Multiscale numerical methods for the simulation of diffusion processes in random heterogeneous media with guaranteed accuracy

The possibility of combining several constituents to obtain properties that cannot be obtained with any of them alone, explains the growing proliferation of composites in mechanical structures. However, the modelling of such heterogeneous systems poses extreme challenges to computational mechanics. The direct simulation of the aforementioned gives rise to computational models that are extremely expensive if not impossible to solve. Through homogenisation, the excessive computational burden is eliminated by separating the two scales (the scale of the constituents and the scale of the structure). Nonetheless, the hypotheses under which homogenisation applies are usually violated. Traditional homogenisation schemes provide no means to quantify this error. The �rst contribution of this thesis is the development of a method to quantify the homogenisation error. In this method, the heterogeneous medium is represented by a stochastic partial di�erential equation where each realisation corresponds to a particle layout. This representation allows us to derive guaranteed error estimates with a low computational cost. The e�ectivity (ratio between true error and estimate) is characterised and a relation is established between the error estimates and classical results in micromechanics. Moreover, a strategy to reduce the homogenisation error is presented. The second contribution of this thesis is made by developing a numerical method with guaranteed error bounds that directly approximates the solution of heterogeneous models by using shape functions that incorporate information of the microscale. The construction of those shape functions resembles the methods of computational homogenisation where microscale boundary value problems are solved to obtain homogenised properties

Multi-scale methods for fracture: model learning across scales, digital twinning and factors of safety

Authors: S. P. A. Bordas, L. A. A. Beex, P. Kerfriden, D. A. Paladim, O. Goury, A. Akbari, H. Rappel  Multi-scale methods for fracture: model learning across scales, digital twinning and factors of safety Fracture and material instabilities originate at spatial scales much smaller than that of the structure of interest: delamination, debonding, fibre breakage, cell-wall buckling, are examples of nano/micro or meso-scale mechanisms which can lead to global failure of the material and structure. Such mechanisms cannot, for computational and practical reasons, be accounted at structural scale, so that acceleration methods are necessary.  We review in this presentation recently proposed approaches to reduce the computational expense associated with multi-scale modelling of fracture. In light of two particular examples, we show connections between algebraic reduction (model order reduction and quasi-continuum methods) and homogenisation-based reduction. We open the discussion towards suitable approaches for machine-learning and Bayesian statistical based multi-scale model selection. Such approaches could fuel a digital-twin concept enabling models to learn from real-time data acquired during the life of the structure, accounting for “real” environmental conditions during predictions, and, eventually, moving beyond the “factors of safety” era

Implementation of extended finite element method in the commercial library diffpack

Proyecto ConfidencialAlves Paladim, D. (2012). Implementation of extended finite element method in the commercial library diffpack. http://hdl.handle.net/10251/28930.Archivo delegad

Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fast evolving diffusion coefficients. We typically consider linear elliptic diffusion problems in randomly packed particulate composites. Our work extends the pioneering work presented in [26,32] in order to bound the error in the expectation and second moment of quantities of interest, without ever solving the fine-scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine-scale features. Our computations are purely macroscopic, deterministic, and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager-Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss-Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation

Advances in error estimation for homogenisation

In this paper, the concept of modeling error is extended to the homogenisation of elliptic PDEs. The main difficulty is the lack of a full description of the diffusion coefficients. We overcome this obstacle by representing them as a random a field. Under this framework, it is possible to quantify the accuracy of the surrogate model (the homogenised model) in terms of first moments of the energy norm and quantities of interest. This work builds on the seminal work of [1]. The methodology here presented rely on the Constitutive Relation Error (CRE) which states that a certain measures of the primal and dual surrogate model upper bound the exact error. The surrogate model, in agreement with homogenisation, is deterministic. This property exploited to obtain bounds whose computation is also deterministic. It is also shown that minimising the CRE in the set of homogenisation schemes leads us to an optimal surrogate that is closely related to the classical Voigt and Reuss models. Numerical examples demonstrate that the bounds are easy and affordable to compute, and useful as long as the mismatch between he diffusion coefficients of the microstructure remain small. In the case of high mismatch, extensions are proposed, through the introduction of stochastic surrogate models.. [1]Romkes, Albert, J. Tinsley Oden, and Kumar Vemaganti."Multi-scale goal-oriented adaptive modeling of random heterogeneous materials." Mechanics of materials 38.8(2006): 859-872