Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.73, No.4, pp. 395-432, 2011, DOI:10.3970/cmes.2011.073.395

Abstract In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x^· = λ[αF + (1−α)B^TF], where λ and α are scalars and B_ij = ∂F_i/∂x_j. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian… 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 >

Hong-Hua Dai^1,2, Jeom Kee Paik³, S. N. Atluri²

CMC-Computers, Materials & Continua, Vol.23, No.2, pp. 155-186, 2011, DOI:10.3970/cmc.2011.023.155

Abstract The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin… More >

Hong-Hua Dai^1,2, Jeom Kee Paik³, Satya N. Atluri²

CMC-Computers, Materials & Continua, Vol.23, No.1, pp. 69-100, 2011, DOI:10.3970/cmc.2011.023.069

Abstract In this paper, the global nonlinear Galerkin method is used to perform an accurate and efficient analysis of the large deflection behavior of a simply-supported rectangular plate under combined loads. Through applying the Galerkin method to the governing nonlinear partial differential equations (PDEs) of the plate, we derive a system of coupled third order nonlinear algebraic equations (NAEs). However, the resultant system of NAEs is thought to be hard to tackle because one has to find the one physical solution from among the possible multiple solutions. Therefore, a suitable initial guess is required to lead… More >

Chia-Ming Fan^1,2, Chein-Shan Liu³, Weichung Yeih¹, Hsin-Fang Chan¹

CMC-Computers, Materials & Continua, Vol.15, No.1, pp. 67-86, 2010, DOI:10.3970/cmc.2010.015.067

Abstract In this study, the nonlinear obstacle problems, which are also known as the nonlinear free boundary problems, are analyzed by the scalar homotopy method (SHM) and the finite difference method. The one- and two-dimensional nonlinear obstacle problems, formulated as the nonlinear complementarity problems (NCPs), are discretized by the finite difference method and form a system of nonlinear algebraic equations (NAEs) with the aid of Fischer-Burmeister NCP-function. Additionally, the system of NAEs is solved by the SHM, which is globally convergent and can get rid of calculating the inverse of Jacobian matrix. In SHM, by introducing More >

Chein-Shan Liu¹, Weichung Yeih², Chung-Lun Kuo³, Satya N. Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.1, pp. 47-72, 2009, DOI:10.3970/cmes.2009.053.047

Abstract Iterative algorithms for solving a system of nonlinear algebraic equations (NAEs): F_i(x_j) = 0, i, j = 1,... ,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one to solve the NAEs, due to the ease of its numerical implementation. However, this type of algorithm is sensitive to the initial guess of solution, and is expensive in terms of the computations of the Jacobian matrix ∂F_i/∂x_j and its inverse at each iterative step. In addition, the Newton-like methods restrict one to construct an iteration procedure for n-variables… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.31, No.2, pp. 71-84, 2008, DOI:10.3970/cmes.2008.031.071

Abstract Iterative algorithms for solving a nonlinear system of algebraic equations of the type: F_i(x_j) = 0, i,j = 1,…,n date back to the seminal work of Issac Newton. Nowadays a Newton-like algorithm is still the most popular one due to its easy numerical implementation. However, this type of algorithm is sensitive to the initial guess of the solution and is expensive in the computations of the Jacobian matrix ∂ F_i/ ∂ x_j and its inverse at each iterative step. In a time-integration of a system of nonlinear Ordinary Differential Equations (ODEs) of the type B_ijx_j + F_i = 0… More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.8, No.2, pp. 53-66, 2008, DOI:10.3970/cmc.2008.008.053

Abstract Proposed is a time-marching algorithm to solve a nonlinear system of complementarity equations: P_i(x_j) ≥ 0, Q_i(x_j) ≥ 0 , P_i(x_j)Q_i(x_j) = 0, i, j = 1,...,n, resulting from a discretization of nonlinear obstacle problem. We transform the above nonlinear complementarity problem (NCP) into a nonlinear algebraic equations (NAEs) system: F_i(x_j) = 0 with the aid of the Fischer-Burmeister NCP-function. Such NAEs are semi-smooth, highly nonlinear and usually implicit, being hard to handle by the Newton-like method. Instead of, a first-order system of ODEs is derived through a fictitious time equation. The time-stepping equations are obtained by applying More >