Domain Derivation in Continuum Mechanics

Abstract

In this chapter the formal shape sensitivity analysis is performed in the setting of continuum mechanics. The word formal is used in the sense that all objects are assumed to be sufficiently smooth. It means that we do not state smoothness hypotheses, since standard differentiability assumptions sufficient to make an argument rigorous are generally obvious to mathematicians and of little interest to engineers and physicists. In addition, we do not employ any function spaces setting for all the developments. The case of elliptic boundary value problems is considered separately in Chapter 3 in the framework of the variational solutions in functional spaces. Therefore the required results on the shape and material derivatives are proved e.g. in [210] by an application of the so-called speed method. The demand on shape sensitivity analysis for our purposes of the topological differentiability is not involved, since we restrict ourselves to the shape gradients of the energy type shape functionals for the elliptic boundary value problems with the absence of singularities. We refer to Chapter 3 for a presentation of the complete arguments on shape differentiability for the solutions of the Poisson, Kirchhoff plate, linear elasticity. These results are of independent interest and the proofs are borrowed from the monograph [210]. The results on boundary value problems of fluid mechanics are formally obtained by the method proposed in [196].