An analytical solution exists for a circular bilayer cloak against a background of linearly varying temperature distribution and under steady state. In particular, the ratio of the outer-layer and background conductivity is fixed, given the inner and outer radius of the outer layer. This conductivity ratio could be large, thus limiting the selection of bulk materials for the outer layer. This study reports a different design by adding a third layer of known conductivity between the inner and the outer layer of the bilayer cloak. The analysis extends the analytical solution of the bilayer cloak. For the same outer layer geometry between the bilayer and trilayer cloak, the proposed trilayer design gives a less limiting conductivity ratio between the outer layer and the background material than the bilayer design. Consequently, the trilayer design can enhance the selection of natural materials for the outer layer of the thermal cloak.

中文翻译：

---

三层热斗篷的分析

对于圆形双层斗篷，存在一种在线性变化的温度分布和稳定状态下的解析解决方案。特别地，给定外层的内半径和外半径，外层和背景电导率的比率是固定的。该导电率可能很大，从而限制了外层的块状材料的选择。这项研究报告了另一种设计，方法是在双层披风的内层和外层之间添加第三层已知电导率。该分析扩展了双层披风的分析解决方案。对于双层和三层披风之间的相同外层几何形状，建议的三层设计在外层和背景材料之间的极限电导率比双层设计要小。所以，