Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.3, pp. 279-304, 2011, DOI:10.3970/cmes.2011.071.279

Abstract For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or F_i(x_j) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂F_i/∂x_j. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. … More >

Ying-Hsiu Shen¹, Chein-Shan Liu^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.2, pp. 157-178, 2011, DOI:10.3970/cmes.2011.071.157

Abstract When the local differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by requiring to solve the inversions of ill-posed matrices. Previously, when one uses (n-1)th order polynomial test functions to determine the weighting coefficients with n grid points, the resultant n ×n Vandermonde matrix is highly ill-conditioned and its inversion is hard to solve. Now we use (m-1)th order polynomial test functions by n grid points that the size of Vandermonde matrix is m×n, of which m is much less than n. We More >

S. Abbasbandy^1,2, V. Sladek³, A. Shirzadi¹, J. Sladek³

CMES-Computer Modeling in Engineering & Sciences, Vol.71, No.1, pp. 15-38, 2011, DOI:10.3970/cmes.2011.071.015

Abstract This paper deals with the development of a new method for solution of initial-boundary value problems governed by a couple of nonlinear diffusion equations occurring in the theory of self-organization in non-equilibrium systems. The time dependence is treated by linear interpolation using the finite difference method and the semi-discrete partial differential equations are considered in a weak sense by using the local integral equation method with approximating 2-d spatial variations of the field variables by the Moving Least Squares. The evaluation techniques are discussed and the applicability of the presented method is demonstrated on two More >

A. Grebennikov¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.15, No.3, pp. 85-90, 2010, DOI:10.3970/icces.2010.015.085

Abstract New General Ray (GR) method for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. GR-method consists in application of the Radon transform directly to the PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). The class of the PDE includes the Laplace, Poisson and Helmgoltz equations. GR-method presents the solution of the Dirichlet boundary value problem for this type of equations by explicit analytical formulas that use the direct and inverse Radon transform. Proposed version of GR-method justified theoretically, realized by fast algorithms and More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.3, pp. 301-324, 2010, DOI:10.3970/cmes.2010.059.301

Abstract When problems in engineering and science are discretized, algebraic equations appear naturally. In a recent paper by Liu and Atluri, non-linear algebraic equations (NAEs) were transformed into a nonlinear system of ODEs, which were then integrated by a method labelled as the Fictitious Time Integration Method (FTIM). In this paper, the FTIM is enhanced, by using the concept of arepellorin the theory ofnonlinear dynamical systems, to situations where multiple-solutions exist. We label this enhanced method as MSFTIM. MSFTIM is applied and illustrated in this paper through solving boundary-layer problems, boundary-value problems, and eigenvalue problems with More >

Caibin Zeng¹, Qigui Yang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.1, pp. 65-78, 2010, DOI:10.3970/cmes.2010.059.065

Abstract In this paper, a fractional order model is established to describe HIV internal viral dynamics involving HAART effect. First, the model is proved to possess non-negative solutions as desired in any population dynamics. Then, a detailed analysis is carried out to study the stability of equilibrium points. Numerical simulations are presented to illustrate the stability analysis. More >

Yunhua Li^1,2, Yunze Li³, Chieh-Li Chen⁴, Cha’o-Kuang Chen⁵

CMES-Computer Modeling in Engineering & Sciences, Vol.58, No.1, pp. 1-14, 2010, DOI:10.3970/cmes.2010.058.001

Abstract Addressing the identification problem of the general Lagrange multiplier in the He's variational iteration method, this paper proposes a new kind of method based on Duhamel's principle for the dynamic system response analysis. In this method, we have constructed an analytical iteration formula in terms of the convolution for the residual error at the nth iteration, and have given a new interpretation to He's variational iteration method. The analysis illustrates that the computational result of this method is equal to that of He's variational iteration method on the assumption of considering the impulse response of More >

Y. T. Gu¹, P. Zhuang², F. Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.3, pp. 303-334, 2010, DOI:10.3970/cmes.2010.056.303

Abstract Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation(FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by… More >

Chia-Cheng Tsai¹, Chein-Shan Liu², Wei-Chung Yeih³

CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.2, pp. 131-152, 2010, DOI:10.3970/cmes.2010.056.131

Abstract The fictitious time integration method (FTIM) previously developed by Liu and Atluri (2008a) is combined with the method of fundamental solutions and the Chebyshev polynomials to solve Poisson-type nonlinear PDEs. The method of fundamental solutions with Chebyshev polynomials (MFS-CP) is an exponentially-convergent meshless numerical method which is able to solving nonhomogeneous partial differential equations if the fundamental solution and the analytical particular solutions of the considered operator are known. In this study, the MFS-CP is extended to solve Poisson-type nonlinear PDEs by using the FTIM. In the solution procedure, the FTIM is introduced to convert More >

R. Martin¹, D. Komatitsch^1,2, S. D. Gedney³, E. Bruthiaux^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.1, pp. 17-42, 2010, DOI:10.3970/cmes.2010.056.017

Abstract Unsplit convolutional perfectly matched layers (CPML) for the velocity and stress formulation of the seismic wave equation are classically computed based on a second-order finite-difference time scheme. However it is often of interest to increase the order of the time-stepping scheme in order to increase the accuracy of the algorithm. This is important for instance in the case of very long simulations. We study how to define and implement a new unsplit non-convolutional PML called the Auxiliary Differential Equation PML (ADE-PML), based on a high-order Runge-Kutta time-stepping scheme and optimized at grazing incidence. We demonstrate More >