Solution of Helmholtz equation by Trefftz method

Cheung, Y.K.; Jin, Wenhe; Zienkiewicz, O. C.

doi:10.1002/nme.1620320105

Cited by 74 publications

(46 citation statements)

References 14 publications

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“…This category of hybrid elements are also known as hybridTrefftz or Trefftz elements linked by boundary displacement or displacement-frame [6][7][8][9]. The underlying reasons are that the domain displacement is truncated from a Trefftz solution set, which is the basis of various Trefftz methods [11][12][13][14][15], and the boundary or frame variable u b linking adjacent elements is a displacement. With the displacement compatibility and boundary conditions satisfied by u b as a priori, the variational functional of the hybrid-displacement finite element method in the absence of body force can be expressed as [2][3][4]: …”

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confidence: 99%

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Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

Sze

Liu

2010

Comput Mech

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In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.

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confidence: 99%

“…Following the success of the Trefftz boundary element methods for Helmholtz problems [11][12][13][14][15][16], a number of Trefftz finite element methods can be noted in literature. A core issue of the Trefftz finite element methods is the enforcement of inter-element continuity of the domain or intra-element variable.…”

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confidence: 99%

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

Sze

Liu

2010

Comput Mech

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“…The application to wave propagation has been originally proposed by Zienkiewicz et al [11]. However, at this stage of development, the efficiency and the accuracy of the T(refftz)-element are still problemdependent.…”

Section: State Of the Artmentioning

confidence: 99%

“…High order approximations were then proposed, based on the finite element method (hp formulations) [20] or on meshless methods [8,26]. By now, everybody seems to agree that it is even more advantageous to incorporate information from the solution itself into the numerical discrete subspace, as performed for instance in the wave envelope approach, the Discontinuous Galerkin FEM [18], the Trefftz based approach [11,17], the Variational Theory of Complex Rays [27,28] or generalised FEM [5,12].…”

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Some challenges in computational vibro-acoustics: verification, validation and medium frequencies

Bouillard

Almeida²,

Decouvreur

et al. 2007

Comput Mech

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The simulation of the wave propagation phenomenon is one of the most challenging issue in the computational mechanics field. Many questions are still open among which we quote: the need for reliable error estimators, preferably in local or engineering quantities, the need for reliable updating methods with respect to material absorption parameters, and, finally, the need for more accurate numerical solutions, particularly to extend the frequency range that can be properly simulated by deterministic approaches. This paper gives an overview showing that the key issue to face all these challenges is the control of the pollution error, mainly caused by the dispersion effect.

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“…Trefftz collocation methods have essentially been applied to the case of acoustics, that is, the scalar Helmholtz case (see Li [11]). Galerkin type Trefftz methods have been applied to the acoustic case (see Pluymers et al [12], Cheung et al [13]) but also to the elastodynamic case (Freitas and Cismasiu [14]) amongst others.…”

Section: Introductionmentioning

confidence: 99%

Trefftz collocation for frequency domain elastodynamic problems

Leitão

Sensale

Rodriguez

2009

Mesh Reduction Methods

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A Trefftz collocation method is proposed for the analysis of two-dimensional elastodynamic problems subjected to steady-state time-harmonic loads. Trefftz methods are characterized by the use of a superposition of a number of actual solutions of the homogeneous part of governing equations, the so-called T(Trefftz)-functions. This leads to high quality approximations where the unknowns are the weights of each of the T-functions considered. The unknowns are found by matching the approximations to the boundary conditions by following a standard collocation approach.The numerical implementation of the method is briefly described and results are obtained for two anti-plane elastodynamic problems. The results compare quite favourably to other results available in the literature whereby Galerkin and collocation boundary element methods were used.

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Solution of Helmholtz equation by Trefftz method

Cited by 74 publications

References 14 publications

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

Some challenges in computational vibro-acoustics: verification, validation and medium frequencies

Trefftz collocation for frequency domain elastodynamic problems

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