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
 | (6–33) |
and
 | (6–34) |
Equation 6–33 and Equation 6–34 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 6–34 forms a set of linear algebraic equations for the unknown,
outgoing radiative flux (radiosity) at each surface. Equation 6–34 can
be rewritten as
 | (6–35) |
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 6–35 is solved using a Newton-Raphson procedure for the radiosity flux {qo}.
When the qo values are available, Equation 6–34 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
 | (6–36) |
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 6–35 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 6–35 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.
When condensed view factor is turned off (VFCO
command is not used (default) or issued with LEVEL = 0 ), the
radiosity equations are Equation 6–35 expressed in matrix
form:
 | (6–37) |
where
(assuming emissivity, 
, is constant for simplicity)
is the identity matrix.
For a model with symmetry, 
, where 
and 
are the loads on independent and dependent facets, respectively. Since loading
is symmetric, 
, and
 | (6–38) |
Here, 
matrix is decomposed into independent and dependent blocks. See View Factor Matrix for a Model with
Symmetry for more details.
Solving Equation 6–38 for
independent and dependent fluxes, 
and 
, it is typically found that they are close in value.
When condensed view factor is turned on (VFCO
command is issued with LEVEL = 1 or 2), the constraint
can be imposed by adding dependent row to independent row and dependent column
to independent column, and recalling that 
(Equation 6–19) and 
(Equation 6–20), Equation 6–38 becomes:
 | (6–39) |
 | (6–40) |
Thus, the problem is reduced to solving only for 
.
Note: Similar results are true for models with multiple planes of symmetry.
For solving radiation problems in 3D, the radiosity solution method calculates the view factors using the hemicube method as compared to the traditional double area integration method for 3D geometry. Details related to using the hemicube method for view factor calculation are given in Glass ([272]) and Cohen and Greenberg ([276]). For 2D and axisymmetric models, see View Factor Calculation (2D): 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:
 | (6–41) |
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.
Two methods in the AUX 12 processor are available for calculating 2D view factors: the Non-Hidden and Hidden methods.
The non-hidden procedure calculates a view factor for every surface to every other surface whether the view is blocked by an element or not. In this procedure, the following equation is used and the integration is performed adaptively.
For a finite element discretized model, Equation 6–15 for the view factor Fij between two surfaces i and j can be written as:
 | (6–42) |
where:
| m = number of integration points on surface i |
| n = number of integration points on surface j |
When the dimensionless distance between two viewing surfaces D, defined in Equation 6–43, is less than 0.1, the accuracy of computed view factors is known to be poor (Siegal and Howell([89])).
 | (6–43) |
where:
| dmin = minimum distance between the viewing surfaces A1 and A2 |
| Amax = max (A1, A2) |
So, the order of surface integration is adaptively increased from order one to higher orders as the value of D falls below 8. The area integration is changed to contour integration when D becomes less than 0.5 to maintain the accuracy. The contour integration order is adaptively increased as D approaches zero.
The hidden procedure is a simplified method which uses Equation 6–15 and assumes that all the variables are constant, so that the equation becomes:
 | (6–44) |
The hidden procedure numerically calculates the view factor in the following conceptual manner. The hidden-line algorithm is first used to determine which surfaces are visible to every other surface. Then, each radiating, or "viewing", surface (i) is enclosed with a hemisphere of unit radius. This hemisphere is oriented in a local coordinate system (x' y' z'), whose center is at the centroid of the surface with the z axis normal to the surface, the x axis is from node I to node J, and the y axis orthogonal to the other axes. The receiving, or "viewed", surface (j) is projected onto the hemisphere exactly as it would appear to an observer on surface i.
As shown in Figure 6.10: Receiving Surface Projection, the projected area is defined by first extending a line from the center of the hemisphere to each node defining the surface or element. That node is then projected to the point where the line intersects the hemisphere and transformed into the local system x' y' z', as described in Kreyszig([24])
The view factor, Fij, is determined by counting the number of rays striking the projected surface j and dividing by the total number of rays (Nr) emitted by surface i. This method may violate the radiation reciprocity rule, that is, AiFi-j ≠ Aj Fj-i.
When the radiation view factors between the surfaces of axisymmetric bodies are calculated (GEOM,1,n command), special logic is used. In this logic, the axisymmetric nature of the body is exploited to reduce the amount of computations. The user, therefore, needs only to build a model in plane 2D representing the axisymmetric bodies as line "elements."
Consider two axisymmetric bodies A and B as shown in Figure 6.11: Axisymmetric Geometry.
The view factor of body A to body B is calculated by expanding the line "element" model into a full 3D model of n circumferential segments (GEOM,1,n command) as shown in Figure 6.12: End View of Showing n = 8 Segments.
View factor of body A to B is given by
 | (6–45) |
where:
Fk -  = view factor of segment k on body A to segment  on body B |
The form factors between the segments of the axisymmetric bodies are computed using the method described in the previous section. Since the coefficients are symmetric, the summation Equation 6–45 may be simplified as:
 | (6–46) |
Both hidden and non-hidden methods are applicable in the computation of axisymmetric view factors. However, the non-hidden method should be used if and only if there are no blocking surfaces. For example, if radiation between concentric cylinders are considered, the outer cylinder can not see part of itself without obstruction from the inner cylinder. For this case, the hidden method must be used, as the non-hidden method would definitely give rise to inaccurate view factor calculations.
A space node may be defined (SPACE command) to absorb all energy not radiated to other elements. Any radiant energy not incident on any other part of the model will be directed to the space node. If the model is not a closed system, then the user must define a space node with its appropriate boundary conditions.