Bidirectional reflectance distribution functions (BRDF)

With the Monte Carlo method, the thermal solver launches rays onto an element in random directions from random locations. When a ray hits a surface, the behavior of the ray depends on the properties of that surface. All surfaces have the following radiative properties:

• Absorptivity, A

• Diffuse reflectivity, Rd

• Specular reflectivity, Rs

• Diffuse transmissivity, Td

• Transmissivity, T

These properties are related by the following equation:

A+Rd+Rs+Td+T=1

The behavior of the ray is determined by sampling the random variable, ξ, and comparing its value to the conditions in the following table.

Condition	Behavior of the ray
ξ ≤ A	Absorption
A < ξ ≤ A+Rd	Diffuse reflection
A+Rd < ξ ≤ A+Rd+Rs	Specular reflection
A+Rd+Rs < ξ ≤ A+Rd+Rs+Td	Diffuse transmission
A+Rd+Rs+Td < ξ ≤ A+Rd+Rs+Td+T	Transmission

In real materials, reflection is rarely purely diffuse. For the Monte Carlo method, the thermal solver substitutes the diffuse reflectivity of the material by a bidirectional reflectance distribution function (BRDF), thereby allowing you to simulate these complicated reflectance properties.

In general, BRDF gives the probability distribution of a ray reflecting in a direction (θr, Φr) for each incoming direction (θi, Φi). Thus, BRDF is a four-dimensional function expressed as ρ(θi, Φi,θr, Φr)

In the thermal solver, BRDF is simplified by neglecting the dependence on Φi and Φr. This approximation assumes that BRDF is a two-dimensional table function ρ(θi,θr) where:

• θi is the angle of incidence of the ray

• θr is the angle of refection of the ray

The following figure illustrates a simplified BRDF for one incoming angle, θi. The distance from the intercept point to the curve is the value of the probability density function, ρ(θi, Φi,θr, Φr).

BDRF implementation in the thermal solver

The directional reflectance, Rd, is given by:

and the hemispherical reflectance, Rd-h, is given by:

Note that for a purely diffusive case BRDF is equal to:

One consequence of using the BRDF model instead of a purely diffusive reflection model is that the reflectivity of a material becomes dependent on the incoming ray direction. Thus, Rd depends on the incoming direction. However, since the laws of physics require that A+Rd+Rs+Td+T=1, the absorptivity, A, is also adjusted so that:

A(θi, Φi)=1–Rd(θi, Φi)–Rs—Td–T

Similarly, the directional emissivity is given by:

ε(θi, Φi)=1–Rd(θi, Φi)–Rs–Td–T

and the total effective emissivity is adjusted according to:

εtot=1–Rd-h–Rs–Td–T

Note that the purely specular component of reflection, Rs, is retained as a separate optical property. If the full four-dimensional BRDF representation were used, the specular component could be incorporated into the BRDF.

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