1. Introduction

The goal of MonteCarlo method is to calculate the heat transfert between two surface. MonteCarlo method use finite element and probabilist theory.

First, the MonteCarlo method calculates the view factor between two surface.

The MonteCarlo method consist to decompose the surfaces (emissive surface \(A_i\) and absortive surface \(A_j\)) in little finite element. Here the finite element is triangle.

Figure 1. Rays generate by \(A_i\) and intercept by \(A_j\)

We generate \(N_r\) random ray from \(N_s\) random source point, from emissive surface \(A_i\). Those rays touch the finite element of absorbent surface \(A_j\).

The local view factor between element \(e^i_k\) of \(A_i\) and element \(e^j_l\) of \(A_j\) is calculate by subsurface summation :

\[F_{e^i_k \rightarrow e^j_l} = \frac{n}{N} = \frac{n}{N_s N_r}\]

With :

  • \(N=N_s N_r\) the total number of rays generate by \(A_i\)

  • \(n\) the number of rays generate by \(A_i\) intercept by \(A_j\)

After calculating all local view factor, we can compute the global view factor with :

\[F_{A_i \rightarrow A_j} = \frac{1}{A_i} \sum_{k=1}^{K}{ \left( Area(e^i_k) \sum_{l=1}^{L}{F_{e^i_k \rightarrow e^j_l}} \right) }\]

After calculating the view factor, it calculates the heat transfert between surfaces \(A_i\) and \(A_j\), with the formula :

\[q_{A_i \rightarrow A_j} = \sigma F_{A_i \rightarrow A_j} (T_j^4 - T_i^4)\]

With \(T_i\) the absolute temperature of \(A_i\) and \(T_j\) the absolute temperature of \(A_j\).

2. Tools

2.1. Visibility of surface

Before calculating the view factor, it is interesting to know if the to surface are visible to each other.

For that, we use the oriented normal associated to element.

Set oriented surfaces element \(S_1\) and \(S_2\) (with oriented normal : \(\mathbf{n_1}\) and \(\mathbf{n_2}\)) :

\[S_1 \text{ and } S_2 \text{ are visible } \Longleftrightarrow \mathbf{n_1} \cdot \mathbf{n_2} < 0\]
Figure 2. Visibility between two surface

2.2. Distribution of random ray

For the emissive element, we generate some \(N_s\) source point and \(N_r\) ray from one source point.

For that we choose good supposition on matters :

  • homogeneous (the emission is the same for any position)

  • anisotropic (the emission is the same for any direction)

The distributions :

  • Source point \(O\) : \(\left\{\begin{eqnarray} x_O = x_C + R \sqrt{\xi_2} cos(2\pi \xi_1) \\ y_O = x_C + R \sqrt{\xi_2} sin(2\pi \xi_1) \end{eqnarray}\right.\) with \(\xi_1\) and \(\xi_2\) wich follow uniformely distribution in \([0,1]\) until in finite element.

  • Incident angle \(\phi\) of ray follow uniformely distribution in \([0,2\pi]\)

  • Azimuthal angle \(\theta\) of ray follow uniformely distribution in \([0,\pi]\)

For more detail, see the section Distribution of random rays.

2.3. Interception of ray by square

We have the conditon to intersection between right and square \(\mathcal{R}\) (of origin point \(P\) and directional vector \(\mathbf{e_1}\) and \(\mathbf{e_2}\)).

ray square
Figure 3. Interception of ray by square
\[\text{The ray } \mathcal{C} \text{ is intersept by } \mathcal{T} \Longleftrightarrow \left\{ \begin{eqnarray} \left( \mathbf{v} \wedge \mathbf{e_2} \right) \cdot \mathbf{e_1} \ne 0 \\ 0 \leq \frac{ \vec{OP} \cdot \left( \mathbf{e_2} \wedge \mathbf{v} \right)}{\left( \mathbf{e_2} \wedge \mathbf{v} \right) \cdot \mathbf{e_1} } \leq 1 \\ 0 \leq \frac{ \vec{OP} \cdot \left( \mathbf{e_1} \wedge \mathbf{v} \right)}{\left( \mathbf{e_1} \wedge \mathbf{v} \right) \cdot \mathbf{e_2} } \leq 1 \\ \frac{ \vec{OP} \cdot \left( \mathbf{e_1} \wedge \mathbf{v} \right)}{\left( \mathbf{e_1} \wedge \mathbf{v} \right) \cdot \mathbf{e_2} } + \frac{ \vec{OP} \cdot \left( \mathbf{e_2} \wedge \mathbf{v} \right)}{\left( \mathbf{e_1} \wedge \mathbf{v} \right) \cdot \mathbf{e_2} } \leq 1 \end{eqnarray} \right.\]

3. Algorithm

4. Efficiency of method

Discussion of correctness, convergence of method


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