Solution of the three‐dimension wave equation by using wave polynomials

Macig, Artur; Wauer, Jörg

doi:10.1002/pamm.200410334

Cited by 5 publications

(3 citation statements)

References 3 publications

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“…For the embedded Trefftz DG method we plot also the time spent finding the local kernels of the operator using SVD, as described in Section 3.4. Note that in the case of the embedded Trefftz DG method the solving step includes the matrix multiplications needed in (7), therefore the time spent is not equal to that of the standard Trefftz method. In 2d performing the SVD sequentially is time consuming, as shown in Figure 2, a QR decomposition could improve the performance.…”

Section: F I G U R Ementioning

confidence: 99%

“…Polynomial Trefftz functions have been obtained for several linear partial differential operators with constant coefficients such as Laplace equation, 5,6 acoustic wave equation, 7‐9 heat equation, 10 plate vibration and beam vibration equation, 11,12 and time‐dependent Maxwell's equation 13 . Efforts to generate Trefftz polynomials in a general case have been undertaken, see References 14 and 15.…”

Section: Introductionmentioning

confidence: 99%

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Embedded Trefftz discontinuous Galerkin methods

Lehrenfeld

Stocker

2023

Numerical Meth Engineering

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In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz‐type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non‐constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.

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Section: F I G U R Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Embedded Trefftz discontinuous Galerkin methods

Lehrenfeld

Stocker

2023

Numerical Meth Engineering

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“…The first paper devoted to the Trefftz functions in which the time is considered as a continuous variable, discussed a one-dimensional (one spatial variable) heat conduction equation (Rosenbloom and Widder 1956). This aspect of the Trefftz functions method was developed for the heat conduction problems in the papers (Ciałkowski et al 1999(Ciałkowski et al , 2007Yano et al 1983) for the wave equation and thermoelasticity problems in the papers (Grysa and Maciag 2011;Maciag 2004Maciag , 2005Maciag , 2007Maciag and Wauer 2005a, b) and for the equation of a plate vibration in the paper . So far, also monographs concerning the Trefftz method have been published (Ciałkowski and Frackowiak 2000;Grysa 2010; Kołodziej and Zieliński 2009;Li et al 2008;Maciag 2009;Qing-Hua 2000).…”

Section: Introductionmentioning

confidence: 99%

Solution of the direct and inverse problems for beam

Maciąg

Pawińska

2014

Comp. Appl. Math.

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The article presents an approximate method of solving direct and inverse problems described by Bernoulli-Euler inhomogeneous equation of vibrations of a beam. A semianalytical solution is approximated by a linear combination of the Trefftz functions (T-functions, solving functions), which satisfies identically the homogenous equation describing the vibrations of a beam. In the paper, the properties of the solving functions have been investigated, theorems concerning their linear independence have been formulated and proved. A method of obtaining the particular solution of the inhomogeneous equation has been shown. To get this solution, recurrent formulas enabling us to determine the inverse operator for monomials have been derived. The paper discusses two kinds of inverse problems. The first one is a boundary inverse problem, in which the boundary conditions are to be determined, based on known displacements within the area. In the second one, the load on the beam needs to be found (identification of the source). The solving functions can be used as a finite element method base functions. This approach is tested for solving inverse problems. The paper includes examples which illustrate the usefulness of the method.

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The usage of wave polynomials in solving direct and inverse problems for two-dimensional wave equation

Maciąg¹

2009

Int. J. Numer. Meth. Biomed. Engng.

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SUMMARYThe paper presents a new relatively simple yet very effective method to obtain an approximate solution of the direct and inverse problems for two-dimensional wave equation (two space variables and time). Such a equation describes, for example, the vibration of a membrane. To obtain an approximate solution, the wave polynomials (Trefftz functions for wave equation) were used. It is shown how to get these polynomials and their derivatives. The method of solving the functions is described and it is proved that the approximation error decreases when taking more polynomials in approximation. A new approach for solving 2D direct and inverse problems of elasticity is described. In order to improve the quality of the solution, a physical regularization was proposed. Moreover, the paper shows a new technique of smoothing the noisy data by using wave polynomials. The quality of the approximate solutions was verified on test examples. In these cases the direct and inverse problems were taken into consideration.

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Solution of the three‐dimension wave equation by using wave polynomials

Cited by 5 publications

References 3 publications

Embedded Trefftz discontinuous Galerkin methods

Embedded Trefftz discontinuous Galerkin methods

Solution of the direct and inverse problems for beam

The usage of wave polynomials in solving direct and inverse problems for two-dimensional wave equation

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