A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems

Shirron, Joseph J.; Babuška, Ivo

doi:10.1016/s0045-7825(98)00050-4

Cited by 99 publications

(77 citation statements)

References 38 publications

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“…The subsequent considerations use the concept of the so-called (mapped and conjugated) Astley-Leis infinite elements in reference to works by Astley et al 2,3,31,32 as well as by Shirron and Babuška. 5 These elements are attached to the outer boundary of the finite elements to extend the fluid to infinity in the radial direction. The coordinate mapping is given in detail by Marques and Owen.…”

Section: Acoustics and Discretization Of The Unbounded Domainmentioning

confidence: 99%

“…The sound pressure field in the radial direction in the domain with the infinite elements is interpolated by polynomials such as Lagrange polynomials, Legendre polynomials or Jacobi polynomials, which lead to differences in the matrix condition number of the discrete, global system matrices. 5,6 The subject of this paper is the investigation of the influence of the choice of finite elements and the polynomial type for infinite element interpolation in the radial direction as well as the choice of the order of these polynomials on modes in exterior acoustics. The authors consider acoustic radiation modes (ARMs) [7][8][9] and normal modes (NMs), [10][11][12] which are based on eigenvalue problems of the acoustic impedance matrix Z R and of a statespace formulation consisting of the discrete system matrices of stiffness, damping and mass, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

Moheit

Marburg

2017

J. Comp. Acous.

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Acoustic radiation modes (ARMs) and normal modes (NMs) are calculated at the surface of a fluid-filled domain around a solid structure and inside the domain, respectively. In order to compute the exterior acoustic problem and modes, both the finite element method (FEM) and the infinite element method (IFEM) are applied. More accurate results can be obtained by using finer meshes in the FEM or higher-order radial interpolation polynomials in the IFEM, which causes additional degrees of freedom (DOF). As such, more computational cost is required. For this reason, knowledge about convergence behavior of the modes for different mesh cases is desirable, and is the aim of this paper. It is shown that the acoustic impedance matrix for the calculation of the radiation modes can be also constructed from the system matrices of finite and infinite elements instead of boundary element matrices, as is usually done. Grouping behavior of the eigenvalues of the radiation modes can be observed. Finally, both kinds of modes in exterior acoustics are compared in the example of the cross-section of a recorder in air. When the number of DOF is increased by using higher-order radial interpolation polynomials, different eigenvalue convergences can be observed for interpolation polynomials of even and odd order.

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Section: Acoustics and Discretization Of The Unbounded Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

Moheit

Marburg

2017

J. Comp. Acous.

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“…Those of Bayliss and Turkel [5], Engquist and Majda [19], and Feng [20] are among the most widely used. However, in spite of the simple implementation of lowest order ABCs, good accuracy is only achieved for higher order ones [37], because these conditions are not fully non-reflecting on the truncated boundary of …”

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confidence: 99%

An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems

Bermúdez¹,

Hervella-Nieto²,

Prieto³

et al. 2008

SIAM J. Sci. Comput.

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Abstract. The aim of this paper is to introduce an "exact" bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function. "Exactness" must be understood in the sense that this technique allows exact recovering of the solution to time-harmonic scattering problems in unbounded domains. In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space. The resulting variational formulation can be numerically dealt with standard finite elements. The high accuracy of this approach is numerically demonstrated as compared with a classical PML technique. 1. Introduction. The typical first step for the numerical solution by finite elements or finite differences of any scattering problem in an unbounded domain is to truncate the computational domain, which entails an inherent difficulty: to choose boundary conditions to replace the Sommerfeld radiation condition at infinity (see for instance [21]).There are several techniques to deal with this: boundary element methods, infinite element methods, Dirichlet-to-Neumann methods based on Fourier expansions, or the use of absorbing boundary conditions. The potential advantages of each of them have been widely studied in the literature [25,35,18].If the domain of the original problem is truncated with a sphere, then the Dirichletto-Neumann (DtN) boundary condition is exactly known (see [21,32]). However, this boundary condition involves an infinite series which must be truncated for numerical use, thus introducing errors. Moreover, the exact DtN condition is non local, leading to dense blocks in the linear system to be solved, when a finite element method is used.As a partial solution to these drawbacks, local absorbing boundary conditions (ABCs) can be introduced to preserve the computational efficiency of the numerical method. Those of Bayliss and Turkel [5], Engquist and Majda [19], and Feng [20] are among the most widely used. However, in spite of the simple implementation of lowest order ABCs, good accuracy is only achieved for higher order ones [37], because these conditions are not fully non-reflecting on the truncated boundary of

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“…In fact, J is obtained by replacing the diagonal coefficients of H by their modulus. We have proved in [19] that the solution of the weak problem (85) is equal to the solution of the original problem (70)- (72), if function σ is not integrable and with the unique restriction of belonging to the Holder space C 1,2 locally in the PML domain. Let us remark that the homogeneous Dirichlet boundary condition on the outer boundary of the PML domain Ω A is implicitly contained in the definition of the space V. This boundary condition will be actually used in the discretization of the weak problem associated with the coupled problem.…”

Section: Numerical Accuracy Of Cartesian Pml For the Helmholtz Equationmentioning

confidence: 99%

“…Those by Bayliss and Turkel [9], Givoli [38], Engquist and Majda [33], and Feng [35] are among the most widely used. However, while computer implementation of lowest order ABCs is very easy, good accuracy is only achieved for higher order ones [72], because ABCs conditions are not fully non-reflecting on the truncated boundary of the computational domain. As a consequence, high accuracy using ABCs leads to a substantial computa-tional cost and increases the difficulty of implementation.…”

Section: Introductionmentioning

confidence: 99%

Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

Bermúdez

Hervella-Nieto

Prieto

et al. 2010

Arch Computat Methods Eng

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The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for time-harmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the two-dimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using non-integrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.

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A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems

Cited by 99 publications

References 38 publications

Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems

Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

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