When seeking a solution to the interface problem, the potential theory to represent solution has been widely used. As the solution representation of the interface problem is only well-known for domains with simple inclusion, such as a disk, conformal mapping is often used to transform the arbitrary inclusion to a manageable inclusion to utilize the known solution representation. This paper proposes the construction of a conformal mapping based on Moser’s grid deformation method that generates a transformation with positive Jacobian determinant. By establishing an optimal control problem that minimizes the dissimilarity between transformed and reference images with desired constraints, we constructed an appropriate conformal mapping. Several numerical results prove the validity of our approach.

在寻求界面问题的解决方案时，代表解决方案的势能理论已被广泛使用。由于接口问题的解决方案表示仅对于具有简单包含的域（例如磁盘）才是众所周知的，因此通常使用保形映射将任意包含转换为可管理的包含，以利用已知的解决方案表示。本文提出了一种基于Moser网格变形方法的共形映射的构造方法，该方法生成具有正Jacobian行列式的变换。通过建立一个最佳控制问题，该问题使具有所需约束的变换图像和参考图像之间的差异最小化，我们构建了适当的共形映射。几个数值结果证明了我们方法的有效性。