inria-00073191
https://inria.hal.science/inria-00073191
https://inria.hal.science/inria-00073191/document
https://inria.hal.science/inria-00073191/file/RR-3495.pdf
[INRIA] INRIA - Institut National de Recherche en Informatique et en Automatique
[INRIA-RRRT] Rapports de recherche et Technique de l'Inria
[INRIA-LORRAINE] INRIA Nancy - Grand Est
[INRIA-NANCY-GRAND-EST] INRIA Nancy - Grand Est
[TESTALAIN1] TESTALAIN1
[INRIA2] INRIA 2
[LARA] LARA
Optimal Shells Formed on a Sphere. The Topological Derivative Method
Lewinski, Tomasz
Sokolowski, Jan
[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH]
REPORT
shape optimization
shape derivative
topological derivative
asymptotic expansion
inverse problem
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.
1998
2006-05-24
en