https://inria.hal.science/inria-00073191Lewinski, TomaszTomaszLewinskiSokolowski, JanJanSokolowskiNUMATH - Mathematical Analysis and Numerical Simulation of Non-Linear Models - INRIA Lorraine - Inria - Institut National de Recherche en Informatique et en AutomatiqueOptimal Shells Formed on a Sphere. The Topological Derivative MethodHAL CCSD1998shape optimizationshape derivativetopological derivativeasymptotic expansioninverse problem[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH]Inria, Rapport De Recherche2006-05-24 12:07:362023-02-07 03:39:482006-05-31 14:24:27enReportsapplication/pdf1The subject of the paper is the analysis of sensitivity of a thin elastic spherical shell to the change of its shape associated with forming a small circular opening, far from the loading applied. The analysis concerns the elastic potential of the shell. The sensitivity of this functional is measured as a topological derivative, introduced for the plane elasticity problem by Sokolowski and $\buildrel . \over {\hbox{Z}}$ochowski (1997) and extended here to the case of a spherical shell. A proof is given that : i) the first derivative of the functional with respect to the radius of the opening vanishes, and : ii) the second derivative does not blow up. A partially constructive formula for the second derivative or for the topological derivative is put forward. The theoretical considerations are confirmed by the analysis of a special case of a shell loaded rotationally symmetric, weakened by an opening at its north-pole. The whole treatment is based on the Niordson-Koiter theory of spherical shells, belonging to the family of correct first order shell models of Love.