Two-scale regularity theory of homogenization problems and two scale FEM convergence theory

Description

The present project is part of the Schwerpunktprogramm Analysis, Modellbildung und Simulation von Mehrskalenproblemen of the German Research Foundation (DFG) and is closely related to the Swiss National Science Foundation Project # 21-58754.99 on Hierarchic Finite Element Models for Periodic Lattice and Honeycomb Materials.

The design and analysis of new advanced materials such as Lattice Block Materials and honeycomb structures is becoming an increasingly important task in today quest for strong, yet lightweight materials. Such materials can be constructed by replicating a small unit-cell of length epsilon. Due to appropriate element proportions (as length to witdh ratio, matrix density) the mechanical properties of such materials are significantly improved. Analysis and design of reliable numerical schemes on such geometries is also of interest for better understanding of failure of new materials with microstructure as regions of non-smoothness of the geometry are areas of stress concentration.

The project intends to develop and to analyze new analytical tools and cost effective computational schemes based on high order two-scale Finite Element Methods (FEM) that are able to reflect the behavior on the small scale epsilon reliably and accurately. A key ingredient of the two-scale FEM is the numerical solution of unit-cell problems that are intimately linked to the repetitive pattern of the structure. Essential for the design and understanding of efficient numerical solution techniques for these unit cell problems, especially for unit-cell problems with piecewise smooth geometries, are precise elliptic regularity assertions for their solutions. Based on this refined regularity theory two-scale FE spaces that incorporate the microstructure are constructed. Investigation of the elliptic regularity for general piecewise analytic unit-cell geometries or coefficients as e.g. unit-cell domains with interior polygonal cavities is based on new tools, namely parameter dependent regularity analysis for homogenization problems where the unit-cell problem does not admit full elliptic regularity. Singularities of the unit-cell solutions due to, e.g., corners of the unit-cell problem, can be taken care of by shift theorems in weighted Sobolev spaces.

We design corresponding two-scale FE approximation theory which gives again optimal robust convergence rates, provided proper mesh-refinements are used within the unit cell (to resolve the corner singularities) and which once again are independent of the a-priori computation of unit-cell solutions.

Contacts

Prof. Christoph Schwab, Dr. Markus Melenk and Dr. Ana-Maria Matache.

Dr. Markus Melenk
MPI Leipzig
Inselstrasse 22-26
D-04103 Leipzig
Germany

Tel.: +49 341 9959 756
Fax.: +49 341 9959 999
Email: markus.melenk@mis.mpg.de