A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis

2017 ◽  
Vol 10 (3) ◽  
pp. 541-561
Author(s):  
Yirang Yuan ◽  
Qing Yang ◽  
Changfeng Li ◽  
Tongjun Sun
Keyword(s):  
Finite Element ◽  
Heat Conduction ◽  
Numerical Analysis ◽  
Partial Differential Equations ◽  
Heat Conduction Equation ◽  
Semiconductor Device ◽  
Mixed Finite Element ◽  
Three Dimensional ◽  
Fractional Step ◽  
Dimensional Problem

AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

scholarly journals Trilinear Finite Element Solution of Three Dimensional Heat Conduction Partial Differential Equations

2018 ◽  
Vol 7 (4.36) ◽  
pp. 379
Author(s):  
Erwin Sulaeman ◽  
S. M. Afzal Hoq ◽  
Abdurahim Okhunov ◽  
Marwan Badran
Keyword(s):  
Finite Element ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Heat Conduction Problem ◽  
Promising Result ◽  
Three Dimensional ◽  
Integration Scheme ◽  
Partial Differential

Solution of partial differential equations (PDEs) of three dimensional steady state heat conduction and its error analysis are elaborated in the present paper by using a Trilinear Galerkin Finite Element method (TGFEM). An eight-node hexahedron element model is developed for the TGFEM based on a trilinear basis function where physical domain is meshed by structured grid.  The stiffness matrix of the hexahedron element is formulated by using a direct integration scheme without the necessity to use the Jacobian matrix.  To check the accuracy of the established scheme, comparisons of the results using error analysis between the present TGFEM and exact solution is conducted for various number of the elements.  For this purpose, analytical solution is derived in detailed for a particular heat conduction problem.  The comparison shows promising result where its convergence is approximately O(h²) for matrix norms L1, L2 and L¥.  

scholarly journals A Modified Characteristics-mixed Finite Element for Semiconductor Device of Heat Conduction and Numerical Analysis

2018 ◽  
Vol 10 (3) ◽  
pp. 97
Author(s):  
Yirang Yuan
Keyword(s):  
Numerical Simulation ◽  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Electric Field ◽  
Information Science ◽  
Semiconductor Device ◽  
Mixed Finite Element ◽  
Diffusion Equations ◽  
Element Method

Numerical simulation of a three-dimensional semiconductor device of heat conduction is a fundamental problem in modern information science. The mathematical model is formulated by a nonlinear system of initial-boundary problem, which is interpreted by four partial differential equations: an elliptic equation for electrostatic potential, two convection-diffusion equations for electron concentration and hole concentration, a heat conduction equation for temperature. The electrostatic potential appears within the latter three equations, and the electric field strength controls the concentrations and the temperature. The electric field potential is solved by a mixed finite element method, and the electric field strength is obtained simultaneously. The first order of the accuracy is improved for the latter. The concentrations and temperature are computed by the characteristics-finite element method, where the characteristic approximation is adopted for the hyperbolic term and finite element method is use to treat the diffusion. The composite computational scheme can solve the convection-dominated diffusion equations well because it can cancel numerical dispersion and nonphysical oscillation. The temperature is computed by finite element method, and an interesting simulation tool is proposed for solving semiconductor device problem numerically. By using the technique of a priori estimates of differential equations, an optimal order error estimates is obtained. A theoretical work is shown for numerical simulation of information science, and the actual problem is solved well.

Performance characteristics of two-axial groove journal bearing for different groove angles

2018 ◽  
Vol 70 (2) ◽  
pp. 432-443
Author(s):  
K.R. Kadam ◽  
S.S. Banwait
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Pressure Distribution ◽  
Heat Conduction Equation ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Journal Bearing ◽  
Three Dimensional ◽  
Content Type

Purpose Different groove angles are used to study performance characteristics of two-axial groove journal bearing. In this study two grooves are located at ±90º to the load line. The various angles of grooves have been taken as 10° to 40° in the interval of 5°. Different equations such as Reynolds equation, three-dimensional energy equation and heat conduction equation have been solved using finite element method and finite difference method. Pressure distribution in fluid is found by using Reynolds equation. The three-dimensional energy equation is used for temperature distribution in the fluid film and bush. One-dimensional heat conduction equation is used for finding temperature in axial direction for journal. There is a very small effect of groove angle on film thickness, eccentricity ratio and pressure. There is a drastic change in attitude angle and side flow. Result shows that there is maximum power loss at large groove angle. So the smaller groove angle is recommended for two-axial groove journal bearing. Design/methodology/approach The finite element method is used for solving Reynolds equation for pressure distribution in fluid. The finite difference method is adopted for finding temperature distribution in bush, fluid and journal. Findings Pressure distribution in fluid is found out. Temperature distribution in bush, fluid and journal is found out. There is a very small effect of groove angle on film thickness, eccentricity ratio and pressure. Research limitations/implications The groove angle used is from 10 to 40 degree. The power loss is more when angle of groove increases, so smaller groove angle is recommended for this study. Practical implications The location of groove angle predicts the distribution of pressure and temperature in journal bearing. It will show the performance characteristics. ±90° angle we will prefer that will get before manufacturing of bearing. Social implications Due to this study, we will get predict how the pressure and temperature distribute in the journal. It will give the running condition of bearing as to at what speed and load we will get the maximum temperature and pressure in the bearing. Originality/value The finite element method is used for solving the Reynolds equation. Three-dimensional energy equation is solved using the finite difference method. Heat conduction equation is also solved for journal. The C language is used. The code is developed in C language. There are different equations which depend on each other. The temperature is dependent on pressure viscosity of fluid, etc. so C code is preferred.

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