%0 Report %T Optimal Shells Formed on a Sphere. The Topological Derivative Method %+ Mathematical Analysis and Numerical Simulation of Non-Linear Models (NUMATH) %A Lewinski, Tomasz %A Sokolowski, Jan %N RR-3495 %P 62 %I INRIA %8 1998 %D 1998 %K shape optimization %K shape derivative %K topological derivative %K asymptotic expansion %K inverse problem %Z Computer Science [cs]/Other [cs.OH]Reports %X The 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. %G English %2 https://inria.hal.science/inria-00073191/document %2 https://inria.hal.science/inria-00073191/file/RR-3495.pdf %L inria-00073191 %U https://inria.hal.science/inria-00073191 %~ INRIA %~ INRIA-RRRT %~ INRIA-LORRAINE %~ INRIA-NANCY-GRAND-EST %~ TESTALAIN1 %~ INRIA2 %~ LARA