System Of Nonhomogeneous Linear Equations Research Articles

A Closed-Form Analytical Conversion between Zernike and Gatinel–Malet Basis Polynomials to Present Relevant Aberrations in Ophthalmology and Refractive Surgery

The Zernike representation of wavefronts interlinks low- and high-order aberrations, which may result in imprecise clinical estimates. Recently, the Gatinel–Malet wavefront representation has been introduced to resolve this problem by deriving a new, unlinked basis originating from Zernike polynomials. This new basis preserves the classical low and high aberration subgroups’ structure, as well as the orthogonality within each subgroup, but not the orthogonality between low and high aberrations. This feature has led to conversions relying on separate wavefront reconstructions for each subgroup, which may increase the associated numerical errors. This study proposes a robust, minimised-error (lossless) analytical approach for conversion between the Zernike and Gatinel–Malet spaces. This method analytically reformulates the conversion as a nonhomogeneous system of linear equations and computationally solves it using matrix factorisation and decomposition techniques with high-level accuracy. This work fundamentally demonstrates the lossless expression of complex wavefronts in a format that is more clinically interpretable, with potential applications in various areas of ophthalmology, such as refractive surgery.

Fast and Generic Energy Flow Analysis of the Integrated Electric Power and Heating Networks

Energy flow analysis is a fundamental tool to determine the network states of the integrated energy systems (IES). For the widely deployed IES with coupled power grids (PG) and heating networks (HN), we still lack a generic tool that can be easily employed to calculate the quasi-dynamic energy flows of it. In this paper, we have developed a generic code package (named as MATHN) for energy flow analysis of the quality-regulated HN. After that, a generic solution framework that leverages MATPOWER and MATHN to realize decomposed energy flow calculation of PG and HN is proposed. Behind the MATHN tool, a model reformulation technique is proposed to eliminate the massive indirect variables of the basic HN model discretized by finite difference, which finally derives a compact matrix formulation (similar to the network equation of power grid: I=YU) for generic description of any HN and also slashes the original model scale by around 20 times. Moreover, the solution strategy behind the MATHN tool further converts this matrix formulation into a standard system of non-homogeneous linear equations, with which some existing well-recognized algorithms can be directly applied for efficient and accurate solution. Case studies on three different scales of test systems demonstrate the generality, efficiency and accuracy of our methods. All the codes and input data are packaged into the MATHN tool and are made open-source for non-commercial use.

Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.

Heatline and Massline Analysis Due to Magnetohydrodynamic Double Diffusive Natural Convection in a Trapezoidal Enclosure with Various Aspect Ratios

As streamlines are the utmost visualization tools of fluid flow and not the isobars, heatlines and masslines are the sufficient visualization tools of heat and mass transfers and not the isotherms and isoconcentrations respectively. Motivated by various applications of conjugate heat and mass transfer in drying, cooling, sterilization process etc., the present work aims to find out the detailed analysis of conjugate heat and mass transfer using heatlines and masslines in a trapezoidal enclosure with different heating and various massive walls in the presence of magnetic field. The main impact of the paper is the analysis of heatlines and masslines with different aspect ratios of the cavity, which has not been considered by any earlier researcher. The second order accurate finite difference approximation streamfunction—velocity formulation of full Navier Stokes equations has been used to find out the numerical solutions. The discretised non-homogeneous system of linear equations are solved by Biconjugate Gradient Stabilized method.

APPLICATIONS OF GENERALIZED INVERSES BOTH IN HOMOGENEOUS AND NONHOMOGENEOUS SYSTEM OF LINEAR EQUATIONS.

28Aug 2018 APPLICATIONS OF GENERALIZED INVERSES BOTH IN HOMOGENEOUS AND NON-HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS. Mohammad Maruf Ahmed. Assistant Professor, Department of Mathematics and Natural Sciences, BRAC University, 66 Mohakhali Dhaka ? 1212, Bangladesh.

Inverse of Matrices and Solution of Linear Equations

In this paper, an algebraic method has been given to find the inverse of a non-singular square matrix and on the basis of this method, a short cut to find inverse of a non-singular square matrix has been given. A method is given to solve a system of non-homogeneous linear equations in n unknowns by reducing it into system of non-homogeneous linear equations in n-1 unknowns.

A Graph-theoretic Pipe Network Method for water flow simulation in discrete fracture networks: GPNM

In order to simulate water flow in discrete fracture networks, a Graph-theoretic Pipe Network Method (GPNM) is proposed. Firstly, identification of water flow pathways is considered and a tree cutting technique is adopted. Then each fracture in a discrete fracture network is treated as a weighted pipe with a starting node and an ending node in an oriented graph. A node law of flow rate and a pipe law of pressure in discrete fracture networks are derived based on the conservation of mass and energy, respectively. Boundaries and fractures are unified with the same form of a unified governing equation. Solutions of water pressures and flow rates in discrete fracture networks are obtained by solving a system of nonhomogeneous linear equations. Since no discretization is needed, GPNM is demonstrated with high efficiency. In addition, a few case studies are implemented and compared with those from analytical solutions or numerical analysis using the software, Universal Distinct Element Code (UDEC). It shows that the proposed Graph-theoretic Pipe Network Method (GPNM) is effective in analyzing water flow in discrete fracture networks. Moreover, GPNM is promising for more engineering applications, and can be used for large scales of water simulation problems with numerous fractures.

A Graph-theoretic Pipe Network Method for water flow simulation in a porous medium: GPNM

A new numerical simulation method for water flow in a porous medium is proposed. A porous medium is discretized graph-theoretically into a discrete pipe network. Each pipe in the oriented network is defined as a weighted element with a starting node and an ending node. Equivalent hydraulic parameters are derived based on the Darcy’s Law. A node law of flow rate and a pipe law of pressure are derived based on the conservation of mass and energy, as well as the graph-theoretic network theory. A unified governing equation for both the inner pipes and the boundary pipes are deduced. A conversion law of flow rate/velocity is proposed and discussed. A few case studies are analyzed and compared with those from analytical solutions and finite element analysis. It shows that the proposed Graph-theoretic Pipe Network Method (GPNM) is effective in analyzing water flow in a porous medium. The advantage of the proposed GPNM is that a continuous porous medium is discretized into a discrete pipe network, which is analyzed same as for a discrete fracture network. Solutions of water pressures and flow rates in the discrete pipe network are obtained by solving a system of nonhomogeneous linear equations. It is demonstrated with high efficiency and accuracy. The developed method can be extended to analyzing water flow in fractured and porous media in 3-D conditions.

Fracture analysis near the interface crack tip for mode I of orthotropic bimaterial

The mechanical behaviors near the interface crack tip for mode I of orthotropic bimaterial are researched. With the help of the complex function method and the undetermined coefficient method, non-oscillatory field if the singularity exponent is a real number, and oscillatory field if the singularity exponent is a complex number are discussed, respectively. For each case, the stress functions are constructed which contain twelve undetermined coefficients and an unknown singularity exponent. Based on the boundary conditions, the system of non-homogeneous linear equations is obtained. According to the necessary and sufficient condition for the existence of solution for the system of non-homogeneous linear equations, the singularity exponent is determined under appropriate condition using bimaterial parameters. Both the theoretical formulae of stress intensity factors and analytic solutions of stress or displacement field near the interface crack tip are given. When the two orthotropic materials are the same, the classical results for orthotropic single material are deduced.

Oscillatory Singularity Behaviors Near Interface Crack Tip for Mode II of Orthotropic Bimaterial

The fracture behaviors near the interface crack tip for mode II of orthotropic bimaterial are discussed. The oscillatory singularity fields are researched. The stress functions are chosen which contain twelve undetermined coefficients and an unknown singularity exponent. Based on the boundary conditions and linear independence, the system of twelve nonhomogeneous linear equations is derived. According to the condition for the system of nonhomogeneous linear equations which has a solution, the singularity exponent is determined. Total coefficients are found by means of successive elimination of the unknowns. The theoretical formulae of stress intensity factors and analytic solutions of stress field near the interface crack tip are obtained. The crack tip field is shown by figures.

Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem

In this article, we first reformulate the generalized nonlinear complementarity problem (GNCP) over a polyhedral cone as a smoothing system of equations and then suggest a smoothing Broyden-like method for solving it. The proposed algorithm has to solve only one system of nonhomogeneous linear equations, perform only one line search and update only one matrix per iteration. We show that the iteration sequence generated by the proposed algorithm converges globally and superlinearly under suitable conditions. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.

Interface crack problems for mode II of double dissimilar orthotropic composite materials

The fracture problems near the interface crack tip for mode II of double dissimilar orthotropic composite materials are studied. The mechanical models of interface crack for mode II are given. By translating the governing equations into the generalized bi-harmonic equations, the stress functions containing two stress singularity exponents are derived with the help of a complex function method. Based on the boundary conditions, a system of non-homogeneous linear equations is found. Two real stress singularity exponents are determined be solving this system under appropriate conditions about bimaterial engineering parameters. According to the uniqueness theorem of limit, both the formulae of stress intensity factors and theoretical solutions of stress field near the interface crack tip are derived. When the two orthotropic materials are the same, the stress singularity exponents, stress intensity factors and stresses for mode II crack of the orthotropic single material are obtained.

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r( ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r −λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.