In the radiosity solution method for the analysis of gray diffuse radiation between N surfaces, Equation 6–12 is solved in conjunction with the basic conduction problem.
For the purpose of computation it is convenient to rearrange Equation 6–12 into the following series of equations
 | (6–40) |
and
 | (6–41) |
Equation 6–40 and Equation 6–41 are
expressed in terms of the outgoing radiative fluxes (radiosity) for each surface, 
, and the net flux from each surface qi. For known
surface temperatures, Ti, in the enclosure, Equation 6–41 forms a set of linear algebraic equations for the unknown,
outgoing radiative flux (radiosity) at each surface. Equation 6–41 can
be rewritten as
 | (6–42) |
where:
 |
|
 |
[A] is a full matrix due to the surface to surface coupling represented by the view factors and is a function of temperature due to the possible dependence of surface emissivities on temperature. Equation 6–42 is solved using a Newton-Raphson procedure for the radiosity flux {qo}.
When the qo values are available, Equation 6–41 then allows the net flux at each surface to be evaluated. The net flux calculated during each iteration cycle is under-relaxed, before being updated using
 | (6–43) |
where:
| φ = radiosity flux relaxation factor |
| k = iteration number |
The net surface fluxes provide boundary conditions to the finite element model for the conduction process. The radiosity Equation 6–42 is solved coupled with the conduction Equation 6–12 using a segregated solution procedure until convergence of the radiosity flux and temperature for each time step or load step.
The surface temperatures used in the above computation must be uniform over each surface in order to satisfy conditions of the radiation model. In the finite element model, each surface in the radiation problem corresponds to a face or edge of a finite element. The uniform surface temperatures needed for use in Equation 6–42 are obtained by averaging the nodal point temperatures on the appropriate element face.
For open enclosure problems using the radiosity method, an ambient temperature needs to be specified using a space temperature (SPCTEMP command) or a space node (SPCNOD command), to account for energy balance between the radiating surfaces and the ambient.
For solving radiation problems in 3-D, the radiosity solution method calculates the view factors using the hemicube method as compared to the traditional double area integration method for 3-D geometry. Details related to using the hemicube method for view factor calculation are given in Glass ([273]) and Cohen and Greenberg ([277]). For 2-D and axisymmetric models, see View Factor Calculation (2-D): Radiation Matrix Method and View Factors of Axisymmetric Bodies.
The hemicube method is based upon Nusselt's hemisphere analogy. Nusselt's analogy shows that any surface, which covers the same area on the hemisphere, has the same view factor. From this it is evident that any intermediate surface geometry can be used without changing the value of the view factors. In the hemicube method, instead of projecting onto a sphere, an imaginary cube is constructed around the center of the receiving patch. A patch in a finite element model corresponds to an element face of a radiating surface in an enclosure. The environment is transformed to set the center of the patch at the origin with the normal to the patch coinciding with the positive Z axis. In this orientation, the imaginary cube is the upper half of the surface of a cube, the lower half being below the 'horizon' of the patch. One full face is facing in the Z direction and four half faces are facing in the +X, -X, +Y, and -Y directions. These faces are divided into square 'pixels' at a given resolution, and the environment is then projected onto the five planar surfaces. Figure 6.8: The Hemicube shows the hemicube discretized over a receiving patch from the environment.
The contribution of each pixel on the cube's surface to the form-factor value varies and is dependent on the pixel location and orientation as shown in Figure 6.9: Derivation of Delta-View Factors for Hemicube Method . A specific delta form-factor value for each pixel on the cube is found from modified form of Equation 6–15 for the differential area to differential area form-factor. If two patches project on the same pixel on the cube, a depth determination is made as to which patch is seen in that particular direction by comparing distances to each patch and selecting the nearer one. After determining which patch (j) is visible at each pixel on the hemicube, a summation of the delta form-factors for each pixel occupied by patch (j) determines the form-factor from patch (i) at the center of the cube to patch (j). This summation is performed for each patch (j) and a complete row of N form-factors is found.
At this point the hemicube is positioned around the center of another patch and the process is repeated for each patch in the environment. The result is a complete set of form-factors for complex environments containing occluded surfaces. The overall view factor for each surface on the hemicube is given by:
 | (6–44) |
where:
| N = number of pixels |
| ΔF = delta-view factor for each pixel |
The hemicube resolution (input on the HEMIOPT command) determines the accuracy of the view factor calculation and the speed at which they are calculated using the hemicube method. Default is set to 10. Higher values increase accuracy of the view factor calculation.