Laurain Antoine, Lopes Pedro T P
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße , 45127 Essen, Germany.
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010 , São Paulo 05508-090, Brazil.
Philos Trans A Math Phys Eng Sci. 2024 Aug 23;382(2277):20230300. doi: 10.1098/rsta.2023.0300. Epub 2024 Jul 15.
In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.
在本文中，我们研究了允许二阶体积张量表示的分布形状导数的一般性质。我们得到了一个一般性结果，给出了形状导数的一系列表达式，一端是分布形状导数，另一端是标准的哈达玛公式。我们进一步将此结果应用于依赖于四阶椭圆方程解的代价泛函，得到了开集情况下的分布形状导数以及[公式：见正文]类集合的哈达玛公式。我们还考虑了多边形的情况，为了得到哈达玛公式，需要描述顶点邻域中解的弱奇点。本文是主题为“力学中的非光滑变分问题及其应用”这一特刊的一部分。
