## 1 Introduction  
In the case of multi-pass surfacing, the application of subsequent welds causes previous welds to heat up and melt. During subsequent thermal cycles, the material in a heat-affected zone can undergo multiple-phase transformations leading to the diversification of the structure between welds and the heat-affected zone, at the same surfacing parameters. It results in diversified states of strains and stresses in the heat-affected zone, as well as outside it. Tracing changes in temperature allows one to determine the phase transformations and stress state. The knowledge of the distribution and the value of permanent deformations and internal stresses after welding is necessary due to their superposition with the strains and stresses generated during the welding process. The knowledge of thermo-mechanical state changes in the deposited steel elements is the basis for the rational design of these processes. The modeling of the temperature field in welding processes is dominated by two approaches. The first--numerical--applies the finite difference method, elementary heat balance and finite element method (FEM). The second approach consists of analytical solutions, which most commonly use integral transformation methods and Green's functions. The literature on the calculation of the temperature field and the welding thermal cycles during single-welding is extensive. Modeling the temperature field during welding was initiated by works by [PERSON] [1] and [PERSON] [2], who assumed point and linear models of the heat source, respectively. [PERSON] and [PERSON] [3] modified [PERSON]'s model including the two-dimensional (2D) Gaussian distributed heat source and developed a solution to the travelling heat source in a semi-infinite steel plate. Since the adoption of a point heat source located near the center of the weld results in significantly deviating from the actual temperature values, many researchers have tried to obtain the solution closest to the real temperature distribution, using different models of heat sources. The model of a double-ellipsoidal three-dimensional (3D) heat source was first introduced by [PERSON] et al. [PERSON] et al. (2016). [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) used an annular heat source where the intensity of the heat flux within the center of the arc welding was determined according to the parameter defined as the radius at which the intensity of the arc power was 5% of the maximum intensity. [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) derived an analytical solution for transient temperature field in finite thickness plate during single-pass arc welding, assuming Gaussian distribution of heat source. An analytical solution for temporary temperature field of fillet weld has been presented by [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) and wasapplied by [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) to compare the thermal cycles during multi-pass gas metal arc welding (GMAW) which was computed for [PERSON]'s models of point heat sources and one-dimensional (1D) Gaussian. The solution is presented in the form of temperature distribution perpendicular to the welding direction. On the other hand, the solution of the temperature field in a massive body caused by a half-elliptical double-ellipsoidal volumetric heat source has been presented by [PERSON] et al. [PERSON] et al. (2017). [PERSON] and [PERSON] [PERSON] and [PERSON] (2017) carried out the thermal analysis in double-sided arc welding of an aluminum alloy plate. [PERSON] at al. [PERSON] et al. (2017) analyzed thermal behaviour during SAW (submerged arc welding) considering an oval-shaped heat source. [PERSON] et al. (2017) proposed closed form solutions of the transient heat diffusion problem during the energy deposition. [PERSON] et al. [PERSON] et al. (2017) presented a novel analytical solution to the 3D transient temperature field during dissimilar material welding, with the use of the Green's function.