Remarks on variational shape sensitivity analysis based on local coordinates

Abstract

Shape and topological sensitivity analysis are two closely related research fields of both theoretical and computational mechanics with a high impact on any analytical and numerical approach in structural optimisation. There are close connections to configurational mechanics describing cracks and dislocations as well as to biomechanics observing growth and morphogenesis. Different approaches exist to compute the gradients needed by nonlinear programming algorithms. But it is of utmost importance to acknowledge that mainly a rigorous analysis of the sensitivities provides the deep insight into the nature of the mechanical problems needed to model and to solve inverse problems efficiently. This paper outlines the author's concept of an intrinsic formulation in local coordinates of continuum mechanics which extends Noll's intrinsic concept to variable material bodies. This viewpoint is derived by a thorough analysis of their manifold properties and yields the separation of the phenomena in material space from the motion in physical space. The subsequent variational shape sensitivity analysis is formulated and compared to known approaches. The interactions with computational techniques such as computer aided geometrical design (CAGD), the finite element method (FEM) and the boundary element method (BEM) are highlighted. Furthermore, the implications on the numerical algorithms for the discrete sensitivity analysis are outlined. Finally, the challenges of a singular value decomposition (SVD) of the resulting sensitivities are discussed.