Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN Author: Prof. Farid. Amirouche University of Illinois-Chicago Chapter 6 Optimization Techniques

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER System Modeling 6.1 INTRODUCTION 6.2 SYSTEM MODELING Analysis and design Figure 6.1 Simple beam (6.2) (6.3)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER System Modeling Example 6.1 Consider a tree trunk, which can be modeled by a beam as shown in figure 6.2. On a windy day, one can determine the wind speed or force that would cause the tree to break. (6.4)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER System Modeling Example 6.2 A second example involves the modeling of an automobile wheel and tire. Figure 6.3 (a) Automobile wheel and tire Figure 6.3 (b) Model representing the tire and shaft (6.5)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER System Modeling M = MO sin (ω t) (6.6) (6.7) (6.8)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Design Optimization 6.3 DESIGN OPTIMIZATION Formulation of an optimum design There are four steps to the formulation of an optimum design: 1. Identifying the design parameters. 2. Defining the design constraints. 3. Defining the objective functions 4. Evaluating alternatives. Design Parameters Constraints (6.9) (6.10) (6.11)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Evaluation Example 6.3 Consider a rectangular box used for storing important documents. Define the objective function if C denotes the cost per unit area of the metal used for fabrication of the box. Define the constraint equations and the limits on their design variables. CHAPTER Design Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Solution: A = 2LH + 2HW + 2WL T = C (2LH + 2HW + 2WL) V =LWH  = V – LWH = 0 As shown in the example, the steps that generally follow in formulation of a design problem are Identification of the design variables. Selection of a cost function and developing an expression for it in terms of the design variables. Identification of constraints and developing expressions for them in terms of design variables. CHAPTER Design Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Optimal Design Concept 6.4 OPTIMAL DESIGN CONCEPT Questions: How is each design described? What is the criterion for best “design”? What are the available means? Design optimization Select a set of variables. Select an objective function. Determine a set of constraints. Solutions will be based on finding the values for the variables that would minimize or maximize the objective function and satisfy the constraints at the same time.

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Optimal Design Concept

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.4 Figure 6.5 A hollow cylinder with thickness t CHAPTER Optimal Design Concept

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.12) (6.13) (6.14) (6.15) (6.16) (6.17) (6.18) CHAPTER Optimal Design Concept

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Unconstrained Optimization 6.5 UNCONSTRAINED OPTIMIZATION Single Variable minimization Figure 6.6 Function f showing local, global and strong minimum (6.20) (6.19)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.5 Find whether the function has a minimum or maximum for CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Solution: Figure 6.7 Tank CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago xL (6.21) (6.22) (6.23) (6.24) (6.25) (6.26) CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.7 (6.27) CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.28) (6.29) (6.30) (6.31) CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.32) (6.33) CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.34) (6.35) CHAPTER Unconstrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago 6.6 CONSTRAINED OPTIMIZATION Example 6.9 Solve for the optimal design problem of example 6.6 by the method of Lagrange multipliers. (6.36) (6.37) CHAPTER Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.38) (6.39) CHAPTER Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization (6.40) (6.41) (6.42) (6.43) (6.44) (6.41) (6.42) (6.43)(6.44)(6.40) (6.43)(6.44)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization (6.45) (6.46) (6.47) (6.48) (6.49)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization (6.50) (6.51) (6.52) (6.53) (6.54) (6.55)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Constrained Optimization

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Fibonacci Method 6.7 FIBONACCI METHOD Fibonacci algorithm Figure 6.10 Rabbit’s multiplication at each mature period

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Figure 6.11 Initial setup of the uncertainty interval CHAPTER Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago gx 2 v 0 sinΘt CHAPTER Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago 80 CHAPTER Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Fibonacci Method

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Newton’s Method 6.8 NEWTON’S METHOD (6.72)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Linear Programming Figure 6.13 Newton’s method 6.9 LINEAR PROGRAMMING (6.74)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.75) CHAPTER Linear Programming

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Linear Programming

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago Figure 6.14 Graphical solution CHAPTER Linear Programming

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Geometric Programming 6.10 GEOMETRIC PROGRAMMING where the are chosen properly to minimize T. (6.76) (6.77) (6.78)

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER Other Optimization Techniques Search methods Exhaustive Search. Grid search. Random search. Simplex search. Gradient methods Steepest descent Conjugate gradients Second-derivative methods Non-linear programming Johnson’s method Powell’s method for unconstrained situations OTHER OPTIMIZATION TECHNIQUES