Weichung Yeih^1,2, I-Yao Chan¹, Cheng-Yu Ku¹, Chia-Ming Fan¹, Pai-Chen Guan³

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.2, pp. 123-149, 2014, DOI:10.3970/cmes.2014.099.123

Abstract In this paper, a novel double iteration process for solving the nonlinear algebraic equations is developed. In this process, the outer iteration controls the evolution path of the unknown vector x in the selected direction u which is determined from the inner iteration process. For the inner iteration, the direction of evolution u is determined by solving a linear algebraic equation: B^TBu = B^TF where B is the Jacobian matrix, F is the residual vector and the superscript ''T'' denotes the matrix transpose. For an ill-posed system, this linear algebraic equation is very difficult to solve since the resulting… More >

T.A. Elgohary¹, L. Dong², J.L. Junkins³, S.N. Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.6, pp. 543-563, 2014, DOI:10.3970/cmes.2014.098.543

Abstract In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of [Kondoh and Atluri (1986)]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton’s method (and… More >

D. Ishihara¹, T. Horie¹

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.6, pp. 421-440, 2014, DOI:10.3970/cmes.2014.101.421

Abstract In this study, a projection method for the monolithic interaction system of an incompressible fluid and a structure using a new algebraic splitting is proposed. The proposed method splits the monolithic equation system into the equilibrium equations and the pressure Poisson equation (PPE) algebraically using the intermediate velocity in the nonlinear iterations. Since the proposed equilibrium equation satisfies the interface condition, the proposed method is strongly coupled. Moreover, the proposed PPE enforces the incompressibility constraint. Different from previous studies, the proposed algebraic splitting never generates any Schur complement. The proposed method is applied to a channel with a flexible flap,… More >

George A. Gravvanis¹, Christos K. Filelis-Papadopoulos¹, Paschalis I.Matskanidis¹

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.4, pp. 323-345, 2014, DOI:10.3970/cmes.2014.100.323

Abstract Since the introduction of the Algebraic MultiGrid algorithm (AMG) over twenty years ago, significant progress has been made in improving the coarsening and the convergence behavior of the method. In this paper, an AMG method is introduced that utilizes a new generic approximate inverse algorithm as a smoother in conjunction with common coarsening techniques, such as classical Ruge-Stüben coarsening, CLJP and PMIS coarsening. The proposed approximate inverse scheme, namely Generic Approximate Banded Inverse (GenAbI), is a banded approximate inverse based on Incomplete LU factorization with zero fill–in (ILU(0)). The new class of Generic Approximate Banded Inverse can be computed for… More >

Cheng-Yu Ku¹

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.6, pp. 409-434, 2013, DOI:10.3970/cmes.2013.096.409

Abstract This paper proposes a novel method, named the dynamical Jacobianinverse free method (DJIFM), with the incorporation of a two-sided equilibrium algorithm for solving ill-conditioned systems of linear equations with extreme physical property contrasts. The DJIFM is based on the construction of a scalar homotopy function for transforming the vector function of linear or nonlinear algebraic equations into a time-dependent scalar function by introducing a fictitious time-like variable. The DJIFM demonstrated great numerical stability for solving linear or nonlinear algebraic equations, particularly for systems involving ill-conditioned Jacobian or poor initial values that cause convergence problems. With the incorporation of a newly… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.2, pp. 213-239, 2013, DOI:10.3970/cmes.2013.092.213

Abstract We show that the problem "real-time numerical differentiation" of a given noisy signal in time, by supplementing a compensated controller in the second-order robust exact differentiator, the tracking differentiator or the continuous hybrid differentiator, can be viewed as a set of differential algebraic equations (DAEs) to enhance a precise tracking of the given noisy signal. Thus, we are able to solve the highly ill-posed problem of numerical differentiation of noisy signal by using the Lie-group differential algebraic differentiators (LGDADs) based on the Lie-group GL(n,R), whose accuracy and tracking performance are better than before. The "index-two" differentiators (ITDs), which do not… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.90, No.4, pp. 255-282, 2013, DOI:10.3970/cmes.2013.090.255

Abstract In this paper, a scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations is proposed. To conduct the proposed method, we first convert the vector residual function to a scalar function by taking the square norm of the vector function and then, introduce a fictitious time variable to form a scalar homotopy function. To improve the convergence and the accuracy of the proposed method, a vector with multiple search directions and an iterative algorithm are introduced into the evolution dynamics of the solutions. Further, for obtaining the optimal search direction, linear and nonlinear optimization algorithms are… More >

Chih-Wen Chang^1,2, Chein-Shan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.4, pp. 269-292, 2012, DOI:10.3970/cmes.2012.088.269

Abstract In this article, we utilize an optimal vector driven algorithm (OVDA) to cope with the nonlinear heat conduction problems (HCPs). From this set of nonlinear ordinary differential equations, we propose a purely iterative scheme and the spatial-discretization of finite difference method for revealing the solution vector x, without having to invert the Jacobian matrix D. Furthermore, we introduce three new ideas of bifurcation, attracting set and optimal combination, which are restrained by two parameters g and a. Several numerical instances of nonlinear systems under noise are examined, finding that the OVDA has a fast convergence rate, great computation accuracy and… More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.6, pp. 575-602, 2012, DOI:10.3970/cmes.2012.084.575

Abstract An iterative algorithm based on the concept of best descent vector u in x^· = λu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x) = 0. In terms of the residual vector F and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm evolves. A new method to approximate the best descent vector is developed, and we find a critical value of the weighting parameter α_c in the best descent vector u = α_cF + B^TF,… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.81, No.3&4, pp. 335-363, 2011, DOI:10.3970/cmes.2011.081.335

Abstract To solve an ill- (or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h / ∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x^· = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x)… More >