To calculate the temperature history, the researchers employed Fourier’s equation for three-dimensional heat conduction [12] . This equation is a partial differential equation that includes temperature-dependent material properties and can be expressed as follows:

k δ 2 T δ x 2 + k δ 2 T δ y 2 + k δ 2 T δ z 2 + δ Q δ t = ρ C δ T δ t (2.1)

The equation above includes several variables and coefficients. The variable Q represents the volumetric internal energy generation, while T represents temperature. The spatial coordinates are denoted by x, y, and z, and time is represented by t. Additionally, the equation incorporates several coefficients: ρ represents the density of the material, C represents specific heat, and k represents the thermal conductivity coefficient.

During welding, the electrode is moving along a direction. Therefore a moving heat source has to be employed. The modified Gaussian heat source model, which its frame is represented by the Gaussian function shown in Figure 1 is adopted and it is used to simulate the thermal history at different points along the welded zone [13] .

Q s ( x , y , t ) = 3 q s π r 0 2 e ( − 3 r 2 d 0 2 ) (2.2)

Q denotes the rate of heat input. The effective radius of the welding arc is

represented by r 0 , and the diameter of the welding arc is represented by d 0 . r is calculated by:

r = ( x − x 0 ) 2 + ( y − v t − y 0 ) 2 (2.3)

where x 0 and y 0 represent the distance of the heat source from the reference coordinate system’s origin along the x- and y-axes, respectively. The welding arc travel speed is given by v, and the travel time is given by t.

In this analysis, the geometrical model was divided into two parts and the hexahedral element was used. The first part is the fusion zone and its surrounding area. The second is for the rest of the area where no high temperature occurred (see Figure 3). Due to the high temperature subjected to the first area, a mesh convergence study is based on the maximum temperature attained after the simulation process at a node. The node located 7 mm from the origin along the reinforced welded joint (see Figure 6) was chosen for the mesh convergence study. Different maximum temperatures were obtained in the simulation as the mesh size was varied. As the size kept varying, the maximum temperature became constant, as is shown in Figure 2. Subsequently, the mesh size of 0.8 mm was chosen to reduce the computational time taken during the simulation. From the mesh convergence results, the first area was sized 0.8 mm, and the remaining area was sized 3 mm. The geometrical model was discretized into 18,072 elements and 90,730 nodes (see Figure 3).

Boundary conditions specify the relationship between the domain and the external environment. In welding, boundary conditions are the external specific heat flux which is applied on the top of the specimen and heat losses due to radiation and convection are applied at all faces of the specimen. Each boundary can be described with mathematical description of thermal boundary conditions.

The specific heat flux was considered to be Modified Gaussian heat source model discussed in previous section. The equation for convection q c and radiation q e are written as [15] :

q c = h ⋅ ( T − T 0 ) (2.4)

q e = ε ˙ ⋅ ξ ⋅ ( T 4 − T 0 4 ) (2.5)

The temperature of the room is denoted as T 0 , while the convection coefficient and emissivity coefficient for all surfaces of the plate are represented by h and ε , respectively. ξ is the Stefan Boltzmann constant. In thermal analysis, thermal material properties also called the temperature-dependent material properties contribute to the distribution of temperature during the welding process. Those properties are density, thermal conductivity, and specific heat. As each material has their thermal property, Table 1 shows the properties of mild steel, and Table 2 shows that titanium alloy powder was considered to be the same material properties of Ti-6Al-4V. The thermal properties of the electrode were considered to be the same as mild steel because they have small differences in chemical composition. The ANSYS workbench solver did the interpolation to determine the intermediate values of the two tables.

The transient thermal analysis was used to predict the weld pool shape of weldments. The simulated weld pool geometry can be compared to the actual weld geometry obtained through optical microscopy tests to validate the simulation method’s effectiveness [28] .

To determine the simulated weld pool geometry, a cross-section of the weld joint is cut, and the temperature fields in this cross-section are examined to record the temperature contours [29] . The area where the temperature was above the melting point of mild steel for unreinforced joints and titanium alloy for the reinforced joint was defined as the weld pool geometry.

Figure 5(a) compares the simulation result and experiment result for the unreinforced joint. The red area is for the simulation result, and the white area is for the experiment. The two areas were the weld pool geometry, due to the observed temperature being above the melting point of mild steel, 1430˚C. They were compared to each other and found a good agreement between them. Figure 5(b) compares the simulation and experiment results for the reinforced joint. The same areas were identified as it was done in the unreinforced joint by considering the temperature higher than melting point of titanium, 1600˚C. They were also compared to each other and found a good agreement between them. The inaccuracy of the weld bead height shown in those figures is affected by the predetermined bead height used in the simulated model. The bead height was taken as 2 mm.

The dimensions of interest are the depth of penetration and width of the weld pool, measured along the x and y axes of the cross-section. The weld pool is defined as areas with temperatures above titanium’s melting point.

Table 4 presents the measured dimensions of the simulated and experimental butt joint geometries, indicating good agreement between the experimental and

simulated values for the depth of penetration and the weld pool width. The high relative error was 10%. This error may have occurred because it was challenging to accurately replicate the actual conditions in which the process takes place. Factors such as material physical parameters may have contributed to the difficulty in matching real conditions [28] .

To collect the thermal history in the joint, a temperature probe was inserted at different distances along the weld joint, as shown in Figure 6. Thermal history was measured at four different points to observe the variation in temperature in the reinforced welded zone and unreinforced welded joint. In Figure 8, the curve shows the peak temperature has been above the melting point of mild steel, 1430˚C. It can be observed that the peak of temperature at each point is different. The observation means that as the welding heat source moves, it causes the heat to transfer from one point to another.

This movement of heat can be observed by the high temperature of the next adjacent node after the previous node has reached its peak temperature. Furthermore, the peak temperature of the next adjacent node is higher than that of the previous one, which suggests that heat is being accumulated. In Figure 7, the curve shows that the peak temperature has been above the melting point of titanium alloy powder, 1600˚C. Comparing Figure 7 and Figure 8 shows that by adding the titanium inside the groove, the titanium alloy powder slows the cooling. Titanium has a low thermal conductivity, making it retain heat for a long time compared to mild steel. It is observed even in the figures where the cooling rate from 2000˚C to 500˚C was 39.985 s and 41.768 s for unreinforced and reinforced welded joints, respectively.