Numerical Solution Techniques Research Articles

Theoretical optimization of double dielectric back reflector layer for thin c-Si based advanced solar cells with notable enhancement in MAPD

Dielectric layers on the back surface of a solar cell not only enhance the back reflection but also contribute to better light management by minimizing the absorption loss that happens for conventional back metal contacts. Considering the critical applications of back dielectric layers in solar photovoltaic (PV) technology, their physical characteristics must be optimized, which is the goal of this present investigation. Thin (~30 μm) c-Sibased advanced solar cells that are gradually drawing the attention of modern PV technology; have been chosen as the typical case here. Theoretical optimization of thickness-dependent light reflection capabilities for two different types of dielectric stacks, namely, SiO2/Al2O3 and HfO2/Al2O3 have been carried out. COMSOL Multiphysics™ simulator based on the finite element method (FEM) numerical solution technique has been used. The electromagnetic wave frequency domain (EWFD) module has been used for computational purposes. The back layer engineering using double dielectric back reflector layers has been carried out to overcome the light transmission losses that are usually prevalent in thin (~30 μm) c-Si solar cells at longer wavelengths. Such analyses are of immense importance for thin c-Si based PERC solar cells also. The carrier generation rate, external quantum efficiency (EQE), and maximum achievable photocurrent density (MAPD) have been measured for the optimized cell structures that resulted in a significant increment in each of the parameters. This makes such back reflecting structures worth implementing in thin c-Si based advanced solar cells to minimize the transmission losses at longer wavelengths.

Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations

In this article, a new collocation technique for numerical solution of Fredholm, Volterra and mixed Volterra-Fredholm integral equations of the second kind is introduced and also developed a numerical integration formula on the basis of linear Legendre multi-wavelets. We also use the linear Legendre multi-wavelets basis for the proposed method. In this technique, the unknown function is approximated by truncated linear Legendre multi-wavelets series. The newly developed numerical technique is applied to both linear and nonlinear benchmark test models from the literature including models with discontinues and non-differentiable exact solutions. The numerical results are compared with the other existing numerical techniques from literature. The comparison among the discussed methods with Monte Carlo method, rationalized Haar functions method, Operational matrix with block-pulse functions method, Bernstein operational matrices method and P-order spline direct method divulge that the present technique is authentic and valid for other physical and engineering problems. The proposed numerical method handle the discontinuity and non-differentiability in the exact solutions very well, whereas the other existence numerical methods are failed to capture these problems. Further, more the comparison of exact and approximate in term of L∞ norms which demonstrate the accuracy, flexibility and robustness of the newly proposed numerical method.

Numerical solution technique for solving isoperimetric variational problems

In this paper, we have a zeal for fulfilling the estimated scientific answers for the calculus of variations by using the Sumudu transform method (STM). The main target is to search the numerical arrangement of ordinary differential equations (ODEs) which emerge from the variational problems where first the fundamental condition for the arrangement of the issue is to fulfill the Euler–Lagrange condition and then solve the equations using STM. The valuable properties of the Sumudu change technique are used to downsize the calculation of the issue to a gathering of straight arithmetical conditions. We introduce four variational problems and discover the numerical solution of those problems using STM and plot the curves of those solutions. These models are picked such that there exist systematic answers for them to offer a reasonable diagram and show the effectiveness of the proposed strategy. Numerical outcomes are registered utilizing Maple programming.

An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases

This work introduces a direct method based on orthonormal Bernstein polynomials wavelet bases, to present a stable algorithm for numerical inversion of a system of generalized Abel integral equations. The application of all the currently existing numerical inversion methods was strictly limited to only one portion of the generalized Abel integral equations. The proposed method is quite accurate, and several numerical illustrations demonstrate the convergence and utilization of the proposed method compared to some of the preexisting numerical solution techniques. The permanence of the numerical result under the effect of small perturbation in input data has been examined, which is depicted with the use of numerical illustrations.

Heuristic computing technique for numerical solutions of nonlinear fourth order Emden–Fowler equation

The present study presents numerical solution of the fourth order, singular, nonlinear Emden–Fowler equation integrated computational intelligence by exploiting the efficiency of feedforward artificial neural networks (ANNs), global optimization strength of genetic algorithms (GAs) and rapid local search of sequential quadratic programming (SQP). ANN is applied to discretize the nonlinear Emden–Fowler differential equation and consequentially construct a fitness/merit function on mean squared error sense. The optimization of the decision variables of ANN to solve the multi-singular, fourth order, nonlinear Emden–Fowler equation is performed by integrating capability GAs and SQP. The proficiency of the presented approach is certified on precision, convergence and stability indices, which are further enhanced through effectiveness, significance and reliability measures on statistical data for solving three different cases of the singular systems.

Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires

Numerical techniques for approximate solution of a system of reaction-diffusion-convection partial differential equations modeling the evolution of temperature and fuel density in a wildfire are proposed. These schemes combine linearly implicit-explicit Runge–Kutta (IMEX-RK) methods and Strang-type splitting technique to adequately handle the non-linear parabolic term and the stiffness in the reactive part. Weighted essentially non-oscillatory (WENO) reconstructions are applied to the discretization of the nonlinear convection term. Examples are focused on the applicative problem of determining the width of a firebreak to prevent the propagation of forest fires. Results illustrate that the model and numerical scheme provide an effective tool for defining that width and the parameters for control strategies of wildland fires.

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

PurposeThe purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory.Design/methodology/approachThe system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously.FindingsA different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations.Originality/valueThe current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method

In this paper, we propose an efficient numerical technique for numerical solutions of the equivalent integral form of Emden–Fowler type boundary value problems (BVPs), which model many phenomena in mathematical physics and astrophysics. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. The iterative method is applied to solve the system numerically. The error analysis of the proposed method is provided. Several examples are provided to demonstrate the accuracy, applicability, and efficiency of the present method.

Effect of time-dependent inhomogeneous magnetic fields on the particle momentum spectrum in electron-positron pair production

Electron-positron pair production in spatially and temporally inhomogeneous electric and magnetic fields is studied within the Dirac-Heisenberg-Wigner formalism (quantum kinetic theory) through computing the corresponding Wigner functions. The focus is on discussing the particle momentum spectrum regarding signatures of Schwinger and multiphoton pair production. Special emphasis is put on studying the impact of a strong dynamical magnetic field on the particle distribution functions. As the equal-time Wigner approach is formulated in terms of partial integro-differential equations an entire section of the manuscript is dedicated to present numerical solution techniques applicable to Wigner function approaches in general.

Atmospheric dispersion modelling in ecological engineering problems

The applied problems of hydrodynamics, heat and mass transfer often claim the use of numerical solution techniques in curvilinear coordinates. Such an approach is quite natural and common for flows modelling, particularly in atmosphere dynamics. The paper deals with development of atmospheric dispersion model and numerical algorithm using approximations in curvilinear coordinate systems. The approach allowed significantly simplifying the formulas and reducing the amount of computations during experiments on forecasting the air pollutants concentration nearby existing industrial plants.

Approximate Moment Methods for Population Balance Equations in Particulate and Bioengineering Processes

Population balance modeling is an established framework to describe the dynamics of particle populations in disperse phase systems found in a broad field of industrial, civil, and medical applications. The resulting population balance equations account for the dynamics of the number density distribution functions and represent (systems of) partial differential equations which require sophisticated numerical solution techniques due to the general lack of analytical solutions. A specific class of solution algorithms, so-called moment methods, is based on the reduction of complex models to a set of ordinary differential equations characterizing dynamics of integral quantities of the number density distribution function. However, in general, a closed set of moment equations is not found and one has to rely on approximate closure methods. In this contribution, a concise overview of the most prominent approximate moment methods is given.

Nonlinear Forced Vibration Analysis of FG Cylindrical Nanopanels Based on Mindlin’s Strain Gradient Theory and 3D Elasticity

Abstract A numerical approach is used herein to study the primary resonant dynamics of functionally graded (FG) cylindrical nanoscale panels taking the strain gradient effects into consideration. The basic relations of the paper are written based upon Mindlin’s strain gradient theory (SGT) and three-dimensional (3D) elasticity. Since the formulation is developed using Mindlin’s SGT, it is possible to reduce it to simpler size-dependent theories including modified forms of couple stress and strain gradient theories (MCST & MSGT). The governing equations is derived and directly discretized via the variational differential quadrature technique. Then, a numerical solution technique is employed to study the nonlinear resonance response of nanopanels with various edge conditions under a harmonic load. The impacts of length scale parameter, material and geometrical parameters on the frequency–response curves of nanopanels are investigated. In addition, comparisons are provided between the predictions of MSGT, MCST and the classical elasticity theory.

A new class of A-stable numerical techniques for ordinary differential equations: Application to boundary-layer flow

The present attempt is made to propose a new class of numerical techniques for finding numerical solutions of ODE. The proposed numerical techniques are based on interpolation of a polynomial. Currently constructed numerical techniques use the additional information(s) of derivative(s) on particular grid point(s). The advantage of the presently proposed numerical techniques is that these techniques are implemented in one step and can provide highly accurate solution and can be constructed on fewer amounts of grid points but has the disadvantage of finding derivative(s). It is to be noted that the high order techniques can be constructed using just two grid points. Presently proposed fourth order technique is A-stable but not L-stable. The order and maximum absolute error are found for a fourth order technique. The fourth order technique is employed to solve the Darcy-Forchheimer fluid-flow problem which is transformed further to a third-order non-linear boundary value problem on the semi-infinite domain.

Application Progress of Computational Fluid Dynamic Techniques for Complex Viscous Flows in Ship and Ocean Engineering

Complex flow around floating structures is a highly nonlinear problem, and it is a typical feature in ship and ocean engineering. Traditional experimental methods and potential flow theory have limitations in predicting complex viscous flows. With the improvement of high-performance computing and the development of numerical techniques, computational fluid dynamics (CFD) has become increasingly powerful in predicting the complex viscous flow around floating structures. This paper reviews the recent progress in CFD techniques for numerical solutions of typical complex viscous flows in ship and ocean engineering. Applications to free-surface flows, breaking bow waves of high-speed ship, ship hull–propeller–rudder interaction, vortex-induced vibration of risers, vortex-induced motions of deep-draft platforms, and floating offshore wind turbines are discussed. Typical techniques, including volume of fluid for sharp interface, dynamic overset grid, detached eddy simulation, and fluid–structure coupling, are reviewed along with their applications. Some novel techniques, such as high-efficiency Cartesian grid method and GPU acceleration technique, are discussed in the last part as the future perspective for further enhancement of accuracy and efficiency for CFD simulations of complex flow in ship and ocean engineering.

English

The present work is devoted to study the effect of rotation and gravity field in a homogeneous, isotropic elastic semi- infinite, which is at initial temperature, and subjected to heat source and load moving with finite velocity. In view of calculating general problems, a numerical solution technique is to be used. For this purpose, the normal mode analysis method is used. The results for the displacement, stress components and temperature distribution are illustrated graphically with some comparisons. The numerical results are given and presented graphically for two different theories, L-S theory and G-L theory. Influence of rotation and gravity on temperature, displacement and stresses components is observed through a numerical example. Key words: Internal heat source, isotropic, thermoelasticity, two thermal relaxation times, gravity, rotation. &nbsp

Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

Random Walk Particle Tracking Embedded Cellular Automata Model for Predicting Temporospatial Variations of Chlorine in Water Distribution Systems

Cellular Automata (CA) is an evolutionary computing technique that makes discrete idealizations of differential equations and represents the physical system at the mesoscopic scale. A novel CA approach for predicting the temporal and spatial variations of chlorine in Water Distribution Systems (WDSs) is presented in this paper. Random Walk Particle Tracking (RWPT), a stochastic Lagrangian technique, is used to represent the advection and dispersion processes. A one-dimensional CA-based reactive-transport model for chlorine, named as RWPT_CA model, incorporating advective-dispersive transport mechanism is developed and demonstrated. The significance of the cell dimension in the model algorithm is ascertained, and a deterministic approach is formulated for its selection. An indirect numerical solution technique is developed to improve the computational efficiency of the CA algorithm and to minimize the restrictions in the process of discretization of mass into equivalent particles. The numerical accuracy of the proposed RWPT_CA model is verified by applying it on to a benchmark problem. The RWPT_CA model provided excellent representations of the chlorine concentration profiles for low to medium range dispersion in WDSs. The model testing on a benchmark problem from the literature, well tested by researchers, revealed its effectiveness to derive the chlorine concentration patterns under dynamic hydraulic conditions. The dispersion mechanism was found significant in controlling the temporospatial distribution of chlorine at the nodes farther from the source nodes. The models which consider only advective transport mechanism were found over-predicting the chlorine concentrations, and thereby, establishing untrue representations of the quality of the delivered water.

Решение задачи распределения ресурсов дискретного типа методами линейного программирования

Introduction. When planning projects, work packages, one often has to solve distribution-type problems associated with the optimal allocation of resources. Existing methods for solving such problems suggest the presence of an analytical dependence of a continuous type between the volumes of the distributed resource and performance indicators. However, optimization problems become inapplicable when the resource is discrete and dependencies are specified in tabular ways. Aim. To develop a mathematical model for solving the problem of the optimal distribution of resources of a discrete type with table-defined optimality criteria using linear programming methods. Describe the method of numerical solution of the problem using computer technology. Materials and methods. It is possible to solve the problem by forming an analytical dependence of a piecewise-continuous type between the volume of the distributed resource and the optimality criterion. This allows us to formulate an optimization problem solved by mathematical programming methods. It is possible to construct an analytical objective function by introducing additional parameters and restrictions. A numerical solution to the problem can be obtained both using mathematical packages of applied programs, such as, for example, “Mathcad”, and using programming systems. The paper describes the methodology for solving problems in the MS Excel environment using the add-on “Solver”. Results. A mathematical model is developed for solving a discrete distribution problem for the optimality criterion given in a table by integer linear programming methods. The technique of numerical solution in the MS Excel environment is described. Conclusion. Previously, such problems were solved by dynamic programming methods, which is more difficult in the computational plan. Conducted computational experiments showed high accuracy of model calculations and resistance to changes in the source data.

Generalized operational matrices and error bounds for polynomial basis

This work presents a direct way to obtain operational matrices for all complete polynomial basis, considering limited intervals, by using similarity relations. The algebraic procedure can be applied to any linear operator, particularly to the integration and derivative operations. Direct and spectral representations of functions are shown to be equivalent by using the compactness of Dirac notation, permitting the simultaneous use of the numerical solution techniques developed for both cases. The similarity methodology is computer oriented, emphasizing aspects concerning matrices operations, compacting the notation and lowering the computational costs and code elaboration times. To illustrate the operational aspects, some integral and differential equations are solved, including non-orthogonal basis examples, showing the generality of the method and the compatibility of the results with those obtained by other recent research works. As the Dirac notation is used, the projector operator allows to calculate the precision of the obtained solutions and the error superior limit, even if the exact solution is not available.

IMPROVEMENT ON NUMERICAL SOLUTION OF 1-D FLOW WITH HYPER-CONCENTRATED SEDIMENT

The flow phenomena in a mountainous area are often dominated by the situation where the streams have steep slopes and potential to carry the high sediment concentration. After the initiation of the flow with certain characteristic including sediment concentration within the flow, the flow would then erode and deposit the sediment in the next reach of the stream. Such phenomena would take place until the equilibrium state is met. The gradient of flow velocity, depth of flow, and sediment concentration are studied considering the equation of mass and momentum conservation, followed by the establishment of a linear equation. The improvement of the numerical solution persists in the form of the utilization of MS-Excel program as a technique of solving a linear three system of equations which was found to be practical. Such a technique was found to be effective to investigate the sensitivity analysis of the results due to some changes on various dependent parameters including the slope parameter. The three systems of equation and the proposed numerical solution technique offer further works associated with the hyper-concentrated flow through both numerical and physical model.