Shape Optimization of Piecewise-Homogeneous Bodies

In this Chapter we will deliberate problems of shape optimization for piecewise-homogeneous elastic solids from the local Noetherian viewpoint. The optimization criteria are provided by integral and local functionals. A procedure is appropriate for deriving the necessary optimality conditions for the problem with unknown boundaries under arbitrary boundary conditions. The derivation makes essential use of a formula that expresses the variation of a functional in terms of the variations of the boundaries and of the state functions. The necessary optimality condition expresses as the constancy of the jump trace of Noether’s current tensor over the varying interface boundary.