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  • Open Access

    ARTICLE

    Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x· = λ[αR + βP] or x· = λ[αF + βP] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F

    Chein-Shan Liu1,2, 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 >

  • Open Access

    ARTICLE

    An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method

    Chein-Shan Liu1,2, Satya N. Atluri1

    CMES-Computer Modeling in Engineering & Sciences, Vol.80, No.3&4, pp. 275-298, 2011, DOI:10.3970/cmes.2011.080.275

    Abstract To solve an ill-conditioned system of linear algebraic equations (LAEs): Bx - b = 0, we define an invariant-manifold in terms of r := Bx - b, and a monotonically increasing function Q(t) of a time-like variable t. Using this, we derive an evolution equation for dx / dt, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of t. Using the concept of discrete dynamics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involving an Optimal Descent VectorMore >

  • Open Access

    ARTICLE

    A Revision of Relaxed Steepest Descent Method from the Dynamics on an Invariant Manifold

    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 >

  • Open Access

    ARTICLE

    Stable Manifolds of Saddles in Piecewise Smooth Systems

    A. Colombo1, U. Galvanetto2

    CMES-Computer Modeling in Engineering & Sciences, Vol.53, No.3, pp. 235-254, 2009, DOI:10.3970/cmes.2009.053.235

    Abstract The paper addresses the problem of computing the stable manifolds of equilibria and limit cycles of saddle type in piecewise smooth dynamical systems. All singular points that are generically present along one-dimensional or two-dimensional manifolds are classified and such a classification is then used to define a method for the numerical computation of the stable manifolds. Finally the proposed method is applied to the case of a stick-slip oscillator. More >

  • Open Access

    ARTICLE

    Five Different Formulations of the Finite Strain Perfectly Plastic Equations

    Chein-Shan Liu 1,2

    CMES-Computer Modeling in Engineering & Sciences, Vol.17, No.2, pp. 73-94, 2007, DOI:10.3970/cmes.2007.017.073

    Abstract The primary objectives of the present exposition focus on five different types of representations of the plastic equations obtained from an elastic-perfectly plastic model by employing different corotational stress rates. They are (a) an affine nonlinear system with a finite-dimensional Lie algebra, (b) a canonical linear system in the Minkowski space, (c) a non-canonical linear system in the Minkowski space, (d) the Lie-Poisson bracket formulation, and (e) a two-generator and two-bracket formulation. For the affine nonlinear system we prove that the Lie algebra of the vector fields is so(5,1), which has dimensions fifteen, and by the Lie theory the superposition… More >

  • Open Access

    ARTICLE

    Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

    Muhammad Shoaib Arif1, Mairaj Bibi2, Adnan Jhangir3

    CMC-Computers, Materials & Continua, Vol.54, No.2, pp. 181-195, 2018, DOI:10.3970/cmc.2018.054.181

    Abstract This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices. We choose the geodesic distance between -AHX - XA and P as the cost function, and put forward the Extended Hamiltonian algorithm (EHA) and Natural gradient algorithm (NGA) for the solution. Finally, several numerical experiments give you an idea about the effectiveness of the proposed algorithms. We also show the comparison between these two algorithms EHA and NGA. Obtained results are provided and analyzed graphically. We also conclude that the extended Hamiltonian algorithm has better convergence speed than the… More >

  • Open Access

    ARTICLE

    A Sliding Mode Control Algorithm for Solving an Ill-posed Positive Linear System

    Chein-Shan Liu1

    CMC-Computers, Materials & Continua, Vol.39, No.2, pp. 153-178, 2014, DOI:10.3970/cmc.2014.039.153

    Abstract For the numerical solution of an ill-posed positive linear system we combine the methods from invariant manifold theory and sliding mode control theory, developing an affine nonlinear dynamical system with a positive control force and with the residual vector as being a gain vector. This system is proven asymptotically stable to the zero residual vector by using an argument from the Lyapunov stability theory. We find that the system fast tends to the sliding surface and then moves with a sliding mode, such that the resultant sliding mode control algorithm (SMCA) is robust against large noise and stable to find… More >

  • Open Access

    ARTICLE

    A New Optimal Iterative Algorithm for Solving Nonlinear Poisson Problems in Heat Diffusion

    Chih-Wen Chang1,2, Chein-Shan Liu3

    CMC-Computers, Materials & Continua, Vol.34, No.2, pp. 143-175, 2013, DOI:10.3970/cmc.2013.034.143

    Abstract The nonlinear Poisson problems in heat diffusion governed by elliptic type partial differential equations are solved by a modified globally optimal iterative algorithm (MGOIA). The MGOIA is a purely iterative method for searching the solution vector x without using the invert of the Jacobian matrix D. Moreover, we reveal the weighting parameter αc in the best descent vector w = αcE + DTE and derive the convergence rate and find a criterion of the parameter γ. When utilizing αc and γ, we can further accelerate the convergence speed several times. Several numerical experiments are carefully discussed and validated the proposed… More >

  • Open Access

    ARTICLE

    The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using x· = λ[αF + (1 - α)BTF]

    Hong-Hua Dai1,2, 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 method, applied to the piecewise… More >

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