Structural Optimization 6, 194-199 (1993)

StructuralOp zation © Springer-Vcrlag 1993

On

a game

approach

to optimal

structural

design

V. K o b e l e v * Institute for Problems in Mechanics, Russian Academy of Sciences, Vernadskogo 101, Moscow 117526, Russia Abstract A game problem of structural design is formulated as a problem of playing against external circumstances. The compliance plays the role of the pay-off function. Two players - the designer and nature - try to respectively minimize or maximize it, possessing limited resources. Discrete and continual optimization problems are considered. It is proved that the essential game characteristic, the upper game value, is expressed in terms of the minimal eigenvalue of the inverse operator of the system (continual case), or the response matrix (discrete case). For a special case of convex game, this eigenvalue is equal to the value of the game, and the optimal strategies of the designer and nature are uniquely determined.

1

Introduction

The general goal of optimal structural design theory is to find the structure, of which one of the definite mechanical characteristics (aim function) attains its optimal value, while the other obeys the prescribed constraints. In the classical optimal design problem there is only one decision maker, or designer. Its options are the shapes, sizes, material properties and mutual positions of structural members. These options are called design parameters. The conditions of exploitations and external actions on the structure are to be prescribed. Game theory is usually concerned with the logic of decision making in situations where the outcome depends on the decision of two or more autonomous agents (Vorob'ev 1977). An essential feature of such situations is that each decision maker has only partial control over the outcomes. A phrase such as "the theory of interdependent decision making" captures the essence of the theory in a less misleading way. Game theory deals with situations in which the options available to each decision maker and their possible consequences are well defined, and each decision maker has consistent preferences among the potential outcome. The main goal of the game theory is to find the solutions to games. A solution is a specification of the decision to be made and the outcome which will therefore be reached if the decision makers are rationM according to criteria laid down by the theory. One of the possible game approaches in structural optimal design can be formulated as follows: the external actions are worse in the sense that they result in the maximal stress intensity, or maximal deflection, or maximal level of fracture. Instead of the aim function - as the goal of the optimal design problem is called - in game theory it is usual to speak about a pay-off function. Here, for example, the stress intensity, deflection, or the level of fracture play the role of a pay-off function. *Currently: Muhr und Bender, Postfach 360, D-5952 Attendorn, Germany

Contrary to classical optimal design, the external loads are not prescribed in advance, but should be chosen from the set of admissible external actions. Moreover, if the external loads are applied, one can expect that they are applied in the most inappropriate manner for the designer load. The external actions can be considered as the second player, who is acting against the designer, and is called nature or circumstances. Its strategy consists, therefore, in the choice of the external action which maximizes the pay-off function in our case, the most severe exploitation conditions, leading to the maximal deformation of the structure. The strategy of the designer is to find the most effective response to the action of nature and to neutralize, as far as possible, the sequences of external action in their worst realization. Thus, we have a classical antagonistic game with two decision makers and one pay-off function. 2

F o r m u l a t i o n of a g a m e p r o b l e m

Consider the mechanical system described by the equilibrium equations in the form L[v] u = p ,

(1)

where v is the vector of design variables, u is the vector of state variables and p is the vector of external loads. The selfadjoint positive definite operator £[v] describes the state of the system. The operator coefficients parametrically depend upon the number of design variables. The vector of design variables v belongs to a set of admissible controls

v e v,

(2)

while the vector of external loads should belong to a set of admissible external loads

p e P.

(3)

Introduce now a pay-off functional G = G[v, p], which characterizes the essential mechanical characteristic of the structure, for example, compliance, period of vibration, maximal stress, etc. Putting it roughly, the natural aim of the design is to minimize the functional G. The game design problem can be formulated as a twoperson game. Namely, the first player (the "designer") selects the vector of control variables, while the second player ("nature") can apply an arbitrary admissible external load. The control variables v and p are usually called strategies of the first and second player, respectively, while the admissible sets V and P are usually called the strategy sets. The aim of the designer is to minimize the pay-off functional G[v,p], while nature tries to maximize it. Consequently, the interests of designer and nature are in conflict,

195 and this game is called antagonistic. It is important that the design is characterized by a pair of control variables (v, p), which represent the actions of both players. The upper and lower game values are defined as follows: G*=min v6U

max PEP

G[v,p],

(4a)

G**=max rain G[v,p]. (4b) pEP vEU The minimax theorem states that, in general, G** ~ G*. If the upper value of the game is equal to the lower value, the common value of minimaxes G* and G** is called the value of the game, Gopt -- G* = G**. (5) The process of determining the value of the game and the pairs of optimM strategies is a solution of the antagonistic game, Consider the game where the pay-off functional G[v, p] has a physical meaning of integral compliance and is written as

G[v,p] = < p , v > . Here < p, v > is a bilinear form, or scalar product, satisfying =,

>

0.

The strategy set P is assumed in the form HP[[ < P0, (6) where Ilpll-- x / < p , p > is a norm associated with the bilinear form. The determination of the upper value of the game is reduced to an ordinary optimization problem. To prove this, write the solution of (1) in the form u = £ - l [ v ] p (where £ - t [ v ] is an operator, reciprocal to £[v]) and substitute it into (4a), c [ v , p ] = < p,

-l[v]p > .

(7)

According to (4a), we have G* = rain max G[v, p] = rain G[v, p*] = rain Girl, vEU pEP vEU vEU where p* is an optimal strategy for nature, and G[v] -- < p*, /:-1iv]p* > = max < p, £ - l [ v ] p > .(8) IIpIl
G a m e p r o b l e m for a t h r e e - b a r truss

The application of the considerations discussed in Section 1 is illustrated by the optimization problem for the plane threebar truss shown in Fig. 1. An arbitrary directed load 15 is applied at point A. The norm of vector 15 is restricted by

0 x/m I b Fig. 1. Three-bar truss. Design variables: $1, 5:2 and Sa I1 1t = < p20. (9) The design variables are the cross-sectional areas S1, $2 and S 3 of bars 1, 2 and 3, respectively. The mass of the truss is constrained, ~ a 2 J- b2 (S 1 J-S3) J- bS 2 < _ M , where the density of the material is assumed to be unity. The matrix form of the equilibrium equation is L ~ = 15,

(10)

or, equivalently, (S1 + $ 3 ) ~

,

($1 - $3) ~

($1 -S3)(a2+b~)~ ab "'S - S ~ ,t 1~-

b2

Ux , S_z

uy

Py

The compliance minimization for a class of loads satisfying (9) is equivalent to the maximization of the minimal eigenvalue of the matrix L. The solution of t~e optimization problem can easily be obtained analytically. Graphs illustrating the optimal solution are shown in Figs. 2 and 3. It is assumed that parameters M and a are fixed (M = 1, a = 1), and value b varies. The graph in Fig. 2 represents the dependence of design variables S1 and $2, corresponding to the optimal solution, upon b. Note that due to symmetry $1 = $3. The behaviour of minimal (value of the game, solid line) and maximal eigenvalues (dashed line) of matrix L of the optimal truss is shown in Fig. 3. The value 5 = (A2 - A1)/(A 1 + A2) characterizes the relative difference between the eigenvalues. If b < 1, both eigenvalues are equal. This is the so-called bimodal solution (Olhoff and Rasmussen 1977). The eigenvector of matrix L in this case can be written in the form pz = cos0, py = sin0, where 0 is

196 p and for any admissible pair v 1 and v2, and 0 < n < 1, the function v 0 = ~ v 1 + (1 - n)v 2 belongs to the admissible set (v0 E V) and the following inequality is valid:

e . 58-



~. 4 g -

(l-n) -o-

,,',,, ',,

.

< ~ , L[v0] ~ > < , ~ < ~ , L[vl]~> +

/ O.Og

', . . . . . . . . ~. . . . . . . . . ~. . . . . . . . . L. . . . . . . . . , . . . . . . . . . 4 . . . . . . . . . , g. ge 1 . gO 2 . gO 3. g g 4 . gO 5. gO 6 . Og

b Fig. 2. Parametrical dependence of the optimal cross-sections upon the size b (in the x-direction) I. @0-

g. BO

+

One example of a complex pay-off function is the discrete system described by a linear algebraic equation L[v] t~ = 15, where fi, 15 are the displacement and load vectors, and L[v] is a stiffness matrix of finite dimension. For most matrices appearing in finite-element programs, if membrane, plate or shell elements are considered and the element thicknesses are taken as design variables, the coefficients of matrix L are convex functions ofv. The quadratic f o r m / . / = < fi, L[v] fi > is, therefore, a convex function of the design variables v,

60 8.28-

/I

£-l[v2]p>

£-l[vl] p>

S2

~09 ~.'3g

g. le


< n
(1 -

<

r,[v2]

It can be shown that the quadratic f o r m P = < 15, L - l [ v ] 15 > is also a convex function of v. To show this, introduce the vectors ~0, t~l, ~2 as the solutions of three linear algebraic equations, L[v0] u0 = 15,

l

>

L[vl] t~l = 15,

L[v2] 112 = 15.

The following identities are valid:

o...°°-

< 15, L - l [ v 0 ] 15 > = < 170, L[v0] t~0 > , "

),l

< 15, L - l [ v l ] 15 > -- < 1~l, ,L[Vl] Ul > ,

< 15, L - l [ v J 15 > = < ~2, L [ v j a2 > .

O.60

The sequence of inequalities proves the convexity of 7), < 15, L - l [ v 0 ] 15 > = < t~0, L[vo] a0 > <

,< e. 40:

g. 2g-

g. gO

< ~ 0 , L[Vl]~0 > + ( 1 - ~ )

<,70, L[v2]t~0 > - <


< ~ 2 , L[v2]t12 > - -

L[Vl]Ul > + ( 1 - ~ )

<15, L[Vl]15> + ( 1 - ~ )

i i'1

g. gg

jilt,ll[,i

1 • gO

2 . gO

3.

ge

4.

gg

S. gg

I

6. gg

b Fig. 3. Dependence of the game value (lowest eigenvalue of the stiffness matrix )`1, solid line), the second eigenvalue ),2, and the specific difference 6 = (A2 - )u)/(),2 + ),1) upon the size b arbitrary. This means that the compliance remains constant for all directions of force /Y. If b > 1, the eigenvalues of matrix L differ. The minimal eigenvalue corresponds to the eigenvector pz = 1, py = 0, or to the force acting in the a-direction. 4 Convexity and person game

optimal

s t r a t e g i e s in t h e two-

For some practically important mechanical problems, the game could be solved completely and (5) established. This happens in the case when the pay-off function G[v, p] is a convex function of variable v. This means that for every function

<15, L[v2] 15> .

The first inequality is a sequence of convexity of/,/, while the second follows from the variational principle for quadratic functions. The other examples of convex pay-off functions are shown below. The game with a convex pay-off function is called convex and the players in the game possess optimal strategies (Vorob'ev 1977). Namely, the convex antagonistic game is equivalent to the optimization problem with a single functional, which is formulated as follows. Find the optimal vector of control variables which maximizes guaranteed pay-off of the second player G,

Copt

=

max

G[v].

(11)

v6V This optimal design problem is equivalent to the problem of the maximization of the principal eigenvalue of the operator £, )`opt : max )`min[V], vEV

and

Gopt :p0 )`Ut

(12)

197 5

Game method for a discrete structure

The truss structure shown in Figs. 4 and 5 is an example of the convex problem. "Nature" can freely distribute the external loads, applied to five lower nodes. The possibilities of "nature" are restricted by the quadratic sum of forces. The mean squared force is equal to 1/2. The "designer" manages the cross-sections of 25 bars, keeping the weight of the structure unchanged and equal to the weight of the reference truss. Young's modulus is constant for all bars, so the axial stiffness of a bar is directly proportional to the cross-sectional area. The linear dependence of the cross-sections of all bars in the reference truss is equal to unity; the design vector with unit components, corresponding to the reference truss, is denoted by v*. For comparison one needs the uniform load distribution, which is denoted by 15 *

O

O, O

O

.

O

O'L

¢5 0

Fig. 5. Optimal design. Truss under the action of a uniform load (left diagram, specific stiffness 1.23) and under the load that caused the maxima] energy of deformation of a truss with optima] elements (right diagram, specific stiffness 1.00). Specific stiffness is related to the value of the game

= C[v, 15] / Gopt. O

Fig. 4. Reference design (all bars possess unit cross-section). Truss under the action of a uniform load (left diagram, specific stiffness 0.92) and under the load that caused the maximal energy of deformation of a truss with uniform elements (right diagram, specific stiffness 1.23) The numerical procedure was realized using the method of sequential quadratic programming. The optimization procedure works iteratively. In each iteration the vector 15, which causes the maximal compliance, is determined as the eigenvector of the matrix L. This load distribution is referred to as the design load. The gradient of lowest eigenvalue is determined using sensitivity analysis formulae. The optimal design and the load distribution which causes the extremal energy of deformation are shown in Fig. 5. The value of compliance of the optimal structure under the action of the design load is equal to Gop t. The specific stiffness of the design (v, 15) is determined as

Using this value, one can characterize the quality of different structural decisions. The specific stiffness of the reference design, subjected to uniform load distribution is equal to /~1 = 0.92. The specific stiffness of the reference design, but subjected to the design load, is/~2 -- 0.73. Thus, the inappropriate distribution of the applied loads leads to a 19% increase of compliance. The specific stiffness of the optimal design, which is subjected to design load/~3 = 120, corresponds to a saddle point of the functional G. Any other load leads to an increase in the specific stiffness; for example, the specific stiffness of uniformly loaded optimal design is equal to R 4 = 1.23. This example shows the importance of the correct definition of the design load, which causes the maximal deformation of the structure. 6

Game method for a continuous structure

As an example of the application of the above method to a structure described by an ordinary differential equation, a straight shaft with variable cross-sections is considered. The distribution of the cross-sectional area S -- S(x) satisfies the constraints 1

s(x) > 0,

] s(x) d~ = V0,

(13)

0 where V0 is the volume of material. The admissible external loads p ~- p ( x ) belong to a set

198 1

6.2 Bending of a bar with variable cross-section

[ip[]L2 = / p 2 dx <_p~.

(14)

0 The gain functional G[S, p] characterizes the compliance of a shaft 1

a[s, p] = f p(x) y(~)

dx

(15)

Consider the extension of a shaft by axial loads. The equation for axial displacement, y(x), is written as x e [0, 1].

(16)

The operator £ acts on the functions satisfying the boundary conditions

ESy'(1) = y(O) = 0. The game problem consists in finding the distribution of the cross-sections S = S(x), for which the compliance is minimal for any admissible external load p. The operator £ - 1 , reciprocal to £, is written as

y(x) = £ - l p ( x ) =

x 1 1 - /- - ~ / p ( ~ )

d~ d~.

(17)

0

Because for all ~(0 < ~ < 1) the function xS](~) + (1

-

~)$2(~) satisfies (11) and 1

~

,¢ES1 (~) + (1 - a) ES2(~)

1-~

< - + - ESI(~) ES2(~)'

(18)

< - + - (23) [aSl(~) + (1 - t~)S2(~)] n [SI(~)] n [$2(~)] n ' the compliance functional (13) is convex in S. The game problem thus reduces to the maximization of the principal eigenvalue Aopt ---=max A[S], S of the problem

(ES y')' + Ay -- 0, ESy'(1) -- y(0) = 0.

(20)

The minimal eigenvalue of (20) is simple. The solution of the optimization problem (19)-(20) can be obtained using the Lagrange multiplier method. The necessary optimality condition assumes the form J 2 = const. The substitution of the necessary optimality condition in (20) yields the solution of the optimization problem (19) and, therefore, the optimal strategies of the game problem (13)-(16)

Sopt = ~Y0(1 - x2), Aopt = 3EVo,

popt =

Gopt = P20(3EVo)-].

(21)

It is worth-while mentioning that the problem (19)-(20) is formally equivalent to the problem of the optimization of a critical divergence speed of a wing in gas flow studied by Ashley and McIntosh (1969).

(25)

This optimization problem can be easily solved using the method of necessary optimality conditions. Necessary optimality conditions assume the form

OJ .2 E-~y = const.

(26)

Particularly, for n - 1, (26) is written as y,2 = const. The substitution of the necessary optimality condition into (25) allows us to exclude the variable y(x) and express the solution explicitly, S o p t ( X ) : ~V0(x 2 - 1 ) ( x 2 - 5 ) ,

7

of the eigenvalue problem

yt(O) = (EJyU)'(O) = O,

y(1) = EJy"(1) = 0.

.~opt

(19)

(24)

( E J y " ) " - Ay = 0,

Aopt = 2EVo/3,

m~x A[S],

x ~ [0,1],

y'(O) = (EZy")'(O) = 0, y'(1) = EJyU(1) = 0, (22) where J = kS n is the bending rigidity. The coefficients k and

the functional (13) is convex in S. Thus, the game possesses the optimal strategies for both players, and the value of the game is equal to the maximum of the principal eigenvalue ----

= p,

-

Axial tension of a bar with variable cross-section

£[S]y = (ES y')' = - p ,

£[s]y = (~jJ')"

n depend upon the cross-section shape (Banichuk 1990). It can be shown that because for all x(0 < g < 1) and n > 0 the function J[xS1 (~) + (1 x)S 2 (~)] = [xS] (~) + (1 - x)S2 (~)]n satisfies (11),

o 6.i

The problem of the bending of a bar with variable crosssections can be treated similarly. The equation for normal displacement y(x) is written as

Popt = i S p 0 ( l - x 2 ) ,

Gopt --= 3p2(2EVo) -1 .

Conclusions

The game method for structural optimization problems can be applied in situations when the external loads are not definitely prescribed. The essence of the method consists in the search of the structure that optimally resists the worse external load. The method can be applied in situations when the shape of a structure optimized using standard techniques differs sufficiently from the initial one, and the maximal deformations of this new structure are caused by another external forces distribution. The essential game characteristic - the upper game value - is expressed in terms of the minimal eigenvalue of the inverse operator of the system, or response matrix. If the game is convex, the minimal eigenvalue is equal to the value of the game, and the optimal strategies of the designer and nature are uniquely determined. Acknowledgement

The author wishes to thank the Alexander von Humboldt Foundation (Germany) for its support of this study.

199 References Ashley, H.; McIntosh, S.C., Jr. 1969: Applications of aeroplastic constraints in structural optimization. Proc. 12th Int. Cong. o] Applied Mechanics (held at Stanford Univ. ]968). Berlin, Heidelberg, New York: Springer

Courant, R.; Hilbert, D. 1968: Methoden der mathematischen Physik. Berlin, Heidelberg, New York: Springer Olhoff, N.; Rasmussen, S.H. 1977: On single and bimodal optimum bucldi~g loads of clamped columns. Int. J. Solids ~ Struct. 13, 605-614

Banichuk, N.V. 1976: About one game problem of optimization of elastic bodies. D A N USSR 226, 497-499

Love, A.E.H. 1944: A treatise on the mathematical theory of elasticity. New York: Dover Publ.

Banichuk, N.V. 1990: Introduction to optimization o] structures. Berlin, Heidelberg, New-York: Springer

Vorob'ev, N.N. 1977: Game theory. Lectures for economist and systems scientists. Berlin, Heidelberg, New York: Springer

Received May P9, 1992

Forthcoming Papers Sobieszczanski-Sobieski, J. (USA): Two alternative ways for solving the coordination problem in multilevel optimization

Fadel, G.M.; Cimtalay, S. (USA): Automatic evaluation of move-limits in structural optimization

Chang, K.-H.; Choi, K.K. (USA): Shape design sensitivity analysis and optimization of spatially rotating objects

Asnake, A.; Karihaloo, B.L. (Australia): Optimal design of reinforced concrete beams continuum-type optimality criteria

Bend:sce~ M.P.; Ben-Tal, A.; Zowe, J. (Denmark/Israel/Germany): Optimization methods for truss geometry and topology design

Kleiber, M.; ttien, T.D. (Poland): Variational basis for adjoint sensitivity analysis of arbitrarily nonlinear structural systems

Pietrzak, J. (Portugal): Multicriteria optimization of structures with stability constraints P£czelt, I.; Szab6, T. (Hungary): Optimal shape design for contact problems Jendo, S.; Paczkowski, W.M. (Poland): Multicriteria discrete optimization of large scale truss systems Morris, A.J.; Ponzi, S. (U.K.): Weight/shape structural optimization exploiting rigid movement

using

Giambanco, F.; Palizzolo, L.; Polizzotto, C. (Italy): Optimal shakedown design of beam structures Svanberg, K. (Sweden): On the convexity and concavity of compliances Bendsce, M.P.; Haber, R.B. (Denmark/USA): The Michell truss problem as a low volume fraction limit of the perforated plate topology optimization problem: an asymptotic study Larsson, T.; R6nnqvist, M. (Sweden): Simultaneous structural analysis and design based on augmented Lagrangian duality