Structure of shape derivatives
Structure of shape derivatives
复制标题
形状导数的结构
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
M. Pierre
中科院分区：
文献类型：
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作者：
A. Novruzi;M. Pierre
抽象。在本文中，我们描述了第二个“形状导数”的精确结构，即其参数是$ mathbb{R}^N $的变量子集的函数的导数。这是在适当的Banach空间的Fréchet导数。除了结构本身，兴趣在于它是如何得出的：出发点是一个众所周知的事实，即一个给定的正则域的小正则扰动可以通过该域的边界的正常变形“唯一”表示的“功能分析”声明。该方法涉及一个方便的功能空间中的隐函数定理。这种“正规表示”性质的结果是，任何形状泛函都可以通过仅在给定域的边界上定义的函数来描述。对这个表示进行两次微分，就得到了结构定理。我们恢复的事实是，在临界形状，在给定的域周围的二阶导数仅取决于在其边界处的变形矢量场的正常分量。一些例子显式计算。
Abstract. In this paper, we describe the precise structure of second "shape derivatives", that is derivatives of functions whose argument is a variable subset of $ mathbb{R}^N $. This is done for Fréchet derivatives in adequate Banach spaces. Besides the structure itself, interest lies in the way it is derived: the starting point is a "functional analytic" statement of the well-known fact that small regular perturbations of a given regular domain may be "uniquely" represented through normal deformations of the boundary of this domain. The approach involves the implicit function theorem in a convenient functional space. A consequence of this "normal representation" property is that any shape functional may be described through a functional depending on functions defined only on the boundary of the given domain. Differentiating twice this representation leads to the structure theorem. We recover the fact that, at critical shapes, the second derivative around the given domain depends only on the normal component of the deformation vector-field at its boundary. Some examples are explicitly computed.