In the radiation matrix method, for a system of two radiating surfaces, Equation 6–14 can be expanded as:

(6–47)

or

(6–48)

where:

K' cannot be calculated directly since it is a function of the unknowns T_i and T_j. The temperatures from previous iterations are used to calculate K' and the solution is computed iteratively.

For a more general case, Equation 6–12 can be used to construct a single row in the following matrix equation:

(6–49)

such that:

(6–50)

(6–51)

Solving for {Q}:

(6–52)

and therefore:

(6–53)

Equation 6–52 is analogous to Equation 6–12 and can be set up for standard matrix equation solution by the process similar to the steps shown in Equation 6–47 and Equation 6–48.

(6–54)

[K'] now includes T³ terms and is calculated in the same manner as in Equation 6–48). To be able to include radiation effects in elements other than LINK31, MATRIX50 (the substructure element) is used to bring in the radiation matrix. MATRIX50 has an option that instructs the solution phase to calculate [K']. The AUX12 utility is used to create the substructure radiation matrix. AUX12 calculates the effective conductivity matrix, [K^ts], in Equation 6–52, as well as the view factors required for finding [K^ts]. The user defines flat surfaces to be used in AUX12 by overlaying nodes and elements on the radiating edge of a 2D model or the radiating face of a 3D model.