Relaxation of Optimization Problems with Equations Containing the Operator ∇ · ⅅ · ∇: An Application to the Problem of Elastic Torsion
In the preceding chapter, we investigated the problem of optimal distribution of the specific resistance of the working fluid in a channel of an MHD power generator. This problem represents a typical example of optimization problems which arise when a control function ⅅ (either scalar or tensor) enters the main part of the elliptic operator ∇ · ⅅ · ∇: of the second order. As was pointed out in the Introduction, the first step in such problems consists of building the G-closure of the initially given set U of controls ⅅ; i.e., an extension of this set to some wider set GU including, in addition to U, all possible composites assembled of the initially given components—elements of the set U. The problem of obtaining an invariant description of this wider set is also essential for the theory of composite materials. For a given microstructure of a composite, its effective characteristics may be obtained by the procedure of homogenization (see Refs. 403, 405, 406, 421, 427, 441, 525, and 526) which required the solution of some fairly complicated auxiliary boundary-value problems. On the other hand, for many applications, particularly for optimal control and optimal design problems, a description of the whole set of materials which may be obtained from some initially given compounds with the aid of the process of mixing is of central interest.476–479,499,514.
KeywordsLayered Composite Straight Segment Torsional Rigidity Switching Line Layered Microstructure
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