Fig. 1. Parabolic dendrite tip of curvature ρtip and growth velocity vtip truncated by a cylinder of radius rcyl. The yellow area of surface Σ is parametrized by the length a defining a distance behind the tip of the parabola. Blue lines correspond to isovalues of the concentration field in the liquid phase $w_{PTN}^{l}$ computed by solving Eq. (1).

Fig. 2. Illustration of the coupling between the CA and the PTN method. Initialization of a polygon after the capture of a cell μ by the polygon of cell ν.

Fig. 3. Illustration of the CAPTN-FE coupling for a single grain growing at the center of a disc at given time. Arrows correspond to transferred fields. PTN: w P T N l (color), edge of δ P T N s (white line), edge of area where gm=0 (pink line), edge of area where g m = g l i m m (black line). Zoom: PTN mesh at a dendritic tip. CA: state of cells. The white zone in the center corresponds to deallocated cells. FE: all FE fields are displayed on separate images.

Fig. 4. (a, b) FE mesh of the PTN model for two sets of numerical parameters. Theoretical parabolae are reported in green and integration zone in cyan. Iso-concentration lines are highlighted in light red (c, d) Ratios v t i p / v I v and ρ t i p / ρ I v according to a / h min for various values of h min / ρ I v and for Ω=0.1.

Fig. 5. Ratios (a) v t i p / v I v and (b) ρ t i p / ρ I v according to the supersaturation Ω for a/hmin=5 and (red) hmin/ρIv=0.1 and (blue) hmin/ρIv=2.

Fig. 6. Snapshot at time t=9.6s of the (CA) cell state, with (white) deallocated cells and (FE) volume average variables for the CA-FE simulation.

Fig. 7. Average temperature (a), fraction of mushy zone and fraction of solid (b) over time for the CAPTN-FE and the CA-FE simulations.

Fig. 8. Length (a) velocity (b) and normalized composition (Eq. (12)) (c) of the four primary branches of the grain in the CAPTN-FE (red) and CA-FE (blue) simulations.

Fig. 9. Snapshot at time t=9.6s of the (PTN) composition field w P T N l, (CA) mush fraction g + m and cell state, with (white) deallocated cells and (FE) volume average variables.

Fig. 10. Evolution of the (a) (PTN) composition field w P T N l, (b) (CA) mush fraction g + m and cell state, with (white) deallocated cells and (c) (FE) volume fraction of solid at four different times. (PTN) The white contour is the border of the Dirichlet condition using parabolae and the black line is the edge of area where g m = g l i m m.

Fig. 11. Parabola of curvature radius ρtip with cartesian and parabolic coordinates.

Fig. 12. Initial polygon (blue) associated to the central cell ν. The branch of summit S ν 1 is considered as the internal branch of the polygon. Circumscribed circles of neighbor cells are represented with dotted lines. The grey polygon in dashed lines has an area A ν max. This polygon captures all neighbor cells.