Tracking Neumann Data for Stationary Free Boundary Problems
Tracking Neumann Data for Stationary Free Boundary Problems
复制标题
跟踪稳态自由边界问题的诺依曼数据
DOI：
10.1137/080733760
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
H. Harbrecht
中科院分区：
文献类型：
--
作者：
K. Eppler;H. Harbrecht
本文致力于验证由平稳自由边界问题引起的形状优化问题的充分二阶条件。我们假设状态满足泊松方程的Dirichlet问题，并在自由边界处跟踪Neumann数据。计算了所考虑的形状泛函的梯度和Hessian。通过对匹配数据情况下形状黑森的分析，导出了其严格矫顽力的充分判据。严格矫顽力意味着稳定的极小值，在Ritz-Galerkin方法中，意味着近似形状的存在性和收敛性。通过快速边界元法，实现了求解自由边界问题的一种有效的数值算法。在三维空间上进行了数值实验。
The present paper is dedicated to the verification of sufficient second order conditions for shape optimization problems that arise from stationary free boundary problems. We assume that the state satisfies the Dirichlet problem for the Poisson equation and track the Neumann data at the free boundary. The gradient and Hessian of the shape functional under consideration are computed. By analyzing the shape Hessian in case of matching data a sufficient criterion for its strict coercivity is derived. Strict coercivity implies stable minimizers and, in case of a Ritz-Galerkin method, existence and convergence of approximate shapes. By a fast boundary element method we realize an efficient numerical algorithm to solve the free boundary problem. Numerical experiments are carried out in three spatial dimensions.