The one dimensional Stefan problem is reformulated as a shape optimization problem for the position of the phase transition as a function of time. The functional to be minimized is the mismatch of the Dirichlet to Neumann map at the moving interface. We show that the minimizer is the only stationary point of the shape functional. A gradient based optimization method is derived using shape calculus. The state and adjoint equations of the heat equation are solved with integral equation techniques which avoid a discretization in the domain. A Nyström quadrature method is analyzed and numerical results are presented.

中文翻译：

---

热方程控制的运动界面问题的形状优化方法

将一维Stefan问题重新构造为形状优化问题，以解决随时间变化的相变位置。要最小化的功能是在移动界面处Dirichlet与Neumann映射的不匹配。我们显示最小化器是形状函数的唯一固定点。使用形状演算推导基于梯度的优化方法。用积分方程技术求解热方程的状态方程和伴随方程，该积分方程技术避免了域中的离散化。分析了Nyström正交方法，并给出了数值结果。