Optimum composite material design

— The microstructure identification problem is treated : given certain phases in given volume fractions, how to mix them in aperiodic cell so that the effective material constants of the periodic composite lie the closest possible to certain prescribed values ? The problem is studied for the linear conduction équation. It is stated in terms of optimal con trol theory ; the admissible micro geometrie s are single inclusion ones. Existence of solution is proved under suitable hypotheses, as well as the convergence of numerical approximations. Numerical exampies are presented. In the conduction case, the f uil characterization of the GQ -clos ure set (the set of all effective conductivities that result from taking the given phases in the given volume fraction mixed in any feasible microgeometry) is known. One carried out numerical experiments how well can its boundaries be attained using the subclass of single inclusion microgeometries. Results of these experiments are shown as well The concept of composite media not only comes directly from the physical world but also provides a theoretically sound means for relaxation of variational problems — the problem of optimum topology design (see [5], [22], [12], [2] or [14]) in the first rank of importance. It is a classical result of the homogenization theory that composites can be replaced by a macroscopically homogeneous medium whose material constants — the so called effective constants or effective moduli — depend on the microgeometry in which the (*) Manuscrit received March 10, 94. C) Dept. of Métal Physics, Faculty of Mathematics and Physics, Charles University of Prague, Ke Karlovu 5, CZ-121 16 Praha, Czech Republic. E-mail address : [email protected] [email protected]. M AN Modélisation mathématique et Analyse numérique 0764-583X/95/06/$ 4.00 Mathematical Modelling and Numerical Analysis © AFCET Gauthier-Villars 658 JHASLINGER, J. DVORAK constituent phases form the mixture. The set of all effective constants of mixtures of a given number of phases taken in a given proportion is called the Go -closure set and its knowledge is essential for the relaxation procedure. In the case of a scalar linear elliptic partial differential équation (the steady heat transfer équation), the Go -closure sets are known for mixtures of two phases ; one of the phases may be degenerate, i.e., a void ; see [14], [23], [16]. Ho wever, for the case of the System of PDE's of linear elasticity, only a partial information about the Go -closure sets is available so far ; namely we know how to minimize the complementary energy for a given single macroscopic stress field (see [1]). This is enough for the minimum compliance design of single loaded elastic structures (see [2]) ; however, for the case with multiple loadings as well as for design with other (non-compliance) objective functions, the full knowledge of the GQ -closure set seems inévitable. But also the scalar case has its difficulties : the construction of extremal microstructures — i.e., those that maximize/minimize the effective energy — is usually a nice theory while practically immanufacturable. So far, the following constructions are known : Multiple rank laminâtes. The microstructure is a laminate (= layered composite) whose one or both components are again laminâtes that in turn can consist of laminâtes, etc. The layered microstructure has the advantage that one can calculate the effective constants analytically. However, the scale levels of the subséquent laminations must be well separated which prohibits practical realization of these microstructures. For an overview cf. [3]. Coated ellipsoids construction. This construction is based on the fact that having a medium with the material constants that are equal to those of our desired microstructure, one can insert an ellipsoid of one phase with an ellipsoidal inclusion of the other phase where the ellipsoids have appropriately balanced dimensions, and the effective properties of the medium are not changed upon this insertion. Thus, one fills up the whole body with coated ellipsoids, but using infinitely many length scales, this time not even separated from one another. As a conséquence, one cannot manufacture but a rough approximation of such a microstructure. For details see [9]. Vidgergauz' microstructure. The only known extremal microstructure that stays on a single length scale is the Vidgergauz' microstructure. It has the form of a properly shaped (oval-like) inclusion of one phase within the matrix of the other phase. The shape of the inclusion is found from the optimality conditions that in this setting have the form that « the inclusions be equally strong », see [24] or [10]. However, the shapes of the inclusion have to be evaluated using elliptic intégrais or other non-elementary functions. We note that although it is presented in the elasticity setting, similar results hold for the scalar équation. M AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis OPTIMUM COMPOSITE MATERIAL DESIGN 659 The aim of this work is two fold : 1. Establish a numerical technique for the microstructure identification problem. We use the optimal control theory to formulate the problem of microstructure identification : given a target matrix of effective moduli, what microstructure has its effective moduli the closest to this target ? — ideally, what microstructure attains the target ? As we are using the classical optimal shape design approach based on the boundary variation technique, the class of admissible microgeometries is restricted to single inclusion ones. We present an approximation of the optimal control problem. We note that from a different point of view, the problem of microstructure identification has been treated also by O, Sigmund : [20], [21]. That approach works in the linear elasticity setting and uses discrete structures (trusses). 2. Investigate numerieally what parts of the Go -closure set are covered by the chosen class of mïcrogeometries. Since the single inclusion microgeometries belong to the « reasonable » (meaning : more or less manufacturable) classes» it is interesting to see how well they manage to cover the full Go -closure sets. Specifically, in [5] and [22], one uses only sub-optimal microstructures — rectangular inclusions — for relaxation of the optimum shape design problem. We study how big are the différences among the rectangular inclusion composites, the single inclusion composites, and the full Go -closure sets. 1. FORMULATION OF THE IDENTIFICATION PROBLEM Let us have a body in a plane represented by a domain Q 0 is a scaling factor. The steady heat transfer in Q is described by the scalar elliptic équation and by appropriate boundary conditions, say of the Dirichlet type