Optimal shells formed on a sphere. The topological derivative method
Tomasz Lewiński , Jan Sokolowski
AbstractThe 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 \$\\textbackslash\buildrel . \\textbackslash\over \\\textbackslash\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.
|Journal series||Rapports de recherche, ISSN 0249-6399|
|Keywords in English||asymptotic expansion, inverse problem, shape derivative, shape optimization, topological derivative|
|Citation count*||11 (2018-06-22)|
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.