## 3 Model Description
### Governing Equations  
In this study, the workpiece domain was modeled as a Eulerian framework, while the welding tool was considered as a Lagrangian rigid body. It is well identified that the conservation equations in the Eulerian description are written using the spatial time derivative. On the contrary, the Lagrangian description for solid and structure analysis conservation equations are a material time derivative. Spatial and material time derivatives can be interconnected with each other as in Equation (1) [33]:  
\[\mathrm{D}\phi/\mathrm{D}t=\partial\phi/\partial t+\mathrm{v}\cdot(\
abla\phi) \tag{1}\]  
where \(\phi\) is the arbitrary solution variable, \(\mathrm{v}\) is the material velocity, and \(\mathrm{D}\phi/\mathrm{D}t\) and \(\partial\phi/\partial t\) are the material and spatial time derivatives of \(\phi\), respectively.  
The mass, momentum, and energy can be written as follows:  
\[\mathrm{D}\rho/\mathrm{D}t+\rho\
abla\cdot\mathrm{v}=0 \tag{2}\]  
\[\rho(\mathrm{D}\mathrm{v}/\mathrm{D}t)=\
abla\cdot\sigma+\rho\mathrm{b} \tag{3}\]  
\[\mathrm{DE}/\mathrm{D}t=\
abla\cdot(\sigma\cdot\mathrm{v})+\rho\mathrm{b}\cdot \mathrm{v} \tag{4}\]  
where \(\rho\) is the density, \(\sigma\) is the Cauchy stress, \(\mathrm{b}\) is the body force, and \(\mathrm{E}\) is the total energy per unit volume.  
As the total energy is the sum of kinetic energy (\(0.5\rho\mathrm{v}\cdot\mathrm{v}\)) and the internal energy (e), Equation (4) can be rewritten with the velocity strain \(\mathrm{D}\) as shown in Equation (5):  
\[\mathrm{De}/\mathrm{D}t=\sigma\mathrm{:D}. \tag{5}\]  
Figure 1: Geometric dimensions of the experimental friction stir welding (FSW) tool.  
The three Lagrangian conservation Equations, (2), (3) and (5), can be translated into the spatial time derivative; thus, the Eulerian conservation equations can be written as follows:  
\[\partial\varrho/\partial\mathrm{t}+\
abla\cdot(\varrho\mathrm{v})=0 \tag{6}\]  
\[\partial\rho\mathrm{v}/\partial\mathrm{t}+\
abla\cdot(\varrho\mathrm{v}\mathrm{ \otimes}\mathrm{v})=\
abla\cdot\sigma+\rho\mathrm{b} \tag{7}\]  
\[\partial\mathrm{e}/\partial\mathrm{t}+\
abla\cdot(\mathrm{ev})=\sigma\mathrm{ :D} \tag{8}\]  
To solve the problem, Equations (6)-(8) can be written in the general conservation form, as shown in Equation (9):  
\[\partial\mathrm{\phi}/\partial\mathrm{t}+\
abla\cdot\mathbf{\Phi}(\mathrm{ \phi},\mathrm{v}\mathrm{x},\mathrm{t})=\mathrm{S} \tag{9}\]  
where \(\mathbf{\Phi}\) is the flux function and \(\mathrm{S}\) is the source term. Operator splitting divides Equation (9) into two equations, as shown in Equations (10) and (11) [34]:  
\[\partial\mathrm{\phi}/\partial\mathrm{t}=\mathrm{S} \tag{10}\]  
\[\partial\mathrm{\phi}/\partial\mathrm{t}+\
abla\cdot\Phi(\mathrm{\phi},\mathrm{ v}\mathrm{x},\mathrm{t})=0 \tag{11}\]  
where Equation (10) is referred to as the Lagrangian step, which contains the source term, while Equation (11) is the Eulerian step and contains the convective term.  
Figure 2 schematically represents the split operator for each step of the Coupled Eulerian Lagrangian method. Equation (10) corresponds to the standard Lagrangian governing formulation if the spatial time derivative is replaced by the material time derivative on the fixed mesh. In order to solve Equation (11), the deformed mesh is moved to the original one and the volume of the material transported between adjacent elements is calculated. Mass, momentum, energy, and other Lagrangian variables are adapted to account for the material flow between adjacent elements by the transport algorithms.