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Cooling Systems
Published in S.V. Kulkarni, S.A. Khaparde, Transformer Engineering, 2017
CFD Model: This allows comprehensive 3-D analysis in contrast with the previous approach which assumes circular symmetry. 3D models generated in a typical CAD software can be imported into a mesh generator where geometry is discretized into a number of computational cells (or grid cells). There are many discretization methods, but the most commonly used are the finite difference method, the finite volume method, and the finite element method. It is cumbersome to apply the finite difference approach for irregular boundaries. Traditionally, commercial solvers were mostly FEM based. Most of the commercial CFD solvers, now, are based on the finite volume method of discretization of governing equations. Basic laws of physics are used to ensure conservation of fluxes at discretized smaller control volumes as well as over the complete computational domain. Necessary boundary conditions such as pressure, velocity, mass flow, temperature and/or heat fluxes are set at the boundaries.
Mixed-Mode TCAD Tools
Published in John D. Cressler, H. Alan Mantooth, Extreme Environment Electronics, 2017
In order to numerically solve for the electric potential and the carrier concentrations and currents, first the semiconductor device partial differential equations (31.1 through 31.10) are discretized (i.e., approximated into equivalent algebraic expressions) over each subdomain of the implemented computational mesh. The matrix of algebraic expressions thus generated is then solved until a preset convergence criterion is satisfied. Commonly used discretization schemes include finite-difference, finite-element, and finite-volume methods. The reader is referred to Refs. [18,29,30] for a detailed discussion of numerical solution procedures, and the relative merits of each.
Method of Finite Differences and Self-Energy of the Leads
Published in Vinod Kumar Khanna, Introductory Nanoelectronics, 2020
Discretization is the process of converting continuous variables, functions, and equations into distinctly disconnected numerical data that can be fed into digital computers as numbers. Thus it is the first step toward making the equations amenable to digital processing for implementation on computers.
Mechanism underlying initiation of migration of film-like residual oil
Published in Journal of Dispersion Science and Technology, 2022
Xu Han, Lihui Wang, Huifen Xia, Peihui Han, Ruibo Cao, Lili Liu
After the mesh is divided, it is necessary to find the approximate solution of the partial differential equation within the error range allowed by the discretization of the equation in the domain of the solution. Therefore, it is necessary to use numerical methods to treat the dependent variable values at a limited number of positions (grid nodes and grid center points) in the computational domain as basic unknowns and to establish a set of algebraic equations for these unknowns. Then, the system of algebraic equations finally yields these node values, and the values of other positions in the calculation domain can be determined via the method of difference according to the values of the node positions. Because of the introduced strain variables, the distribution assumptions between nodes and the method of deriving the discretization equation are different. Different discretization methods, such as the finite difference method, finite element method, and finite volume method, are formed.[27–30] In this study, the finite element method was used to discretize the control equations.
Thermal performance escalation of cross flow heat exchanger using in-line elliptical tubes
Published in Experimental Heat Transfer, 2020
Chidanand K. Mangrulkar, Ashwinkumar S. Dhoble, Pawan Kumar Pant, Nitin Kumar, Ashutosh Gupta, Sunil Chamoli
The meshed computational domain is processed by importing in the CFD software ANSYS FLUENT. All the governing equations for the fluid flow are solved by the finite-volume method. The second-order upwind technique is used for the discretization to achieve high accuracy for momentum and energy equations. The discretization is a process by which partial differential equations (PDE) are converted into algebraic equations in order to achieve a numerical solution. The boundary layer development over the wall surface is captured by maintaining near-wall Y plus as 1.0, with enhanced wall function. The pressure and velocity components are coupled by the Semi-Implicit Method for the pressure-linked equation (SIMPLE) algorithm. The convergence criterion is usually determined by the order of magnitude of the residuals, along with the fact that the results do not change with the number of iteration performed. So the mass flow rate and momentum change for the convergence of the numerical simulation are maintained at 10−4 and 10−8, respectively. The working fluid is air with the thermophysical properties evaluated at bulk mean temperature. The Re within the test section is varied from 5000 to 21000. The heat input in the form of constant heat flux is by circulating hot water at 353 K through the tandem tubes.
Numerical simulation of collision-free near-shortest path generation for Dubins vehicle via Hamilton–Jacobi–Bellman equation: A case study
Published in Cogent Engineering, 2020
Han-Jung Chou, Jing-Sin Liu, Wen-Chieh Tung
A HJB PDE can be solved numerically in different ways. It is well understood that there is no general numerical method for all kinds of HJB equations in terms of accuracy and efficiency. FDM and finite element method (FEM) are two dominant numerical methods for solving PDEs. The former is via direct discretization called finite differences for approximating gradient of value function and the latter uses a linear combination of basic functions such as polynomials or trigonometric polynomials to approximate solutions of variational formulations. Because the HJB equations and hyperbolic PDEs of conservation laws have many mathematical similarities in common, advances in the conceptual principles and higher-order methods for conservation laws are borrowed as effective solvers for the HJB equations (Kao et al., 2004; Tsai et al., 2017; Zhao, 2005, 2016). Firstly, Crandall and Lions (1984) and Michael (1983) first proposed not only an adequate weak solution for HJB equations, namely, viscosity solution, but also along with a principle that any monotone, differenced form with consistent numerical Hamiltonian must converges to its viscosity solution (Crandall & Lions, 1984). Now it is well understood that the convex HJB equation can be solved with monotone upwind scheme on general meshes, such as discussed in (Wang et al., 2000).