Computational multiscale modeling of fracture problems and its model order reduction

Abstract

This work focuses on the numerical modeling of fracture and its propagationin heterogeneous materials by means of hierarchical multiscale models basedon the FE2 method, addressing at the same time, the problem of the excessivecomputational cost through the development, implementation and validation ofa set of computational tools based on reduced order modeling techniques.For fracture problems, a novel multiscale model for propagating fracture has beendeveloped, implemented and validated. This multiscale model is characterized bythe following features:? At the macroscale level, were adapted the last advances of the ContinuumStrong Discontinuity Approach (CSDA), developed for monoscale models,devising a new finite element exhibiting good ability to capture and modelstrain localization in bands which can be intersect the finite element inrandom directions; for failure propagation purposes, the adapted Crack-pathfield technique (Oliver et al., 2014), was used.? At the microscale level, for the sake of simplicity, and thinking on the developmentof the reduced order model, the use of cohesive-band elements,endowed with a regularized isotropic continuum damage model aimingat representing the material decohesion, is proposed. These cohesive-bandelements are distributed within the microscale components, and theirboundaries.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, itsconsistency with respect to Direct Numerical Simulations (DNS), was also testedand verified.For model order reduction purposes, the microscale Boundary Value Problem(VBP), is rephrased using Model Order Reduction techniques. The use of twosubsequent reduction techniques, known as: Reduced Order Model (ROM) andHyPer Reduced Order Model (HPROM or HROM), respectively, is proposed.First, the standard microscale finite element model High Fidelity (HF), is projectedand 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 et al., 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 modeling materials exhibiting hardening behavior, the microscale displacementfluctuations and stresses have been taken as primal variables for the first andsecond reductions, respectively. In this case, the second reduction was carried outby means of the stress field interpolation. However, it can be shown that the stressprojection operator, being computed with numerically converged snapshots,leads to an ill-possed microscale reduced order model. This ill-poseddness isdeeply studied and corrected, yielding a robust and consistent solution.For the model order reduction in fracture problems, the developed multiscaleformulation in this work was proposed as point of departure. As in hardeningproblems, the use of two successive reduced order techniques was preserved.Taking into account the discontinuous pattern of the strain field in problemsexhibiting softening behavior. A domain separation strategy, is proposed. Acohesive domain, which contains the cohesive elements, and the regular domain,composed by the remaining set of finite elements. Each domain has an individualtreatment. The microscale Boundary Value Problem (BVP) is rephrased as asaddle-point problem which minimizes the potential of free-energy, subjected toconstraints fulfilling the basic hypotheses of multiscale models.The strain flucuations are proposed as the primal variable for the first reduction,where the high fidelity model is projected and solved into a low-dimensionalspace via POD. The second reduction is based on integrating the equilibriumequations by means of a Reduced Order Quadrature (ROQ), conformed by a set ofintegration points considerably smaller than the classical Gauss quadrature usedin the high fidelity model.This methodology had been proven to be more robust and efficient than theinterpolation methods, being applicable not only for softening problems, but alsofor hardening problems.For the validation of the reduced order models, multiple test have beenperformed, changing the size of the set of reduced basis functions for bothreductions, showing that convergence to the high fidelity model is achievedwhen the size of reduced basis functions and the set of integration points, areincreased. In the same way, it can be concluded that, for admissible errors (lowerthan 5%), the reduced order model is 110 times faster than the high fidelitymodel, considerably higher than the speedups reported by the literature.