SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1420-1440, January 2020.
For certain shapes of inclusions embedded in a body, the field inside the inclusion is uniform for some boundary condition. We provide a construction scheme for inclusions of general shapes having such a uniformity property in two dimensions based on the conformal mapping technique for the potential problem. Using this complex analysis method, we also design nonelliptical neutral coated inclusions with anisotropic conductivity. Neutral coated inclusions do not perturb a background uniform field when they are inserted into a homogeneous matrix. Although coated inclusions of various shapes are neutral to a single field, only concentric ellipses or confocal ellipsoids can be neutral to all uniform fields. This paper presents our work relating to the construction of nonelliptical coated inclusions with anisotropic conductivity in two dimensions that are neutral to all uniform fields, where the assignment of the flux condition on the boundary of the core depends on the applied background field. Using these neutral inclusions, we obtain cylindrical neutral inclusions in three dimensions, with no flux applied to the boundary of the core and with the anisotropic conductivity function of the shell given in accordance with the background uniform field.

中文翻译：

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核心内部具有恒定场的一般形状的夹杂物和具有各向异性导电性的非椭圆中性涂层夹杂物

SIAM应用数学杂志，第80卷，第3期，第1420-1440页，2020年1月。
对于嵌入在体内的某些形状的夹杂物，对于某些边界条件，夹杂物内部的场是均匀的。我们基于潜在问题的共形映射技术，提供了一种在二维中具有这种均匀性的一般形状的夹杂物的构造方案。使用这种复杂的分析方法，我们还设计了具有各向异性导电性的非椭圆形中性涂层夹杂物。中性涂层夹杂物插入均质基质时不会干扰背景均匀场。尽管各种形状的涂层夹杂物对单个场都是中性的，但只有同心椭圆或共聚焦椭球对所有均匀场都是中性的。本文介绍了我们的工作，该工作涉及二维各向异性电导率非椭圆形涂层夹杂物的构造，这对于所有均匀场都是中性的，其中磁芯边界上的通量条件的分配取决于所施加的背景场。使用这些中性夹杂物，我们获得了三维的圆柱形中性夹杂物，没有磁通施加到核的边界，并且根据背景均匀场给出了壳的各向异性电导率函数。