## 1 Introduction  
The solidification morphology of the liquid metal in the weld pool can be adequately described by three parameters: the crystal axis, the growth rate, and the cross-sectional area of the crystal. These three parameters determine some of the most important properties of the weld metal, such as its primary microstructure and strength. For example, the orientation of the crystal axis near the weld centerline influences the susceptibility to hot cracking to a large extent. On the other hand, the rate of crystallization and the temperature gradient determine the crystallization mode [1; 2; 3].  
The solidification process in welding is extremely complicated as it depends on many factors such as the weld pool shape, the local temperature field and chemical composition (impurity concentration), the crystallographic orientation of the crystals, etc. [1; 2; 4; 5; 6; 7]. Despite the fact that mathematical methods have been used to model and analyze the crystallization of the weld metal for more than 50 years, see, e.g., [4; 7; 8; 9; 10; 11], the dependence of the crystallization parameters on the complex weld pool shape has not been studied sufficiently and no general solution has been obtained so far [4; 8; 9]. Thereby, two of the main difficulties are the determination of the weld pool boundary, especially when more complex weld pool shapes are formed, see, e.g., [12; 13], and its precise mathematical description [9; 14; 15]. Furthermore, to provide a generalized analysis of the trajectory of the crystal axis and the corresponding growth rate and cross-sectional area, the solution should be obtained in the form of analytical expressions [4; 8; 9], which remains a non-trivial task to this day. To the best of the authors' knowledge, most of the available literature deals with numerical algorithms, see, e.g., [10; 11], and only a few analytical solutions have been obtained [4; 9]. Nonetheless, these solutions are limited to a specific group of weld pool shapes and thus do not represent a general solution of the solidification problem, allowing consideration of any dimensions and shape of the weld pool boundary.  
In recent years, however, a new algorithm providing a time and calibration efficient way to reconstruct the weld pool boundary by local Lame curves, well known as superellipes, was developed [15]. By making use of this algorithm, an analytical equation can be constructed that can describe any weld pool shape, thus allowing further attempts at a generalized analysis of the solidification process.  
The present work aims to obtain a general solution for the most important parameters of the crystallization process, namely the crystal axis, the growth rate, and the cross-sectional area of the crystal, in the case of two-dimensional columnar crystal growth during complete penetration laser beam keyhole welding of thin steel sheets. In addition, the dependence of the solidification parameters on the shape and dimensions of the rear part of the weld pool boundary is studied by dimensionless analysis. Thereby, the weld pool boundary is assumed to be known, either obtained from experiments or as the solution of the corresponding thermo-fluid dynamics problem and described by analytical expressions. The validity of the proposed technique and the derived analytical solutions is demonstrated in examples of laser beam welding of 2 mm thick 316L austenitic chromium-nickel steel sheets considering complete penetration laser beam keyhole welding. The derived analytical solutions are compared to known analytical solutions and experimentally determined the shape and size of the crystals on the top surface of the steel sheet.