A Discontinuous Finite Element Method for Non-Fourier Heat Conduction Problems

2003 ◽  
Author(s):  
X. Ai ◽  
B. Q. Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Numerical Solutions ◽  
Mathematical Formulation ◽  
Discontinuous Galerkin Finite Element ◽  
Discontinuous Finite Element ◽  
Fourier Heat Conduction ◽  
Computational Procedures ◽  
Element Method

A discontinuous Galerkin finite element method is presented for the solution of non-Fourier heat conduction problems that arise from the thermal processing of thin films using the ultra-short pulsed lasers. Mathematical formulation is described in detail and computational procedures are given. Numerical example are given and compare with available solutions where available. The numerical solutions exhibit strong wave behavior and reflection and interactions of thermal waves at the boundaries in multi-dimensions.

Use of the Finite-Element Method in the Solution of Diffusion-Convection Equations

1971 ◽  
Vol 11 (02) ◽  
pp. 139-144 ◽  
Author(s):  
Y.M. Shum
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Variational Principle ◽  
Numerical Solutions ◽  
Transient Heat Conduction ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Diffusion Convection ◽  
Element Method

Abstract A variational principle can be applied to the transient heat conduction equation with heat-flux boundary conditions. The finite-element method is employed to reduce the continuous spatial solution into a finite number of time-dependent unknowns. From previous work, it was demonstrated that the method can readily be applied to solve problems involving either linear or nonlinear boundary conditions, or both. In this paper, with a slight modification of the solution technique, the finite-element method is shown to be applicable to diffusion-convection equations. Consideration is given to a one-dimensional transport problem with dispersion in porous media. Results using the finite-element method are compared with several standard finite-difference numerical solutions. The finite-element method is shown to yield satisfactory solutions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat (or mass) by conduction (or diffusion) and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest to petroleum engineers is the classical equation describing the process by which one miscible fluid displaces another in a one-dimensional porous medium. Many authors have presented numerical solutions to this rather simple presented numerical solutions to this rather simple diffusion-convection problem using standard finite-difference methods, method of characteristics, and variational methods. In this paper another numerical method is employed. A finite-element method in conjunction with a variational principle for transient heat conduction analysis is briefly reviewed. It is appropriate here to mention the recent successful application of the finite-element method to solve transient heat conduction problems involving either linear, nonlinear, or both boundary conditions. The finite-element method was also applied to transient flow in porous media in a recent paper by Javandel and Witherspoon. Prime references for the method are the papers by Gurtin and Wilson and Nickell. With a slight modification of the solution procedure for treating the convective term as a source term in the transient heat conduction equation, the method can readily be used to obtain numerical solutions of the diffusion-convection equation. Consideration is given to a one-dimensional mass transport problem with dispersion in a porous medium. Results using the finite-element method yield satisfactory solutions comparable with those reported in the literature. A VARIATIONAL PRINCIPLE FOR TRANSIENT HEAT CONDUCTION AND THE FINITE-ELEMENT METHOD A variational principle can be generated for the transient conduction or diffusion equation. Wilson and Nickell, following Gurtin's discussion of variational principles for linear initial value problems, confirmed that the function of T(x, t) that problems, confirmed that the function of T(x, t) that leads to an extremum of the functional...........(1) is, at the same time, the solution to the transient heat conduction equation SPEJ P. 139

scholarly journals Numerical Solutions of Steady Tensorial Transport Equations Using Discontinuous Galerkin Method Implemented in FreeFem++

2016 ◽  
Vol 8 (1) ◽  
pp. 29-39
Author(s):  
K. M. Helal
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Transport Equations ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Reaction Equations ◽  
Element Method

AbstractThe main purpose of this paper is to approximate the solution of the steady tensorial transport equations using discontinuous Galerkin finite element method implemented with the finite element solver FreeFem++. After introducing the formulations of the tensorial transport equations, the analysis of its componentwise equations, i.e., advection-reaction equations have been discussed. Discretizing the transport problem using discontinuous Galerkin finite element method, the iterative fixed-point method is used to obtain the solutions. We present the numerical simulations of two-dimensional benchmark problem and observe the instability of elasticity. All the simulations are done using the script developed in FreeFem++.

scholarly journals A Comparative Analysis of Neutron Transport Calculations Based on Variational Formulation and Finite Element Approaches

Energies ◽  
2020 ◽  
Vol 13 (20) ◽  
pp. 5424
Author(s):  
Khashayar Sadeghi ◽  
Seyed Hadi Ghazaie ◽  
Ekaterina Sokolova ◽  
Ahmad Zolfaghari ◽  
Mohammad Reza Abbasi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Formulation ◽  
Numerical Solutions ◽  
Neutron Transport ◽  
Computational Error ◽  
Fine Mesh ◽  
Discontinuous Finite Element ◽  
Discontinuous Finite Element Method ◽  
Element Method

The application of continuous and discontinuous approaches of the finite element method (FEM) to the neutron transport equation (NTE) has been investigated. A comparative algorithm for analyzing the capability of various types of numerical solutions to the NTE based on variational formulation and discontinuous finite element method (DFEM) has been developed. The developed module is coupled to the program discontinuous finite element method for neutron (DISFENT). Each variational principle (VP) is applied to an example with drastic changes in the distribution of neutron flux density, and the obtained results of the continuous and discontinuous finite element (DFE) have been compared. The comparison between the level of accuracy of each approach using new module of DISFENT program has been performed based on the fine mesh solutions of the multi-PN (MPN) approximation. The obtained results of conjoint principles (CPs) have been demonstrated to be very accurate in comparison to other VPs. The reduction in the number of required meshes for solving the problem is considered as the main advantage of this principle. Finally, the spatial additivity to the context of the spherical harmonics has been implemented to the CP, to avoid from computational error accumulation.

Steady State Heat Conduction Modeling by the Generalized Finite Element Method

2018 ◽  
Author(s):  
Humberto Alves da Silveira Monteiro ◽  
Guilherme Garcia Botelho ◽  
Roque Luiz da Silva Pitangueira ◽  
Rodrigo Peixoto ◽  
FELICIO BARROS
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Steady State ◽  
Generalized Finite Element Method ◽  
State Heat ◽  
Steady State Heat ◽  
Generalized Finite Element ◽  
Element Method
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