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arXiv 1702.07579math.OC
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Optimization形状的适用空间

Suitable Spaces for Shape Optimization

Kathrin Welker

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摘要：

本文将光滑形状流形的微分几何结构应用于形状最优化问题理论。特别地，定义了关于第一Sobolev度量和Steklov-Poincaré度量的黎曼形状梯度。此外，本文还导出了与第一Sobolev度量相关的共变导数.的明确表达的共变导数导致的定义的黎曼形状海森关于第一个Sobolev度量。在本文中，我们简要概述了各种优化技术的基础上的梯度和海森。由于光滑形状空间限制了优化技术的应用，本文将光滑形状的定义推广到形状最优化问题中自然出现的$H^{1/2}$ -形状。我们在新的$H^{1/2}$ -形空间上定义了一个拓扑结构。这可以被看作是在拓扑空间上制定优化技术的第一步。

英文摘要：

The differential-geometric structure of the manifold of smooth shapes is applied to the theory of shape optimization problems. In particular, a Riemannian shape gradient with respect to the first Sobolev metric and the Steklov-Poincaré metric are defined. Moreover, the covariant derivative associated with the first Sobolev metric is deduced in this paper. The explicit expression of the covariant derivative leads to a definition of the Riemannian shape Hessian with respect to the first Sobolev metric. In this paper, we give a brief overview of various optimization techniques based on the gradients and the Hessian. Since the space of smooth shapes limits the application of the optimization techniques, this paper extends the definition of smooth shapes to $H^{1/2}$-shapes, which arise naturally in shape optimization problems. We define a diffeological structure on the new space of $H^{1/2}$-shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces.

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