Solution of Pure Scattering Radiation Transport Equation (RTE) Using Finite Difference Method (FDM)

Solution of Pure Scattering Radiation Transport Equation (RTE) using Finite Difference Method (FDM)

Abstract. Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering. Radiative Transfer Equation (RTE) have been applied in a many subjects including optics, astrophysics, atmospheric science, remote sensing, etc. Analytic solutions for RTE exist for simple cases, but, for more realistic media with complex multiple scattering effects, numerical methods are required. In the RTE, six different independent variables define the radiance at any spatial and temporal point. By making appropriate assumptions about the behavior of photons in a scattering medium, the number of independent variables can be reduced. These assumptions lead to the diffusion theory (or diffusion equation) for photon transport. In this work, the diffusive form of RTE is discretized, using a Forward-Time Central-Space (FTCS) Finite Difference Method (FDM). The results reveal the radiance penetration according to Beer-Lambert law.

Keywords: Radiation Transport Modelling (RTM), Radiation Transport Equation (RTE) Finite Difference Method (FDM), Forward-Time Central-Space (FTCS)

Hassan A. Khawaja

Department of Engineering and Safety (IIS-IVT),

UiT The Arctic University of Norway, 9037, Tromsø, Norway [email protected]

SCIA 2017 - SCANDINAVIAN CONFERENCE ON IMAGE ANALYSIS, 12-14 JUNE 2017, TROMSØ

Radiation Transport Equation.

𝜕𝜕𝐼𝐼_𝑣𝑣 𝒓𝒓, �𝒏𝒏, 𝑡𝑡

𝑐𝑐𝜕𝜕𝑡𝑡 + �Ω. 𝛻𝛻𝐼𝐼_𝑣𝑣 𝒓𝒓, �𝒏𝒏, 𝑡𝑡 + 𝑘𝑘_{𝑣𝑣,𝑠𝑠} + 𝑘𝑘_{𝑣𝑣,𝑎𝑎} 𝐼𝐼_𝑣𝑣 𝒓𝒓, �𝒏𝒏, 𝑡𝑡 = 𝑗𝑗_𝑣𝑣 𝒓𝒓, 𝑡𝑡 + 1

4𝜋𝜋 𝑘𝑘^{𝑣𝑣,𝑠𝑠} �

Ω

𝐼𝐼_𝑣𝑣 𝒓𝒓, �𝒏𝒏, 𝑡𝑡 𝑑𝑑Ω

𝐼𝐼_𝑣𝑣 is spectral radiance of electromagnetic waves 𝑐𝑐 is speed of light

�Ω is the vectorial position of a solid angle

𝑘𝑘_{𝑣𝑣,𝑠𝑠} is the scattering opacity of the medium

𝑘𝑘_{𝑣𝑣,𝑎𝑎} is the absorption opacity of the medium

𝑗𝑗_𝑣𝑣 is the emission coefficient of the medium 𝑡𝑡 is time variable

Two Dimensional Pure Scattering Radiation Transport Equation.

𝜕𝜕𝜙𝜙_𝑣𝑣 𝒓𝒓, 𝑡𝑡

𝑐𝑐 𝜕𝜕𝑡𝑡 = 𝐷𝐷𝛻𝛻^𝟐𝟐𝜙𝜙_𝑣𝑣 𝒓𝒓, 𝑡𝑡 = 𝐷𝐷 𝜕𝜕^𝟐𝟐𝜙𝜙_𝑣𝑣 𝒓𝒓, 𝑡𝑡

𝜕𝜕𝑥𝑥^𝟐𝟐 + 𝜕𝜕^𝟐𝟐𝜙𝜙_𝑣𝑣 𝒓𝒓, 𝑡𝑡

𝜕𝜕𝑦𝑦^𝟐𝟐 𝜙𝜙_𝑣𝑣(𝒓𝒓, 𝑡𝑡) is radiance intensity

𝑐𝑐 is speed of light

𝑥𝑥, 𝑦𝑦 are the space dimensions 𝐷𝐷 is diffusion coefficient

𝑡𝑡 is time variable

Forward-Time Central-Space (FTCS) Finite Difference Method.

𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_{𝑖𝑖,𝑗𝑗}^𝑡𝑡+1

= 𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_{𝑖𝑖,𝑗𝑗}^𝑡𝑡

+ 𝐷𝐷 𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_{𝑖𝑖+1,𝑗𝑗}^𝑡𝑡

− 2𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_{𝑖𝑖,𝑗𝑗}^𝑡𝑡

+ 𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_{𝑖𝑖−1,𝑗𝑗}^𝑡𝑡

∆𝑥𝑥

²

∆𝑡𝑡 + 𝐷𝐷 𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_𝑖𝑖^𝑡𝑡_,𝑗𝑗+1

− 2𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_𝑖𝑖^𝑡𝑡_,𝑗𝑗

+ 𝜕𝜕𝜙𝜙

_𝑣𝑣

𝒓𝒓, 𝑡𝑡

_𝑖𝑖^𝑡𝑡_{,𝑗𝑗−1}

∆𝑦𝑦

²

∆𝑡𝑡

subscripts i and j represent the computational points in the space domain superscript t represents the transient state

Figure.

Contour plots with various radiance intensities

.

Conclusion. The Radiation Transport Equation (RTE) can be solved using a Forward-Time Central-Space (FTCS) Finite Difference Method (FDM) in special cases such as a highly diffusive mediums (as discussed above). The results show the radiance intensity in space, which reflects on the radiance penetration.

Future Work. This work can be further extended by varying the diffusion coefficient and observing its impact on the radiance penetration. It will help to find the validity limits of Radiation Transport Equation (RTE) solution using Forward-Time Central-Space (FTCS) Finite Difference Method (FDM).

Figure.

Radiance intensities in the two- dimensional discretized domain. Indices i and j refer to the nodal position of radiance intensity.

Figure.

Flow chart of the method of solution.

Loop over time and space!

No!

Start

Define total time, total space variables…

Allocate size, dimensions to radiance intensity variable…

Define boundary conditions…

Solve equation (over time and space) … Define time steps, space steps variables…

Define speed of light DQG spectral diffusion

Meet CFL criteria…

Yes!

Define Lnitial conditions…

Loop over time and space finishes!

Check Uesults…

݅ = 1 ݅ = 2 ݅ = 3 ݅ = 4 ݅ = 5 ݅ = 6 … ݅ =݊

݆ = 1

݆ = 2

݆ = 3

݆ = 4

݆ = 5

݆ = 6 ڭ

݆ =݉

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