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 =… More >

Chein-Shan Liu^1,2, 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 Vector… More >

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 (tangent stiffness matrix)… 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. We can prove… More >

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 where… More >

K. Bououni¹, T. Jaber¹, S. Elbehissy¹

FDMP-Fluid Dynamics & Materials Processing, Vol.13, No.2, pp. 127-141, 2017, DOI:10.3970/fdmp.2017.013.127

Abstract The economic progress in the United Arab Emirates (UAE) induces to a significant increase in the demand for agricultural development. In Emirates the majority of the farms are irrigated by underground water, characterized by a high level of salinity. Liwa, Al Ain and Al Khatem areas are suffering from high water well salinity that exceeds 20,000 ppm. This work focuses on this problem and suggests a suitable solution allowing the use of renewable energy (Solar Photovoltaic) to drive RO desalination units. An optimal design of RO/PV unit adapted to a typical farm in Abu Dhabi was suggested using a model… More >

ShadiaJ. Ikhmayies¹, Naseem M. Abu El-Haija¹, Riyad N. Ahmad-Bitar¹

FDMP-Fluid Dynamics & Materials Processing, Vol.6, No.2, pp. 165-178, 2010, DOI:10.3970/fdmp.2010.006.165

Abstract Undoped polycrystalline ZnO thin films were produced on glass substrates at a substrate temperature Ts= 450 C by the spray pyrolysis (SP) technique. The films were characterized by analyzing their I-V curves, transmittance, X-ray diffractograms (XRD) and their scanning electron microscope (SEM) images. The I-V plots are all linear and the resistivity was found to be about 200W.cm. The transmittance in the visible and near infrared regions is as high as 85% which is suitable for solar cell applications. The absorption coefficient which is deduced from the transmittance measurements is continuously increasing with the photon's energy and it rapidly increases… More >