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  • Open Access

    ABSTRACT

    The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Optimization Problem

    H.F. Chan, C.M. Fan

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.19, No.1, pp. 29-30, 2011, DOI:10.3970/icces.2011.019.029

    Abstract The inverse boundary optimization problem, which is governed by Helmholtz equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the MCTM, one kind of boundary-type meshless methods, will be adopted in this study, since… More >

  • Open Access

    ABSTRACT

    A moving modified Trefftz method for inverse Laplace problems in two dimensional multiply-connected domain

    C.-L. Kuo, C.-S. Liu

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.11, No.3, pp. 85-86, 2009, DOI:10.3970/icces.2009.011.085

    Abstract In this paper, the inverse problems in a multiply connected domain governed by the Laplace equation have been investigated numerically by the developed moving modified Trefftz method. When solving the direct Laplace problem with the conventional Trefftz method, one may treat the ill-posed linear algebraic equations because the solution is obtained by expanding the diverging series; while when the inverse Laplace problem is encountered, it is more difficult to treat the more seriously ill-posed behaviors because the incomplete boundary data, and its solution, if exists, does not depend on the given boundary data continuously. Even many researchers have proposed lots… More >

  • Open Access

    ARTICLE

    On the use of a wave based prediction technique for steady-state structural-acoustic radiation analysis

    B. Pluymers1, W. Desmet1, D. Vandepitte1, P. Sas1

    CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 173-184, 2005, DOI:10.3970/cmes.2005.007.173

    Abstract Conventional element based methods for modelling structural-acoustic radiation problems are limited to low-frequency applications. Recently, a novel prediction technique has been developed based on the indirect Trefftz approach. This new wave based method is computationally more efficient than the element based methods and, as a consequence, can tackle problems also at higher frequencies. This paper discusses the basic principles of the new method and illustrates its performance for the two-dimensional radiation analysis of a bass-reflex loudspeaker. More >

  • Open Access

    REVIEW

    Trefftz Methods for Time Dependent Partial Differential Equations

    Hokwon A. Cho1, M. A. Golberg2, A. S. Muleshkov1, Xin Li1

    CMC-Computers, Materials & Continua, Vol.1, No.1, pp. 1-38, 2004, DOI:10.3970/cmc.2004.001.001

    Abstract In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then… More >

  • Open Access

    ARTICLE

    Development of 3D Trefftz Voronoi Cells with Ellipsoidal Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials

    Leiting Dong1, Satya N. Atluri11

    CMC-Computers, Materials & Continua, Vol.30, No.1, pp. 39-82, 2012, DOI:10.3970/cmc.2012.030.039

    Abstract In this paper, as an extension to the authors's work in [Dong and Atluri (2011a,b, 2012a,b,c)], three-dimensional Trefftz Voronoi Cells (TVCs) with ellipsoidal voids/inclusions are developed for micromechanical modeling of heterogeneous materials. Several types of TVCs are developed, depending on the types of heterogeneity in each Voronoi Cell(VC). Each TVC can include alternatively an ellipsoidal void, an ellipsoidal elastic inclusion, or an ellipsoidal rigid inclusion. In all of these cases, an inter-VC compatible displacement field is assumed at each surface of the polyhedral VC, with Barycentric coordinates as nodal shape functions. The Trefftz trial displacement fields in each VC are… More >

  • Open Access

    ARTICLE

    Development of 3D T-Trefftz Voronoi Cell Finite Elements with/without Spherical Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials

    L. Dong1, S. N. Atluri1

    CMC-Computers, Materials & Continua, Vol.29, No.2, pp. 169-212, 2012, DOI:10.3970/cmc.2012.029.169

    Abstract In this paper, three-dimensionalT-Trefftz Voronoi Cell Finite Elements (VCFEM-TTs) are developed for micromechanical modeling of heterogeneous materials. Several types of VCFEMs are developed, depending on the types of heterogeneity in each element. Each VCFEM can include alternatively a spherical void, a spherical elastic inclusion, a spherical rigid inclusion, or no voids/inclusions at all.In all of these cases, an inter-element compatible displacement field is assumed at each surface of the polyhedral element, with Barycentric coordinates as nodal shape functions.The T-Trefftz trial displacement fields in each element are expressed in terms of the Papkovich-Neuber solution. Spherical harmonics are used as the Papkovich-Neuber… More >

  • Open Access

    ARTICLE

    A Multiple-Precision Study on the Modified Collocation Trefftz Method

    Chia-Cheng Tsai1, Po-Ho Lin2

    CMC-Computers, Materials & Continua, Vol.28, No.3, pp. 231-260, 2012, DOI:10.3970/cmc.2012.028.231

    Abstract Recently, Liu (CMES 21(2007), 53) developed the modified collocation Trefftz method (MCTM) by setting a characteristic length slightly larger than the maximum radius of the computational domain. In this study, we find that the range of admissible characteristic length can be significantly enlarged if the LU decomposition is applied for solving the resulted dense unsymmetric matrix. Furthermore, we discover a range formula for admissible characteristic length, in which the number of the T-complete functions, the shape of the computation domain, and the exponent bits of the involved floating-point arithmetic have been taken into consideration. In order to validate the prescribed… More >

  • Open Access

    ARTICLE

    Simulation of Stress Concentration Problems by Hexahedral Hybrid-Trefftz Finite Element Models

    F.L.S. Bussamra1, E.Lucena Neto1, W.M. Ponciano1

    CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.3, pp. 255-272, 2014, DOI:10.3970/cmes.2014.099.255

    Abstract Hybrid-Trefftz stress finite elements have been applied with success to the analysis of linear and non-linear problems in structural mechanics. Two independent fields are approximated: stresses within the elements and displacements on their boundary. The stress field satisfies the Trefftz constraint a priori, i.e., it is extracted from the Navier equation solution. This type of element has provided remarkable improvement in stress predictions compared to the standard displacement-based finite elements. In this work, solution of stress concentration problems is carried out by hexahedral hybrid-Trefftz stress element models. Stress concentration factors and stress intensity factors are then identified and compared with… More >

  • Open Access

    ARTICLE

    On Solving Three-dimensional Laplacian Problems in a Multiply Connected Domain Using the Multiple Scale Trefftz Method

    Cheng-Yu Ku 1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.5, pp. 509-541, 2014, DOI:10.3970/cmes.2014.098.509

    Abstract This paper proposes the numerical solution of three-dimensional Laplacian problems in a multiply connected domain using the collocation Trefftz method with multiple source points. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 36 independent functions satisfying the governing equation in the cylindrical coordinate system. To deal with complicated problems for multiply connected domain, we adopted the generalized multiple source point boundary collocation Trefftz method which allows many source points in the Trefftz formulation without using the decomposition of the problem domain. In addition, to mitigate a severely ill-conditioned system of linear equations, this… More >

  • Open Access

    ARTICLE

    Error Analysis of Trefftz Methods for Laplace's Equations and Its Applications

    Z. C. Li2, T. T. Lu3, H. T. Huang4, A. H.-D. Cheng5

    CMES-Computer Modeling in Engineering & Sciences, Vol.52, No.1, pp. 39-82, 2009, DOI:10.3970/cmes.2009.052.039

    Abstract For Laplace's equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coefficients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Workshops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there… More >

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