Model Order Reduction in computational multiscale fracture mechanics

Nowadays, the model order reduction techniques have become an intensive research field because of the increasing interest in the computational modeling of complex phenomena in multiphysic problems, and its consequent increment in high-computing demanding processes; it is well known that the availability of high-performance computing capacity is, in most of cases limited, therefore, the model order reduction becomes a novelty tool to overcome this paradigm, that represents an immediately challenge in our research community. In computational multiscale modeling for instance, in order to study the interaction between components, a different numerical model has to be solved in each scale, this feature increases radically the computational cost. We present a reduced model based on a multi-scale framework for numerical modeling of the structural failure of heterogeneous quasibrittle materials using the Strong Discontinuity Approach (CSD). Themodel is assessed by application to cementitious materials. The Proper Orthogonal Decomposition (POD) and the Reduced Order Integration Cubature are the proposed techniques to develop the reduced model, these two techniques work together to reduce both, the complexity and computational time of the high-fidelity model, in our case the FE standard model. Introduction The present model departs from the multiscale framework developed in [2] for the numerical modeling of failure via hierarchical multi-scale models, taking advantage of the reduced order techniques developed in [1], the theoretical framework used in this work is based on the so-called (FE)methods via first order computational homogenization for the coupling between scales, in which homogenized quantities at the lower scale, represented by a so-called failure-cell, are therefore transferred, in a one-way fashion, to material points (Gauss points) of the macroscopic structure. The formulation is presented in terms of strains in a non-conventional format imposing the natural multiscale boundary conditions via Lagrange multipliers. This work attempts to solve the problematic of excessive computational time in multi-scale models, in our case an additional complexity is induced by the discontinuous displacement field produced by the strain localization at both scales. Nonetheless, the methodology can also be straightforward extended to problems with continuous fields. Model description Generalities of FE method applied to multiscale fracture problems: This approach is developed under a small strain framework, the equality of internal power at both scales is guaranteed via HillMandell Macro-Homogeneity principle. In this approach, the macroscopic constitutive response is proven to be point-wise equivalent to an inelastic law (in an incremental fashion) as a function of the homogenized elastic tangent tensor, C, and the incremental homogenized inelastic strain rate ε̇ i.e.: Key Engineering Materials Online: 2016-09-30 ISSN: 1662-9795, Vol. 713, pp 248-253 doi:10.4028/www.scientific.net/KEM.713.248 © 2016 Trans Tech Publications, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (#69108547-18/08/16,17:02:19) σ̇ = C : (ε̇(x)− ε̇) ε̇ = 1 lμ (n⊗ β̇) (1) Where, the inelastic strain component ε̇ is expressed as a function of the homogenized variables taken from the lower scale, and represent the average value of the symmetrical tensor product between the strong discontinuity normal n, and the rate of displacement jump β̇ of each cohesive band, belonging to the manifold of the mesoscopic failure mechanism Sμ, i.e. the mesoscopic crack. In addition, the so-called material characteristic length lμ is defined as the ratio between the measure (volume or area) of the representative volume and the measure (surface or length) of the mesoscopic failure mechanism. The equations that govern the lower scales are the next: Problem I: Given a macroscale strain ε, Find ũμ such that εμ = ε+∇ũμ and: ∫ Bμ σμ(εμ) : ∇ũμ dBμ = 0 ; ∀ũμ ∈ V μ := {ũμ | ∫ Bμ ∇ũμ dBμ = 0}; (2) Model Order Reduction techniques: The reduction process is divided into two sequential stages. The first stage consists of a common Galerkin projection, via Proper Orthogonal Decomposition POD for the meso-scale strain field, onto a small space (reduced-order space). For the second stage, the main goal is to reduce the number of integration points given by the standard Gauss quadrature, by defining a new scheme that efficiently determines optimal points and its corresponding weights so that the error in the integration of the reduced model is minimized (Reduced Order Cubature ROC). In order to provide the reduced model with the input parameters and entities, the general procedure is also divided into two parts, the first one (offline part) in which the projection operators for the mesoscale strain field and the parameters of the new integration cubature are computed. These data, together with the material and geometrical parameters, define the set of input parameters for the first and second stage (online part). By comparison with the standard (FE) scheme, the proposed model in (2) can be redefined in term of strains in a generalized fashion, imposing the kinematic constraint (2-b) in an explicit way via Lagrange multipliers. Problem IB: Given a macro-scale strain ε, find ε̃μ and λ satisfying: (ε̃μ(ε, dμ),λ(ε, dμ)) = arg{minε̃μ max λ Π(ε̃μ,λ)}; such that ḋμ(y, εμ) = g(εμ, dμ) (3) Where Π is the homogenized potential of energy at the meso-scale. Projection of strain field via POD: The reduction of the meso-scale strain field is based on the projection of the weak form of the discrete mechanical problem into a reduced manifold (reducedorder space), this reduced space is spanned by Ritz (globally supported) basis functions obtained via Singular Value Decomposition (SVD) of a set of snapshots taken from training tests computed during the offline part. Following this reasoning, the meso-scale strain fluctuation can be expressed as: