Yi-Hsien Lin¹, Chien-Ching Ma^1,2

CMC-Computers, Materials & Continua, Vol.24, No.1, pp. 15-42, 2011, DOI:10.3970/cmc.2011.024.015

Abstract In this article, explicit transient solutions for one-dimensional wave propagation behavior in multi-layered structures are presented. One of the objectives of this study is to develop an effective analytical method for constructing solutions in multilayered media. Numerical calculations are performed by three methods: the generalized ray method, numerical Laplace inversion method (Durbin's formula), and finite element method (FEM). The analytical result of the generalized ray solution for multilayered structures is composed of a matrix-form Bromwich expansion in the transform domain. Every term represents a group of waves, which are transmitted or reflected through the interface. More >

A.V. Ekhlakov², O.M. Khay², Ch. Zhang², J. Sladek³, V. Sladek³

Structural Durability & Health Monitoring, Vol.6, No.3&4, pp. 329-350, 2010, DOI:10.3970/sdhm.2010.006.329

Abstract In this paper, transient crack analysis in two-dimensional, isotropic, continuously non-homo -ge -neous and linear elastic functionally graded materials is presented. A boundary-domain element method based on boundary-domain integral representations is developed. The Laplace-transform technique is utilized to eliminate the dependence on time. Laplace-transformed fundamental solutions of linear coupled thermoelasticity for isotropic, homogeneous and linear elastic solids are applied to derive boundary-domain integral equations. The numerical implementation is performed by using a collocation method for the spatial discretization. The time-dependent numerical solutions are obtained by the Stehfest's inversion algorithm. For an edge crack in a More >

Weichung Yeih¹, Chein-Shan Liu², Chung-Lun Kuo³, Satya N. Atluri⁴

CMC-Computers, Materials & Continua, Vol.17, No.3, pp. 275-302, 2010, DOI:10.3970/cmc.2010.017.275

Abstract In this paper, a multiple-source-point boundary-collocation Trefftz method, with characteristic lengths being introduced in the basis functions, is proposed to solve the direct, as well as inverse Cauchy problems of the Laplace equation for a multiply connected domain. When a multiply connected domain with genus p (p>1) is considered, the conventional Trefftz method (T-Trefftz method) will fail since it allows only one source point, but the representation of solution using only one source point is impossible. We propose to relax this constraint by allowing many source points in the formulation. To set up a complete… More >

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… More >

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 More >

J. Sladek¹, V. Sladek¹, P.H. Wen², Y.C. Hon³

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.2, pp. 191-218, 2009, DOI:10.3970/cmes.2009.048.191

Abstract The meshless local Petrov-Galerkin (MLPG) method is used to solve the inverse heat conduction problem of predicting the distribution of the heat transfer coefficient on the boundary of 2-D and axisymmetric bodies. Using this method, nodes are randomly distributed over the numerical solution domain, and surrounding each of these nodes, a circular sub-domain is introduced. By choosing a unit step function as the test function, the local integral equations (LIE) on the boundaries of these sub-domains are derived. To eliminate the time variation in the governing equation, the Laplace transform technique is applied. The local… More >

Liviu Marin¹

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.2, pp. 121-154, 2009, DOI:10.3970/cmes.2009.048.121

Abstract We investigate the numerical implementation of the alternating iterative algorithm originally proposed by ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 in the case of the Cauchy problem for the two-dimensional Laplace equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. An efficient More >

Chih-Chang Chi¹, Weichung Yeih^1,2, Chein-Shan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.47, No.2, pp. 167-190, 2009, DOI:10.3970/cmes.2009.047.167

Abstract In this study, a novel method for solving the Cauchy problem of Laplace equation is developed. Through the fictitious time integration method (FTIM), the finding of the root of the resulting linear equations can be transformed into for finding the fixed point of a system of first order ordinary differential equations, in which a fictitious time variable is introduced. In such a sense, the inverse of ill-posed leading matrix is not necessary for the FTIM. This method uses the residual of each equation to control the evolution of unknowns in the fictitious time, and it More >

Chein-Shan Liu¹, Weichung Yeih², Satya N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.3, pp. 281-312, 2009, DOI:10.3970/cmes.2009.044.281

Abstract Here we develop a general purpose pre/post conditionerT, to solve an ill-posed system of linear equations,Ax=b. The conditionerTis obtained in the course of the solution of the Laplace equation, through a boundary-collocation Trefftz method, leading to:Ty=x, whereyis the vector of coefficients in the Trefftz expansion, andxis the boundary data at the discrete points on a unit circle. We show that the quality of the conditionerTis greatly enhanced by using multiple characteristic lengths (Multiple Length Scales) in the Trefftz expansion. We further show thatTcan be multiplicatively decomposed into a dilationTDand a rotationTR. For an odd-orderedA, we More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.3, pp. 253-276, 2009, DOI:10.3970/cmes.2009.043.253

Abstract Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a₀ + a₁x + ... + a_nxⁿ expanded by the monomials is the simplest one, which is easy to handle mathematically. For higher accuracy, one always attempts to use a higher-order polynomial as an interpolant. But, Runge gave a counterexample, demonstrating that the polynomial interpolation problem may be ill-posed. Very high-order polynomial interpolation is very hard to realize by numerical computations. In this paper we propose a… More >