This Thesis focuses on the numerical modeling of fracture and its propagation in heterogeneous materials by means of hierarchical multiscale models based on the FE2 method, addressing at the same time, the problem of the excessive computational cost through the development, implementation and validation of a set of computational tools based on reduced order modeling techniques.

For fracture problems, a novel multiscale model for propagating fracture has been developed, implemented and validated. This multiscale model is characterized by the following features:

At the macroscale level, were adapted the last advances of the Continuum Strong Discontinuity Approach (CSDA), developed for monoscale models, devising a new finite element exhibiting good ability to capture and model strain localization in bands which can be intersect the finite element in random directions; for failure propagation purposes, the adapted Crack-path field technique (oliver/2014), was used.
At the microscale level, for the sake of simplicity, and thinking on the development of the reduced order model, the use of cohesive-band elements, endowed with a regularized isotropic continuum damage model aiming at representing the material decohesion, is proposed. These cohesive-band elements are distributed within the microscale components, and their boundaries.

The objectivity of the solution with respect to the failure cell size at the microscale, and the finite element size at the macroscale, was checked. In the same way, its consistency with respect to Direct Numerical Simulations (DNS), was also tested and verified.

For model order reduction purposes, the microscale Boundary Value Problem (VBP), is rephrased using Model Order Reduction techniques. The use of two subsequent reduction techniques, known as: Reduced Order Model (ROM) and HyPer Reduced Order Model (HPROM or HROM), respectively, is proposed.

First, the standard microscale finite element model High Fidelity (HF), is projected and solved in a low-dimensional space via Proper Orthogonal Decomposition (POD). Second, two techniques have been developed and studied for multiscale models, namely: a) interpolation methods, and b) Reduced Order Cubature (ROQ) methods (An/2009). The reduced bases for the projection of the primal variables, are computed by means of a judiciously training, defining a set of pre-defined training trajectories.

For the model order reduction in fracture problems, the developed multiscale formulation in this Thesis was proposed as point of departure. As in hardening problems, the use of two successive reduced order techniques was preserved.

Taking into account the discontinuous pattern of the strain field in problems exhibiting softening behavior. A domain separation strategy, is proposed. A cohesive domain, which contains the cohesive elements, and the regular domain, composed by the remaining set of finite elements. Each domain has an individual treatment. The microscale Boundary Value Problem (BVP) is rephrased as a saddle-point problem which minimizes the potential of free-energy, subjected to constraints fulfilling the basic hypotheses of multiscale models.

For the validation of the reduced order models, multiple test have been performed, changing the size of the set of reduced basis functions for both reductions, showing that convergence to the high fidelity model is achieved when the size of reduced basis functions and the set of integration points, are increased. In the same way, it can be concluded that, for admissible errors (lower than 5%), the reduced order model is 110 times faster than the high fidelity model, considerably higher than the speedups reported by the literature.

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