T-elements: State of the art and future trends

Jiroušek, J.; Wroblewski, Adam C.

doi:10.1007/bf02818934

Cited by 88 publications

(80 citation statements)

References 62 publications

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“…These unknowns are then obtained from the approximate satisfaction of the boundary conditions by means of collocation or in a weighted residual sense. Trefftz-based formulations have been studied by several authors, such as Jirousek and co-workers (of which Jirousek and Wrobleski [5] give a thorough account), Cheung et al [6], Jin et al [7], Zielinski and Zienkiewicz [8], Zielinski and Herrera [9], and Freitas and Ji [10] amongst others. Reviews on the subject may be found in Zielinski [11], Kita and Kamiya [12] and Jirousek and Wrobleski [5].…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of thin circular plates by the indirect Trefftz method

Ghannadiasl

Noorzad

2009

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The vibration of plates is important in many applications pertaining to mechanical, civil and aerospace engineering. Therefore, the vibration of plates is an important research area that has been studied by many researchers. To date, there are abundant plate vibration solutions available in the open literature based on Kirchhoff assumptions. The purpose of the present paper is free vibration analysis of thin circular plates by the indirect Trefftz method. In thin plate vibration problems, we will deal with the governing equation with the homogeneous boundary conditions. The Trefftz method basically employs the complete set of solutions satisfying the governing equation as the first step. To derive the boundary integral equation, either the reciprocity law, which is similar to those used in conventional BEMs, or the weight residual method can be used. The proposed approach has some merits when compared with other regular boundary element formulations reported so far. A main benefit for the Trefftz method is that it does not involve singular integrals due to the properties of its solution basis functions (T functions); thus, it can be categorized into the regular boundary element method. Besides, this advocated approach yields a solution that offers simultaneously the advantages of the classical FEM and BEM solutions, without having their drawbacks. Finally, several numerical examples are demonstrated to show the validity of the current approach.

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Section: Introductionmentioning

confidence: 99%

Free vibration analysis of thin circular plates by the indirect Trefftz method

Ghannadiasl

Noorzad

2009

Mesh Reduction Methods

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“…The unknown coefficients are determined by satisfying the boundary condition with the approximate solution. Many applications to the Helmholtz equation [4], the Navier equation [15,17] and the biharmonic equation [6] had been done. Until recent years, the ill-posed nature in the method was noticed [8,19] increasingly.…”

Section: Introductionmentioning

confidence: 99%

Adaptive error estimation of the Trefftz method for solving the Cauchy problem

Chen

Lee³

et al. 2007

WIT Transactions on Modelling and Simulation, Vol 44

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In this paper, the Laplace problem with overspecified boundary conditions is investigated by using the Trefftz method. The main difficulty will appear an obvious divergent result when the boundary condition on an overspecified boundary contaminates artificial errors. The occurring mechanism of the unreasonable result originates from an ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method respectively, to reconstruct the influence matrix. The optimal parameters of the Tikhonov technique and linear regularization method are determined by adopting the adaptive error estimation technique. The numerical evidence of the Trefftz method is given to verify the accuracy of the solutions after comparison with the results of analytical solution and to demonstrate the validity and instructions of the proposed adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the example.

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“…A main bene"t for the Tre!tz method is that it does not involve singular integrals due to its solution basis functions (T functions); thus, it can be categorized into the regular boundary element method. A review article about the Tre!tz method can be found in reference [28]. The above-mentioned boundary element formulations are all regular; however, they are not complete formulations as we will explain this later in this paper.…”

Section: Introductionmentioning

confidence: 99%

Applications of the Generalized Singular-Value Decomposition Method on the Eigenproblem Using the Incomplete Boundary Element Formulation

Kuo

Yeih

2000

Journal of Sound and Vibration

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In this paper, four incomplete boundary element formulations, including the real-part singular boundary element, the real-part hypersingular boundary element, the imaginary-part boundary element and the plane-wave element methods, are used to solve the free vibration problem. Among these incomplete boundary element formulations, the real-part singular and the hypersingular boundary elements are of the singular type and the other two are of the regular type. When the incomplete formulation is used, the spurious eigensolution may be encountered. An auxiliary system, whose boundary conditions are linearly independent of those of the original system, is required in the proposed method in order to eliminate the spurious eigensolution. A mathematical proof is given to show that the spurious eigensolution will appear in both the original and auxiliary systems. As a result, one can eliminate spurious eigensolution by means of the generalized singular-value decomposition method. In addition to the spurious eigensolution problem, the regular boundary element formulation further su!ers ill-conditioned behaviors in the problem solver. It is explained analytically using a circular domain why ill-conditioned behaviors exist. Two main ill-conditioned problems are the numerical instability of solution and the nonexistence of solution. To solve the numerical instability of solution problem existing in these proposed regular formulations, the Tikhonov's regularization technique is used to improve the condition number of the leading coe$cient matrix; further, the generalized singular-value decomposition method is used to eliminate spurious eigensolutions. Numerical examples are given to verify the performance of these proposed methods, and the results match the analytical values well.

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T-elements: State of the art and future trends

Cited by 88 publications

References 62 publications

Free vibration analysis of thin circular plates by the indirect Trefftz method

Free vibration analysis of thin circular plates by the indirect Trefftz method

Adaptive error estimation of the Trefftz method for solving the Cauchy problem

Applications of the Generalized Singular-Value Decomposition Method on the Eigenproblem Using the Incomplete Boundary Element Formulation

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