Shape calculus for the efficient solution of shape optimization problems with elliptic and parabolic state equation

在过去的 25-30 年里，优化形状设计在工程应用中变得越来越重要。结构力学、流体动力学和电磁学中出现的许多问题导致在一类容许域上定义的泛函的最小化，这些泛函由边值问题或初始边值问题的解控制。本申请的方法基于数​​学形状敏感性分析，并且利用形状梯度的分析计算，以及如果可用的话，甚至更高的导数。之后，形状（以及形状梯度）以及基础状态方程被离散化，这与完全离散的形状优化方法相反。本项目的范围是本着这一政策的精神，基于有限元技术，特别是针对三维应用，实现高效的求解器。

Throughout the last 25-30 years, optimal shape design has become more and more important in engineering applications. Many problems that arise in structural mechanics, fluid dynamics and electromagnetics lead to the minimization of functionals defined over a class of admissible domains, governed by the solutions of boundary value problems or initial boundary value problems.The methodology of the present application is based on a mathematical shape sensitivity analysis and exploits the analytic computation of the shape gradient and, if available, even higher derivatives. Afterwords, the shape (and thus, the shape gradient) as well as the underlying state equation are discretized, which is in contrast to fully discrete shape optimization methods. The scope of the present project is the realization of an efficient solver in spirit of this policy, based on finite element techniques, in particular for three dimensional applications.