Multiobjective insensitive design of structures

004S-7949/92SS.00 + 0.00

Compulers & Smcrures Vol. 45, No. 2, pp. 349-359, 1992 Printed

0 1992 Pergamon Press Ltd

in Great Britain.

MULTIOBJECTIVE INSENSITIVE STRUCTURES

DESIGN

OF

S. S. RAO,~ K. SIJNDARARAJU,$ C. BALAKRISHNA~ and B. G. PRAKA~H$ tSchoo1 of Mechanical Engineering, Purdue University, West Lafayette, IN 47906, U.S.A. SAeronautical Development Agency, Bangalore, India

(Received 24 July 1991)

Abstract--Design parameters are subject to variability in many optimization models. Since an optimal point often lies on the boundary of the feasible domain, any variation in the design parameters is likely to make the design infeasible and useless. A general approach for the insensitive design of structures is presented within the framework of multiobjective optimization. Suitable sensitivity-based functions arising from constraints and objective functions are constructed for inclusion in the optimization process. The designs obtained from the present methodology yield highly insensitive, robust and reliable designs. The insensitive design procedure is illustrated through the design of two truss examples.

lNTRODUCTION

Traditional structural optimization methods assume that the design parameters are known precisely by ignoring the variations that are common in material properties, applied loads, geometrical dimensions and many other parameters. If we consider the real-life situation, it is likely that the permissible stress is not exactly defined. As we observe in any material testing scheme, a wide range of values is found for the permissible stress. The same situation holds for the applied load. Similarly, the geometric dimensions of the structure are not known precisely. For any manufacturing or construction problem the designer needs to provide tolerances. These variations, together with possible assembly errors, will constitute the variability of the uncertain parameters. Although the designers are aware of the variations in parameters, they are not considered in the analysis due to the increased level of difficulty they pose even for the most simple design problems. The parameter variations have traditionally been accounted for in the factor of safety of the overall design. The optimal design obtained using this approach may be quite acceptable in terms of the objective function value and the constraint satisfaction but the design may be extremely sensitive to small variations in the parameter which was assumed to be a constant in the problem formulation. It is well known that in most constrained optimum design problems, the optimum design will lie on the boundary of the feasible space. This implies that the design will become infeasible and useless if the actual constraint value such as the permissible stress or buckling load were to have a slightly lower value compared to the theoretical value assumed in the original formulation. To avoid such a problem, we would like to make the optimum design insensitive to 349

variations in the design parameters. When the design parameters change, the optimum value of the objective function achieved through optimization may also be disturbed. Again to avoid any undesirable changes in the optimum objective function value, we would like to minimize the sensitivity of the objective function along with those of the constraints. Such design procedures can be thought of as approaches which in a sense tend to ‘freeze’ the optimum design. The sensitivity of performance characteristics in optimum design have been considered by several investigators. The maximization of a performance measure over a region of parameter space was accomplished by including a suitable constraint in the minimax framework. The constraint required a certain minimum level of performance given a specified range of errors or disturbances in the parameter values. A brief historical review of minimax methods in the context of robust optimal control problems was given by Dorato [l]. The mean and standard deviation of the performance measures, caused by uncertain parameters, were used to incorporate chance constraints within nonlinear programming models for the reliability-based design of structures by Rao [2]. A general introduction to system sensitivity theory was given by Frank [3]. Kwak and Haug [4] considered the optimum design problem in the presence of parametric uncertainty. In [5], the problem of design of a class of welded beam structures for low-cost with minimal sensitivities was considered. The dominant constraints were chosen which contain variations in the uncontrollable parameters. The optimization was achieved through the generalized reduced gradient method. Trade-off curves were used to select the final solutions. An approach, called sensitivity constrained nonlinear programming, was presented by Uber and Brill[6] by including functions that depend on the

350

S. S.

R.40

system sensitivity to changes in model parameters and independent variable values. The ‘off-line quality control’ method presented by Taguchi and Wu [I is also closely related to the sensitivity analysis. The present work considers insensitive design in the setting of multiobjective optimization of structures. The following problems are addressed in this work: 1. Minimization of sensitivities of the constraints. 2. Minimization of the objective function and its associated sensitivity along with the sensitivities of the constraints. 3. Minimization of multiple objective functions, their associated sensitivities and the sensitivities of the constraints. Two truss examples are presented to illustrate the insensitive design procedures. The first is a two-bar truss for which graphical solutions are presented. The graphical solutions, whenever feasible, are desirable in providing a qualitative insight into the behavior of the optimum solutions. The second one is a 25bar truss for which the solutions are obtained using numerical optimization methods. APPROACHES FOR INSENSITIVE DESIGN

Basically two approaches can be used to produce insensitive designs; one involves developing additional constraints and the other aims at including additional objective functions to the original optimum design problem. Development of sensitivity-based constraints

To illustrate the development of insensitivity-based constraints, consider a single objective optimization problem:

el

al.

2. By constraining the maximum sensitivity of the objective function to be less than a specified value (&) as

3. By including an upper bound (/3r) on the sum of squared sensitivities

N af’

j=,

I( ) az,

Gb2.

4. By restricting the weighted sum of the absolute sensitivities of the objective function to be less than a specified value (/?,) N

af 0) GBS,

cI I

where oj is a weighting factor. 5. By requiring the probability of a measure of the sensitivity of the objective function exceeding a specified value to be less than a certain value, t (chance constraint). This can be stated as PLf-72

kU/l< t,

Af x 5 dfAzj=,$,$Ayj+

i afAx,
axj

f(x, y)

minimize subject to

(7)

g,(x, y) < 0; j = 1,2,. . . , m

h,(x,y)=O;

j = 1,2,. ..,p,

(1)

wherex={x,,x, ,,.., x.}’ is the design vector and Y={Y,,Y*, . . . , y,}’ is the preassigned parameter vector. It is assumed that the design variables and the preassigned parameters are subject to variability (uncertainty). Let

denote

(6)

where f and 6, denote the mean and standard deviations off and k is a constant. 6. By letting the linear estimate of the change in the objective function (Af) to be not greater than a predetermined value (&) as

j,laZj

and

(5)

azj

j-l

the vector

of uncertain

parameters

with

where AZ’, Ayj and Axj denote small variations in zj, yj and xj, respectively. 7. By placing an upper bound (&) on the objective function over a specified region of the uncertain variables maxf (5 x) 6 Bs YES,,

dr -cu. aZ. ’ J'

II J

j=l,2

,...,

N.

(2)

(8)

where S, (S,) represents the feasible region in the y(x) space. 8. By constraining the standard deviation of the objective function as

N=n+q.

The following constraints can be used to achieve an insensitivity design: 1. By placing upper bounds (a,) on the sensitivities of the objective function as

XES2,

co N

a,2

1

‘=I

112

af2

uf + 2 5 =pjpJ,u, r/L azj azjazk

-

j_lk_l

k#i

1

(9)

where /?6is a constant, by is the standard deviation of zj and p# is the correlation coefficient between zI and zk.

351

Multiobjective insensitive design of structures Development of sensitivity-based objective functions

To illustrate the development of insensitivity-based objective functions, consider a multiobjective nonlinear programming problem [8] f, (x, y), . . . ,fk(x, y)

minimize subject to and

gj(x, y) < 0, j = 1,2, .

h,(x,y)=O,

j=l,2

,111

p,

,...,

MULTIOBJECTIVE OPTIMIZATlON

STRATEGIES

The insensitivity design problem stated in eqns (13) can be solved using several multiobjective optimization methods such as the weighting, global criterion, ratios, and game theory methods. In order to have a common basis for comparison and to avoid working with different objectives in different units, the objective functions & are normalized as follows:

(10)

F,(x,y)=

Er(x*Y) -E ~_~

,

r=l,2,...,k

(14)

with where c is the minimum value of E, and E: is the maximum value of E, so that F, varies between 0 and 1 at most points of the feasible design space.

X

2=

11 Y

denoting the N = n + q parameters subject to variability. The following sensitivity-based objective functions can be defined

Weighting method

In this method, the multiobjective problem is stated as minimize

F(x, y) = i w,F,(x, y) r=l

subjectto

gj(x,y)
and

h,(x,y)=O,

j=l,2

j=l,2

,...,

optimization

m

p,

,...,

(15)

where o, are the weighting factors associated with the objective functions, F,, indicating their relative importances and 0 < w, < 1 with Zf=, o, = 1. It can be proved that the solution obtained by solving the problem stated in eqns (15) for any set of weights will be Pareto-optimal. Global criterion method

where the constants c, , c2, and cj are selected such that the three terms on the right-hand side of eqn (12) contribute equally at some reference point (x, y) and AZ, is a small variation in zj. The multiobjective optimization problem stated in eqn (10) can be reformulated in a comprehensive manner to minimize the combined effect of the original objective functions and their associated sensitivities

E,tx, Y) =f,(x, minimize

y) + d,Ff, (x, y);

and

gj(x,y)
hj(x,y)=O,

j=l,2,...,P,

subjectto and

r=l,2,...,k subject to

In this method, the multiple objectives are expressed into a single criterion with the aim of minimizing a power of the deviation of the objectives from their respective ideal feasible solutions. The common formulation of the problem is

(13)

where the constants d, (r = 1,2, . . . , k) are chosen such that the contributions of the two parts of the new objective function E, are same at some reference design.

gj(x,y)
h,(x,y)=O,

j=1,2

j=l,2 ,...,

,..., p.

m (16)

The value of p corresponds to the utility function of the designer a;id is taken as two in this work. Further, Ft denotes the ideal feasible solution corresponding to the ith objective function and can be seen to be zero in view of the normalization scheme used in eqn (14). Thus the objective F(x, y) reduces to F(x, y) = {F:(x, y) + F:(x, y) + . . . + F;(x, Y)}“~. (17)

352

S. S.

RA0

et al.

Ratios method In the general ratios method, if J is to be minimized and fi is to be maximized, a composite function for minimization is constructed as

m,

Y)

=;$$ >

where the exponents a, and a2 are chosen depending on the relative importances off, andf,. In the present case, since F,(x, y) given by eqn (14) are to be minimized, the composite function is constructed as

m

Y> = f-j

mx, y),

(19) P

where a, are constants. This function is minimized over the feasible space to find the solution of the multiobjective optimization problem. Game theory method In the cooperative version of game theory, the multiobjective optimization problem is viewed as a game problem involving several players, one corresponding to each of the objective functions[9]. It is assumed that all the players wish to maximize the deviation of their objective functions from the respective worst possible values. Let y denote the worst values of F, obtained by finding

where x,* is the function when ation problem. ning model to

optimum solution of thejth objective solved as a single objective optimizThen the supercriterion or the beginbe maximized can be constructed as

F(x, Y) = fi ]F,(x, Y) - Yl,

,=I

(20)

where Fy is the maximum value of F, and is equal to one in view of the normalization scheme used in eqn (14). Thus eqn (20) reduces to

F(x, Y) = fi FAX, Y) - 11.

Fig. 1. Two-bar truss. where A,, is the minimum permissible value of A and h is the height of the truss. The truss is assumed to be symmetric about the y-axis. The coordinates of joint 3 are held constant. The weight of the truss and the total displacement of the joint 3 under the given load are considered as the original objective functions f, andf, . The stresses induced in the two members are constrained to be smaller than the permissible stress, crO.In addition, upper and lower bounds are placed on the design variables. Thus the original multiobjective optimization problem becomes

which minimizes f, (x) = Zphx, Jm

h(x) =

Ph( 1 + x;)“( 1 + x;)“’ 24Ex;

(23)

x2 Amin

subject to g1(x) =

g2 (4

(21)

Amin and

=

pu +x,)(1 +x:)o.s_ 2JZxlX2Aminao ’ 1

P(-% - l)(l+x:)“~’ 2fiX1X2Aminao


_ 1
(24)

(25)

’

,=I

g,=x’,-x,
(26)

EXAMPLES

g4(x) = x:-x2

Example 1 The two bar truss shown in Fig. 1 is considered as the first example to illustrate the multiobjective insensitive optimization procedure. The non-dimensional cross-sectional areas of the members and the position of the joints 1 and 2 are treated as design variables A XI=Amin,

X x2=;,

<0

(27)

gs(x) =x,

-x; < 0

(28)

&(X)

-

xy,<0,

(2%

=x2

where E = Young’s modulus, p = weight density of the material, P = applied load, xi = lower bound on xi (i = 1,2), and xr = upper bound on xi (i = 1,2). The (nominal) values of E, p, P, xi and x7 are taken

353

Multiobjective insensitive design of structures

I 00

0.2

IA

0.4

I

I

0.6

0.6

I

I

I

I

I

I

1.0

1.2

1.4

I6

I6

2.0

Fig. 2. Contours of structural weight cf,) and displacement of joint 3 cf7).

as 30 x 106psi, 0.283 lb/in3, 10,000 lb, 0.1 and 1.0, respectively. The values of the other parameters are selected as u0 = 20,000 psi, h = 100 in, A,, = 1 in*. In order to observe the qualitative behavior of the optimum point, the solution of the problem stated in eqns (22)-(29) is found by plotting the contours of the two objective functions and identifying the feasible region in the x,-x2 space as shown in Fig. 2. The individual minima of the two objective functions and the corresponding characteristics are indicated in Table 1. In this example, the parameters a,,, p, h, x, A, E, P, and A,, are considered to be uncertain with a variability of 5% about their stated nominal values. Denoting these parameters as z, , z2, . . . , z8, respectively, the measures of insensitivity corresponding to the behavior (stress) constraints and the objective functions can be obtained from eqns (11) and (12) where the partial derivatives of g,, g,, fi and fr appearing in eqns (11) and (12) can be readily found by differentiating eqns (22x29) with respect to a

particular z,. These sensitivity measures are minimixed by formulating the objective functions &(x7 YL

F/,(x, Y)

and

F/,(x, Y)

as indicated in eqns (11) and (12). The constants c, and cr are chosen such that each term on the righthand side of eqn (12) is equal to 5.0 at the minimum of the objective function f,(r = 1,2). The variations of the sensitivity objective functions F,, F,, , and F,* in the feasible design space are found and the minimum values of these functions are identified as indicated in Table 2. For the multiobjective formulation of the problem, the optimum solutions of the individual objective functions E, and &, defined as in eqn (13), are determined graphically as indicated in Figs 3 and 4, respectively. The characteristics of these solutions are given in Table 3. The solution of the problem stated in the weighting method with Ftx, Y) = 01 F, (x, Y) + eGr(x,

Y)

(30)

Table 1. Characteristics of individual optimum solutions Minimization off,

Minimization

36.1493t 0.0943

186.7361 0.0182t

oI @si)

19,994.9$

4033.6

a2 (Psi)

3889.7

300.9

Quantity

Off*

and equal weights, o, = wr = 0.5, is given in Table 4. For the global criterion method, the objective function is formulated as F(x,

7 Optimum value. $ Active constraint. 0, = stress in member i (i = 1,2).

Y)= [F:(x,Y)+ F:(x,y)l”*.

(31)

The characteristics of the optimum solution of F are indicated in Table 4. In the ratios method, the composite function is constructed (with a, = a2 = 1) as F(x, Y) = F, (x, y)F,(x, Y).

(32)

S. S. FL40 et al.

354

Table 2. Minimum values of F’, Ffi and F,* Minimization of F,

Quantity

Minimization

Minimization of Fh

of %

Design vector

x={::> {;:I}

{Z}

{~~}

Original objectives f={]

{ZG54}

{Ei~45}

F,

{ZZ9} 0.961

4.61

0.9625

F/I

28.3

9.96

21.6

FJZ

5.39

28.5

5.32

This function is minimized over the feasible space as illustrated in Fig. 5. It can be seen that the contours of the objective function F(x, y) given by eqn (32) exhibit a different behavior compared to those given by eqns (30), (31), and (33). There are two relative minima, one at M,, with F* = -0.000546 and the other at M2 with F* = 0.000104. There is a saddle point at S, IO.80, 1.25}, with F = 0.0819. The characteristic features of a saddle point can be observed at S also. For example, the point S can be seen to behave as a relative maximum in the x, direction and as a relative minimum in the x, direction, that is F(0.80, x2) < F(0.8, 1.25) and

The characteristics of the two relative minima, M, and M2 are summarized in Table 4. The objective function to be maximized in the game theory is constructed as

F(x, Y) = 14 6, Y) - ll[F,(x, Y) - 11.

The optimum solution of F and its characteristics are summarized in Table 4. It can be observed from Table 4 that the solutions given by the weighting method, global criterion formulation and game theory approach are quite similar and are very close to one another. However, the solutions given by the ratios method as well as the characteristics exhibited by the objective function of eqn (32) are totally different. Example

F(x,,

\

‘\ 2.25

2

The second example deals with the insensitivity optimization of the 25-bar space truss shown in

1.25) > F(0.8, 1.25).

250-

-

(33)

\\

Fig. 3. Optimization of r,.

‘\

‘.\

\

355

Multiobjective insensitive design of structures

0.0

02

04

06

06

1.0

1.2

1.4

1.6

1.6

2.0

XI

Fig. 4. Optimization of &.

Fig. 6. Eight independent areas of cross-section of the members are taken as design variables

XI=A,, x2=A2=A3=A4=AS,

xq=A,o=A,,,

x6 =

the permissible stress as 40,000 psi. The weight of the structure, the deflection of joint 1 in both the load conditions and the negative of the fundamental natural frequency of vibration are treated as the original objectives for minimization. The stresses induced in the members are restricted to be smaller than a maximum permissible stress and are also required not to cause buckling. The members are assumed to be tubular with a diameter to thickness ratio of 100 so that the buckling stress of ith member (G,~) can be. expressed as bib =

A,, = A,, = A,, = A,,,

x, = A,, = A,,, = AZ0= A,, ,

xg = A,, = A,, = A, = A,,, where Ai denotes the area of cross-section of ith member. The truss is subjected to the two load conditions given in Table 5. The Young’s modulus is taken as 10’ psi, the material density as 0.1 lb/in3 and Table 3. Characteristics of minima of f, and &

- lOO.OlnEA, 81;

,

i = 1,2,. . . ,25,

(34)

where E = Young’s modulus and li = length of ith member. Lower and upper bounds are specified on the design variables as x)0 and xl”; i = 1,2,. . . ,8. Thus the original multiobjective optimization problem can be stated as Minimize

A(x) = f pAJ, i-l

(35)

f2(X)=(6:,+6:y+6:r)l~~~l

Design vector

x* = (:I}

{FE:}

h(x)

Original objectives

f= fi (1h New objectives

+ (S:, + S:, + S:,,,‘L

(36)

=

(37)

{E} -01

subject to

g,(X)d 0; l, 50+l”~(x)IG aO, i=l,2,...

,25,

j = 1,2

(38)

356

S. S.

et al.

Rho

Table 4. Solutions given by multiobjective optimization methods

Quantity

g,(X) < 0,

Ratios method Point IU, Point M,

i = 51,1OO+c~~(x) 2 sib(x),

i=l,2,... gi(x) < 0,

Global criterion method

Weighting method

i = lO1,116+xj<

measure of insensitivity of the behavior constraints is expressed as

,25, j = 1,2

(39)

xi
,...,

8,

(40)

where p is the weight density, 6,,, a,,, and S,, are the x, y, and z components of deflection of joint 1, o, is the fundamental natural frequency and oii is the stress in member i in load condition j. The individual objective functions are optimized over the feasible design space and the optimum values are found as fl = 233.07 lb, f: = 0.3083 in, and fl = 108.6224 Hz [9]. The eleven parameters E, p, a,, xi (i = 1,2, . . . ,8) are considered as the uncertain parameters with a variability of 5% about their nominal values. Denoting these parameters as zI, z2, . . . , zl,, respectively, a

00

0.2

Game theory method

0.4

0.6

08

i = 1,2,3.

(42)

The partial derivatives (agj/azi) and (aA/az,) appearing in eqns (41) and (42) are evaluated using the forward finite difference formula. The sensitivities given by eqns (41) and (42) are minimized over the feasible design space at the starting design vector, x = %. The results of minimization are given in Table 6. It can be seen that the minimization of constraint sensitivities, F’, resulted in values, for all design variables except x, and x4, which are close to their upper bounds.

I .c

Fig. 5. Graphical minimization of F in ratios method.

Multiobjective insensitive design of structures

75 in

100 in

71 I6

Xx) in

Fig. 6. Twenty-five bar truss.

Table 5. Loads acting on the 25bar truss Load condition No.

Load component

6

0

0

0

F,

20,000 -5,ooO

-20,000 - 5,000

0 0

0 0

F, F, FZ

1,000 10,000 -5,000

0 10,000 -5,000

500 0 0

500 0 0

F,

2

Value at joint (lb) 3 2

0

FX

1

1

Table 6. Minimization of F, and F/,(i = 1,2,3) for 25-bar truss Minimization of Quantity

Starting point

F, = F,

F2 =

2.3209 4.5255 4.6324 1.9047 4.9684t 4.7197 4.6479 4.7729

0.50733 1.3087 1.3345 0.20063 0.48252 0.89165 1.4039 1.2061

F/I

4

=

51

F,

=

Fr,

Design vector 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

XI x=

T X8

1.1535 4.83987 4.8513? 1.5298 2.1667 4.4011 4.7559 4.9997t

2.1985 4.6483 4.5445 0.62883 2.0567 1.5775 4.1374 0.94560

Original objectives $ i 2 & Active constraints

330.7197 69.0253 1.5417 0.313221 35.3949 0.168809 5.296571 g,, g,, *Buckling in members 19 and 20

1487.80 72.6705 0.33221 0.0254303 81.4050 0.025050 3.024 -

tBound value. c, = 56.50132; c, = 0.269429; c, = 8.45363.

370.094 69.0815 1.1918 0.200083 30.11904 1.3278 4.4170 -

1454.38 73.7216 0.3181

958.091 40.5021 0.535249

0.02452 79.8832 0.024017

0.07799 56.8245

3.08304 -

0.05019 2.31387 -

358

S. S. RAo et al. Table 7. Minimization of &, & and & for 25bar Truss Quantity

Starting point

Minimization of EZ

El

E3

Design vector 0.22626 0.91384 0.91926 0.13143 0.33552 0.61863 1.0161 0.85268

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

2.2440 4.9912t 4.9990t

2.3241 4.9379 5.0005t 4.9807t 4.9838t

0.11827 0.97722 1.0821 0.60233 0.65868 2.4293 2.0984 4.7942

Original objectives 259.6435 330.7197 69.0253 1.5417

5

; F;,

1.698190 70.22624 0.388091 35.99629 0.197402 6.05916

0.313221 35.3949

$Ih

0.168809 5.296571

Active constraints

1589.46 0.308951

72.06807 0.023 1396 86.7611 0.023145 2.98899

701.0437 1.038018 106.7091 0.197309 50.0016 0.112045 5.505087

g,, g,,+Buckling in members 19 and 20

7Uppe.r bound value.

Next, the sensitivity-based objectives are appended to the original objective functions to define the new objectives, E, as indicated in eqn (13) where the constants d, are selected such that E, = 2fi (r = 1,2,3) at the starting design vector x,, . The results of optimization of E, subject to the constraints stated in eqns (38)-(40) are shown in Table 7. It can be observed that the individual minimizations of the three sensitivity-based objectives resulted in values of weight, deflection and frequency as 25964lb, 0.31 in, and 106.71 Hz which are quite comparable, respectively, to the values of 233.07 lb, 0.31 in, and 108.62 Hz obtained without a consideration of sensitivities. This illustrates that the present formulation yields designs which are very close to the optima

achieved through conventional optimization while making the results insensitive to the uncertain parameters. The numerical results given by the multiobjective optimization approaches, stated in eqns (1 Q-o-(l), are given in Table 8. It can be seen that the minimizations of F in the global criterion, ratios and weighting methods yielded essentially the same results which correspond, approximately, to a weight of 7001b, deflection of 0.72in and frequency of 95 Hz. The game theory approach, on the other hand, gave a different result with a weight of 258.41 lb, deflection of 1.70 in and frequency of 70.32 Hz. These multiobjective optimization results indicate different compromise solutions, based on

Table 8. Results of multiobjective optimization Minimization of

Minimization of

Maximization of F= fi (1 -E,)

(Global criterion method)

Quantity

i=S (Ratios method)

,=5 (Game theory method)

Minimization of F=

i o,E, r=5 (all w, = l/3) (Weighting method)

Design vector 0.23080

0.21757

0.22035

1.7175 1.8427 0.30557 0.75294 1.4468 2.1012 4.6534

0.90538 0.92057 0.12986 0.3272 0.61795 1.0097 0.85036

1.6356 1.7410 0.3322 0.5078 1.6448 2.1599 4.9970t

0.22091 1.5065 1.6026 0.40657 0.60351 1.7430 2.22246 4.8875

Original objectives

weight

691.4841 0.70837 92.7523

Deflection Frtquew’ t Upper

bound

value

258.407 1.7042 70.3181

716.451 0.721965 95.7931

712.1580 0.76246 97.4607

Multiobjective insensitive design of structures different philosophies, achieved among the three objective functions, objective sensitivities, and constraint sensitivities.

359 REFERENCES

1. P. Dorato, A historical review of robust control. IEEE Control systems Magazine 7, 4441 (1987). S. S. Rao, Structural optimization by chance constrained programming techniques. Comput. Struct. 12, 777-781 (1980). 3. P. M. Frank, Introduction to System Sensitivity Theory. Academic Press, New York (1978).

2. CONCLUSION

An approach is presented for the insensitive design of structures within the framework of multiobjective optimization. Since conventional optimal designs often lie on the boundaries of the feasible domains, any variations in the design parameters are likely to make the optimal designs infeasible and useless. The present methodology makes an optimum design insensitive and robust to the uncertain parameters. The sensitivity-based functions formulated in this work are applicable not only to structural problems but also to mechanical and other system design problems. The numerical results indicate that the insensitive design procedures outlined in this work lead to robust designs without much compromise in the values of the objective functions.

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