# Modelling of short wave diffraction problems using approximating systems of plane waves

10 Aug 2002-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 54, Iss: 10, pp 1501-1533

TL;DR: In this article, a finite element model for the solution of Helmholtz problems at higher frequencies is described, which offers the possibility of computing many wavelengths in a single finite element.

Abstract: This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed.

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TL;DR: State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed and Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite- element methods are described.

Abstract: State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given.

368 citations

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TL;DR: Recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles is described.

Abstract: In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods

242 citations

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TL;DR: In this paper, an enriched finite element method is presented to solve various wave propagation problems, which combines advantages of finite element and spectral techniques, but an important point is that it preserves the fundamental properties of the FEM method.

187 citations

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TL;DR: In this article, a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation has been proposed, which requires a minimal resolution of the mesh beyond what it takes to resolve the wavelength.

Abstract: We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give ap riori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.

185 citations

### Cites methods from "Modelling of short wave diffraction..."

...This has been attempted in the partition of unity (PUM) finite element method [3,20,25,26,28], the discontinuous enrichment approach [14,15,33], in the context of least squares approaches [27,32], and in the so-called “Variational Theory of Complex Rays” (VTCR) approach [29]....

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TL;DR: In this paper, the generalized finite element method for the Helmholtz equation is applied on Cartesian meshes, which may overlap the boundaries of the problem domain, and enriched the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method.

181 citations

##### References

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TL;DR: In this article, the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM) are presented and a detailed and illustrative analysis is given for a one-dimensional model problem.

3,276 citations

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TL;DR: In this article, a new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved, which can therefore be more efficient than the usual finite element methods.

Abstract: A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved This new method can therefore be more efficient than the usual finite element methods An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers The basic estimates for a posteriori error estimation for this new method are also proved © 1997 by John Wiley & Sons, Ltd

2,387 citations

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TL;DR: In this article, the authors developed a method that inverts seismic body waves to determine the mechanism and rupture pattern of earthquakes, where the rupture pattern is represented as a sequence of subevents distributed on the fault plane.

Abstract: We have developed a method that inverts seismic body waves to determine the mechanism and rupture pattern of earthquakes. The rupture pattern is represented as a sequence of subevents distributed on the fault plane. This method is an extension of our earlier method in which the subevent mechanisms were fixed. In the new method, the subevent mechanisms are determined from the data and are allowed to vary during the sequence. When subevent mechanisms are allowed to vary, however, the inversion often becomes unstable because of the complex trade-offs between the mechanism, the timing, and the location of the subevents. Many different subevent sequences can explain the same data equally well, and it is important to determine the range of allowable solutions. Some constraints must be imposed on the solution to stabilize the inversion. We have developed a procedure to explore the range of allowable solutions and appropriate constraints. In this procedure, a network of grid points is constructed on the τ - I plane, where τ and I are, respectively, the onset time and the distance from the epicenter of a subevent; the best-fit subevent is determined at all grid points. Then the correlation is computed between the synthetic waveform for each subevent and the observed waveform. The correlation as a function of τ and I and the best-fit mechanisms computed at each τ - I grid point depict the character of allowable solutions and facilitate a decision on the appropriate constraints to be imposed on the solution. The method is illustrated using the data for the 1976 Guatemala earthquake.

989 citations