PROPERTIES OF THE COST FUNCTIONAL IN FREE MATERIAL DESIGN
Cristian Barbarosie, Sérgio Lopes, Gama Pinto · 2010
We study several properties of integral functionals arising in free material optimization as cost functions. The framework could be that of linearly elastic solids, but for simplicity of presentation purposes we consider scalar equations that model other physical phenomena such as heat or electrical conductivity (thus making the dependance of the cost functions less complicated - a matrix variable instead of an problem. The lower semicontinuity of the cost function (together with the compactness of the design space) ensures the well-posedness of the optimization problem. This latter aspect raises the ques- tion of what notion of convergence should be considered in the design space, a problem often mistreated in the literature, usually by means of introducing a topology that, despite being mathematically sound, does not reflect well the physical reality. It is known for several years that the H-convergence (under which we develop our analysis) models correctly the mechanical behaviour of fine mixtures of materials. The lower semicontinuity of the cost functional is related (but not equivalent) to the convexity of the integrand. A particular functional emerging from the homogenization theory, equal to the minimum amount of material needed to build a certain composite, is given special attention and its relevancy is discussed regarding subadditivity and lower semicontinuity. The majority of the results here presented can be found in (1). 2. Setting of the Problem Let › be a bounded domain in R n ; let fi and fl be two real constants such that 0 < fi < fl. Denote by M fi,fl s the set of symmetric n ◊ n matrices A such that fiIAflI (that is, A i fiI and flI i A