Three-dimensional (3D) axisymmetric invisibility cloaks with arbitrary shaped in layered-media background are presented using the transformation optics. The inner and outer boundaries of the cloaks can be non-conformal with arbitrary shapes, which considerably improve the flexibility of the cloaking applications. However, such kinds of 3D cloaks cannot be simulated using the commercial softwares due to the tremendous memory requirements and CPU time. By taking advantage of the rotationally symmetrical property, we propose an efficient finite-element method (FEM) to simulate and analyze the 3D cloaks, which can greatly reduce the CPU time and memory requirements. The method is based on the electric-field formulation, in which the transverse fields are expanded in terms of second-order edge-based vector basis functions and the azimuth components are expanded using second-order nodal-based scalar basis functions. The FEM mesh is truncated using the absorbing boundary condition. Excellent cloaking performance of the 3D cloaks in layered-media background has been verified by the proposed method.
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