Cheng-Yu Ku^1,2,3,Weichung Yeih^1,2, Chein-Shan Liu⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.76, No.2, pp. 83-108, 2011, DOI:10.3970/cmes.2011.076.083

Abstract In this paper, a general dynamical method based on the construction of a scalar homotopy function to transform a vector function of Non-Linear Algebraic Equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of a transformation matrix, the proposed general dynamical method can be transformed into several dynamical Newton-like methods including the Dynamical Newton Method (DNM), the Dynamical Jacobian-Inverse Free Method (DJIFM), and the Manifold-Based Exponentially Convergent Algorithm (MBECA). From the general dynamical method, we can also derive the conventional Newton method using a certain fictitious time-like function. The formulation presented… More >

Anil Misra¹, Shiping Huang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.52, No.2, pp. 197-216, 2009, DOI:10.3970/cmes.2009.052.197

Abstract The behavior of contact between solid bodies with rough surfaces under combined normal and shear loading remains a problem of interest in many areas of engineering. In this paper, we have utilized a micromechanical methodology to derive an expression of stress-displacement relationship applicable to combined normal and shear loading conditions. The micromechanical methodology considers the mechanics of asperity contacts and the interface roughness in terms of asperity height and asperity contact orientation distribution. A numerical procedure is implemented to evaluate the derived expressions under complex and mixed loading conditions using an incremental approach. We find that the proposed numerical procedure… More >

Xuefei He¹, Kian-Meng Lim^1,2,3, Siak-Piang Lim^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.35, No.1, pp. 21-48, 2008, DOI:10.3970/cmes.2008.035.021

Abstract The boundary element method (BEM) is known to have the advantage of reducing the dimension of problem by discretizing only the boundary of the domain. But it becomes less attractive for solving Poisson-type equations, due to the need to evaluate the domain integral which is computationally expensive. In this paper, we present the extension of a recently developed fast algorithm for Laplace equation, based on fast Fourier transform on multipoles (FFTM), to solve large scale 3D Poisson-type equations. We combined the Laplace solver with two fast methods for handling the domain integral based on fast Fourier transform (FFT). The first… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.1, pp. 17-42, 2013, DOI:10.3970/cmes.2013.091.017

Abstract In the present paper, we propose a novel two-side equilibration method to properly reduce the condition number of a given non-singular matrix only through a few operations. Then, two different conditioners together with the conjugate gradient method (CGM) are developed, which can overcome the defect of CGM, being not vulnerable to noisy disturbance exerted on an ill-posed linear system. The twoside CGM (TSCGM) and the pre-conditioning CGM (PrCGM) are convergent fast and accurate in solving linear inverse problems and the linear Hilbert problem under a large random noise. More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.80, No.1, pp. 57-86, 2011, DOI:10.3970/cmes.2011.080.057

Abstract Based-on the ordinary differential equations defined on an invariant manifold, we propose a theoretical procedure to derive a Relaxed Steepest Descent Method (RSDM) for numerically solving an ill-posed system of linear equations when the data are polluted by random noise. The invariant manifold is defined in terms of a squared-residual-norm and a fictitious time-like variable, and in the final stage we can derive an iterative algorithm including a parameter, which is known as the relaxation parameter. Through a Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation with faster… More >

Hsin-Ping Chu¹, Cheng-Ying Lo²

CMES-Computer Modeling in Engineering & Sciences, Vol.77, No.3&4, pp. 161-172, 2011, DOI:10.3970/cmes.2011.077.161

Abstract This paper presents the application of the differential transform method to solve strongly nonlinear equations with cubic nonlinearities and self-excitation terms. First, the equations are transformed by the differential transform method into the algebra equations in terms of the transformed functions. Secondly, the higher-order transformed functions are calculated in terms of other lower-order transformed functions through the iterative procedure. Finally, the solutions are approximated by the n-th partial sum of the infinite series obtained by the inverse differential transform. Two strongly nonlinear equations with different coefficients and initial conditions are given as illustrative examples. More >