Volumetric Photon Monte Carlo (VPMC) Radiation

The Volumetric Photon Monte Carlo (VPMC) method is a statistical method for solving the radiative transfer equation (RTE) given by Eqn. (1721). In this method, the radiative processes (such as emission, absorption, scattering, or boundary interaction) are modeled explicitly in a stochastic sense, as opposed to obtaining a numerical solution of the RTE or its simplified form. The radiative processes in the VPMC method are modeled by tracing the history of a large number of photon bundles, representing samples of radiative energy, in the computational domain. The generation of these photon bundles mimics the emission of photons, and the interaction of these photon bundles with the medium and the surfaces as they travel in the computational domain represents absorption, scattering, and the radiation boundary treatment.

The tracing procedure, and thus the evolution of the photon bundles in the VPMC method, is governed by the underlying probability density functions (PDFs) of the radiative processes. These PDFs provide a statistical characterization of the radiative processes, and are used during ray tracing to prescribe the emission of photon bundles and their subsequent interaction with the media or surfaces. To give an example, for an arbitrary photon bundle that is to be emitted, the PDFs characterizing emission are randomly sampled and the starting location and the direction are specified. Then, as that photon bundle is traveling, the boundary and media interaction PDFs are randomly sampled for specifying boundary and media treatment. An adequate number of photon bundles need to be considered for accurate representation of the PDFs and, consequently, for obtaining meaningful statistics during the ray tracing procedure. The statistics are recorded in the mesh cells and faces and patches to obtain the radiative quantities of interest, such as incident radiation, absorption, and radiative heat flux, which are then used to compute source terms for the energy equation. For more information on Photon Monte Carlo method for radiation, see Modest [398].

VPMC Modeling in Simcenter STAR-CCM+

The radiative processes listed below are modeled stochastically in the VPMC model:

Surface emission
Boundary treatment
Volumetric emission (emission from media)
Tracing/movement of photon bundles
Absorption and scattering in participating media

Modeling of surface-specific processes in VPMC is similar to that in SPMC. See Emission Modeling and Boundary Treatment Modeling in Surface Photon Monte Carlo (SPMC) Radiation for detailed modeling information of these processes.

Volumetric Emission (Emission from Media)

The emission of radiative energy from media is modeled as the generation of photon bundles. A number of photon bundles are generated in a cell and each of these bundles have a direction vector and a starting location in the cell.

Number of Photon Bundles from a Given Cell

The total number of photon bundles to be emitted from cells (which is given by user-specified rays per cell times the total number of participating cells) is distributed among cells such that the cells with higher emissive power or volume emit more photon bundles than cells with lesser emissive power or volume. This power and size weighting in the photon bundle sampling allows for better representation of volumetric emission and facilitates lower statistical noise in the results.

Direction of Emission

An arbitrary direction is sampled in the spherical space about the location of emission, using the following expressions:

Figure 1. EQUATION_DISPLAY

ψ = 2 π R_{ψ}; θ = \cos^{-1} (1.0 - 2.0 R_{θ})

(1756)

where:

$ψ$ and $θ$ are the azimuthal and polar angles, respectively, for the emission direction.
$R_{ψ}$ and $R_{θ}$ are the uniform random numbers ( $\in [0, 1]$ ) used in the sampling.

Emission Location in a Cell

The emission process occurs inside the entire cell and is correspondingly modeled by randomly choosing the location of emission inside the cell. For an arbitrary cell, the emission location can be sampled using the expressions below:

Figure 2. EQUATION_DISPLAY

\begin{matrix} \begin{matrix} \begin{matrix} R_{x} = \frac{1}{E_{c}} \int_{V_{c} \forall x^{'} < x}^{} 4.0 κ σ T^{4} d V \end{matrix} \\ \begin{matrix} R_{y} = \frac{1}{E_{c}} \int_{V_{c} \forall y^{'} < y}^{} 4.0 κ σ T^{4} d V \end{matrix} \end{matrix} \\ \begin{matrix} R_{z} = \frac{1}{E_{c}} \int_{V_{c} \forall z^{'} < z}^{} 4.0 κ σ T^{4} d V \end{matrix} \end{matrix}

(1757)

where:

$V_{c}$ is the cell volume.
$E_{c}$ is the emissive power of the cell ( $E_{c} = \int_{V_{c}}^{} 4.0 κ σ T^{4} d V$ ).
$κ$ is the absorption coefficient.
$R_{x}$ , $R_{y}$ , and $R_{z}$ are the random numbers between 0 and 1.

The above expressions are inverted to get the (x, y, z) emission location.

Multi-Band Modeling

The spectral band is not randomly sampled and each photon bundle carries energy from the various spectral bands considered in the simulation.

Tracing and Movement of Photon Bundles

The radiative energy generated in the media in the form of photon bundles is transported in the simulation along their direction vectors. The photon bundles are traced along the direction of emission, and absorption and scattering are modeled as these bundles travel through the cells.

Absorption and Scattering in Participating Media

Photon bundles traveling through cells undergo absorption and scattering as they interact with the medium. These processes are modeled by computing the probability of a photon bundle undergoing absorption or scattering as it travels through an arbitrary cell. For a photon bundle that is traveling a distance l inside the cell, the probability of the photon bundle undergoing absorption or scattering is given by:

Figure 3. EQUATION_DISPLAY

P {(E)}_{} = e^{- (κ_{a} + σ_{s}) l}

(1758)

where:

$P (E)$ is the probability of an absorption or scattering event.
$κ_{a}$ is the absorption coefficient.
$σ_{s}$ is the scattering coefficient in the cell.

Inverting the above expression and using a uniform random number to represent the probability of the attenuation event (absorption or scattering), the distance that a photon bundle travels before getting attenuated can be computed as:

Figure 4. EQUATION_DISPLAY

d_{E} = - \frac{\log (R_{E})}{(κ_{a} + σ_{s})}

(1759)

where $R_{E}$ is a uniform random number that represents the occurrence of an attenuation event.

The attenuation event is modeled at this distance $d_{E}$ if it is less than the distance $l_{c}$ that the photon bundle travels inside the cell. If the distance is greater than $l_{c}$ , then tracing of the photon bundle continues through the neighboring cell. In the event that the attenuation of the photon bundle occurs, the mode of attenuation is selected appropriately between absorption and scattering.

The absorption attenuation event is modeled by depositing the photon bundle energy in the cell. For scattering events, the photon bundle is moved to the location corresponding to distance $d_{E}$ inside the cell and the direction vector of the photon bundle is modified.

Direction Change Upon Scattering

For an isotropic scattering specification, the new direction is computed by randomly sampling from a unit sphere in a way similar to how the direction is defined for volumetric emission modeling above. For an anisotropic specification, which is prescribed by a non-zero input for the asymmetry parameter, the Henyey-Greenstein phase function ([398]) is used to sample a random direction for anisotropic scattering. The Henyey-Greenstein phase function is given by:

Figure 5. EQUATION_DISPLAY

P (θ) = \frac{1}{4 π} \frac{1 - g^{2}}{{[1 + g^{2} - 2 g \cos (θ)]}^{3 / 2}}

(1760)

where:

$P (θ)$ is the probability of scattering at a polar angle $θ$ .
$g$ is the asymmetry parameter for anisotropic scattering.

Various radiative processes in VPMC (such as emission, absorption and scattering in participating media, boundary emission and boundary treatment) are modeled by generation and tracing of photon bundles as described above. During the tracing of photon bundles, incident radiation and absorption in cells are recorded along with the boundary irradiation and absorption on the boundary patches, and these quantities are treated to lower the statistical noise before computing the cell and boundary source terms for the energy equation. The treatment is applied by the application of the statistical sampling factor, which is a user-specified quantity. The statistical sampling factor is described in detail in the boundary treatment for SPMC and that description is applicable for boundary treatment in VPMC. The treatment at cell level in VPMC employs a similar approach for incident radiation. The sampling factor is a controlling parameter for the update of incident radiation in Volumetric PMC. The incident radiation computed from a single PMC ray trace calculation at an arbitrary iteration contains statistical noise due to the use of a finite number of photon bundles. To address this statistical noise, only a portion of this newly computed iteration-specific incident radiation is used to update the main incident radiation field, as controlled by the sampling factor in the following manner:

Figure 6. EQUATION_DISPLAY

G = f_{s s f} G_{P M C} + (1.0 - f_{s s f}) G

(1761)

where:

$G$ is the incident radiation.
$f_{s s f}$ is the sampling factor.
$G_{P M C}$ is the incident radiation computed from the latest PMC compute call.

The $G$ field is used in computing the cell-wise radiative sources for the energy equation.

You are advised to use a high sampling factor at the start of a simulation or during any transient, to allow for faster feedback of effect of radiation on the energy solution. As the solution evolves to a statistically stationary state, the sampling factor can be gradually reduced to a lower value to collect the statistics over a number of iterations.