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

在过去的25-30年中，最优形状设计在工程应用中变得越来越重要。在结构力学、流体动力学和电磁学中出现的许多问题导致在一类可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所控制。本应用程序的方法是基于数学形状敏感性分析，并利用形状梯度的解析计算，如果可用，甚至更高的导数。随后，形状（以及形状梯度）和底层状态方程被离散化，这与完全离散的形状优化方法不同。本项目的范围是在这一政策精神的基础上，基于有限元技术，特别是三维应用，实现一个有效的求解器。

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.