## 2 Global-Local Problem Formulation
### Implemented Non-Intrusive Method
\begin{Bmatrix}f_{L}^{\text{T}}\\ f_{L}^{\text{I}}\end{Bmatrix}  \tag{11}\]  
where \(K_{L}\) denotes the tangent stiffness matrix of the local problem. This problem can be solved by using the Newton method. After the convergence of the Newton method, the interface reactions, \(\lambda_{L}\), are obtained following a procedure similar to that for the auxiliary problem Equation (9), which can be written as  
\[\lambda_{L}=S_{L}(u_{L})\cdot(\psi_{GL}\cdot u_{G}^{\text{T}})+K_{L}^{\text{IT }}(u_{L})\cdot[K_{L}^{\text{II}}(u_{L})]^{-1}\cdot f_{L}^{\text{I}}-f_{L}^{ \text{T}} \tag{12}\]  
where \(S_{L}(u_{L})\) corresponds to the Schur complement for the current state of the local problem. Both auxiliary and local problems are solved with the prescribed displacement, corresponding to the primal part of the strategy. The global-local strategy consists of subtracting the auxiliary response, \(\lambda_{A}\), from the global problem, while adding the local response, \(\lambda_{L}\). Finally, the strategy can be written as follows:  
\[K_{G}\cdot u_{G}-C_{\text{TG}}\cdot\{(K_{A}\cdot u_{A})\mid_{\text{T}}\}+C_{ \text{TG}}\cdot\{\psi_{LG}\cdot(K_{L}\cdot u_{L})\mid_{\text{T}}\}=f_{G}^{ \text{d}} \tag{13}\]  
where \(C_{\text{TG}}\) is the operator that relates the interface degrees of freedom to the global degrees of freedom and \(\psi_{LG}\) (as for \(\psi_{GL}\)) corresponds to a projector from the local model interface degrees of freedom (3D) to the global model interface degrees of freedom (2D). The terms \(C_{\text{TG}}\{(K_{A}\cdot u_{A})\mid_{\text{T}}\}\) and \(C_{\text{TG}}\{\psi_{LG}\cdot(K_{L}\cdot u_{L})\mid_{\text{T}}\}\) represents the response of the substructures (auxiliary or local) at the interface. Thus, Equation (13) can be rewritten as  
\[K_{G}\cdot u_{G}-C_{\text{TG}}\cdot(\lambda_{A}+f_{A}^{\text{T}})+C_{\text{TG }}\cdot\psi_{LG}\cdot(\lambda_{L}+f_{L}^{\text{T}})=f_{G}^{\text{d}} \tag{14}\]  
Rearranging the terms:  
\[K_{G}\cdot u_{G}=f_{G}^{\text{d}}+C_{\text{TG}}\cdot(f_{A}^{\text{T}}-\psi_{ LG}\cdot f_{L}^{\text{T}})+C_{\text{TG}}\cdot(\lambda_{A}-\psi_{LG}\cdot \lambda_{L}) \tag{15}\]  
since the expression \(C_{\text{TG}}\cdot(f_{A}^{\text{T}}-\psi_{LG}\cdot f_{L}^{\text{T}})\) cancels to zero because \(f_{A}^{\text{T}}\) and \(\psi_{LG}\cdot f_{L}^{\text{T}}\) both correspond to the same external load applied to the interface. Finally, the global problem can be written as  
\[K_{G}\cdot u_{G}=f_{G}^{\text{d}}+C_{\text{TG}}\cdot(\lambda_{A}-\psi_{LG} \cdot\lambda_{L}) \tag{16}\]  
The expression \(C_{\text{TG}}\cdot(\lambda_{A}-\psi_{LG}\cdot\lambda_{L})\) corresponds to the interface load correction applied to the global model, to include the effect of the development of nonlinearity in the local model. This interface load correction for the global problem corresponds to the dual part of the strategy. A residual is computed between two iterations by performing the following:  
\[r^{n}=C_{\text{TG}}\cdot\{(\lambda_{A}^{n}-\psi_{LG}\cdot\lambda_{L}^{n})-( \lambda_{A}^{n-1}-\psi_{LG}\cdot\lambda_{L}^{n-1})\} \tag{17}\]  
where super-index \({}^{n}\) corresponds to the current iteration \(n\). To evaluate the convergence of the strategy, the norm of the current residual is compared with the norm of the first residual:  
\[\eta=\frac{||r^{n}||_{2}}{||r^{1}||_{2}} \tag{18}\]