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The accurate analysis of periodic surface acoustic wave (SAW) structures by combined finite element method and boundary element method (FEM/BEM) is important for SAW design, especially in the extraction of couple-of-mode (COM) parameters. However, the time cost is very large. With the aim to accelerate the calculation of SAW FEM/BEM analysis, some optimization algorithms for the FEM and BEM calculation have been reported, while the optimization for the solution to the final FEM/BEM equations which is also with a large amount of calculation is hardly reported. In this paper, it was observed that the coefficient matrix of the final FEM/BEM equations for the periodic SAW structures was similar to a Toeplitz matrix. A fast algorithm based on the Trench recursive algorithm for the Toeplitz matrix inversion was proposed to speed up the solution of the final FEM/BEM equations. The result showed that both the time and memory cost of FEM/BEM was reduced furtherly.

Cathodic protection (CP) is a technique that prevents corrosion of underground metallic structures. Design of any CP system first requires defining the protection of current density and potential distribution, which should meet the given criterion. It also needs to provide, as uniform as possible, current density distribution on the protected object surface. Determination of current density and potential distribution of CP system is based on solving the Laplace partial differential equation. Mathematical model, along with the Laplace equation, is represented by two additional equations that define boundary conditions. These two equations are non-linear and they represent the polarization curves that define the relationship between current density and potential on electrode surfaces. Nowadays, the only reliable way to determine current density and potential distribution is by applying numerical techniques. This paper presents efficient numerical techniques for the calculation of current density and potential distribution of CP system based on the coupled boundary element method (BEM) and finite element method (FEM).