The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the \(L^2\)-distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.

中文翻译：

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使用新的边界代价函数求解外部伯努利自由边界问题的二阶形状优化算法

使用一种称为Dirichlet-data-gap成本函数的新型目标函数，将外部Bernoulli问题改写为形状优化问题，该函数测量两个状态函数的Dirichlet数据之间的\（L ^ 2 \）-距离。成本函数的一阶形状导数通过链式规则方法明确确定。使用相同的技术，还可以计算自由边界问题解时成本函数的二阶形状导数。然后，使用梯度和Hessian信息来制定有效的基于二阶梯度的下降算法，以数值方式解决最小化问题。通过各种数值例子说明了该方法的可行性。