论文标题
在新的耦合复杂边界方法上，以解决固定自由边界问题的形状优化框架
On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems
论文作者
论文摘要
我们在这里揭示了所谓的耦合复杂边界方法的新应用 - 首先由Cheng等人提出。 （2014年）处理逆源问题 - 在用于解决外部伯努利问题的形状优化框架中，这是一种固定自由边界问题的原型模型。该方法的想法是将过度确定的问题转换为复杂的边界价值问题，复杂的罗宾边界条件将dirichlet和neumann边界条件耦合在自由边界上。然后，我们优化了整个域中解决方案的假想部分构建的成本函数，以识别自由边界。我们还证明了相对于域的复杂状态的形状衍生物的存在。之后，我们计算成本功能的形状梯度，并将其形状的Hessian表征在强大的最佳结构域，然后在域上进行轻度的规则性假设。然后，我们通过证明后一种表达的紧凑性来检查所提出的方法的不稳定性。另外，我们通过有限元方法基于SOBOLEV梯度方案设计了一种迭代算法，以解决最小化问题。最后，我们通过在两个和三个空间维度中通过几个数值示例说明了该方法的适用性。
We expose here a novel application of the so-called coupled complex boundary method -- first put forward by Cheng et al. (2014) to deal with inverse source problems -- in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then examine the instability of the proposed method by proving the compactness of the latter expression. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.
