Arbitrarily Shaped Hollow Waveguide Analysis by the $\alpha $-Interpolation Method

Ikeuchi, Masatoshi; Inoue, Kazuo; Sawami, Hideo; Niki, Hiroshi

doi:10.1137/0140007

Cited by 8 publications

(6 citation statements)

References 5 publications

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“…Nevertheless, just like for the GLS-FEM, the coefficient a 0 can be chosen so as to arrive at a higher-order modification of the interior stencil of the Galerkin FEM. Similar studies for eigenvalue problems using the AIM with simplicial FEs was carried out in [4,5,19,20].…”

Section: Simplicial Finite Elementsmentioning

confidence: 93%

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A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Nadukandi

Oñate

García

2011

Numerical Meth Engineering

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SUMMARYA new Petrov-Galerkin (PG) method involving two parameters, namely˛1 and˛2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is,˛1 D˛2 D˛; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18-46) for the Helmholtz equation if the parameters are distinct, that is,˛1 ¤˛2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by 1 ) and the mass terms (specified by˛2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18-46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions.

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Section: Simplicial Finite Elementsmentioning

confidence: 93%

“…The fact that the shape functions N a being a partition of unity is used to arrive at the last part of Equation (19). For every element K, we define a local transformation matrix W as follows:…”

Section: Trial and Test Spacesmentioning

confidence: 99%

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Nadukandi

Oñate

García

2011

Numerical Meth Engineering

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“…The finite-element method has been used in [98] for Land H-shapes, in [20] for circular and square waveguides with quadruple ridges, in 126] for various crosses, in [65] for "bent" waveguides (L-shapes at angles other then 90), in [82] for rectangles with rounded corners, in [32 for shapes given by polar coordinates such as portions of spirals, and in [89] for limaons and cardioids. It is also used in the interesting paper [84] where mode shapes are followed as parameters are varied to change rectangles into ellipses via hyperellipses and into parabolas via superellipses.…”

Section: I=1mentioning

confidence: 99%

Eigenvalues of the Laplacian in Two Dimensions

Kuttler¹,

Sigillito²

1984

SIAM Rev.

231

159

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The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem. 3. Elementary solutions. We quote Rayleigh 128 who says, "The theory of the free vibrations of a membrane was first successfully considered by Poisson 116]. His theory in the case of the rectangle left little to be desired." For the rectangle 0 _< x _< a, 0 <_ y <_ b, the eigenfunctions are

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“…M α : = α M + (1 − α) M L . This GMM scheme was later baptized as the alpha‐interpolation method (AIM) 62 and was extended to the hollow waveguide analysis in 63 and the Schrodinger equation in 64. For the simple 1D case, our scheme mimics the AIM and in 2D making the choice α = 0.5 we recover the generalized fourth‐order compact Padé approximation 38, 39 (therein using the parameter γ = 2).…”

Section: Introductionmentioning

confidence: 99%

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

Nadukandi

Oñate

García

2010

Numerical Meth Engineering

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We propose a fourth-order compact scheme on structured meshes for the Helmholtz equation given by R():= f (x)+D+n 2 = 0. The scheme consists of taking the alpha-interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha-interpolation method (J. Comput. Appl. Math. 1982; 8(1):15-19) and in 2D making the choice = 0.5 we recover the generalized fourth-order compact Padé approximation (J. 128:325-359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325-359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((n ) 4 ), where n, represent the wavenumber and the mesh size, respectively. An expression for the parameter is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L 2 norm, the H 1 semi-norm and the l ∞ Euclidean norm are done and the pollution effect is found to be small. 19As the current problem admits a variational principle, naturally, discretization methods based on variational formulations viz. the Galerkin and the Trefftz-Galerkin-type methods have been preferred to other methods. The Galerkin-type methods are domain-based wherein the integral statement involves only the weak form of the governing differential equation and the sub-space of test-functions are assumed to satisfy a priori the kinematic compatibility and essential boundary conditions. The Trefftz-Galerkin-type methods are boundary-based and are formulated using the reciprocal principle wherein the integral statement involves only the kinematic compatibility and essential boundary conditions of the problem and the sub-space of test-functions are assumed to satisfy a priori the governing differential equation [1,2].In the context of the Galerkin-type methods, the finite element method (FEM) is a powerful technique to systematically generate subspaces of test-functions (classically piecewise polynomial spaces). Some of the earlier works on the use of FEM for the numerical solution of the Helmholtz equation can be found in [3-10] and the references cited therein. In [5,7] error estimates were given for the asymptotic (n 2 assumed sufficiently small) and pre-asymptotic (n assumed sufficiently small) cases, respectively. It was shown that for the discrete problem the LBB ‡ constant can be expressed as h = min{| h m −n 2 |/ h m } [6]. Thus, for the continuous problem (visualized as → 0) the LBB constant can be expressed as = min{| m −n 2 |/ m }, which in an average sense implies that is inversely proportional to the wavenumber n, i.e. ∝ n −1 [6,7]. Thus for high wavenumbers and for the case of degeneracy (n → h m ), the LBB constant for the discrete problem tends to be small which in turn leads to a loss of stability. The loss of stability...

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Arbitrarily Shaped Hollow Waveguide Analysis by the $\alpha $-Interpolation Method

Cited by 8 publications

References 5 publications

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

Eigenvalues of the Laplacian in Two Dimensions

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

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