In this work, a class of bilateral free boundary problem is considered. This identification problem is formulated as a shape optimization problem via the definition of a cost functional. The existence of an optimal solution for the optimization problem is proved. An augmented Lagrangian approach is used to facilitate the computation of the shape derivatives. A numerical approach is proposed to solve iteratively the shape optimization problem, also we prove its convergence. Various type of domains are investigated to confirm the validity of the presented approach. For the numerical stability we investigate the shape without and with noisy data.

中文翻译：

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双边自由边界问题的增强拉格朗日方法

在这项工作中，考虑了一类双边自由边界问题。通过成本函数的定义，将该识别问题表述为形状优化问题。证明了优化问题的最优解的存在。增强的拉格朗日方法用于促进形状导数的计算。提出了一种数值方法来迭代求解形状优化问题，并证明了其收敛性。研究了各种类型的域，以确认所提出方法的有效性。为了获得数值稳定性，我们在没有噪声数据的情况下研究形状。