## 2 Materials and Methods  
Given the complex nature of the solidification process, it is inevitable to simplify its mathematical description to the most influential effects. Therefore, the derivation of the analytical expressions and all corresponding calculations are based on the following assumptions, simplifications, and methods:  
* The dimensions of the weld pool boundary change only to a minor extent along the thickness, which is the case when welding thin plates in complete penetration keyhole mode. Thus, the crystallization problem becomes two-dimensional, i.e., the crystals do not grow in the thickness direction. * The crystallization process reaches a steady state, i.e., the shape and size of the weld pool boundary do not vary with time. * The weld pool boundary is not affected by the increasing impurity concentration ahead of the solid-liquid interface. Furthermore, the boundary is defined by the isosurface of the liquidus temperature, \(T_{\text{liq}}\). * The weld pool boundary, obtained experimentally or from a thermo-fluid dynamics simulation, is mathematically reconstructed with Lame curves according to [15]. * The formation of substructures, e.g., planar, cellular, or dendritic, inside the crystals is not taken into account as it has a rather insignificant effect on the macroscopic crystal structure in polycrystalline materials [2]. * The crystallographic orientation and competitive growth are neglected as their consideration would significantly complicate the task [7; 16]. As a consequence, the crystal is assumed to grow along the temperature gradient which coincides with the inward normal direction to the crystallization front [4; 7; 8; 9]. The technique proposed in this paper refers to previously developed concepts for the analysis of the crystallization process; however, these concepts were based on elliptical functions and thus limited the analysis to weld pool boundaries of certain shapes, see, e.g., [4; 5; 6; 8]. The crystal axis, \(x(y)\), crystal growth rate, \(R\), and crystal cross-sectional area (or width in the two-dimensional case), \(W\), are uniquely determined by the shape of the rear part of the weld pool boundary, see Figure 1. According to the reconstruction technique of the weld pool boundary presented in [15] and the metallographic analysis from [4; 17], the movement of the solidification front, i.e., the rear part of the weld pool boundary, can be precisely described by a set of shifted Lame curves, as follows:  
\[\varphi(x,\ x_{c},\ y)=\left(\frac{x-x_{c}}{L_{r}}\right)^{m}+\left(\frac{y}{ b}\right)^{n}-1=0,\ \text{for}\ y\geq 0,\ x\leq\ x_{c},\ m>1,\ n>1. \tag{1}\]  
Here \(\varphi\) represents the set of shifted Lame curves in a coordinate system with origin at point 0, see Figure 1. \(x\) and \(y\) are the Cartesian coordinates; \(x_{c}\) is the coordinate of the center of the shifted Lame curve. Note that the shifting distance between the curves is denoted as \(\mathrm{d}x\), and should not be confused with the function of the crystal axis, \(x(y)\), derived below. \(L_{r}\), \(b\), \(m\), and \(n\) are parameters of the Lame curve, describing the rear part of the weld pool boundary. Thereby, \(L_{r}\) and \(b\) are used to define the semi-major axes of the superellipses, i.e., the length and the half-width of the rear part of the weld pool; \(m\) and \(n\) determine the shape of the boundary and the corresponding condition in Equation (1) is required to ensure smoothness of the continuous functions reconstructing the complete weld pool boundary (front and rear part). Note, however, that since only solidification and not melting is considered in the present paper, there is no need to reconstruct the front part of the weld pool boundary. As seen in Figure 1, the crystal axis starts growing from point I, located on the weld pool boundary described by the curve \(\varphi(x,0,y)\).