The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

Peake, M.J.; Trevelyan, J.; Coates, Graham

doi:10.1016/j.enganabound.2013.11.020

Cited by 13 publications

(13 citation statements)

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“…Care was taken so that at least 1.4 times as many collocation points were used as degrees of freedom. By doing this, the errors continue to decrease as τ is increased; this is more like the behaviour expected of both the PU-BEM and XIBEM and observed in earlier works [24,32,33].…”

Section: Torussupporting

confidence: 62%

“…The case in three-dimensions is not as straightforward. The authors adopt the Coulomb Force Method described in [32] to produce an quasi-uniform distribution of wave directions about the unit-sphere. Care is taken so that one of the wave directions is the same as the incident wave propagation direction d inc [8].…”

Section: In Conventional Bem Simulations N Dof Is Typically Increasementioning

confidence: 99%

“…For the conventional BEM mesh, the cube-to-sphere mapping described in [32] is used. The sphere is discretised initially into six elements.…”

Section: Unit Spherementioning

confidence: 99%

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Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

Peake

Trevelyan

Coates

2015

Computer Methods in Applied Mechanics and Engineering

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Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. In this paper, the collocation XIBEM formulation is described and numerical results given. The numerical results are compared against closed-form or converged solutions. Comparisons are made against the conventional boundary element method and the non-enriched isogeometric BEM (IGABEM).When compared to non-enriched boundary element simulations, using XIBEM significantly reduces the number of degrees of freedom required to obtain a solution of a given error; thus, with a fixed computational resource, problems of a shorter wavelength can be solved.

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Section: Torussupporting

confidence: 62%

Section: In Conventional Bem Simulations N Dof Is Typically Increasementioning

confidence: 99%

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Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

Peake

Trevelyan

Coates

2015

Computer Methods in Applied Mechanics and Engineering

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“…To obtain isotropic approximations, in 2D, uniformly-spaced directions on the unit circle can be chosen (i.e. d ℓ = (cos(2πℓ/p K ), sin(2πℓ/p K ))); in 3D, [103] and [94] provide directions that are "almost equally spaced" (see [1, §3.4] for a simpler version). In these cases, the PW spaces are not hierarchical.…”

Section: Plane Waves (Pws)mentioning

confidence: 99%

A Survey of Trefftz Methods for the Helmholtz Equation

Hiptmair

Moiola

Perugia

2016

Lecture Notes in Computational Science and Engineering

125

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Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces.We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties.One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.

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“…6. If K is not star-shaped but can be decomposed in parts that are star-shaped with respect to some balls, a bound similar to (38) holds with at the denominator the radius of the smallest of those balls.…”

Section: We Introduce An Anisotropic Sobolev Seminorm In An Openmentioning

confidence: 99%

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

Moiola

Perugia

2017

Numer. Math.

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We introduce a space-time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer. Anal., 36, 1599-1635. Test and trial discrete functions are space-time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeletonbased and mesh-independent norms. The space-time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds.1 studied in [3]. Another Trefftz discontinuous Galerkin (DG) formulation for time-dependent electromagnetic problems formulated as first-order systems has been proposed in [25] and analysed in [24] in one space dimension; it has been extended to full three-dimensional Maxwell equations in [10,11,23].We mention here that, independently of the Trefftz approach, space-time finite elements for linear wave propagation problems, originally introduced in [21] (see also [15,22]), have been used in combination with DG formulations e.g. in [7,14,35] and, more recently, in [8,16,17,27].In this paper we extend the Trefftz-DG method of [24] to initial boundary value problems for the acoustic wave equations posed on Lipschitz polytopes in arbitrary dimensions. We write the acoustic wave problem as a first-order system, as it is originally derived from the linearised Euler equations, [6, p. 14]; we consider piecewise-constant wave speed, Dirichlet, Neumann and impedance boundary conditions. The main focus of this paper is on the a priori error analysis of the Trefftz-DG scheme.The DG formulation proposed can be understood as the translation to time-domain of the Trefftz-DG formulation for the Helmholtz equation of [18], which in turn is a generalisation of the Ultra Weak Variational Formulation (UWVF) of [5]. The DG numerical fluxes are upwind in time and centred with a special jump penalisation in space. Under a suitable choice of the numerical flux coefficients, combining the proposed formulation with standard discrete spaces and complementing it with suitable volume terms, one recovers the DG formulation of [35], cf. Remark 4.2 below. The Trefftz formulation for Maxwell's equations of [10,11,23,25] corresponds to the "unpenalised" version of that one proposed here (the numerical experiments in [24, §7.5] show that the numerical error depends very mildly on the penalisation parameters).We first describe the IBVP under consideration in §2, the assumptions on the mesh in §3 and the Trefftz-DG formulation in §4. Following the thread of [18,24], in §5.2 and §5.3 we prove that the scheme is well-posed, quasi-optimal, dissipative (quantifying dissipation using the jumps of the discrete solution), and derive error estimates for some traces of the solution on the mesh skeleton. ...

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The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

Abstract: The equal spacing of N points on a sphere with application to partition-of-unity wave diraction problems.', Engineering analysis with boundary elements., 40. pp. 114-122.

Cited by 13 publications

References 26 publications

Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

A Survey of Trefftz Methods for the Helmholtz Equation

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

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