Finite-Element Method Applied to Heat Conduction in Solids with Nonlinear Boundary Conditions

1973 ◽  
Vol 95 (1) ◽  
pp. 126-129 ◽  
Author(s):  
R. E. Beckett ◽  
S.-C. Chu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Transient Heat Conduction ◽  
Nonlinear Boundary Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Element Method

By use of an implicit iteration technique, the finite-element method applied to the heat-conduction problems of solids is no longer restricted to the linear heat-flux boundary conditions, but is extended to include nonlinear radiation–convection boundary conditions. The variation of surface temperatures within each time increment is taken into account; hence a rather large time-step size can be assigned to obtain transient heat-conduction solutions without introducing instability in the surface temperature of a body.

Numerical oscillation in seepage analysis of unsaturated soils

2001 ◽  
Vol 38 (3) ◽  
pp. 639-651 ◽  
Author(s):  
Muthusamy Karthikeyan ◽  
Thiam-Soon Tan ◽  
Kok-Kwang Phoon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Unsaturated Soil ◽  
Unsaturated Soils ◽  
Heat Diffusion ◽  
Step Size ◽  
Time Step ◽  
Seepage Analysis ◽  
Time Step Size ◽  
Element Method

The finite element method provides a popular means of analyzing groundwater flow in an unsaturated soil. In such problems, oscillatory results are often observed in the finite element solution. Such a phenomenon is observed, for example, when a typical finite element program such as Seep/w is used to model water infiltration into unsaturated soils. Numerical oscillations are often found near the wetting front where the hydraulic gradient is the steepest. These oscillations do not always reduce with decreasing or increasing time-step size alone; rather, an appropriate ratio between time-step size and element size is required. As the pore-water pressures predicted from a transient seepage analysis are used as input groundwater conditions for other types of analysis such as slope stability, contaminant transport, and capillary barrier, these oscillations may have important practical ramifications. Since seepage analysis is common in engineering practice, it is important that appropriate criteria are identified to minimize, if not to remove, the oscillations. In this paper, numerical examples are provided to demonstrate that a simple set of criteria, developed in heat diffusion problems with constant properties to control oscillation, is also applicable to one- and two-dimensional unsaturated seepage analyses, for a range of material nonlinearities that are frequently encountered in unsaturated soils.Key words: unsaturated soil, soil-water characteristic curve, seepage analysis, finite element method, numerical oscillation.

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

Sign in / Sign up

Export Citation Format

Share Document