Optimal Shells Formed on a Sphere. The Topological Derivative Method

• Main Authors: Lewinski, Tomasz, Sokolowski, Jan
• Other Authors: Mathematical Analysis and Numerical Simulation of Non-Linear Models (NUMATH), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), INRIA
• Format: Report
• Language: English
• Published: HAL CCSD 1998
• Subjects: shape optimization; shape derivative; topological derivative; asymptotic expansion; inverse problem; [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH]; North Pole
• Online Access: https://hal.inria.fr/inria-00073191; https://hal.inria.fr/inria-00073191/document; https://hal.inria.fr/inria-00073191/file/RR-3495.pdf

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.