Waveguide Boundary Spectral Finite Elements
A waveguide boundary spectral finite element method (SFEM) is developed for the study of acoustical wave propagation in non–uniform waveguide–like geometries. The formulation is based on a variational approach using a mixture of non–internal node element
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ummary. A waveguide boundary spectral finite element method (SFEM) is developed for the study of acoustical wave propagation in non–uniform waveguide–like geometries. The formulation is based on a variational approach using a mixture of non–internal node element shape functions and wave solutions. The numerical method provides solutions to acoustic duct or fluid waveguide environments which may be divided into uniform cross–sectional regions. Trial functions are determined by solution of an eigenvalue problem defined in the cross– section, which in turn, depends upon the boundary data. Illustration of the method through demonstration of transmission loss of acoustic energy through two–dimensional dissipative mufflers is presented and solutions of a three–dimensional elliptical duct problem are shown.
14.1 Introduction The spectral element method is an advanced implementation of the finite element method in which the solution over each element is expressed in terms of a priori unknown values at carefully selected spectral nodes. The advantage of the spectral element method is that stable solution algorithms and high accuracy can be achieved with a low number of elements under a broad range of conditions. Spectral element techniques are high order methods which allow for either obtaining very accurate results or reducing the number of degrees of freedom for fixed standard precision. As explained by Gottlieb and Orszag [20] in the 1970’s spectral methods for boundary value problems rely on high degree polynomial approximations on square and cubic domains. Trefethen [31] also describes the theory behind spectral methods and the connection between Fourier series, Chebyshev polynomials and includes MATLAB code. Both Boyd [7] and Fornberg [17] books are also excellent constructive editions on spectral methods on regular finite domains. Recently though the article by Philipps and Davies [28] proposes a general spectral method for Poissons equation in rectangularly decomposable regions including semi–infinite regions. Eigenfunctions of a differential operator are chosen as trial functions within each element and are nonconforming. The solution is determined by matching the solution and its derivative across element interfaces. A similar approach of using eigenfunctions of a
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differential operator in the trial function space is adopted in here where the principal field equation is the Helmholtz equation. This is the underpinning principle for the waveguide spectral element method. Handling complex geometries by spectral finite element methods are now an established alternative to finite difference and finite element methods to solve elliptic Partial Differential Equations (PDE). Spectral methods are naturally chosen to solve problems in regular rectangular, cylindrical or spherical regions. However in a general irregular region it would be unwise to turn away from the finite element method since models defined in such regions are extremely difficult to implement and solve with a spectral method. The examples h
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