Dr. Lukas Pflug

In the area of nonlocal balance equations, integro-differential equations, in which the process dynamics depends in an integral sense on the solution, we have been able to achieve fundamental results in recent years. From a mathematical point of view, the existence and uniqueness theory without entropy conditions in the one and multidimensional case, on restricted domains, for irregular convolution kernels, and discontinuous dynamics on the one hand, and the consideration of local limits of nonlocal balance equations, the so-called sinuglar limit problem, on the other hand, should be emphasized. We were able to prove this rigorously for the first time for monotone initial data and extend it to more general classes of transport and balance equations.
Similarly, we were able to analyze classical controllability issues for nonlocal balance equations. Nonlocal balance equations are the basis of much macroscopic modeling. Among others, we focus on nanoparticle synthesis, traffic flow modeling, and pandemic modeling.

This research area is concerned with both the accurate description of structure-property relationships and the development of matched optimization algorithms.
With the application focus on the optimization of optical properties of particulate systems, we developed suitable shape optimization approaches as well as a sequential global optimization (SGP) algorithm for solving multi-material optimization problems.

Furthermore, the analysis of the structure-property relationship of optical properties of particulate systems allowed for the first time the automated determination of multidimensional particle shape distributions by means of analytical ultracentrifuge (AUZ).

The development of hybrid stochastic-continuous gradient methods is another focus of my research, which is applicable to the first mentioned research areas. The goal of these algorithms is to solve PDE-restricted optimization problems with a large number or even infinite number of scenarios with as few numerical simulations as possible. Such problems arise, for example, in optical or acoustic application problems as well as in distributionally robust optimization. We were able to show the convergence for a class of stochastic-continuous gradient methods in a mathematically rigorous way and to extend them with respect to flexible numerical integration rules and with respect to step-size controls. Only this allowed the successful optimization of optical properties by means of structural optimization. In the field of robust optimization of composite materials with respect to crack propagation, we were able to successfully apply comparable concepts.

The key objective and long-term vision of Collaborative Research Centre 1411 is the targeted design of particulate products by rigorous optimisation based on predictive structure-property and process-structure functions.
We target scientific breakthroughs in the product engineering of nanoparticles with optimised optical properties produced by continuous synthesis directly coupled to property-specific classification of nanoparticles by chromatography. These challenges are addressed from different perspectives in four strongly interlinked research areas. These will be underpinned by the development of joint methodologies in synthesis, classification, characterisation as well as modelling, simulation, and optimisation.

Subproject: D03 – Unifying mathematical framework for synthesis and chromatographic separation of nanoparticles
The objective is the model-based optimization of the synthesis and the chromatographic separation of nanoparticles. Building on the analysis of a unifying nonlocal balance law, efficient numerical solution schemes will be developed. The basis for these is the mathematical structure of the equation, which allows the construction of semi-analytic solutions. For the optimization of processes, optimality conditions will be analytically derived, which are the starting point for gradient-based optimization methods. In cooperation with the synthesis and chromatographic separation projects these will be applied to technical processes.

Integrating Mechanics, Materials Science, Mathematics, Chemistry, and Physics, FRASCAL aims to improve understanding of fracture behaviour in brittle heterogeneous materials by developing simulation methods that are able to capture the multiscale nature of failure. With (i) its rooting in different scientific disciplines, (ii) its focus on the influence of heterogeneity on fracture behaviour at different length and time scales as well as (iii) its integration of highly specialised approaches into a “holistic” concept, FRASCAL addresses a truly challenging interdisciplinary topic in mechanics of materials.

Although various simulation approaches describing fracture behaviour exist for particular types of materials and specific time and length scales, an integrated and all-encompassing approach that is able to capture fracture processes in different – and in particular heterogeneous – materials at various length and time resolutions is still lacking. Thus, the objective of this interdisciplinary Research Training Group (RTG) consisting of experts from mechanics, materials science, mathematics, chemistry, and physics is to develop the necessary methodology to investigate the mechanisms underlying brittle fracture and how they are influenced by heterogeneity in various materials.

The insights obtained together with the methodological framework will allow tailoring and optimising materials with regard to fracture behaviour. FRASCAL covers a representative spectrum of brittle materials and their composites, together with granular and porous materials. We study these at length and time scales relevant to science and engineering, ranging from sub-atomic via atomic and molecular over mesoscale to macroscopic dimensions. Our modelling approaches and simulation tools are based on concepts from quantum mechanics, molecular mechanics, mesoscopic approaches, and continuum mechanics. These are integrated into an overall framework, which represents an important step towards a virtual laboratory eventually complementing and minimising extensive and expensive experimental testing of materials and components.

Within FRASCAL, young researchers under the supervision of experienced PAs perform cutting-edge research on challenging scientific aspects of fracture. FRASCAL fosters synergies in research and advanced education and is intended to become a key element in FAU‘s interdisciplinary research areas “New Materials and Processes” and “Modelling–Simulation–Optimisation”

Subproject: P11 – Fracture Control by Material Optimization
In previous works, e.g. (Prechtel, et al., 2011), the dependence of failure mechanisms in composite materials like debonding of the matrix-fibre interface or fibre breakage have been discussed. The underlying model was based on specific cohesive zone elements, whose macroscopic properties could be derived from DFT. It has been shown that the dissipated energy could be increased by appropriate choices of cohesive parameters of the interface as well as aspects of the fibre. However due to the numerical complexity of applied simulation methods the crack path had to be fixed a priori. Only recently models allow computing the full crack properties at macroscopic scale in a quasi-static scenario by the solution of a single nonlinear variational inequality (Leugering, Prechtel, Steinmann, & Stingl, 2013) for a given set of material parameters and thus model based optimization of the fracture properties can be approached.

The goal of the project is to develop an optimization method, in the framework of which crack properties (e.g. the crack path) can be optimized in a mathematically rigorous way. Thereby material properties of matrix, fibre and interfaces should serve as optimization variables.

The iPMT consortium consists of partners from industry and academia. The aim of the project is the development and validation of methods for the synthesis of training data for Deep Learning-based particle measurement techniques, exemplified by image-based methods for the detection of particles in (microscope ) images for the determination of particle size distribution, as well as time- and location-dependent measurements of the transmission spectra of particle dispersions with commercially available measurement devices as an example of multimodal sensor data. Complementary to this, methods for quantifying the quality of the generated synthetic data sets with respect to their suitability for machine learning will be developed and validated.
To facilitate future collaboration between research and industry with respect to classified particle images, methods for anonymization and abstraction of data sets from industry will be developed.
The project is funded by the German Federal Ministry of Education and Research (BMBF) and is a collaboration with Einar Kruis (Institut für Nanostrukturtechnik Universität Duisburg-Essen), Josef Pauli (Lehrstuhl für intelligente Systeme Universität Duisburg-Essen), Wolfgang Peukert (Lehrstuhl für Feststoff- und Grenzflächenverfahrenstechnik, FAU), Dietmar Lerche (LUM GmbH), Sebastian Maaß (SOPAT GmbH), Rainer Friehmelt (BASF SE, associated) and Lukas Pflug (CSC, FAU)