Chapter 6 by o8UaI6M3

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									      University of Illinois-Chicago




             Chapter 6
   Optimization Techniques



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
                                                    My
                                                                       (6.2)
                                                    I




                                              PL3
                                                                       (6.3)
                                              3EI
            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.


        r 4
     I                                                                          F

         4

                                                                             h
          4 Fh
    max              (6.4)
          r 3                                                     2r




            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.




     ..
   J   K  M r             (6.5)




                                          Figure 6.3 (a) Automobile wheel and tire
                                          Figure 6.3 (b) Model representing the tire and shaft

            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

                      GI p
              K                                           (6.6)
                         l
                    d 4
               Ip                                         (6.7)
                     32
              G  80  109 N / m 2

              K  2.455N.m / rad

              I p  0.006136108 m4

                             K
              n                                            (6.8)
                             J

              M = MO sin (ω t)
            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

        x1 , x 2 ,......... , x n   0            (i  1, m)               (6.9)

                   ( x1 , x2 )  x12 x2  V  0                             (6.10)

                    j  x1 , x2 ,......... , xn   L j                      (6.11)

             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

     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.

            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

     Solution:
                 A = 2LH + 2HW + 2WL

                 T = C (2LH + 2HW + 2WL)
                 V =LWH

                  = V – LWH = 0

                         L  L1           W  W1               H H 1
  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.
             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.
            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




            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

  Example 6.4




                  Figure 6.5 A hollow cylinder with thickness t

            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

      d o  d i  2t
       d o  d i  2t  0

      h(d o , d i , t )  d 0  d i  2t       (6.12)
                             m ax  s         (6.13)
                         M t (d o / 2)
      max   max                            (6.14)
                               J
                         m ax  s  0         (6.15)

       m ax  M t (d o / 2 J )  0            (6.16)

           
                   
       J    d o4  d i4  0                (6.17)
            32 
         16 M t d o
                       s0                    (6.18)
         (d 0  d i )
             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                                                                 6.5 Unconstrained Optimization

  6.5 UNCONSTRAINED OPTIMIZATION
  Single Variable minimization



  f ( x * )  0    (6.19)

  f ( x * )  0
  f ( x * )  0




                                    Figure 6.6 Function f showing local, global and strong minimum

                         
      f  (q1 , q2 ,...., qn , q1 , q2 ,...., qn , t )                           (6.20)

               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

  Example 6.5

 Find whether the function f  x has a minimum or maximum for x  [1,1]
                                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                                                              6.5 Unconstrained Optimization




  Solution:




                                                                              Figure 6.7 Tank

            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


                                                             xL
                   x 2                                  2x 2       
             f  2 * 
                                     * 20  10xL   20
                                                            4      xL 
                                                                                                 (6.21)
                   4                                                 
              f  10 x 2  10 xL  10 x 2  20 xL
              f  20 x 2  30 xL                                                                (6.22)

             x 2
                  * L  120                                                                       (6.23)
              4
                     480                                                                          (6.24)
              L
                     x 2
                                   480                                                          (6.25)
               f  20 x 2  30 x    2 
                                   x 
                          14400
               f  20x       2                                                                  (6.26)
                            x
            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




                                                      f x   40x 
                                                                             14400
                                                                                 2
                                                                                   0
                                                                               x
                                                     40x 3  14400                           (6.27)
 Example 6.7                                         x 3  114.59
                                                     x  4.86m




            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




                                                                              (6.28)


                                                                              (6.29)




                                                                              (6.30)


                                                                              (6.31)


            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




                                                                                                  (6.32)




                                                                                                   (6.33)




            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




            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




                                                                              (6.34)




                                                                               (6.35)




            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                                                                           6.6 Constrained Optimization

  6.6 CONSTRAINED OPTIMIZATION
            m
  T  To    j  j                  (6.36)
            j 1




                                                                                                       (6.37)



    Example 6.9

   Solve for the optimal design problem of example 6.6 by the
method of Lagrange multipliers.

                   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




                                                                               (6.38)




                                                                               (6.39)




            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                                                                     6.6 Constrained Optimization


                                                                               (6.40)

                                                                               (6.41)

                                                                               (6.42)
            (6.41)


                                (6.42)                                         (6.43)

                                      (6.43)     (6.44)                        (6.44)          (6.40)


                                                            (6.43)              (6.44)




             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)




            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)

            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




            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




            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




            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

             Period 1


             Period 2


             Period 3


             Period 4


             Period 5




            Figure 6.10 Rabbit’s multiplication at each mature period

            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




                Figure 6.11 Initial setup of the uncertainty interval




            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




            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




                                     v0sinΘt



                                                gx2




            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




                                                           80




            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




            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
                                           fk
                          x k 1     xk                                    (6.72)
                                           f k

            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



                             f



                                                                Linear approximation
                                                                to f at x
                                                        fk fk
                                                         '


                                                           xk                  x
                                                        ' -1
                                             =
                                         x k+1 x k- (f k) f k




                        Figure 6.13 Newton’s method


   6.9 LINEAR PROGRAMMING
                                    T   k i xi                                   (6.74)
                                                    i

            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




                                                                                                     (6.75)




            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




            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




           X2




                                    C
       5


                                            D
       4



       3



       2



       1



                                                                      X1   Figure 6.14 Graphical solution
                 1      2       3       4       5    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                                                                    6.10 Geometric Programming


  6.10 GEOMETRIC PROGRAMMING

                                                    3
                T  x1  2 x1 x2 
                            1
                                                       x1 x2  x3
                                                                3
                                                                                        (6.76)
                                                     5

                T  u1  u 2  ......  u p                                              (6.77)


                                                               sp
                                        u2   u p 
                                  s1           s2
                    u1 
                G  
                   s                   ... 
                                       s  s                                          (6.78)
                    1                 2  p

                 where the        sp 's     are chosen properly to minimize T.

            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


  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.

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

								
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