 Open Access
 Total Downloads : 685
 Authors : Samir Dey, T. K. Roy
 Paper ID : IJERTV2IS80729
 Volume & Issue : Volume 02, Issue 08 (August 2013)
 Published (First Online): 24082013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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.

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 twobar truss and the analysis results show that the geometric programming method can always converges to the global optimal solution.

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 crosssectional 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 equation3 on the constraints include truss member stresses and joint displacements.

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 nonlinear 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.

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.
Table1: 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:
Table2: Primal Solution
x*
1
x*
2
x*
3
x*
4
g*(x)
0
5.08405
2.68255
7.33232
5.74837
43.998


APPLICATION
A twobar 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 ;Crosssectional area of bar 1( A1 )=
0 mm2 A 1000 mm2 ;Crosssectional 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.
Figure1: Design of the twobar planar truss
The Optimization model of the twobar 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 ,

x ,
1 (2 y
)2 x ,
1 y2 x ,
y x ,
1 1 2 2
1 4×2 x x ,

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 Crosssectional area of bar 1 A* x* 520.68 mm2 ,
Crosssectional 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:
Crosssectional area of bar 1( A* ) = 497.9 mm2 , Crosssectional 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:
Crosssectional area of bar 1( A* ) = 520 mm2 , Crosssectional 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.
Table3: Comparison of the rsults for the twobar 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 )
programming 520.68 (GP) 
640.31 
0.80 
125.7667 
MATLAB genetic 520 algorithm toolbox (MGA) 
680 
0.73 
128.1 
First order method in 497 
671 
0.89 
126.46 
programming 520.68 (GP) 
640.31 
0.80 
125.7667 
MATLAB genetic 520 algorithm toolbox (MGA) 
680 
0.73 
128.1 
First order method in 497 
671 
0.89 
126.46 
ANSYS (FOMA)
It can be seen from the table3. 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 Figure2 .
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
Figure2: 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.
REFERENCE

A. Kaveh, and H. Rahami, Nonlinear Analysis and Optimal Design of Structures via Force Method and Genetic Algorithm, Computers and Structures, Vol.84, pp.770778, 2006.

A.Schmit, and H.Miurai, Approximation Concepts for Efficient Structural Synthesis, NASA CR2552, University of California, Los Angeles, CA, 1976.

A.Kanarachos, P. Markis, and M.Koch, Localization of MultiConstrained Optima and Avoidance of local Optima in Structural Optimizations Problems, Comput.Meth.In Appl.Mech.And Eng., 51, pp.79106, 1985.

C.Zener, Engineering Design by Geometric Programming, Wiley, 1971.

C.S.Beightler, D.T.Phillips and D.J.Wilde, Foundations of Optimization, Prentice Hall, Englewood Cliffs, NJ, 1979.

John S. Gero, S. Sarkar, How to Make Design Optimization More Useful to Designers, Architectural Design, Vol.23, pp.4350, 2006.

M.Saka, and M.Ulker, Optimum design of geometrically non linear space trusses, Computers and Structures, 41(6), 13871396, 1991.

M.Chiang, Geometric Programming for Communication Systems. Hanover: Now Publishers Inc., pp.1132, 2005.

Nicholas Ali, Kumaran Behdinan, Zouheir Fawaz, Applicability and Viability of a GA based Finite Element Analysis Architecture for Structural Design Optimization, Computers and Structures, Vol.81, 2003, pp.22592271.

R.J.Duffin, E.L.Peterson and C.M.zener, Geometric Programming theory and Applications, Wiley, New York, 1967.

S.Boyd, GGPLAB: A simple Matlab Toolbox for Geometric Programming, http://www.stanford.edu/~boyd /ggplab/2006.

S. Boyd, S.J.Kim, L. Vandenberghe and A. Hassibi, A Tutorial on Geometric Programming, Optim. Eng., vol. 8, pp.67127, 2007.

S. Boyd, and L. Vandenberghe, Convex Optimization.United Kingdom: Cambridge University Press, pp.561614, 2004.