**Author details**

26 Numerical Simulations

**Figure 35.** Time distribution of the thermal history time *T*1800*K*(*r*, *z*, 5) for (*r*, *z*) ∈ Ω.

**Figure 36.** Time distribution of the thermal history time *T*2400*K*(*r*, *z*, 5) for (*r*, *z*) ∈ Ω.

*T*2400*K*(*x*, *y*, *z*, .), *Tsyn*(*x*, *y*, *z*, .).

the heat supplied to the six faces of the cubic sample is different, in the sense that the wall temperature that contributes to the preheating of each face of the cubic sample is different. This will induce a specific transient spatial pattern of the combustion front. A meaningful snapshot of the transient temperature profile is depicted in Fig.(37). It is clearly observed that the shape of the front's propagation inside the cube is influenced by the boundary conditions. If a regular propagation is required, providing heat supply over one single face of the cube is enough to obtain quickly titanium-carbide. So combining a different heat supply system for each face of the cube appears complicated from an experimental point of view. Numerical simulation show that the pattern of the propagation in this case is more complicated than in the single heat supply case and requires more computational power to get finely resolved. An analysis, not presented here due space constraints, and using the same methodology as in the previous 2D case show that the same conclusions can be drawn for

A. Aoufi *SMS/RMT, LCG, UMR CNRS 5146, Ecole des Mines de Saint-Etienne, 158 cours Fauriel, 42023 Saint-Etienne Cedex, France*

G. Damamme *CEA, Centre DAM-Île de France, Bruyères le Châtel, 91297 Arpajon cedex, France*
