不适定Bernoulli自由边界问题形状导数的CutFEM比较
Comparison of Shape Derivatives using CutFEM for Ill-posed Bernoulli Free Boundary Problem
摘要：
在本文中，我们讨论了一个水平集的方法来识别一个未知的边界在一个计算域。这个问题采取的形式是一个伯努利问题，只有狄利克雷数据是已知的边界上，这是要确定的，但额外的信息诺依曼条件是可用的已知部分的边界。该方法使用了一个经典的约束最优化问题，其中的成本功能是最小化相对于未知的边界，其位置是隐式定义的水平集函数。为了解决最优化问题，应用了使用形状导数的最速下降算法。在每次迭代中，切割有限元法被用来获得高精度近似的PDE模型的约束，为一个给定的水平集配置，而无需重新网格。我们考虑三种不同的形状导数。首先是经典的，使用连续最优化问题（优化然后离散化）导出。然后，首先使用CutFEM方法对泛函进行离散化，并在有限元泛函上评估形状导数（离散化然后优化）。最后，我们考虑第三种方法，也使用离散功能。在这种情况下，我们不扰动域，但考虑所谓的边界值校正方法，其中对边界位置的小校正可以包括在弱边界条件中。使用这种校正，可以通过扰动离散变分公式中的距离参数来获得形状导数。一系列的数值例子表明，所有这三种方法产生类似的结果建议伯努利问题的理论讨论。
英文摘要：
In this paper we discuss a level set approach for the identification of an unknown boundary in a computational domain. The problem takes the form of a Bernoulli problem where only the Dirichlet datum is known on the boundary that is to be identified, but additional information on the Neumann condition is available on the known part of the boundary. The approach uses a classical constrained optimization problem, where a cost functional is minimized with respect to the unknown boundary, the position of which is defined implicitly by a level set function. To solve the optimization problem a steepest descent algorithm using shape derivatives is applied. In each iteration the cut finite element method is used to obtain high accuracy approximations of the pde-model constraint for a given level set configuration without re-meshing. We consider three different shape derivatives. First the classical one, derived using the continuous optimization problem (optimize then discretize). Then the functional is first discretized using the CutFEM method and the shape derivative is evaluated on the finite element functional (discretize then optimize). Finally we consider a third approach, also using a discretized functional. In this case we do not perturb the domain, but consider a so-called boundary value correction method, where a small correction to the boundary position may be included in the weak boundary condition. Using this correction the shape derivative may be obtained by perturbing a distance parameter in the discrete variational formulation. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.
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