Optimization of Structural Design using Geometric Programming Method

Optimization of Structural Design using Geometric Programming Method

Optimization of Structural Design using Geometric Programming Method

Samir Dey 1*, T. K. Roy 2

1 Department of Mathematics,Asansol Engineering College,Vivekananda Sarani, Asansol, -713305, West Bengal, India.

2Department of Mathematics, Bengal Engineering and Science University,

Shibpur, Howrah: 711104, India.

Abstract: The main objective of structural engineers throughout design history has been to obtain structure under the prescribed design conditions which can not only withstand external loads safety but also achieve an economic solution. This paper focuses on the use of geometric programming solution method to optimum design of plane truss structures. This approach is illustrated on planer truss optimization model and the results are discussed.

Keywords: Structural Optimization, Geometric Programming.

1. INTRODUCTION

   A Geometric Program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. It has useful theoretical and computational properties. Although GP in standard form is apparently a non convex optimization problem, it can be readily turned into a convex optimization problem; hence a local optimum is also global optimum. Here the advantage is that it is usually much simpler to work with the dual than the primal one. Solving a nonlinear programming problem by GP method with degree of difficulty (DD) plays a significant role.

   Since late 1960s Geometric Programming (GP) has been known and used in different field like Operations Research, Engineering designs etc. The general theory of geometric programming and its engineering application was initially developed by Duffin,Peterson and Zener [10] and Zener [4] in their published book. A serious limitation in the application of this theory has been that all the functions involved in the problem are to be posynomials.This shortcoming was overcome by Wild and Beightler

   [5] in 1967 when they generalized the theory to allow the use of negative coefficients in both objective and constraints, and also to permit reversed inequality constraints. Generalized GP refers to minimizing a generalized posynomial subject to upper bound inequality constraints on generalized posynomials. This method is a general form of geometric programming method in which signomal functions are present in objective function and in constraints.

   The main objective of a structural engineering is to design structures which withstand external loads safely and at a minimum cost or weight [2,3 and7].The desire to improve a design without compromising the structural integrity has been a strong driving force behind the development of various optimum design methods.

   Finally this GP method is identified through the numerical example of two-bar truss and the analysis results show that the geometric programming method can always converges to the global optimal solution.

2. Truss Structural Optimization

   The mathematical form of optimization problem for truss structure can be expressed as follows:

   Find

   AT A , A ,……, A

   (2.1)

   1 2 n

   n

   To minimize

   F W ( A) Li Ai

   i1

   (2.2)

   Subject to

   g L g ( A) gU

   j 1, 2,3,….., m

   (2.3)

   and

   j j j

   i i i

   i i i

   Amin A Amax

   i 1, 2,3,….., n

   (2.4)

   Where

   Ai = the design variable i (member i cross-sectional area, n= the number of

   i

   i

   design variables,W ( A) = the objective function ( the structural weight), = the material

   density,

   Li the member of length, m the number of inequality constraints (g),

   Amin and

   A

   A

   max

   i

   are the lower and the upper bounds of the

   ith

   variable respectively. The lower

   bounds posed by equation-3 on the constraints include truss member stresses and joint displacements.

3. Geometric Programming Method:

   A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. GP is a methodology for solving algebraic non-linear optimization problems. Also linear programming is a subset of a geometric programming .The theory of geometric programming was initially developed about three decades ago and culminated in the publication of the seminal text in this area by Duffin, Peterson, and Zener [10]

   The general constrained Primal Geometric Programming problem is as follows:

   Minimize g

   T0

   0 0t n

   0 0t n

   (x) c

   N

   xa0 tn

   (3.1)

   Subject to

   t1

   Tm N

   n1

   m mt n

   m mt n

   g (x) c xamtn 1; m 1, 2, 3,……., M

   (3.2)

   t1 n1

   xn 0, n 1, 2,………., N.

   Here

   c0t 0

   and

   a0tn be any real number. The objective function contains

   T0 terms and

   Tm terms in the inequality constraints. Here the coefficient of each term is positive.So it is

   a constrained posynomial geometric programming problem. Let

   T T0 T1 ……… Tm

   be the total number of terms in the primal program. The degree of difficulty (DD) is defined as DD = Total no. of terms (Total no. of variables -1) =T (N 1) .The dual

   problem (with the objective function d(w) ,where

   w w(wmt ), m 0,1, 2……, M ;t 1, 2,…..Tm is the decision vector) of the geometric programming problem (1) for the general posynomial case is as follows:

   T0 c

   w0 t M

   Tm c w

   wmt

   Maximize d (w) 0t

   mt mt

   (3.3)

   Subject to

   T0

   t1 w0t

   m1

   t1

   wmt

   w0t 1 , (Normality condition)

   t 1

   M Tm

   amtnwmt 0

   m0 t 1

   for n 1, 2,……, N.

   (Othogonality conditions)

   wmt 0 m 0,1,………, M ; t 1, 2,……..Tm .

   For a primal problem with M variables,

   T0 T1 ……… Tm

   terms and N constraints, the

   dual problem consists of T0 T1 ……… Tm

   variables and M+ 1 constraint. The relation

   between these problems, the optimality has been shown […] to satisfy

   N

   c xa0 tn d *(w*) w*

   t 1, 2,3,…,T

   (3.4)

   0t n

   n1

   N w*

   0t m

   c xamtn mt

   m 1, 2, 3,…., M ; t 1, 2,3,…,T

   (3.5)

   w

   w

   mt n Tm m

   n1 *

   mt

   t1

   Taking logarithms in (3.4) and (3.5) and putting tn log xn

   for n 1, 2,………., N. we shall

   get a system of linear equations of

   tn ( n 1, 2,………., N.).We can easily find primal

   variables from the system of linear equations.

   Case I: For T N 1 ,the dual program presents a system of linear equations for the dual variables where the number of linear equations is either less than or equal to the number of dual variables. A solution vector exists for the dual variable (Beightler and Philips [20]).

   Case II: For T N 1,the dual program presents a system of linear equations for the dual variables where the number of linear equation is greater than the number of dual variables. In this case, generally, no solution vector exists for the dual variables. However, one can get an approximate solution vector for this system using either the least squares or the linear programming method.

4. Numerical Example:

   Anumerical problem as follows: The primal problem is

   Minimize g (x) 2x 5x 2x x 0.5x1x1x1

   0 1 2 3 4 1 2 3

   1 1 4 2 4

   1 1 4 2 4

   Subject to g (x) x2 x2 x2 x2 1;

   (4.1)

   2 1 2 3

   2 1 2 3

   g (x) 100x1x1 x1 1;

   x1, x2 , x3 , x4 0;

   This is a posynomial constraints geometric programming problem. This problem is having degree difficulty = 8-(4+1) =3. The problem is solved via dual geometric programming.

   The corresponding dual of geometric programming (DGP) problem is:

   max d (w)

   2

   w01

   5 w02

   2 w03 1

   w04 0.5 w05

   1 w11

   1 w12

   (4.2)

   w w

   w11 w12 100w21

   w01 w02 w03 w04 w05

   Subject to

   w01 w02 w03 w04 w05 1

   w11 w12

   11 12

   For the primal variable x1

   w01 w05 2w11 w21 0;

   (4.4)

   For the primal variable x2

   w02 w05 2w12 w21 0;

   (4.5)

   For the primal variable x3

   w03 w05 w21 0;

   (4.6)

   For the primal variable x4

   w04 2w11 2w12 0;

   w01, w02 , w03, w04 , w05 , w11, w12 , w21 0

   (3.7)

   The dual variables and the corresponding maximum value of dual objective are given in the following table.

   Table-1: Dual Solution

   w01	w02	w03	w04	w05	w11	w12	w21	g0 (x)
   0.23111	0.30484	0.33332	0.13064	0.00011	0.05119	0.01422	0.33318	43.998

   The dual primal relations are 2×1 w01d w;

   5×2 w02d w; 2×3 w03d w; x4 w04d w;

   1 2 3 05

   1 2 3 05

   0.5x1x1x1 w d w;

   1 4

   1 4

   x2 x2

   w

   w11 ;

   • w

     11 12

     x2 x2 w12 ;

     2 4 w w

     11 12

     100x1x1x1 w21 ;

     1 2 3

     w21

     The primal variables and the corresponding minimum value of primal objective are given in the following table:

     Table-2: Primal Solution

     x* 1	x* 2	x* 3	x* 4	g*(x) 0
     5.08405	2.68255	7.33232	5.74837	43.998

5. APPLICATION

A two-bar truss shown in Fig.1 is designed to support the loading condition Consider the following data Nodal load ( P ) =100 KN ; Volume density ( )= 7.7 KN / m3

; Length ( l )= 2000 mm ;Width( xB )=1000 mm ; Allowable tensile stress( t )=150 MPa

;Allowable compressive stress( c )=100 MPa ;Cross-sectional area of bar 1( A1 )=

0 mm2 A 1000 mm2 ;Cross-sectional area of bar 2( A )= 0 mm2 A 1000 mm2 ;Y

1 2 2

coordinate of node B( yB )= 500 mm yB 1500 mm ;The structure is subject to constraints in geometry, area, stress [9]. The maximum tensile stress is restricted to 150MPa, while

the maximum compressive stress is restricted to 100MPa. The three design variables are

A1 ,

A2 and yB . Obviously, this is minimization problem.

Figure-1: Design of the two-bar planar truss

The Optimization model of the two-bar truss is as follows:

minW A x2 (l y )2 A x2 y2

1 B B

2 B B

P x2 (l y )2

subject to.

B B ;

1

1

lA t

P x2 y2

(5.1)

c

c

B B ;

lA2

0.5 yB 1.5 A1 0; A2 0;

Now this optimization model is not in standard form of geometric programming model. First we transfer it into the standard geometric programming problem with suitable

substitution

A x ,

1. x ,

   1 (2 y

   )2 x ,

   1 y2 x ,

   y x ,

   1 1 2 2

   1 4×2 x x ,

2. 3 B 4 B 5

3 5 6

Then the new form of posynomial Geometric Programming (GP) Problem is;

Minimize W 7.7x1x3 7.7×2 x4

subject to 1 x1 x 1

3 1 3

1

x x

x x

1 1

2 2 4

4 4 5

4 4 5

x2 x2 x2 1

6 3 5 6

6 3 5 6

3 6 3 5 6

3 6 3 5 6

5×2 x1 x2 x2 x1 1 x1 4×2 x x1 1

(5.2)

0.5 x5

1.5

x1, x2 , x3, x4 , x5 , x6 0

When the constraint

0.5 x5 1.5

of (5.2) is excluded, then (5.2) is a constrained

posynomial geometric programming problem with degree of difficulty = 10-(6+1) =3. The problem is solved via dual programming.

The corresponding dual of geometric programming (DGP) problem is:

7.7 w01 7.7 w02

1 w11 1

w21

1 w31 1

w32

Maximize d (w)

w w

w31w32

w w 3w

2w w w

31 32

01 02 11

21 31 32

5 w41

1 w42

1 w51

4 w52

w w

w41w42

w w

w51w52

w w

41 42

w w

51 52

Subject to

41 42 51 52

w01 w02 1

For primal variable

For primal variable

x1

w01 w11 0

x2

For primal variable

For primal variable

For primal variable

w02 w21 0

x3

w01 w11 2w41 2w42 2w52 0

x4

w02 w21 2w31 2w32 0

x5

2w32 2w42 w52 0

For primal variable

x6

w41 w42 w51 w52 0

The dual primal relations are

7.7x1x3 w01d (w) 7.7×2 x4 w02d (w)

1 x1x w11

3 1 3

w11

1 x1x w21

2 2 4

w21

4

4

x2

w

w31

• w

  31 32

  x2 x2 w32

  4 5 w w

  31 32

  5×2 x1 w41

  3 6 w w

  41 42

  x2 x2 x1 w42

  3 5 6

  w w

  x2 x2 x1

  41 42

  w42

  3 5 6

  w w

  41 42

  x1 w51

  6 w w

  51 52

  3 5 6

  3 5 6

  4×2 x x1

  w

  w52

• w

51 52

Solving above equations we get optimal solution of primal variables

1 2 3 4 5 6

1 2 3 4 5 6

x* 0.52068, x* 0.640312, x* 1.56205, x* 1.280625, x* 0.80, x* 2.31147 and

W 125.7667 . It is noted that

x5 0.8 0.5,1.5

1 1

1 1

We get the optimal values of Cross-sectional area of bar 1 A* x* 520.68 mm2 ,

Cross-sectional area of bar 2

A* x* 640.31 mm2 , Y coordinate of node B

2 2

2 2

y* x* 0.80 m andW * 125.7667N .

B 5

This parametric model of the two bar planer truss is built in First order method

in software ANSYS 10.0.The solving results are as follows:

Cross-sectional area of bar 1( A* ) = 497.9 mm2 , Cross-sectional area of bar 2( A* ) =

1 2

B

B

671.5 mm2 , Y coordinate of node B ( y* ) = 0.89 m andW * 126.46N .

This parametric model of the two bar planer truss is built in the MATLAB genetic algorithm toolbox .The solving result are as follows:

Cross-sectional area of bar 1( A* ) = 520 mm2 , Cross-sectional area of bar 2( A* ) =

1 2

B

B

680 mm2 , Y coordinate of node B ( y* ) = 0.73 m andW * 128.1N .

A comparison of the results between geometric programming problem (GP) method and other algorithms mentioned before is presented in table 3.

Table-3: Comparison of the rsults for the two-bar planer truss problem

Algorithm

geometric

Design variable

2

2

A1 (mm )

Design variable

2

2

A2 (mm )

Y coordinate of node B YB (m)

Weight W (N )

ANSYS (FOMA)

It can be seen from the table-3. that the firstorder method in ANSYS gives better results than that of the genetic algorithm native to MATLAB, but Geometric Programming (GP) method yields better result than that of the firstorder method in

ANSYS and the genetic algorithm native to MATLAB.The chart of the comparison of results obtained by different algorithms is shown in Figure-2 .

128.5

128

127.5

127

126.5

126

125.5

125

124.5

Geometric Programming Method

MATLAB genetic algorithm toolbox

First order method in ANSYS

Weight

Figure-2: Comparison of the results under different methods

Conclusion: The successful results that are obtained in this study by GP solving method will contribute to further studies whenever the reliability of the structure is specified with respect to several criteria such as deflection, buckling and natural frequency of vibration.

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