This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli’s type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in Boulkhemair (SIAM J Control Optim 55(1):156–171, 2017) and Boulkhemair and Chakib (J Convex Anal 21(1):67–87, 2014), that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with a boundary element method is performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach.

中文翻译：

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一类自由边界问题的数值形状优化方法

本文致力于一种数值方法，用于逼近一类伯努利类型的自由边界问题，该问题被重新公式化为具有适当形状功能的最佳形状设计问题。我们显示一类可允许域上成本函数的形状导数的存在，并通过使用Boulkhemair（SIAM J Control Optim 55（1）：156–171，2017）和Boulkhemair和Chakib中提出的公式来计算其形状导数（J Convex Anal 21（1）：67-87，2014），即通过辅助功能。从数值上讲，这使我们避免了基于矢量场的方法的繁琐计算。为了解决该问题，采用了结合边界元素方法的梯度方法，以克服有限元方法所需的重新网格化任务。最后，