Chein-Shan Liu¹, Jian-Hung Shen², Chung-Lun Kuo¹, Yung-Wei Chen^2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.2, pp. 1317-1335, 2024, DOI:10.32604/cmes.2023.030618 - 29 January 2024

Abstract This study sets up two new merit functions, which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems. For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less, where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector. 1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and… More >

I-Yao CHAN, Chein-Shan Liu, Weichung Yeih

The International Conference on Computational & Experimental Engineering and Sciences, Vol.20, No.2, pp. 53-54, 2011, DOI:10.3970/icces.2011.020.053

Abstract In this study, we adopt the fictitious time integration method to treat the image reconstruction problem. The distorted image is considered as a result of diffused data from the initial perfect image by using a nonlinear diffusion equation. The image reconstruction problem then becomes an inverse problem by using the data in the final time to recover the data in the initial time. This inverse problem is known as the backward in time nonlinear diffusion problem which is highly ill-posed. We propose to use the fictitious time integration method to tackle this highly ill-posed image More >

Chein-Shan Liu

The International Conference on Computational & Experimental Engineering and Sciences, Vol.16, No.2, pp. 53-54, 2011, DOI:10.3970/icces.2011.016.053

Abstract We consider an inverse problem for identifying a time-space-dependent heat transfer coefficient h(x,t) in a two-dimensional heat conduction equation, with the aid of an extra measurement of temperature at the top side of a rectangular plate. Finite differences are used to discretize the governing equation and boundary conditions of Neumann type, and then the Fictitious Time Integration Method (FTIM) is used to solve a large scale linear system of unknown variables. The numerical results show that the FTIM is effective and robust against noise. More >

Chih-Wen Chang

The International Conference on Computational & Experimental Engineering and Sciences, Vol.16, No.1, pp. 19-20, 2011, DOI:10.3970/icces.2011.016.019

Abstract In this study, we propose a new numerical approach for solving the nonhomogeneous backward heat conduction problems (BHCPs). A fictitious time I" is used to transform the dependent variable u(x, t) into a new one by (1+I")u(x, t)=: v(x, t, I"), such that the original nonhomogeneous heat conduction equation is written as a new parabolic type partial differential equation in the space of (x, t, I"). Besides, a fictitious viscous damping coefficient can be employed to strengthen the stability of numerical integration of the discretized equations by utilizing a group preserving scheme. Several numerical instances More >

Chein-Shan Liu^1,2, Hong-Hua Dai¹, Satya N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.2, pp. 195-228, 2011, DOI:10.3970/cmes.2011.081.195

Abstract In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x^· = λ[αR + βP], where P = F − R(F·R) / ||R||² such that P·R = 0; or x^· = λ[αF + βP^∗], where P^∗ = R − F(F·R) / ||F||² such that P^*·F = 0. From these evolution More >

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 >

Chih-Wen Chang¹

CMC-Computers, Materials & Continua, Vol.21, No.2, pp. 87-106, 2011, DOI:10.3970/cmc.2011.021.087

Abstract We address a new numerical approach to deal with these multi-dimensional backward wave problems (BWPs) in this study. A fictitious time τ is utilized to transform the dependent variable u(x, y, z, t) into a new one by (1+τ)u(x, y, z, t)=: v(x, y, z, t, τ), such that the original wave equation is written as a new hyperbolic type partial differential equation in the space of (x, y, z, t, τ). Besides, a fictitious viscous damping coefficient can be employed to strengthen the stability of numerical integration of the discretized equations by using a group preserving scheme. 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 >

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 >