## 2 Method
### Inversion Process  
In summary, several experiments have been done and, given a thermal program, various heat flow-rates have been measured during time. Further, a numerical model has been built so as to compute the numerical heat-flow rate for a given set of input parameters \(\overrightarrow{p}\); that is to say for different values  
Figure 4: Convergence study. (**a**) Spatial discretization; (**b**) Temporal discretization. of the thermodynamic and thermal properties, i.e., for a given \(\overrightarrow{p}=\left(\lambda_{s},\lambda_{l},c_{s},c_{l},T_{M},T_{M}^{*},L_{M },R_{L},R_{R}\right)\). These nine parameters correspond to the thermodynamical ones associated with the phase change, and to the heat transfers ones with the thermal conductivities of the material, as well as the thermal contact resistances on both sides. They are determined during transient heating and cooling modes, involving both fusion and solidification. Given these two data, i.e., the experimental and numerical heat flow-rates, a criterion can be defined in order to assess the correctness of the estimation of \(\overrightarrow{p}\), here thanks to the classic least-square error:  
\[f(\overrightarrow{p})=\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}\left(\varphi_ {L}^{\text{exp}}(t)-\varphi_{L}^{\text{num}}(t,\overrightarrow{p})\right)^{2} +\left(\varphi_{R}^{\text{ext}}(t)-\varphi_{R}^{\text{num}}(t,\overrightarrow{p })\right)^{2}\!\text{d}t \tag{6}\]  
The basic principle of the inversion step is thus to minimize this cost function \(f\) over \(\overrightarrow{p}\), which can be done in different ways, as for instance using a simplex approach [75] or a Levenberg-Marquardt method [76] or genetic algorithms [77; 78]. In the present case, the identifications were performed using the GenOpt free tool from the Lawrence Berkeley National Laboratory [79; 80]. It indeed implements several classic algorithms, including the 'GPSCoordinateSearch' that has been used here: the basic principle of the generalized pattern search (GPS) is to link monodimensional minimizations according to the different directions of the search space (dimension \(N\)). The version implemented in GenOpt automatically adapts the search accuracy to obtain an acceptable solution without requiring too much computation time. As any optimization method, GenOpt requires to define all the \(N\) parameters to be identified (\(\overrightarrow{p}\in\mathbb{R}^{N}\)) so as to minimize the cost function \(f(\overrightarrow{p}):\mathbb{R}^{N}\longrightarrow\mathbb{R}\) given by relation (6). Let us mention here that GenOpt always delegates the evaluation of the cost function \(f\) to an external program, which corresponds here to the code implementing the aforementioned system of PDE. From a practical point of view, the raw text files that ensure the communication between these two threads are configured following the documentation [81]. As a rule of thumb, all identified parameters are described by their name, used only for display purpose, and by an initial value (first guess), as well as by lower and upper bounds. As an example, the latent heat and solid thermal conductivity are initialized, respectively, at 16.5 kJ kg\({}^{-1}\) and 0.4 W K\({}^{-1}\) m\({}^{-1}\), and their minimal and maximal authorized values are 14 and 18 kJ kg\({}^{-1}\), and 0.3 and 0.5 W K\({}^{-1}\) m\({}^{-1}\). Moreover, each run is repeated three times: in one case, the initial seed is set by the user, and in the two others, they are randomly chosen inside the search space. In these latter cases, it is obviously checked that the chosen first guess will lay inside the respective bounds defined hereinbefore. Finally, since the iterative algorithm always starts from an initial estimation \(\overrightarrow{p}^{0}\) arbitrarily chosen, it can occur that the algorithm converges towards a local minimum.