Two variants of the finite-element method have been developed for use in computational simulations of radiative transfers of heat among diffuse gray surfaces. Both variants involve the use of higher-order finite elements, across which temperatures and radiative quantities are assumed to vary according to certain approximations. In this and other applications, higherorder finite elements are used to increase (relative to classical finite elements, which are assumed to be isothermal) the accuracies of final numerical results without having to refine computational meshes excessively and thereby incur excessive computation times.One of the variants is termed the radiation sub-element (RSE) method, which, itself, is subject to a number of variations. This is the simplest and most straightforward approach to representation of spatially variable surface radiation. Any computer code that, heretofore, could model surface-to-surface radiation can incorporate the RSE method without major modifications.

Temperature Versus Position along a plate was computed in a test case exactly and by two variants of the finite-element method.
In the basic form of the RSE method, each finite element selected for use in computing radiative heat transfer is considered to be a parent element and is divided into sub-elements for the purpose of solving the surface-to-surface radiation- exchange problem. The sub-elements are then treated as classical finite elements; that is, they are assumed to be isothermal, and their view factors and absorbed heat fluxes are calculated accordingly. The heat fluxes absorbed by the sub-elements are then transferred back to the parent element to obtain a radiative heat flux that varies spatially across the parent element. Variants of the RSE method involve the use of polynomials to interpolate and/or extrapolate to approximate spatial variations of physical quantities.

The other variant of the finite-element method is termed the integration method (IM). Unlike in the RSE methods, the parent finite elements are not subdivided into smaller elements, and neither isothermality nor other unrealistic physical conditions are assumed. Instead, the equations of radiative heat transfer are integrated numerically over the parent finite elements by use of a computationally efficient Gaussian integration scheme. In this scheme, the radiant heat transfer is computed at discrete points on each surface in the radiation exchange. These points corresponding to the Gauss points are used in evaluating the element matrices.

The IM is implemented in the following iterative procedure:

  1. Initialize unknowns (temperatures and radiative heat fluxes).
  2. Calculate differential form factors between Gauss points on all elements.
  3. Calculate radiative heat fluxes at Gauss points on all elements.
  4. Integrate radiative heat fluxes to obtain a radiative-heating load vector.
  5. Solve for and update temperatures.
  6. Examine the results for convergence. If results have not converged to within acceptably narrow margins, return to step 3.

In a test problem, an upper plate of length L was assumed to be maintained isothermal at a temperature of 1,000 R (≈556 K) and a lower plate of the same length was assumed to be placed at the same horizontal position at a distance 0.1L below the upper plate and allowed to come to thermal equilibrium. The figure depicts the temperature of the lower plate versus position along the plate as calculated by the RSE method with cubic-spline interpolation and by the IM with 16 Gauss points. Also shown is the exact solution. Both methods show reasonably good agreement with the exact solution, with the integration method nearly indistinguishable from the exact solution over most of the plate. In general, the integration method proved to be more accurate with respect to spatial variation; however, it was also more costly (longer run times). Both methods captured temporal variations equally well. These results indicate that the RSE method is preferred for efficient analyses in which temperature variations are mainly temporal, while the integration method is reserved for analyses requiring very accurate resolution of spatial gradients.

This work was done by Dana C. Gould of Langley Research Center. For further information, access the Technical Support Package (TSP) free on-line at under the Information Sciences category. LAR-16101

NASA Tech Briefs Magazine

This article first appeared in the May, 2004 issue of NASA Tech Briefs Magazine.

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