Bridging Scales in Random Materials

Projektinformationen

RandSCALES

ID Finanzhilfevereinbarung: 948819

DOI

10.3030/948819

Projekt abgeschlossen

EK-Unterschriftsdatum 20 August 2020

Startdatum 1 März 2021

Enddatum 28 Februar 2026

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 142 899,00

EU-Beitrag

€ 1 142 899,00

Koordiniert durch

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Austria

Periodic Reporting for period 3 - RandSCALES (Bridging Scales in Random Materials)

Berichtszeitraum: 2024-03-01 bis 2025-08-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

Many phenomena in physics and nature involve a multitude of time and length scales. Homogenization theory attempts to derive simplified macroscopic models that describe the effective large-scale behavior of complex microscopic systems. The project RandSCALES aims to achieve a deeper understanding of the role and impact of randomness in multiscale problems, as it is present for instance in mathematical models for heterogeneous media. The first focus of the project is on problems with nonconvex energies, for which random heterogeneities may give rise to rough energy landscapes and complex macroscopic behavior. The second focus is on effective homogenization theories in the absence of clear scale separation. The third focus of the project concerns the regularizing effect of randomness in interface evolution problems.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

As the key results in the first reporting period, we have developed a first quantitative homogenization theory for energies in fracture mechanics and obtained lower bounds on energies for a model problem in fracture mechanics; furthermore, we have achieved a first result on convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow prior to topology changes. Additional results involve stochastic PDEs and numerical problems in homogenization.
As the key results in the second reporting period, we have finalized results on the weak-strong stability of higher-order curvature-driven flows, begun to study the quantitative homogenization of interface motions through a field of random obstacles as well as the homogenization of fracture based on local energy minimization principles, and obtained a first result on the weak-strong stability of multiphase mean curvature flow beyond shrinking circle type topology changes.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

We aim to develop a quantitative theory of homogenization in settings which were so far out of reach, such as nonconvex problems and problems without clear separation of scales. Regarding nonconvex problems, we have already achieved a first result in the context of fracture energies and have successfully started working on the homogenization of interface motions through a field of random obstacles. Furthermore, we have obtained a lower bound on the energy density for a model problem in fracture mechanics. As another goal, we seek to develop a conditional stability theory for some of the most important interface evolution equations and rates of convergence for their diffuse-interface approximations. Regarding the latter, we have already obtained a positive result for the approximation of multiphase mean curvature flow by the vectorial Allen-Cahn equation as well as a result on the convergence of the Navier-Stokes-Allen-Cahn equation towards two-phase flow with sharp interface.

A schematic depiction of stochastic homogenization