## 2 Global-Local Problem Formulation
### Implemented Non-Intrusive Method  
The nonlinear reference structure \(\Omega_{R}\), representing the DCB test, is shown in Figure 1 for the 3D and 2D cases. Both models can be divided into two parts:  
* For the 3D model: into the local domain \(\Omega_{L}\) that represents the area where the crack growth occurs and the rest of the structure domain \(\Omega_{RS}\), so \(\Omega_{R}^{3D}=\Omega_{RS}\cup\Omega_{L}\). * For the 2D model: into the auxiliary domain \(\Omega_{A}\) that represents the area where the crack growth occurs and the rest of the structure domain \(\Omega_{C}\), define as the complementary domain, so \(\Omega_{R}^{2D}=\Omega_{A}\cup\Omega_{C}\). To connect or communicate both subdomains, an interface, \(\Gamma\), is defined as the surface where both subdomains are in contact (\(\Gamma^{3D}=\Omega_{L}\cap\Omega_{RS}\) and \(\Gamma^{2D}=\Omega_{A}\cap\Omega_{C}\)), which communicates both models at the node level. Because crack growth requires nonlinear resolution and the complete problem must be solved, the computing times are high when the complete model (2D or 3D) are large. The strategy seeks to concentrate the nonlinearities on a specific zone, which is treated nonlinearly, while the rest of the structure is solved linearly. For the implementation of the strategy, both 2D models are of interest, while for the 3D model, only the local model is needed, as shown in Figure 1.  
The first stage of resolution corresponds to solving the global linear 2D problem, Equation (7), in the complete model (\(\Omega_{R}^{2D}\)), which can be written as:  
\[K_{G}\cdot u_{G}=f_{G}^{d} \tag{7}\]  
where \(K_{G}\) denotes the stiffness matrix of the 2D linear global problem, \(u_{G}\) the displacement of the nodes of the 2D structure, and \(f_{G}^{d}\) the external load applied to the 2D structure. Since only the complementary domain is needed for the strategy, Equation (7) must be modified by subtracting the response of the auxiliary domain (\(\Omega_{A}\)). This is performed by solving the linear auxiliary problem with prescribed displacements at the interface (\(u_{G}^{\Gamma}\)), previously obtained by solving Equation (7). The auxiliary problem is expressed as follows. \[ \begin{bmatrix}K_{A}^{\Gamma\Gamma}&K_{A}^{\Gamma\Gamma}\\ K_{A}^{\Gamma\Gamma}&K_{A}^{\Gamma I}\end{bmatrix} \cdot \begin{bmatrix}u_{G}^{ \Gamma}\\ u_{A}^{\Gamma}\end{bmatrix} = \begin{bmatrix}f_{A}^{\Gamma}\\ f_{A}^{\Gamma}\end{bmatrix}  \tag{8}\]  
where \(K_{A}\) is the stiffness matrix of the linear auxiliary problem, \(\{\}^{\Gamma}\) represents the interface degrees of freedom, and \(\{\}^{I}\) the non-interface degrees of freedom. After solving Equation (8), the interface reactions, \(\lambda_{A}\), are computed using the following expression:  
\[\lambda_{A}=S_{A}\cdot u_{G}^{\Gamma}+K_{A}^{\Gamma I}\cdot[K_{A}^{I}]^{-1} \cdot f_{A}^{\Gamma}-f_{A}^{\Gamma} \tag{9}\]  
where \(S_{A}\) correspond to the classical Schur operator obtained by  
\[S_{A}=K_{A}^{\Gamma\Gamma}-K_{A}^{\Gamma I}\cdot[K_{A}^{\Gamma I}]^{-1}\cdot K _{A}^{\Gamma\Gamma} \tag{10}\]  
The nonlinear stage corresponds to the nonlinear local problem (3D model) with the cohesive zone method for the representation of crack growth. This stage corresponds to solving a nonlinear problem with the prescribed displacement at the interface of the model. The local displacements of the interface \(u_{L}^{\Gamma}\) are obtained by projecting the global displacements of the interface onto the 3D mesh \(u_{L}^{\Gamma}=\psi_{GL}u_{G}^{\Gamma}\), where \(\psi_{GL}\) corresponds to a projector from the global model (2D) degrees of freedom of the interface to the local model interface degrees of freedom (3D). The linearized tangent local problem can be written as  
Figure 1: Implemented strategy, where the darkest color indicates cohesive zone. \[ \begin{bmatrix}K_{L}^{\text{IT}}(u_{L})&K_{L}^{\text{IT}}(u_{L})\\ K_{L}^{\text{IT}}(u_{L})&K_{L}^{\text{II}}(u_{L})\end{bmatrix} . \begin{Bmatrix}u_{ L}^{\text{T}}\\ \Delta u_{L}^{\text{I}}\end{Bmatrix} =