Second derivatives of cost functions and H1 Newton method in shape optimization problems
Second derivatives of cost functions and H1 Newton method in shape optimization problems
复制标题
形状优化问题中成本函数的二阶导数和 H1 牛顿法
DOI：
10.1007/978-981-10-6283-4_6
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Hideyuki Azegami
中科院分区：
文献类型：
--
作者：
T. Rabago Julius Fergy;Azegami Hideyuki;Hideyuki Azegami
对于定义了偏微分方程组边值问题的区域的形状优化问题，我们推导了其代价函数的二阶形状导数(Shape Hessians)，并提出了利用Shape Hessians求解问题的牛顿方法。本文建立了一个抽象的形状优化问题，给出了抽象框架下代价函数的一阶和二阶形状导数的计算方法。然后，利用形状梯度和Hessians，我们提出了一种牛顿方法来求解该问题。作为说明，导出了平均柔度和定域度量的形状Hessians，并将其用于数值例子。
We derive the second-order shape derivatives (shape Hessians) of cost functions for shape optimization problems of domains in which boundary value problems of partial differential equations are defined, and propose anNewton method to solve the problems using the shape Hessians. In this paper, we formulate an abstract shape optimization problem and show the computations of the first- and second-order shape derivatives of cost functions under the abstract framework. Then, using the shape gradients and Hessians, we propose anNewton method to solve the given problem. As an illustration, the shape Hessians of a mean compliance and a domain measure are derived and then used for a numerical example.