**1. Introduction**

Laser beam welding has numerous merits owing to high welding velocity, low heat input, high flexibility, high efficiency of production, narrow weld width and large weld depth in comparison to conventional welding processes, such as those based on the electric arc. Thus, the use of laser welding in different applications such as high pressure and vacuum vessels, crane fabrication, and in the marine and aerospace industries is currently a reality. For instance, pulsed Nd:YAG laser is of great importance because the laser parameters, that is, average power, pulse duration, and frequency, are controllable. As for aluminum and its alloys, they are comprehensively consumed in different industries such as aerospace and automobile owing to their high specific strength and corrosion resistance in comparison

to other metallic alloys. Among different kinds of aluminum alloys, 5456 aluminum alloy (AA5456), which is a wrought and non-heat treatable aluminum alloy belonging to the 5xxx Al-Mg series, is focus of attention due to the high amount of magnesium (approximately 5 percent), which improved the material weldability [1].

During welding, process control is fundamental to obtain defect-free joints. All aluminum alloys encompass specific characteristics, namely, low absorption of the laser beam, volatile elements such as magnesium in 5456 aluminum alloy, tendency to create oxide films, and a propensity to form constituents with low melting points [2,3]. Therefore, some defects including excessive porosity, blowholes, and hot cracking can occur in laser welded aluminum alloys [4]. These kinds of flaws can be seriously damaging to the mechanical properties of the welds. To achieve a sound weld of aluminum alloys via laser welding, considering the relationships between microstructure and mechanical properties, determining optimum parameters in the process for different aluminum alloys is of paramount importance. It has been reported that hot cracking phenomenon is eliminated during laser welding of 5456 aluminum alloy by performing preheating just before welding [1]. Researchers have found that increasing pulse overlapping factor induces an increase in Mg loss so that a tendency to formation of excessive solidification cracking occurs [5]. With all these results considered, it is obvious that the solidification process and optimization of laser parameters are determining factors in the performance of laser welds and the resulting microstructures. As far as these imperative concepts are concerned, it is of paramount importance to evaluate the temperature regime during this particular process.

Experimental results play an integral role in characterizing laser welded components; however, these results could be restricted by the technique used, and the time-consuming aspect may also adversely affect the whole process. On the other hand, in order for the significant behavior in laser welding to be investigated and determined, mathematical models are additional ways. It should be considered that numerical and analytical solutions are two major approaches to obtain such insights into the effect of different process parameters. To predict the thermal features of welding, Rosenthal's analytical model can be widely utilized and has a potential to give an abrupt estimation [6]. However, some assumptions need to be performed when using the Rosenthal equation in order that this analytical solution would be applicable in welding process; nevertheless, this will raise concerns about the results' accuracy obtained from this equation. On the other hand, the amount of assumptions used in numerical models is often fewer than the analytical solution, which causes to be more genuine. In contrast to this idea, the calculation time needed for numerical models is drastically longer than that of analytical ones.

It has been reported that the Rosenthal equation was already applied to determine which conditions would induce lack of fusion circumstance in different alloys [7]. They found a prediction of blowhole formation induced by lack of fusion under various processing conditions. In another research, the primary dendrite arm spacing (PDAS) was estimated via a processing map [8]. Wang et al. [9] have conducted the phase-field model to evaluate the dendritic growth of grains along the melt pool boundary. Remano et al. [10] have utilized a finite-element (FE) model to investigate molten and solidified Inconel 718 during the laser-based additive manufacturing. In this research, they observed that the fusion width was overestimated when using high energy densities. In this regard, the absorptivity of laser beam by the substrate was determined using a factor which needs to be performed in numerical modeling so that the predicted results became comparable with experimental ones. Artinov et al. [11] developed a 3D model to calculate the heat source and predict the thermal behavior in fusion welding. Furthermore, they accomplished a relationship between numerical results and experimental ones for weld pool geometries and heat transfer. Satyanarayana et al. [12] have applied the convection mode of heat transfer and Marangoni stresses in the fusion zone and have also calculated heating and cooling rates with which they would then use to study the microstructure of fusion and heat affected zones during laser welding of ZR-1%NB alloy. In another study [13], a finite element model was developed to investigate hot cracking phenomena during laser welding of 6xxx aluminum

alloys. It was shown that the heat flux vector field achieved from the simulation is in a good agreement with the grain orientation observed in the experiments. As to laser welding of 5xxx and 6xxx Al series, research works have been conducted using thermomechanical finite element models to study the temperature-dependent characteristics of the joints so that optimized laser parameters would be obtained [14]. Moreover, an irreversible melting state variable into the thermomechanical simulation was presented to evaluate how the melting state influences the mechanical and thermal features during laser welding of Al alloys [15]. Based on previous studies, numerical modeling and analytical solutions are both capable to produce satisfactory results during the laser welding process. However, they should be investigated separately, and there has been no comprehensive evaluation of these two routes yet, especially for laser welding of the 5456 aluminum alloy.

All in all, this research aims to describe both analytical and numerical solutions and compare them with regards to laser welding of AA5456 using a pulsed Nd:YAG laser. Based on the previous studies in this regard, it should be mentioned that the continuous mode of laser equipment has been widely used such that a comprehensive study in order to show the comparison between numerical and analytical solutions in laser welding of AA5456 using a pulsed Nd:YAG laser is still lacking. To do so, some important parameters like fusion dimensions, partially melted zone thickness, temperature gradient, cooling rate, solidification rate, and microstructure such as primary dendrite arm spacing (PDAS) were predicted and compared using analytical and numerical approaches in addition to experiments. Moreover, the predicted results were affected by the assumptions performed in the numerical modeling; therefore, these effects were elaborated. Furthermore, to validate these predicted results, some experimental results were obtained and compared to them. By doing so, it is expected that the knowledge of the correlation between determining parameters in the process and laser welded results is enhanced. Additionally, in order to investigate whether the analytical Rosenthal equation can supplement or replace the numerical modeling, the capability and limitations of this equation are studied in this research.
