1. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche University of Illinois-Chicago Chapter 6 Optimization Techniques

2. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.2 System Modeling 6.1 INTRODUCTION 6.2 SYSTEM MODELING Analysis and design Figure 6.1 Simple beam (6.2) (6.3)

3. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.2 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)

4. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.2 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)

5. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.2 System Modeling M = MO sin (ω t) (6.6) (6.7) (6.8)

6. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.3 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)

7. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 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 6 6.3 Design Optimization

8. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 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 6 6.3 Design Optimization

9. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.4 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.

10. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.4 Optimal Design Concept

11. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.4 Figure 6.5 A hollow cylinder with thickness t CHAPTER 6 6.4 Optimal Design Concept

12. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.12) (6.13) (6.14) (6.15) (6.16) (6.17) (6.18) CHAPTER 6 6.4 Optimal Design Concept

13. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.5 Unconstrained Optimization 6.5 UNCONSTRAINED OPTIMIZATION Single Variable minimization Figure 6.6 Function f showing local, global and strong minimum (6.20) (6.19)

14. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.5 Find whether the function has a minimum or maximum for CHAPTER 6 6.5 Unconstrained Optimization

15. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Solution: Figure 6.7 Tank CHAPTER 6 6.5 Unconstrained Optimization

16. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago xL (6.21) (6.22) (6.23) (6.24) (6.25) (6.26) CHAPTER 6 6.5 Unconstrained Optimization

17. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Example 6.7 (6.27) CHAPTER 6 6.5 Unconstrained Optimization

18. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.28) (6.29) (6.30) (6.31) CHAPTER 6 6.5 Unconstrained Optimization

19. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.32) (6.33) CHAPTER 6 6.5 Unconstrained Optimization

20. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.5 Unconstrained Optimization

21. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.34) (6.35) CHAPTER 6 6.5 Unconstrained Optimization

22. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 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 6 6.6 Constrained Optimization

23. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.38) (6.39) CHAPTER 6 6.6 Constrained Optimization

24. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 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)

25. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 Constrained Optimization (6.45) (6.46) (6.47) (6.48) (6.49)

26. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 Constrained Optimization (6.50) (6.51) (6.52) (6.53) (6.54) (6.55)

27. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 Constrained Optimization

28. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 Constrained Optimization

29. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.6 Constrained Optimization

30. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.7 Fibonacci Method 6.7 FIBONACCI METHOD 6.7.1 Fibonacci algorithm Figure 6.10 Rabbit’s multiplication at each mature period

31. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Figure 6.11 Initial setup of the uncertainty interval CHAPTER 6 6.7 Fibonacci Method

32. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.7 Fibonacci Method

33. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago gx 2 v 0 sinΘt CHAPTER 6 6.7 Fibonacci Method

34. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago 80 CHAPTER 6 6.7 Fibonacci Method

35. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.7 Fibonacci Method

36. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.8 Newton’s Method 6.8 NEWTON’S METHOD (6.72)

37. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.9 Linear Programming Figure 6.13 Newton’s method 6.9 LINEAR PROGRAMMING (6.74)

38. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago (6.75) CHAPTER 6 6.9 Linear Programming

39. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.9 Linear Programming

40. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago Figure 6.14 Graphical solution CHAPTER 6 6.9 Linear Programming

41. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.10 Geometric Programming 6.10 GEOMETRIC PROGRAMMING where the are chosen properly to minimize T. (6.76) (6.77) (6.78)

42. Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8 Author: Prof. Farid. Amirouche, University of Illinois-Chicago CHAPTER 6 6.11 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. 6.11 OTHER OPTIMIZATION TECHNIQUES