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1 Overview
Advanced Finite Element Waveguide Calculation and Analysis
Abstract—This report demonstrates how to calculate the propagation modes and cutoff frequencies of arbitrarily shaped uniform waveguides using the finite element method. It describes a program that uses high-order interpolation polynomials and compares the results with analytical solutions. Additionally, the convergence rate and relative error are discussed. It is believed that the program applying high-order finite element methods can produce higher precision waveguide analysis.
The Finite Element Method (FEM) is a mature technology widely used as an analysis and design tool in many engineering disciplines such as structural and computational fluid dynamics. In the past 30 years, there has been increasing interest in applying FEM to microwave components such as waveguides and antennas. The literature on the application of FEM to empty waveguides or uniform waveguides can be traced back to the late 1960s. Initially, the order of interpolation polynomials used in FEM was low, requiring a large number of meshes. The application of high-order elements was first proposed by P. Silvester. He suggested using high-order interpolation polynomials, where the number of sub-regions does not need to exceed the number required to describe the boundary shape. By finding the polynomial coefficients in each triangle, an approximate solution to the waveguide eigenvalue problem is obtained through a variational program. In this project, relatively high-order (up to fourth order) elements are applied to solve the eigenvalue problem. Due to MATLAB’s powerful computational capabilities, even with high-order elements, the number of elements does not need to be too small. The implementation of the finite element method in MATLAB is described in detail, and both analytical and numerical solutions are presented. Furthermore, the benefits brought by high-order elements are presented in various ways.
2 Results
% import the mesh data produced by the MATLAB mesh generator. It only% can be used with the first order FEMTotalEle = dlmread('Node Number.DAT'); % global node number matrixTotalCoo = dlmread('Coordinates.DAT'); % node coordinatesBoundaries = dlmread('Boundary.DAT'); % boundary condition matrixTotalEle = TotalEle';TotalEle = TotalEle(:,1:3);TotalCoo = TotalCoo';Boundaries = Boundaries';Boundaries = Boundaries(:,1:2);% calculate the second order mesh elements from the first order meshes[TotalEle, TotalCoo, Boundaries]= addMiddlePoint(TotalEle, TotalCoo, Boundaries); % calculate the coefficients Ka and Kb used in the generalized eigenvalue% equation[Ka, Kb] = derivationOfEquation(TotalEle, TotalCoo);noOfElements = size(TotalEle,1);noOfNodes = size (TotalCoo,1); % total number of nodesnoOfBoundaries = size(Boundaries,1); % total number of nodes on the boundary% TE modes, solving magnetic field [D1,V1] = eig(Ka,Kb); eigenValuesTE = zeros(noOfNodes,1);eigenVectorsTE = zeros(noOfNodes);Conclusion: In this project, the high-order finite element method is studied in depth, and the program is written in MATLAB to solve electromagnetic field problems in rectangular uniform waveguides. Furthermore, the solutions obtained using different order elements indicate that the application of higher-order elements improves the accuracy of the solutions without significantly increasing computational costs. The relationship between numerical solution accuracy and computational cost based on second-order and fourth-order finite element methods is also introduced. The numerical and analytical solutions for the TE32 mode are presented. Additionally, the program written for this project is applicable to any waveguide shape with boundaries composed of straight line segments. With some modifications, the program can analyze non-uniform waveguides.
3References
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4 MATLAB Code, Data, and Articles
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