## 3 Results and Discussion  
We have implemented the mixed time partitioning method in C++ programming language as an extension to the author's simulation framework called NuscaS [26]. We have used finite element method to solve apparent heat capacity formulation of solidification, given by the Equation (4) supplemented by boundary conditions (Equations (2) and (3)). We used indirect model of solid phase growth for binary alloy (Equation (6)). We performed a series of numerical simulation of solidification for Al-2%Cu alloy casting solidifying in the metal form (the popular alloy for which there are known thermo-mechanical properties and behavior during solidification and cooling, it used in many foundries). The analyzed casting together with mold and cores are shown in Figure 5. The values of material properties of the liquid phase, solid phase and mold/core are given in Table 2. The physical properties are implicitly included in the algebraic system of equations (Equation (7)). Figure 4: Double numbering of nodes (with the same coordinates) belonging to two sub-domains. We set the initial temperature of liquid alloy, mold, and core to 960 K 590 K and 540 K, respectively. We considered non-ideal contact between casting and mold (with the conductivity of the separating layer 1000 W/m\({}^{2}\)K) and between casting and the core (with conductivity 800 W/m\({}^{2}\)K). We applied Neumann boundary conditions on all other boundaries. We set environment temperature as 300 K and heat exchange coefficient as 100 W/m\({}^{2}\)K. The time step \(\Delta t=0.0035\) s was used in the computer simulations. Such configuration of initial and boundary conditions along with materials properties allows us to study both solidifying of the cast and subsequent cooling of the entire system. We study temperature evolution and solid phase growth for 450 s. First 145 s corresponds to the solidification. The remaining time corresponds to the solid-state cooling (\(f_{\rm s}=1\) in the entire casting). We depict the geometry and dimensions of the considered system in Figure 5, and the mesh in Figure 6. The computational mesh consists of 16,261 triangular finite elements with 8640 nodes (only 2125 nodes belongs to the casting domain). In further analysis, we focus on the selected node (n1336) that we denote in Figure 5. That node belongs to several finite elements lying in the central and the longest solidifying part of the casting. This choice is dictated by the possibility to trace the results from the beginning to the end of the slow solidification in the central part of the cast. \begin{table}
\begin{tabular}{c c c c c} \hline
**Quantity** & **Unit Symbol** & **Liquid Phase** & **Solid Phase** & **Mold/Cores** \\ \hline density & kg/m\({}^{3}\) & 2498 & 2824 & 7500 \\ specific heat & J/kgK & 1275 & 1077 & 620 \\ thermal conductivity & W/mK & 104 & 262 & 40 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical properties of cast and mold/core material using during computer simulation of Al-2%Cu alloy solidification. Figure 5: The analyzed casting in mold. Solidification in the castings is a complex phenomenon where the behavior of one part may be significantly different from other part's behavior. Solidification process may take place at a constant temperature or temperature range. If the solidification occurs at a constant temperature, then it is considered to be the so-called solidification with zero temperature range [27]. However, solidification of most of the alloys (including one considered in this work) occurs in some temperature ranges. It starts at liquidus temperature (\(T_{I}\)) and ends at solidus temperature (\(T_{s}\)) or at eutectic temperature (\(T_{E}\)). The casting is in a liquid state above the liquidus temperature (\(f_{s}=0\)). When the temperature decreases below the liquidus temperature, but it is above the solidus temperature, casting begins to solidify, and the solid phase appears (\(0<f_{s}<1\)).