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

计算均化需要两个独立的有限元模型：一个是宏观尺度的模型，另一个是微观尺度的材料底层结构模型。计算均匀化包括两个主要部分：将宏观载荷转移到微观尺度，并平均相应的微观响应，以获得有效的宏观性质。计算均化的一个具有挑战性的方面是对微观结构中存在不确定性的材料进行适当的建模，正如本项目中所考虑的那样。非均质材料宏观响应的不确定性有多种来源：微观结构的几何形状及其组成成分的材料性质的自然变异性，以及对微观结构缺乏足够的知识。第一种不确定性被表示为任意不确定性，可以用概率方法来描述。第二种不确定性被表示为认知不确定性，可以用模糊算法来描述。考虑两个不确定性源的模型被表示为多态的，需要随机和模糊方法的某种组合。在第一阶段，我们开发了通过材料的微观结构准确而有效地传播多态不确定性的方法，并将所提出的所有方法应用于基准问题。第二阶段的目标是进一步发展建模技术及其在结构工程设计中的应用。第二阶段的成果将是一种成熟的方法，允许不确定性从材料微观结构的最低级别通过宏观结构模拟传播到工程设计和决策。更准确地说，在第二阶段考虑了以下挑战：-我们继续为非均质材料的微观结构开发先进的模糊-随机基准RVE，从而更真实和准确地描述材料微观结构中的多态不确定性。-频谱非确定性有限元分析的建模技术将丰富到非确定性扩展等距分析。-在存在不确定性的情况下对系统进行全阶大规模模拟的计算成本高得令人无法接受，特别是考虑到多查询或实时应用。因此，降阶建模是加速微尺度模拟的重要工具。-降阶模型和元模型为项目的最后阶段提供了必要的桥梁，在该阶段中，将在宏观尺度上使用合适的元模型来运行工程结构的大尺寸模拟。最后分析了宏观结构的不确定性对工程结构在随机荷载作用下的静力和动力行为的影响。

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.