Research Group of Prof. Dr. Barbara Verfürth

Research Projects

Current

Metamaterials are modern, artificially constructed materials that are tailored to exhibit new, astonishing physical properties. Therefore, they play a decisive role to control and manipulate waves, for instance in laser applications. Metamaterials are characterized by fine structures of different material components. The typical length of these fine structures is much smaller than the length of the whole material bulk. Further important building blocks in applications are nonlinear material responses and time modulation. Finally, the robustness of the material properties with respect to imperfections in the fabrication process is highly relevant. Within mathematical models, these applications lead to partial differential equations with a coexistence of multiple temporal and spatial scales, nonlinearities, and random perturbations. Numerical simulations have very high potential in the material design as they can replace time-consuming and costly experiments. Yet, standard numerical methods need to resolve all fine material structures so that their computational effort is prohibitive even with today’s computer resources. In contrast, computational multiscale methods (CMMs) deliver a macroscopic representation of the solution by suitable local upscaling processes. However, the incorporation of nonlinearities, random perturbations, and multiscale dynamics require new computational paradigms for several reasons. Firstly, CMMs often rely on linear arguments that break down for nonlinear problems. Thus, most approaches propose to couple nonlinear problems on fine and macroscopic scales in a rather complicated manner. Secondly, Monte Carlo techniques require many multiscale simulations with new, rather costly upscaling processes for each random sample. Present approaches at least for the numerical analysis additionally rely on stochastic homogenisation results. Thirdly, CMMs for dynamical problems mostly treat multiple spatial or multiple temporal scales exclusively.In this project, we develop and analyse novel CMMs to tackle nonlinear, randomly perturbed, and dynamical problems. The main goals are connected to fundamental mathematical and computational challenges requiring a revolutionary coalescence of multiscale methods, model reduction, uncertainty quantification, and time integration. We (a) explore adaptive linearised and nonlinear approximation spaces for nonlinear multiscale problems, (b) unite multiscale methods and Monte Carlo approaches for randomly perturbed problems, and (c) bridge spatial and temporal multiscale methods for rapid multiscale dynamics. While the general nature of our approaches allows their application to a wide range of problems, we put special emphasis on wave-related phenomena. Moreover, we rigorously justify all methods by error estimates which is crucial beyond the experimentally validated regime. Ultimately, this project will push forward the frontiers of CMMs for realistic (metamaterial) applications.

Completed