Interior Penalty Functions Unconstrained Minimization with Penalty Functions . Integer Programming with Penalty Functions . 5.8 Multiplier Methods... 5.9 Projected Lagrangian Methods (Sequential Quadratic Prog.) . 5.10 Exercises... 5.11 References Chapter 6. Aspects of the Optimization Process in Practice....... 209 6.1 Generic Approximations Local Approximations Giabal and Midrange Approximations . 6.2 Fast Reanalysis Techniques Lincar Static Response. Eigenvalue Problems... .226 6.3 Sequential Linear Programmin, .228 6.4 Sequential Nonlinear Approximate Optimization 236 6.5 Special Problems Associated with Shape Optimi 6.6 Optimization Packages .....cccereces 242 6.7 Test Problems. Ten-Bar Truss Twenty-Five-Bar Truss Seventy-Two-Bar Truss 6.8 Exercises... 6.9 References Chapter 7. Sensitivity of Discrete Systems. 74 Finite Difference Approximations .. Accuracy and Step Size Selection Iterative Methods...... Effect of Derivative Magnitude on Accuracy 7.2 Sensitivity Derivatives of Static Displacement and Stress Constreints263 Analytical First Derivatives. Second Derivatives The Semi- Analytical Meth Nonlinear Analysis Contents Reciprocal- Approzimation Based Approach Scaling-based Approach Other Formulation: 9.5 Exercises . 9.6 References. Chapter 10. Decomposition and Multilevel Optimization... 10.1 The Relation between Decomposition and Multilevel Formulation .387 10.2 Decomposition.....ce caio 10.3 Coordination and Multilevel Optimizal 10.4 Penalty and Envelope Function Approaches 10.5 Narrow-Tree Multilevel Problems. Simultoncous Analysis and Design. Other Applications .. 10.6 Decomposition in Respon 10.7 Exercises... 10.8 References Chapter 11. Optimum Design of Laminated Composite Materials 415 11.1 Mechanical Response of a Laminate Orihotrapic Lamina ........... Classical Laminated Plate Thcory Bending, Extension, and Shear Coupling. 11.2 Laminate Design ......... Design of Laminates for In-plane Response. Design of Laminates for Flexural Response. 11.3 Stacking Sequence Design ......... Graphical Stacking Sequence Design Penalty Function Formulation . Integer Linear Programming Formulation Probabilistic Search Methods . 11.4 Design Applications... Stiffened Plate Design. Aeroelastic Tailoring. 11.5 Design Uncertainties 11.6 Exercises... 11.7 References Name Index .......ceeererenertresaceecanessenaecacarenerenacarvatacanea 469 xi Preface Schmit, and which are becoming widely uscd, are referred to in this book as sequen- tial approximato optimization techniques. These techniques uso tho analysis package for the purpose of constructing an approximation to the structural design problem, and then cmploy various mathematical optimization techniques to solve the approx imate problem. The optimum of the approximate problem is then used as a basis for performing one or more structural analyses for the purpose of updating or refining the approximate design problem. Most of the approximate design problems are based on derivatives of the structural response with respect to design parameters. In the new environment the structural designer is typically called upon to provide the interface between a commercially available analysis program, and a comnercially available optimization software package. The: three most important ingredients of the interface are: sensitivity derivativc calculation, construction of an approximate problem, and evaluation of results for the purpose of fine-tuning the approximate problem or the optimization method for maximum efficiency and reliability. This textbook is organized so that its middle part—Chapters 6, 7 and 8 deal with the two issues of constructing the approximate problem and obtaining sensitivity derivatives. Evaluating the results of the optimization calls for a basie understanding of optimality conditions and optimization methods. This is dealt with in Chapters 1 through 5. The lost three chapters deal witli the specialized topics of optimality criteria methods, multi-level optimization, and applications to composite materials. The material in the toxtbook can be used in various ways in teaching a graduate course in structural optimization, depending on the available amount of time, and whether students have prior preparation in optimization techniques. Without prior preparation in optimizaticn techniques it is suggested that the minimum time requirement is one semester. Tt is suggested to cover Chapter 1, sections 2.1, 2.2 and 2.3 of Chapter 2, Sections 3.1 and 3,4 of Chapter 3, some material from Chapters 4 and 5 depending om the instructor's favorite optimization methods, most of Chapter 6 and the first two sections of Chapter 7. Wit a two- quarter sequence it is suggested to cover Chapters 1 and 2, selected topics of Chapters 3 to 5 and Chapter 6 in the first quarter, and Chapters 7, 9, 11 and either Chapter 8 or Chapter 10 in the second quarter. Finally, in a two-semester sequence it is recommended to cover Chapters 1 through 6 in the first semester, and Chapters 7 through 11 in the second semester. With a preparatory course in mathematical optimization a one quarter and a one semester versions of the course can be considered. À one-quarter version could include Chapters 1 and 2, sections 3.1, 3.2, 3.3 and 3.7 of Chapter 3, and Chapters 6, the first two sections of Chapter 7, and Chapter 9 or 11., À one-semester version could inelude the same part of Chapters 1 through 7 and then Chapters 9 through 11. The authors gratefully acknowledge the assistance of Drs. H. Adelman, B. Bartbelemy, J-F. Barthelemy, E. Berke, R. Grandhi, D. Grierson, E. Hang, R. Plant, iesti, and J. Starnes in reviewing parts of the manuscript and offering critical | xiv Introduction 1 Optimization is concerned with achieving the best outcome of a given operation while satisfying certain restrictions. Human beings, guided and influenced by their natural surroundings, almost instinctively perform all functions in a manner that economizes energy or minimizes discomfort and pain. The motivation is to exploit the available limited resonrces in a manner that maximizes output or profit. The early inventions of the lever or the pulley mechanisms arc clear manifestations of man's desire to maximize mechanical efficiency. Innumerable other such examples abound in the saga of human history. Douglas Wilde [1] provides an interesting account of the origin of the word optimum and the definition of an optimal design. We will paraphrase Wilde and offer the definition of an optimal design as being “the best fcasible design according to a preselected quantitative measure of effectiveness”. As if is beyond the scope of this text to trace the historical development of op- timization, we list a few of the more recent references on the subject of structural optimization. These references [2-19] trace the development of the field cf structural optimization dating back to the eighteenth century. The importance of minimum weight design of structures was first recognized by the aerospace industry where aircraft structural designs are often controlled more by weight than by cost consider- ations. In other industries dealing with civil, mechanical and automotive engineering systems, cost may be the primary consideration although the weight of the system does affect its cost and performance. À growing realization of the scarcity of raw ma- terials and a rapid depletion of our conventional energy sources is being translated into a demand for lightweight, efficient and low cost structures. This demand in tum emphasizes the need for engineers to be cognizant of techniques for weight and cost optimization of structures. The objective of this text is to acquaint students and practicing engineers with these techniques. 1.1 Function Optimization and Parameter Optimization Before the advent of high speed compntation most of the solutions of structural analysis problems were based on formulations employing differential equations. These 1 Section 1.8: Elements of Problem Formulation in Chapter 2. The class of structural optimization problems that secks an optimum structural function is called function or distributed parameter structural optimization. In the late fifties and early sixtics high speed electronic computers had a profound effect on structural analysis solution procedures. Techniques that were well suited to computer implementation, in particular the finite element method (FEM), became dominant. The finite element method discretizes the structure at the very beginning of the analysis, so that the unknowns in the analysis are discrete values of displace- ments and stresses at nodes of the finite element model, rather than functions. The differential equations solved by earlier analysts are replaced by systems of algebraic equations for the variables that describe the discretized system. The same transformation began to take hold in the carly sixtics in the field of structural optimization. When optimizing a structure discretized by finite clemeuts it is natural to discretize the structural properties which are optimized. Consider again the beam example of Figure 1.1.1. À finite element solution for the displace- ments starts by dividing the beam into a number of constant-property segments or finite elements. An optimization of the same beam would naturally use the moments, of inertia of the segments as design parameters. Thus, instead of searching for an optimum function, we will be looking for the optimum values of a number of param- eters. The mathematical discipline that deals with parameter optimization is called mathematícal programming. The bulk af this text (Chapters 3-7, 9-11) is concerned, therefore, with mathematical programming techniques and their application to struc- tural optimization problems defined by discretized models. In particular, it is often implicitly assumed that the structural analysis is based on the finite element method. 1.2 Elements of Problem Formulation 1.2.1 Design Voriables The notion of improving or optimizing a structure implicitly presupposes some free- dom to change the structure. The potential for change is typically expressed in terms of ranges of permissible changes of a group of parameters. Such parameters are ust- ally called design variables in structural optimization terminology and denoted by a vector x = (27,,X2,...,%n) in this book. Design variables can be cross-sectional dimensions or member sizes, they can be parameters controlling the geometry of the structure, its material propertics, etc. Design variables may take continuous ox dis- crete values. Continuous design variables have a range of variation, and can take any value in that range. For example, in the design problem of Figure 1.1.1 the moment of inertia of any segment of the beam may be considered a continuous design variable. Discrete design variables can take only isolated values, typically from a list of permis- sible values, Material design variables are often discrete. If we consider five materials in the design of the beam, then we can define a design variable that can take any integer valuc from one to five to represent the material choice. Design variables that 3 Chapter 1: Introduction are commonly trcated as continuous are often made diserete due to manufacturing considerations. For example, if the beam of Figure 1.1.1 is designed to minimize cost, then we may need to limit ourselves to commercially available cross sections. The moment of inertia would then cease to be a continuous design variable, and would become a discrete one. In most structural design problems we tend to disregard the discrete nature of the design variables in the solution of the optimization problem. Once the optimum design is obtained, we then adjust the values of the design variables to tle nearest available discrete value. This approach is taken because solving an optimization problem with discrete design variables is usually much more difficult than solving a similar problem with continuons design variables. However, rounding off the design to the closest integer solution works well when the available values of te design variables are spaced reasonably close to one another, so that changing the value of a design variable to the nearest integer does not change the response of the structure substantially. In some cases the discrete valucs of the design variablos arc spaced too far apart, and we have to solve the problem with discrete variables, This is done by employing a branch of mathematical programming called integer programming. In this text jk is assumed that design variables are continuous unless otherwise stated. Figure 1.2.1 Optimal thickness distribution of a plate, The choice of design variables can be critical to the success of the optimization process. In particular it is important to make sure that the choice of design variables is consistent with the analysis model. Consider, for example, the process of discretizing a structure by a finite element model and applying the optimization procedure to the model. If the design variable distribution has a one-to-one correspondenee with the finite ele.aent model we can encounter serious accuracy problems. For example, the plate shown in Figure 1.2.1 was analyzed [20] by a 7 x 7 finite element mesh, with most design variables specifying the thickness of individual elements. While the 7x7 4 Chapter 1: Introduction 100in Figure 1.2.9 Three-bar truss example. the stresses in the three bars as 0;, i = 1,2,3, then a composite objective function f could be F=egm+ao + mxp0a + ago, (1.2.1) where the a; are weighting coefficients selected to reflect the relative importance of the four objective functions. The second intuitive way to reduce the number of objective functions is to select the most important as the only objective function and to impose limits on the others. “Thus we can formulate the three-bar truss design problem as minimization of mass, subject to npper limits on the values of the tree stresses. When it is not intuitively clear how to weight or choose between the objective functions, a systematic approach to the problem is tlrough à branch of mathematical programming called Edgeworth-Pareto optimization that deals with multiple objective functions [22-24]. Stadier [25,26] was probably the first to apply Edgeworth-Pareto optimality to structural design. More recent applications can be found in Refs. 27-31. A vector of design variables xº is said to be Edgeworth-Pareto optimal if, for any other vector x, either the values of all the objective functions remain the same, or at least one of them worsens compared to its value at x*. When it is not possible to specify intuitively the relative importance of the objective functions in an equation such as (1.2.1), the values of the weights q; à =: 0,1,2,3in Eq. (1.2.1) can be decided by studying various Edgeworth-Pareto optimal designs. Thus the design process is an interactive process, and the imposition of constraints is postponcd until knowledge of the optimum performance is gained by studying Edgeworth-Pareto optimal designs. One of the approaches for generating a parcto-optimal solution to multiple ob- jective fanction optimization problems is based on the minimization of the deviation of the individual objective functions from their individual minimum values. If the - independent minimizations of each of the objective functions result in function val- ves OE ff, $5,..., [5 associated with design points x3,x3, ...,X5, then for au arbitrary value of the design variable vector x the normalized distance of each of the objective 6 Section 1.2: Elements of Problem Formulation functions from its individual optimum is given by ditx) = i=l,spo. (1.2.2) dt is then possible to pose the problem either as the minimization of the largest deviation of the objectivo functions from their individual minima (Lo norm), minimize nas [ato] , (1.2.3) or of the distance (i.e. the & or Euclidean norm) from the reference point £” = Citi co fo) to E = (fi faro ? minimize SOdê. (1.2.4) e His also possible to nse weighting coefficients in Eq. (1.2.4) for the contributions of the individual objective functions. A more detailed discussion of tlie methods for solving multicriteria optimization problems and their design applications is given by Eschenaner et al. [31]. Example 1.2.1 Consider the design of cross-sectional dimensions of a rectangular beam so'as to minimize the area. At the same time it is desired to minimize the maximum shear stress in the beam corresponding to a unit shear force. Based on some physical constraints, the two variables, w and A, which are the width and height of the cross-section are limited to be in the range 0.5 < w,h < 5 units. 1 2,3 4.5 1 243 4.5 area contours shear stress contours Figure 1.2.4 Design of a beam cross-section for minimum area and minimum shear stress. 
