J. Sladek¹, V. Sladek¹, Ch. Zhang², P. Solek³

CMES-Computer Modeling in Engineering & Sciences, Vol.22, No.3, pp. 217-234, 2007, DOI:10.3970/cmes.2007.022.217

Abstract A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of boundary value problems for coupled thermo-electro-mechanical fields. Transient dynamic governing equations are considered here. To eliminate the time-dependence in these equations, the Laplace-transform technique is applied. Material properties of piezoelectric materials are influenced by a thermal field. It is leading to an induced nonhomogeneity and the governing equations are more complicated than in a homogeneous counterpart. Two-dimensional analyzed domain is subdivided into small circular subdomains surrounding nodes randomly spread over the whole domain. A unit step function is used as the test functions in the… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.1, pp. 53-66, 2007, DOI:10.3970/cmes.2007.021.053

Abstract A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equation, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accurate as one desired. Then, we use the collocation method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor… More >

Chein-Shan Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.20, No.2, pp. 111-122, 2007, DOI:10.3970/cmes.2007.020.111

Abstract A highly accurate new solver is developed to deal with interior and exterior mixed-boundary value problems for two-dimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circular artificial boundary which is uniquely determined by the physical problem domain, and derive a Dirichlet to Robin mapping on that artificial circle, which is an exact boundary condition described by the first kind Fredholm integral equation. As a consequence, we obtain a modified Trefftz method equipped with a characteristic length factor, ensuring that the new solver is stable because the condition number can be greatly reduced. Then,… More >

D.L. Young¹, K.H. Chen², J.T. Chen³, J.H. Kao⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.19, No.3, pp. 197-222, 2007, DOI:10.3970/cmes.2007.019.197

Abstract A boundary-type method for solving the Laplace problems using the modified method of fundamental solutions (MMFS) is proposed. The present method (MMFS) implements the singular fundamental solutions to evaluate the solutions, and it can locate the source points on the real boundary as contrasted to the conventional MFS, where a fictitious boundary is needed to avoid the singularity of diagonal term of influence matrices. The diagonal term of influence matrices for arbitrary domain can be novelly determined by relating the MFS with the indirect BEM and are also solved for circular domain analytically by using separable kernels and circulants. The… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.19, No.2, pp. 145-162, 2007, DOI:10.3970/cmes.2007.019.145

Abstract A new method is developed to solve the Dirichlet problems for the two-dimensional Laplace equation in the doubly-connected domains, namely the meshless regularized integral equations method (MRIEM), which consists of three portions: Fourier series expansion, the Fredholm integral equations, and linear equations to determine the unknown boundary conditions onartificial circles. The boundary integral equations on artificial circles are singular-free and the kernels are degenerate. When boundary-type methods are inefficient to treat the problems with complicated domains, the new method can be applicable for such problems. The new method by using the Fourier series and the Fourier coefficients can be adopted… More >

Chein-Shan Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.19, No.1, pp. 99-110, 2007, DOI:10.3970/cmes.2007.019.099

Abstract A new meshless regularized integral equation method (MRIEM) is developed to solve the interior and exterior Dirichlet problems for the two-dimensional Laplace equation, which consists of three parts: Fourier series expansion, the second kind Fredholm integral equation and an analytically regularized solution of the unknown boundary condition on an artificial circle. We find that the new method is powerful even for the problem with complex boundary shape and with random noise disturbing the boundary data. More >

A. J. Davies¹, D. Crann¹, S. J. Kane¹, C-H. Lai²

CMES-Computer Modeling in Engineering & Sciences, Vol.18, No.2, pp. 79-86, 2007, DOI:10.3970/cmes.2007.018.079

Abstract The solution process for diffusion problems usually involves the time development separately from the space solution. A finite difference algorithm in time requires a sequential time development in which all previous values must be determined prior to the current value. The Stehfest Laplace transform algorithm, however, allows time solutions without the knowledge of prior values. It is of interest to be able to develop a time-domain decomposition suitable for implementation in a parallel environment. One such possibility is to use the Laplace transform to develop coarse-grained solutions which act as the initial values for a set of fine-grained solutions. The… More >

J. Sladek¹, V. Sladek¹, S.N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.3, pp. 309-318, 2004, DOI:10.3970/cmes.2004.006.309

Abstract Meshless methods based on the local Petrov-Galerkin approach are proposed for solution of steady and transient heat conduction problem in a continuously nonhomogeneous anisotropic medium. Fundamental solution of the governing partial differential equations and the Heaviside step function are used as the test functions in the local weak form. It is leading to derive local boundary integral equations which are given in the Laplace transform domain. The analyzed domain is covered by small subdomains with a simple geometry. To eliminate the number of unknowns on artificial boundaries of subdomains the modified fundamental solution and/or the parametrix with a convenient cut-off… More >

B. Jin^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.3, pp. 253-262, 2004, DOI:10.3970/cmes.2004.006.253

Abstract In this paper, we propose a new numerical scheme for the solution of the Laplace and biharmonic equations subjected to noisy boundary data. The equations are discretized by the method of fundamental solutions. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained using the truncated singular value decomposition, with the regularization parameter given by the L-curve method. Numerical experiments show that the method is stable with respect to the noise in the data, highly accurate and computationally very efficient. More >

M.Ganesh¹, D. Sheen²

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.2, pp. 183-194, 2001, DOI:10.3970/cmes.2001.002.183

Abstract A parallel numerical algorithm is introduced and analyzed for solving inhomogeneous initial-boundary value parabolic problems. The scheme is based on the method recently introduced in Sheen, Sloan, and Thomée (2000) for homogeneous problems. We give a method based on a suitable choice of multiple parameters. Our scheme allows one to compute solutions in a wide range of time. Instead of using a standard time-marching method, which is not easily parallelizable, we take the Laplace transform in time of the parabolic problems. The resulting elliptic problems can be solved in parallel. Solutions are then computed by a discrete inverse Laplace transformation.… More >