Regularization solutions to shape and topology optimization problems of domains for elliptic boundary value problems

定义了弹性体或流场的区域的形状的优化问题称为形状优化问题。在此，作为偏微分方程的边值问题（主要问题）的解，给出例如弹性体的变形场。代价函数被定义为定义主问题的域的泛函和主问题的解。在本研究中，一个容许的设计变量表示域的变化被定义。在此基础上，给出了代价函数形状导数的计算方法和形状整形的梯度法（H1梯度法）。此外，还证明了该方法得到的域序列仍在容许集内。

Optimization problem of shape of domain in which an elastic body or a flow field is defined is called the shape optimization problem. Here, a deformation field of an elastic body for example is given as a solution of a boundary value problem of a partial differential equation (main problem). Cost functions are defined as functionals of the domain in which the main problem is defined and the solution of the main problem. In the present research, an admissible set of a design variable representing domain variation was defined. Based on the definition, an evaluation method of the shape derivatives of cost functions and a gradient method (H1 gradient method) for reshaping were presented. Moreover, it was confirmed that the domain sequence obtained by the method remains in the admissible set.