Structural Optimization 13, 195-198 @ Springer-Verlag 1997 Convergence of an algorithm in optimal design A.M. Toader Department of Mathematics, F.C.U.L. and C.M.A.F., Av. Prof. Gama Pinto, 2, P-1699 Lisbon, Portugal Abstract The convergence of an algorithm in optimal design for problems in which the material properties are described by a second-order tensor is proved in this paper. The heat conductance context has been chosen for the presentation. Numerical results by using this kind of algorithm have already been obtained by Allaire et al. (1996) in elasticity. 1 Introduction One is looking for the optimal material for a solid body sub- mitted to heat flux at the boundary, in order to obtain mini- mum dissipated thermal energy. One disposes of two isotropic materials; one of high conductivity whose volume cannot ex- ceed a given amount, and the other of low conductivity. Mix- tures and "generalized" mixtures of the two materials may be produced and one is looking for the ones that improve the conductive properties of the body in order to decrease the dissipated thermal energy. Optimal materials are characterized by optimality condi- tions. Conversely, that these optimality conditions simul- taneously hold may imply that the minimum is achieved. In this context, one may think of an algorithm which al- ternatively iterates the optimality conditions. The link be- tween the "altnerate directions" methods and optimality cri- teria was described by Rozvany (1989) (see also Rozvany el al. 1995). The same idea was explained, more recently by Bendsr (1995). For such an algorithm the value of the cost function always converges since it decreases at each step. Nevertheless, it can converge to a value greater than the min- imum. This means that at each step the algorithm improves the material, and in many cases this can satisfy the practical necessities. Thus for practical problems, in which the aim is to improve the material properties but is not necessary to at- tain the optimum, the algorithm calculates such approaches. However, if one is interested in approaching a real op- timum material, it becomes necessary to prove that the al- gorithm converges to a global minimum, that is to prove a convergence theorem. 2 Mathematical model A very simple model within the above framework is given be- low. One must construct a solid body whose shape is already known (it occupies the domain E2 C h22) such that the dis- sipated stored energy be minimum while it is submitted to a heat flux (J0) at the boundary (0~2), which verifies the ther- mal equilibrium condition (fo~2/I0-n dcr = 0). For this sequel one disposes of two isotropic materials, one of high conduc- tivity, (/?I) and whose volume cannot exceed a given amount and the other one of low conductivity (~I, 0 < c~ < fl), and one may produce mixtures. That is, the domain/2 is divided in parts made only of the high conductor material and others made of the other one. Denote by X : $2 --~ (0, 1} the func- tion defined by X(X) -- 1 if at the point x one has the high conductor material, and X(x) - 0 elsewhere. Then the con- ductivity of the mixture is a(x) = {c~[1-X(X)] +l?X(x)}I and our model problem has the following classical formulation. Minimize the dissipated thermal energy a(x)VT(x) 9 VT(x) (i) dx, E2 subject to the constraint In X(x) dx : 7 (with 0 _< 7 _< I~I given), with T solution of J(x) = a(x)VT(x) (the constitutive equation), div ,1 = 0 in 12 (the equilibrium equation), (2) J .n=J 0 9 n on 0~, where T is the temperature in the solid and J is the heat flUX. From the mathematical point of view the above problem is not well-posed. In order to describe a minimum point of the above cost functional one needs mixtures made by infinitely small inclusions of one material into the other. Such "general- ized" mixtures are not in the class described above. Then one must enlarge the class of mixtures and consequently reformu- late the problem. The mathematical tool through which this is done is the theory of homogenization. This means that at a given point x of the domain S2 one no longer has a material or fl, but a mixture of both in the proportion 0(x) of the good conductor fl(0 < 0(~) <__ 1). However, the proportion cannot completely characterize this mixture since one can produce many different materials with the same proportion. All the materials constructed with the proportion 0 of mate- rial of conductivity ~I and the proportion 1 - 0 of material of conductivity aI have the conductivity matrix such that its eigenvalues belong to a convex set K 0 defined by I' K0 =/(~1,~2) ~ z~2 : ~_(e) < hi < ~+(e), i= 1,2, 2 1 < 1 + 1 i=I 2 1 1 1 ] j=l - ~- p_(e) ~- ~+(0) where #+(o) : (1 - o)~ + oz, t-(o) : 1---2~ +