## 4 Numerical Model
### Boundary Conditions

Table 2: Heat Transfer coefficients (W/m\({}^{2}\) K). Figure 6: Three–dimensional model. Five kinds of local models have been chosen:  
* Local Model 1--extends between y = 0 and y = 0.5, in such a way that the interface with the Global Model occurs in an area where plastic strains are expected;
* Local Model 2--with the interface placed in correspondence of what presumably is the limit between linear (elastic behavior) and nonlinear (elastic-plastic behavior) areas;
* Local Model 3--such that the interface occurs where no plastic strains are envisaged;
* Local Model 4--extends between y = 0 and y = 1.5;
* Local Model 5--extends between y = 0 and y = 1.8. The aim is to demonstrate that even considering local models with interfaces with the Global Model that are far away from the nonlinear areas, at least one iteration is needed to meet the chosen convergence criterion; then, it will be shown that the iterative approach has advantages with respect to the \"non-iterative approaches\" adopted by finite element commercial codes. Some other parameters have also been defined to measure the level of accuracy of the Global/Local solutions:  
\[\eta_{e}=\frac{\left\|e_{GL}^{pl}-e_{R}^{pl}\right\|_{k}}{\left\|e_{R}^{pl} \right\|_{k}} \tag{12}\]  
\[\eta_{o}=\frac{\left\|e_{GL}^{VM}-e_{R}^{VM}\right\|_{k}}{\left\|e_{R}^{VM} \right\|_{k}} \tag{13}\]
* \(e_{GL}^{pl}\) is the equivalent plastic strain calculated by the Global/Local approach,
* \(e_{R}^{pl}\) is the equivalent plastic strain calculated by the Reference Model,  
Figure 7: Highlight of different upper boundaries of local models. * \(\sigma_{GL}^{VM}\) is the Von Mises stress evaluated by the Global/Local approach,
* \(\sigma_{K}^{VM}\) is the Von Mises stress evaluated by the Reference Model,
* \(k\) can be equal to 2 (Euclidean norm) or \(\infty\) (infinite norm). The Euclidean norm provides useful data to understand the level of accuracy of the entire Local Model, whereas the infinite norm describes the maximum deviation (in absolute value) from the Reference Model. For the sake of simplicity, the results will be reported in terms of infinite norms. In the results section, in order to have homogeneous results, stresses and plastic strains on the interface nodes are not taken into account when calculating \(\eta_{e}\) and \(\eta_{o}\), since their deviation from the results of the Reference Model are expected to be much higher than the deviation evaluated on the internal nodes. As mentioned above, conformal and non-conformal meshes have been adopted: in the former case, the algorithm is expected to work in a very stable way, since no mapping is needed at the interface where displacements and nodal forces should be transferred from one model to another; in the latter case, interpolation errors could occur and, then, the curve representing the residuals might not be monotonically convergent towards the final solution. For that reason, three different levels of \"non-conformity\" have been chosen in order to understand which is the lowest ratio, \(v_{n}\), between the number of nodes at the interface, on the Global Model side, \(N^{G}\), and on the Local Model side, \(N^{L}\), such as to guarantee a monotonic convergence behavior of the algorithm and accurate results:  
\[v_{n}=\frac{N^{G}}{N^{L}} \tag{14}\]  
The evaluation of the lowest value of \(v_{n}\) allows minimizing the discretization level and, then, the computation (CPU) time needed to perform the numerical analyses.