Comparison of two wave element methods for the Helmholtz problem

Huttunen, Tomi; Gamallo, Pablo; Astley, R.J.

doi:10.1002/cnm.1102

Cited by 68 publications

(74 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Various choices of approximating functions are available: propagating plane waves [35,14,17,27], evanescent plane waves [29,45,46], Bessel functions [13], Green's functions [47]. Some elements of comparisons of these methods can be found in [48,36,49,19,37].…”

Section: Interpolation Basismentioning

confidence: 99%

“…While providing insight into the accuracy of the numerical schemes, this plane wave solution is not representative of realistic applications where the sound fields typically contain a wide range of wave directions. In addition, realistic problems can also involve evanescent waves in addition to propagating waves, and these can be difficult to capture by some numerical methods [36,13].…”

Section: Description Of the Test Casesmentioning

confidence: 99%

“…All the benchmark problems described above involve smooth solutions, but the performance of high-order schemes is known to deteriorate significantly when dealing with non-smooth solutions, including the p-FEM [53] and the wavebased DGM [36]. Therefore, the fourth test case involves a corner solution.…”

Section: Description Of the Test Casesmentioning

confidence: 99%

“…Therefore, the fourth test case involves a corner solution. This test case, shown in Figure 1(d), was also used in previous studies [36,9]. The analytical expression for the pressure field is given by…”

Section: Description Of the Test Casesmentioning

confidence: 99%

“…The downside is the calculation of the element matrices, which can be costly because it involves highly-oscillatory integrals with products of exponentials and polynomials. Elements of comparisons of PUFEM with the UWVF can be found in [36,37] and the UWVF is relatively better conditioned and generally more efficient.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

View full text Add to dashboard Cite

The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.

show abstract

Section: Interpolation Basismentioning

confidence: 99%

Section: Description Of the Test Casesmentioning

confidence: 99%

Section: Description Of the Test Casesmentioning

confidence: 99%

Section: Description Of the Test Casesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

Performance analysis of the ultra weak variational formulation to compute electromagnetic fields on nonuniform meshes

Gimpel

Silva

Afonso

2017

Int J Numerical Modelling

View full text Add to dashboard Cite

The purpose of this work is to apply a previously developed ultra weak variational formulation (UWVF) to the solution of the Helmholtz equation in electromagnetic scattering problems. The variational form is based on discontinuous Galerkin with plane wave basis functions in 2-dimensional finite element mesh. We present a general strategy, which allows to adjust the number of wave directions based on both the element size and the local conditioning number. This provides a cost-effective mechanism for application of UWVF using nonuniform meshes. Through simulation of scattering of a dielectric-coated circular cylinder, it is shown that the process significantly improves the UWVF performance. KEYWORDS electrically large domains, electromagnetic scattering, nonuniform mesh, plane wave basis, ultra weak variational formulation

show abstract

Iterative solvers for generalized finite element solution of boundary‐value problems

Mohamed

Seaı̈d

Bouhamidi

2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill‐conditioned linear systems that are associated with these methods. In this study, we investigate the performance of a class of iterative solvers for the generalized finite element solution of time‐dependent boundary‐value problems. A fully implicit time‐stepping scheme is used for the time integration in the finite element framework. As enrichment, we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers, we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared with dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix–vector products are involved in their implementation. Two test examples for boundary‐value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill‐conditioned linear systems resulting from the generalized finite element methods for time domain problems.

show abstract

Comparison of two wave element methods for the Helmholtz problem

Cited by 68 publications

References 24 publications

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Performance analysis of the ultra weak variational formulation to compute electromagnetic fields on nonuniform meshes

Iterative solvers for generalized finite element solution of boundary‐value problems

Contact Info

Product

Resources

About