Perfectly Matched Layers in the Cylindrical and Spherical Coordinates for Elastic Waves in Solids

The material constants of perfectly matched layers (PMLs) in the cylindrical and spherical coordinates in the frequency domain are presented. Using the coordinate transformation laws of tensors on manifolds, the quotient rule, and complex coordinate stretching, we obtain the material parameters of PMLs in the real coordinate. Our results show that PML parameters for elastic waves may be determined by the same procedure in the Cartesian coordinates. However, this rule has been determined for PML material constants derived from the analytic continuation in the cylindrical and spherical coordinates by Zheng and Huang in 2002. Our derivation based on differential forms shows that this rule holds for PML parameters in any orthogonal coordinate system.