## 2 Previous Models
### Non-Local Radiation: Some 3D Effects
We also assumed that the dimensionless solid fuel begins to be lost when it reaches this temperature, \(u_{p}\).This means a simplification of the right-hand side of Equation (18), which represents the loss of solid fuel due to combustion. Now, \(g(u)=0\) when \(u<u_{p}\), and \(g(u)\) is constant when \(u=u_{p}\), where this constant is inversely proportional to the half-life time of the combustion of each type of fuel. It remains to explain how the non-local radiation is computed, that is the \(r\) term on the right-hand side of Equation (16). Since radiation essentially comes from flames, we considered a simplified physical model in which the gases produced by pyrolysis burn above the fuel layer, producing a flame above that layer. The flame may eventually be tilted due to the wind or the slope of the ground and emits radiation, which reaches points ahead of the flame, heating the surrounding unburned fuel and allowing the fire to spread. The term \(r\) describes the thermal radiation, which reaches the surface \(S\) from the flame above the layer  
\[r=\frac{[t]}{MCT_{\infty}}R. \tag{20}\]  
where \([t]\) is a time scale that appears during the dimensionless process. \(R\) represents the incident energy at a point \(\mathbf{x}=(x,y,h(x,y))\) of the surface \(S\), due to radiation from the flame over the surface per unit time and per unit area, obtained by summing up the contribution of all directions \(\mathbf{\Omega}\), i.e.,  
\[R(\mathbf{x})=\int_{\omega=0}^{2\pi}I(\mathbf{x},\mathbf{\Omega})\mathbf{ \Omega}\cdot\mathbf{N}\;d\omega, \tag{21}\]  
where \(\omega\) is the solid angle and \(\mathbf{N}\) is the unit normal vector to the surface \(S\). We considered only the hemisphere above the fuel layer, and each contribution depends, among others, on the flame height \(F\) and the flame temperature \(T_{f}\). \(I\) is the total radiation intensity, described by the following differential equation:  
\[\frac{dI}{ds}+a(s)I(s)=a(s)I_{b}(s), \tag{22}\]  
where \(I_{b}\) is the black body total radiation intensity, and it is governed by the Stefan-Boltzmann law:  
\[I_{b}=\frac{\sigma}{\pi}T^{4}, \tag{23}\]  
where \(\sigma=5.6699\times 10^{-8}\) wm\({}^{-2}\)K\({}^{-4}\) is the Stefan-Boltzmann constant and \(a\) is the radiation absorption coefficient inside the flame. The path \(s\) is within the 3D domain \(D\) reaching any point on the surface \(S\) and eventually passing through a flame. We assume that the temperature \(T\) reaches the flame temperature \(T_{f}\). Details on how to approximate Equation (22) and calculate \(R(x)\) for different types of flame, vertical or tilted, can be found in [32]. We include here some aspects of the calculation of the radiation term for the case of a vertical flame in order to clarify some concepts that were introduced to reduce the computational time and to enhance the efficiency of the simulation process. In our model, the radiation term R for a vertical rectangular flame has the form:  
\[R(\mathbf{x})=\frac{a\sigma T_{f}^{4}}{\pi}\int_{\Omega}\chi(\mathbf{\hat{x}} )f(\mathbf{x}-\mathbf{\hat{x}})d\bar{A}, \tag{24}\]  
where \(\chi\) is the characteristic function of the flames, i.e., \(\chi(\mathbf{x})=1\) if \(T(\mathbf{x})=T_{p}\) and \(\chi(\mathbf{x})=0\) otherwise, and \(T_{p}\) represents the pyrolysis temperature. We used the P1 finite-element method for the spatial discretization, so using the finite-element basis to represent the \(f\) function, the radiation term can be written for each node \(\mathbf{x}_{i}\) of the mesh as follows:  
\[R(\mathbf{x}_{i})=\frac{a\sigma T_{f}^{4}}{\pi}\sum_{j}\mathrm{R}_{ij}A_{j} \chi(\mathbf{x}_{j})\]  
where \(\mathrm{R}=(R_{ij}=f^{i}(\mathbf{x}_{i}))\), is the _Radiation Matrix_, representing the nodes of the finite element mesh, which are reached by the radiation emitted by each node, and it is computed only once (outside the time loop). In practice, the radiation term is only calculated in a neighborhood of the fire front, which we call _Active Nodes_.