J. Sladek¹, V. Sladek, Ch.Zhang²

Structural Durability & Health Monitoring, Vol.1, No.2, pp. 131-144, 2005, DOI:10.3970/sdhm.2005.001.131

Abstract A meshless method based on the local Petrov-Galerkin approach is proposed for crack analysis in two-dimensional (2-d), anisotropic and linear elastic solids with continuously varying material properties. Both quasi-static and transient elastodynamic problems are considered. For time-dependent problems, the Laplace-transform technique is utilized. A unit step function is used as the test function in the local weak-form. It is leading to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems. The analyzed domain is divided into small subdomains with a circular shape. The moving least-squares (MLS) method is adopted for approximating the physical… More >

N. Sukumar¹, J. E. Bolander¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.6, pp. 691-706, 2003, DOI:10.3970/cmes.2003.004.691

Abstract In this paper, we explore the numerical approximation of discrete differential operators on non-uniform grids. The Voronoi cell and the notion of natural neighbors are used to approximate the Laplacian and the gradient operator on irregular grids. The underlying weight measure used in the numerical computations is the {\em Laplace weight function}, which has been previously adopted in meshless Galerkin methods. We develop a difference approximation for the diffusion operator on irregular grids, and present numerical solutions for the Poisson equation. On regular grids, the discrete Laplacian is shown to reduce to the classical finite difference scheme. Two techniques to… 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 many researchers have proposed lots… More >

I. Yoshida ¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.6, No.3, pp. 195-200, 2008, DOI:10.3970/icces.2008.006.195

Abstract A priori information is discussed in order to overcome the ill-posedness of damage detection. We compared Gauss and Laplace distributions to express the uncertainties of a priori information. Uncertainty level of a priori and observation information is related to the balance of a priori and observation terms in the objective function. Maximum likelihood method is used to determine the balance adaptively. The method is examined through numerical simulations of identification problem to detect damage of a bridge based on coupling vibration with moving vehicles. More >

Chein-Shan Liu¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 57-68, 2007, DOI:10.3970/icces.2007.003.057

Abstract A new method is developed to solve the interior and exterior Dirichlet problems for the two-dimensional Laplace equation, namely the meshless regularized integral equation method (MRIEM), 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 very complex boundary shape and with boundary noise. More >

J. Sladek¹, V. Sladek¹, J. Krivacek¹, Ch. Zhang²

CMES-Computer Modeling in Engineering & Sciences, Vol.8, No.3, pp. 259-270, 2005, DOI:10.3970/cmes.2005.008.259

Abstract A meshless method based on the local Petrov-Galerkin approach is presented for stress analysis in three-dimensional (3-d) axisymmetric linear elastic solids with continuously varying material properties. The inertial effects are considered in dynamic problems. A unit step function is used as the test functions in the local weak-form. It is leading to local boundary integral equations (LBIEs). For transient elastodynamic problems the Laplace-transform technique is applied and the LBIEs are given in the Laplace-transformed domain. Axial symmetry of the geometry and the boundary conditions for a 3-d linear elastic solid reduces the original 3-d boundary value problem into a 2-d… More >

J. Sladek¹, V. Sladek¹, S.N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.5, pp. 477-490, 2004, DOI:10.3970/cmes.2004.006.477

Abstract A meshless method based on the local Petrov-Galerkin approach is proposed for solution of static and elastodynamic problems in a homogeneous anisotropic medium. The Heaviside step function is used as the test functions in the local weak form. It is leading to derive local boundary integral equations (LBIEs). For transient elastodynamic problems the Laplace transfor technique is applied and the LBIEs are given in the Laplace transform domain. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-d problems. The final form of local integral equations has a pure contour character only in… More >

S.Yu. Reutskiy¹

CMC-Computers, Materials & Continua, Vol.2, No.3, pp. 177-188, 2005, DOI:10.3970/cmc.2005.002.177

Abstract In this paper a new meshless method for eigenproblems with Laplace and biharmonic operators in simply and multiply connected domains is presented. The solution of an eigenvalue problem is reduced to a sequence of inhomogeneous problems with the differential operator studied. These problems are solved using the method of fundamental solutions. The method presented shows a high precision in simply and multiply connected domains. The results of the numerical experiments justifying the method are presented. More >

Chien-Ching Ma^1,2, Yi-Hsien Lin², Shih-Hao Lin²

CMC-Computers, Materials & Continua, Vol.31, No.1, pp. 37-64, 2012, DOI:10.3970/cmc.2012.031.037

Abstract In this article, the transient response in a functionally graded material (FGM) slab is analyzed by Laplace transform technique. The numerical Laplace inversion (Durbin's formula) is used to calculate the dynamic behavior of the FGM slab. The slab is subjected an uniform loading at the upper surface, and the lower surface are assumed to be traction-free or fixed conditions. The analytical solutions are presented in the transform domain and the numerical Laplace inversion is performed to obtain the transient response in time domain. To take the accuracy and computational efficiency in consideration, Durbin's method is suitable for calculating the long-time… More >

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 >