Solution of inter-reflection equations -- Algorithm 3.24

This algorithm is the Gauss-Seidel iterative method, with initial estimates based on Sumpner's formula for inter-reflected light(1) .

Input

Total surface area, A

Area-weighted mean surface reflectance, `r

Tolerance in numerical solution, e

For every finite area i, where i=1 to n:

Area, Ai

Direct illuminance, Edi

Reflectance, ri

Form factor between area i and every other area j, Fi-j

Equation

For every element i, the final illuminance Ei is given by

Ei = Edi + Fi-1 r1 E1 + Fi-2 r2 E2 + ... + Fi-n rn En

To form an iterative method, the previously calculated value of each Ei is used on the right-hand side of the equation. Hence the estimate of illuminance after m iterations is

Ei(m) = Edi + Fi-1 r1 E1* + Fi-2 r2 E2* + ... + Fi-n rn En*

where Ek* is the most recent value, which is Ek*= Ek(m) for k=1 to i-1, and Ek*= Ek(m-1) for k=i to n.

The iterations are repeated until |Ei(m) - Ei(m-1) | < e for all finite areas.

The initial estimates of illuminance are given by the direct illuminances plus an estimate of inter-reflected light:

Source

Littlefair(2)

References

1. Sumpner W E The diffusion of light The London, Edinburgh, and Dublin Philosphical Magazine and Journal of Science 5th Series 35 (213) 81-97 (1893)

2. Littlefair P J Interreflection calculations: Improving convergence Lighting Research and Technology 23 (4) 175-177 (1991)