Domain Decomposition Methods (DDM) are a set of numerical techniques that efficiently implement parallel computing for the structural analysis of large domains. This work presents the implementation of mixed DDM for linear elasticity problems along with non-linear problems such as crack propagation. In addition, optimization algorithms have been used to find the optimal parameters of the mixed domain decomposition method. Finally, a strategy is proposed in order to implement the optimization methods to obtain a good approximation of the Search Direction parameter at a low computational cost.

The global/local analysis allows to embed a specific local zone of interest with a different behaviour in a global coarse model. In this local model, fine meshes are usually used to model some structural details and potentially non-linear behaviours, such as plastic hardening and crack propagation. The standard global/local approach can be observed as a Dirichlet-Neumann iterative algorithm where a Dirichlet problem on the local model and a Neumann problem on the global one are solved successively. This paper proposes a new approach for the global/local framework as Robin parameters are considered on both local and global models to obtain more flexibility and improvement for convergence. Particularly, Robin parameters are obtained using a pre-defined strip of elements and the results are later improved by means of single-objective optimization, minimizing the number of iterations to achieve convergence. This improvement is illustrated for cracked domains and plastic hardening in 2D problems and 3D elements within a non-intrusive framework, allowing the usage of commercial finite element software along with open-source research finite element software.

Global local analysis is a part of the structural analysis that allows to study, with an iterative solution, a coarse global linear model with a specific zone. This zone is defined as a local model with fine mesh and a non-linear behaviour such as crack propagation. However, the current trend in Global Local analysis is to impose displacements on the fine model to later obtain the reactions that will be applied to the global model for each iteration (Primal to Dual solution algorithm). Therefore, we propose a mixed analysis in the local and global models through the application of Robin conditions on the interface, allowing a higher grade of flexibility for the case of the patch or fine model with crack propagation behaviour. As a result, the algorithm converges successfully, presenting kinematic compatibility and good results with respect to the Monolithic (non-decomposed) model. Finally, a sensitivity analysis is performed on some variables regarding the crack propagation for 2D models. Finally, the proposed methodology also allows to improve the performance of the method for cracked models or other nonlinearities when compared with the current global local analysis, presented in the state of the art.