Cheng-Yu Ku^1,2, Weichung Yeih^1,2, Chein-Shan Liu³, Chih-Chang Chi²

CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.2, pp. 173-190, 2009, DOI:10.3970/cmes.2009.043.173

Abstract In this paper, a new time-like function with the incorporation of the fictitious time integration method (FTIM) is proposed. The new time-like function is modified from the original time-like function in the FTIM by adding a control parameter m, which dramatically improves the performance of the FTIM for solving highly nonlinear boundary value problems (BVPs) and plays as an important controller to assure the convergence of the solution during the time integration process. The requirements and the characteristics of the new time-like function are presented by examining the FTIM through the perspective of the new time-like function in which the… More >

Jin Li¹, Ji-ming Wu², De-hao Yu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.42, No.2, pp. 151-176, 2009, DOI:10.3970/cmes.2009.042.151

Abstract The trapezoidal rule for the computation of Hadamard finite-part integrals in boundary element methods is discussed, and the asymptotic expansion of error function is obtained. A series to approach the singular point is constructed and the convergence rate is proved. Based on the asymptotic expansion of the error functional, algorithm with theoretical analysis of the generalized extrapolation are given. Some examples show that the numerical results coincide with the theoretic analysis very well. More >

Rencheng Song^1,2, Wen Chen^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.42, No.1, pp. 59-70, 2009, DOI:10.3970/cmes.2009.042.059

Abstract The regularized meshless method (RMM) is a novel boundary-type meshless method but by now has mainly been tested successfully to the regular domain problems in reports. This note makes a further investigation on its solution of irregular domain problems. We find that the method fails to produce satisfactory results for some benchmark problems. The reason is due to the inaccurate calculation of the diagonal elements of the numerical discretization matrix in the original RMM, which have strong effect on the resulting solution accuracy. To overcome this severe drawback, this study introduces the weighted diagonal element approach. Our numerical experiments demonstrate… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.3, pp. 243-262, 2009, DOI:10.3970/cmes.2009.041.243

Abstract The Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri (2008a) is employed here to solve a system of ill-posed linear algebraic equations, which may result from the discretization of a first-kind linear Fredholm integral equation. We rationalize the mathematical foundation of the FTIM by relating it to the well-known filter theory. For the linear ordinary differential equations which are obtained through the FTIM (and which are equivalently used in FTIM to solve the ill-posed linear algebraic equations), we find that the fictitous time plays the role of a regularization parameter, and its filtering effect is better than… More >

E. J. Sellountos¹, A. Sequeira¹, D. Polyzos²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.3, pp. 215-242, 2009, DOI:10.3970/cmes.2009.041.215

Abstract A Meshless Local Petrov-Galerkin (MLPG) method based on Local Boundary Integral Equation (LBIE) techniques is employed here for the solution of transient elastic problems with damping. The Radial Basis Functions (RBF) interpolation scheme is exploited for the meshless representation of displacements throughout the computational domain. On the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary interpolation functions. The MLPG(LBIE)/RBF method is applied to both transient and steady-state Fourier transform elastodynamic domains. In both cases the LBIEs employ the simple elastostatic fundamental solution instead of the complicated time and frequency dependent… More >

C. L. Tan¹, Y.C. Shiah², C.W. Lin²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.3, pp. 195-214, 2009, DOI:10.3970/cmes.2009.041.195

Abstract The explicit, closed-form expressions of the Green's functions for generally anisotropic elastic solids in three-dimensions that have been derived using Stroh's formalism are employed in a formulation of the boundary element method (BEM). Unlike several other existing schemes, the evaluation of these fundamental solutions does not require further numerical integration in the BEM algorithm; they have surprisingly not been implemented previously. Three numerical examples are presented to demonstrate the veracity of the implementation and the general applicability of the BEM for the 3D elastic stress analysis of generally anisotropic solids. The results are compared with known solutions in the literature… More >

Hung-Chang Lee¹, Chein-Shan Liu²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.1, pp. 1-26, 2009, DOI:10.3970/cmes.2009.041.001

Abstract The group-preserving schemes developed by Liu (2001) for integrating ordinary differential equations system were adopted the Cayley transform and Padé approximants to formulate the Lie group from its Lie algebra. However, the accuracy of those schemes is not better than second-order. In order to increase the accuracy by employing the group-preserving schemes on ordinary differential equations, according to an efficient technique developed by Runge and Kutta to raise the order of accuracy from the Euler method, we combine the Runge-Kutta method on the group-preserving schemes to obtain the higher-order numerical methods of group-preserving type. They provide single-step explicit time integrators… More >

Sergia Colli¹, Massimiliano Gei¹, Davide Bigoni^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.40, No.1, pp. 29-62, 2009, DOI:10.3970/cmes.2009.040.029

Abstract Incremental plane strain deformations superimposed upon a uniformly stressed and deformed nonlinear elastic (compressible) body are treated by developing {\it ad hoc} boundary integral equations that, discretized, lead to a novel boundary element technique. The approach is a generalization to compressible elasticity of results obtained by Brun, Capuani, and Bigoni (2003, Comput. Methods Appl. Mech. Engrg. 192, 2461-2479), and is based on a Green's function here obtained through the plane-wave expansion method. New expressions for Green's tractions are determined, where singular terms are solved in closed form, a feature permitting the development of a optimized numerical code. An application of… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.40, No.1, pp. 1-28, 2009, DOI:10.3970/cmes.2009.040.001

Abstract We consider an inverse problem of Laplace equation by recoverning boundary value on the inner circle of a two-dimensional annulus from the overdetermined data on the outer circle. The numerical results can be used to determine the Robin coefficient or crack's position inside a disk from the measurements of Cauchy data on the outer boundary. The Fourier series is used to formulate the first kind Fredholm integral equation for the unknown data f(θ) on the inner circle. Then we consider a Lavrentiev regularization, by adding an extra term αf(θ) to obtain the second kind Fredholm integral equation. The termwise separable… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.39, No.2, pp. 125-154, 2009, DOI:10.3970/cmes.2009.039.125

Abstract We propose a new numerical method for solving the boundary value problems of ordinary differential equations (ODEs) under multipoint boundary conditions specified at t = T_i, i = 1,...,m, where T₁ < ... < T_m. The finite difference scheme is used to approximate the ODEs, which together with the m-point boundary conditions constitute a system of nonlinear algebraic equations (NAEs). Then a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the new approach is highly accurate and efficient with a fast convergence. The FTIM can also be used to find the periods of… More >