A hybrid Fuzzy-Stochastic-Finite-Element-Method for polymorphic, microstructural uncertainties in heterogeneous materials

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Computational homogenization requires two separate finite element models: a model at the macroscale and a model of the materials’ underlying structure at the microscale. Computational homogenization involves two main ingredients: the transfer of the macroscopic loading to the microscale and averaging the corresponding response of the microstructure to obtain the effective macroscopic properties. A challenging aspect for computational homogenization is the proper modelling of material with uncertainty in the microstructure, as considered in this project. Uncertainties in the macroscopic response of heterogeneous materials result from various sources: the natural variability in the microstructure’s geometry and its constituent’s material properties and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted as aleatoric uncertainty and may be characterized by probabilistic approaches. The second type of uncertainty is denoted as epistemic uncertainty and may be described using fuzzy arithmetic. Models considering both sources of uncertainty are denoted polymorphic, requiring some combination of stochastic and fuzzy methods.In Phase I we developed methods for the accurate and efficient propagation of polymorphic uncertainty through the material’s microstructure and applied all proposed approaches to a benchmark problem. The objectives of the Phase II are further development of modelling techniques and their application to the engineering design of structures. The outcome of Phase II will be an accomplished methodology allowing the uncertainty propagation from the lowest level of a material microstructure through the macroscopic structure simulation to the engineering design and decision making. More precisely in Phase II the following challenges are considered:- We continue the development of advanced fuzzy-stochastic benchmark RVE for the microstructure of heterogeneous materials, resulting thus in a more realistic and precise description of polymorphic uncertainty in the material’s microstructure. - Modelling techniques for spectral non-deterministic finite element analysis will be enriched to non-deterministic eXtended Isogeometric Analysis.- The computational cost of full-order large scale simulations of systems in the presence of uncertainty is unacceptably high, in particular considering many-query or real-time applications. Thus, reduced order modeling is an essential tool which allows a speed up microscale simulations. - Reduced order models and metamodels provide a necessary bridge to the final stage of the project, in which a suitable metamodel will be used on the macroscale to run large size simulations of engineering structures. - Finally, the influence of uncertainty in the macrostructure on the static and the dynamic behavior of engineering structures under random loading will be analyzed.