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The purpose of this study is to calculate the frequency electric field in substation.

The paper proposes a novel fast multipole method (FMM) called Super-FMM to solve the PFEF problems in substations. The paper substitutes the original approaches for analytic expansions and translations through equivalent density representations.

The paper shows that the Super-FMM is more efficient in terms of the complexity of its storage spaces and computational costs compared with the best-known FMM when placed under scenarios with exactly the same error rates.

Using the fast Fourier transform algorithm can further improve the optimization algorithm and computational efficiency.

A novel FMM called Super-FMM is proposed, which has a structure similar to that of the adaptive FMM algorithm, but the paper substitutes the original approaches for analytic expansions and translations through equivalent density representations.

In the vibration reduction field, constrained stand-off layer damping cylindrical shell plays an important role. However, due to the lack of accurate analysis of its damping characteristics, this hinders its further research and application. Therefore, the purpose of this paper is concerned with an accurate solution for the vibration-damping characteristics of a constrained stand-off-layer damping cylindrical shell (CSDCS) under various classical boundary conditions and conducts a further analysis.

Based on the Rayleigh–Ritz method and the Hamilton principle, a dynamic model of CSDCS is established. Then the loss factor and the frequency of CSDCS are obtained. The correctness and convergence behavior of the present model are verified by comparing the calculation results with the literature. By using for various classical boundary conditions without any special modifications in the solution procedure, the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries are discussed.

The Rayleigh–Ritz method is effective in handling the problem of CSDCS with different boundaries and an accurate solution is obtained. The boundary conditions have an important influence on the vibration and damping behavior of the CSDCS.

Based on the Rayleigh–Ritz method and Hamilton principle, a dynamic model of CSDCS is established for the first time, and then the loss factor and frequency of CSDCS are obtained. In addition, the effectiveness of adding the stand-off layer between the base shell and the viscoelastic layer is confirmed by discussing the characteristics of CSDCS with S-S, C-C, C-S, C-F and S-F boundaries.