Elements of Structural Optimization

1. Introduction.
- 1.1 Function Optimization and Parameter Optimization.
- 1.2 Elements of Problem Formulation.- Design Variables.- Objective Function.- Constraints.- Standard Formulation.
- 1.3 The Solution Process.
- 1.4 Analysis and Design Formulations.
- 1.5 Specific Versus General Methods.
- 1.6 Exercises.
- 1.7 References.
- 2. Classical Tools in Structural Optimization.
- 2.1 Optimization Using Differential Calculus.
- 2.2 Optimization Using Variational Calculus.- to the Calculus of Variations.
- 2.3 Classical Methods for Constrained Problems.- Method of Lagrange Multipliers.- Function Subjected to an Integral Constraint.- Finite Subsidiary Conditions.
- 2.4 Local Constraints and the Minmax Approach.
- 2.5 Necessary and Sufficient Conditions for Optimality.- Elastic Structures of Maximum Stiffness.- Optimal Design of Euler-Bernoulli Columns.- Optimum Vibrating Euler-Bernoulli Beams.
- 2.6 Use of Series Solutions in Structural Optimization.
- 2.7 Exercises.
- 2.8 References.
- 3. Linear Programming.
- 3.1 Limit Analysis and Design of Structures Formulated as LP Problems.
- 3.2 Prestressed Concrete Design by Linear Programming.
- 3.3 Minimum Weight Design of Statically Determinate Trusses.
- 3.4 Graphical Solutions of Simple LP Problems.
- 3.5 A Linear Program in a Standard Form.- Basic Solution.
- 3.6 The Simplex Method.- Changing the Basis.- Improving the Objective Function.- Generating a Basic Feasible Solution-Use of Artificial Variables.
- 3.7 Duality in Linear Programming.
- 3.8 An Interior Method-Karmarkar's Algorithm.- Direction of Move.- Transformation of Coordinates.- Move Distance.
- 3.9 Integer Linear Programming.- Branch-and-Bound Algorithm.
- 3.10 Exercises.
- 3.11 References.
- 4. Unconstrained Optimization.
- 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 Analysis.
- 4.3 Specialized Quasi-Newton Methods.- Exploiting Sparsity.- Coercion of Hessians for Suitability with Quasi-Newton Methods.- Making Quasi-Newton Methods Globally Convergent.
- 4.4 Probabilistic Search Algorithms.- Simulated Annealing.- Genetic Algorithms.
- 4.5 Exercises.
- 4.6 References.
- 5. Constrained Optimization.
- 5.1 The Kuhn-Tucker Conditions.- General Case.- Convex Problems.
- 5.2 Quadratic Programming Problems.
- 5.3 Computing the Lagrange Multipliers.
- 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.
- 6. Aspects of the Optimization Process in Practice.
- 6.1 Generic Approximations.- Local Approximations.- Global and Midrange Approximations.
- 6.2 Fast Reanalysis Techniques.- Linear Static Response.- Eigenvalue Problems.
- 6.3 Sequential Linear Programming.
- 6.4 Sequential Nonlinear Approximate Optimization.
- 6.5 Special Problems Associated with Shape Optimization.
- 6.6 Optimization Packages.
- 6.7 Test Problems.- Ten-Bar Truss.- Twenty-Five-Bar Truss.- Seventy-Two-Bar Truss.
- 6.8 Exercises.
- 6.9 References.
- 7. Sensitivity of Discrete Systems.
- 7.1 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 Constraints.- Analytical First Derivatives.- Second Derivatives.- The Semi-Analytical Method.- Nonlinear Analysis.- Sensitivity of Limit Loads.
- 7.3 Sensitivity Calculations for Eigenvalue Problems.- Sensitivity Derivatives of Vibration and Buckling Constraints.- Sensitivity Derivatives for Non-Hermitian Eigenvalue Problems.- Sensitivity Derivatives for Nonlinear Eigenvalue Problems.
- 7.4 Sensitivity of Constraints on Transient Response.- Equivalent Constraints.- Derivatives of Constraints.- Linear Structural Dynamics.
- 7.5 Exercises.
- 7.6 References.
- 8. Introduction to Variational Sensitivity Analysis.
- 8.1 Linear Static Analysis.- The Direct Method.- The Adjoint Method.- Implementation Notes.
- 8.2 Nonlinear Static Analysis and Limit Loads.- Static Analysis.- Limit Loads.- Implementation Notes.
- 8.3 Vibration and Buckling.- The Direct Method.- The Adjoint Method.
- 8.4 Static Shape Sensitivity.- The Material Derivative.- Domain Parametrization.- The Direct Method.- The Adjoint Method.
- 8.5 Exercise.
- 8.6 References.
- 9. Dual and Optimality Criteria Methods.
- 9.1 Intuitive Optimality Criteria Methods.- Fully Stressed Design.- Other Intuitive Methods.
- 9.2 Dual Methods.- General Formulation.- Application to Separable Problems.- Discrete Design Variables.- Application with First Order Approximations.
- 9.3 Optimality Criteria Methods for a Single Constraint.- The Reciprocal Approximation for a Displacement Constraint.- A Single Displacement Constraint.- Generalization for Other Constraints.- Scaling-based Resizing.
- 9.4 Several Constraints.- Reciprocal-Approximation Based Approach.- Scaling-based Approach.- Other Formulations.
- 9.5 Exercises.
- 9.6 References.
- 10. Decomposition and Multilevel Optimization.
- 10.1 The Relation between Decomposition and Multilevel Formulation.
- 10.2 Decomposition.
- 10.3 Coordination and Multilevel Optimization.
- 10.4 Penalty and Envelope Function Approaches.
- 10.5 Narrow-Tree Multilevel Problems.- Simultaneous Analysis and Design.- Other Applications.
- 10.6 Decomposition in Response and Sensitivity Calculations.
- 10.7 Exercises.
- 10.8 References.
- 11.Optimum Design of Laminated Composite Materials.
- 11.1 Mechanical Response of a Laminate.- Orthotropic Lamina.- Classical Laminated Plate Theory.- 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.