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An efficient finite-element method (FEM) is presented to analyze the scattering from complex bodies of revolution (BOR) made of perfectly electric conductors and/or dielectrics. In the proposed method, high-order edge-based vector basis functions are used to expand the transverse field components, and high-order node-based scalar basis functions are used to expand the angular component. The FEM mesh is truncated using a locally-conformal perfectly matched layer (PML) by the complex coordinate stretching. Such a kind of PML is very easy to implement in the numerical process and is able to enclose an arbitrarily-shaped convex object in the spatial domain. Numerical examples are presented to demonstrate the accuracy and efficiency of the presented method.