Pivovarov, Dmytro, Dr.-Ing.

Die realistische Simulation einer technischen Struktur oder eines dynamischen Systems kann nicht durchgeführt werden, ohne verschiedene Quellen von Unschärfen zu berücksichtigen. Unschärfen entstehen durch unzureichende Messgenauigkeit, Modellannahmen, das Fehlen präziser Daten oder durch natürliche Schwankungen und Zufälligkeiten in einigen Prozessen. Es wurde gezeigt, dass die Auswirkungen von Unschärfen stark nichtlinear und schwer vorhersehbar sein können. Dementsprechend ist die Modellierung von Unschärfen ein wichtiges und aktuelles Thema.Trotz zahlreicher vielversprechender Ergebnisse im Bereich der stochastischen und fuzzy Modellierung gibt es noch keinen allgemeinen Ansatz für Unschärfen. Alle vorhandenen Methoden wurden nur für eine bestimmte Art von Modellen oder eine bestimmte Art von Unschärfen entwickelt und getestet. Es gibt keinen universellen Solver, der für die realistischen technischen Probleme geeignet ist, bei denen alle Arten von Unschärfen das Modell beeinflussen.Eine weitere Herausforderung sind die hohen Rechenkosten für die Modellierung der Unschärfen. Es gibt zwei Gruppen von Methoden. Die erste Gruppe stellt Methoden vor, die auf einigen erheblichen Vereinfachungen beruhen. Sie sind schnell und haben eine hohe Genauigkeit, aber es fehlt ihnen an Allgemeinheit. Die zweite Gruppe kann potenziell auf die allgemeinsten Bedingungen verallgemeinert werden, ist aber extrem teuer. Dieses Problem kann nur mit dem fortschrittlichsten Ansatz zur Ordnungsreduktion gelöst werden - die Low-Rank-Tensor-Zerlegung.Das Hauptziel unserer Forschung ist es daher, einen universellen Solver mit reduzierter Ordnung zu entwickeln, der aus eng miteinander verknüpften und in Synergie arbeitenden Methoden besteht, der effizient und präzise sein wird und sich auf die allgemeinsten Problemstellungen anwenden lässt. Der Kern des neuen Solvers besteht aus der spektralen nicht-deterministischen FEM, die durch adaptives Sampling und die Low-Rank-Tensor-Zerlegung ergänzt wird.

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.

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.

The overarching goal of the proposed project at the methodological side is to establish a computationally tractable numerical method that is suited to capture polymorphic uncertainties in large-scale problems (as arising from the numerical analysis of heterogeneous materials microstructures). On the one hand the method will allow for fuzzy probability distributions of the random parameters (describing a microstructures geometry) and on the other hand the method will be based on only a few reduced basis modes. These ingredients will enable to capture epistemic uncertainties in addition to aleatoric uncertainties in a computationally accessible manner. The overarching goal of the proposed project at the application side is to establish a non-deterministic macroscopic material model. On the one hand the model accounts for the heterogeneity of the underlying material's microstructure by computational homogenization, and on the other hand it captures polymorphic uncertainties in the geometry description of the microstructure. The non-deterministic macroscopic material model then represents the necessary input for the mechanical design of macroscopic (engineering) structures under due consideration of polymorphic uncertainties in the heterogeneous materials microstructure.