Variation and Optimization of Shape

In this Chapter the problems of shape optimization for elastic solids are summarized from the Noetherian viewpoint. As usual, the initially unknown shape serves as the input of the mathematical modelling. The optimization criteria are expressed by integral and local functionals. These functionals are the output. Sensitivity analysis is the study of how the output of a mathematical model or system can be divided and allocated to different sources of its inputs. The presentation starts with the sensitivity analysis of the optimization criteria with respect to the shape variation. The procedure for deriving the necessary optimality conditions for the problem with unknown boundaries under arbitrary boundary conditions is explained. 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. An algorithm for numerical determination of optimal shapes of two-dimensional elastic solids is proposed; it is based on the necessary optimality conditions and on analysis of sensitivity. The necessary optimality condition expresses as the constancy of the trace of Noether’s tensor on the varying part of boundary.