The cloaking method in non-uniform static fields of cylindrical and spherical invisibility cloaks

The cloaking methods of infinite length coaxial cylindrical and concentric spherical invisibility cloaks in the non-uniform static field are investigated. First, the arbitrary potential function at the infinite boundary is expanded into a series of orthogonal potential functions. When there is only the nth order orthogonal potential function distribution at the infinite boundary, the analytical cloaking conditions of both the infinite length coaxial cylindrical and concentric spherical cloaks are derived. Furthermore, based on the analytical cloaking conditions, the approaching cloaking method is proposed in a complex non-uniform static field which composed of multiple orders potential functions. This method can acquire the optimal cloaking condition and avoid the iteration process by using numerical methods such as the finite element method etc. Finally, three examples are given to verify the correctness of the analytical cloaking conditions and the effectiveness of the approaching cloaking method proposed in this paper.