Homogenization by high order FEM

Description

Composites and perforated (lattice) materials are widely used in many practical applications, such as aircraft, civil engineering, electrotechnics, and many others. These are materials with a large number of heterogeneities (inclusions or holes), and in strong contrast to continuum materials, their behavior is definitively influenced by micromechanical events.

Limited by computing resources, many practical problems are still out of reach using direct simulations that require scale resolution. On the other hand, numerical methods based on the "averaging" techniques arising in classical homogenization may give only poor convergence results. Such techniques are based on asymptotic analysis for the small length scales going to 0. In practice, however, the small length scales are given and fixed, and the asymptotic limit may be only a poor description of the phenomena of interest. Even if additional correctors are employed to model some of the small scale features, the asymptotic nature of homogenization techniques may preclude its use in the context of given physical small length scales.

The present project is supported by the Swiss National Science Foundation under Project Number BBW 21-58754.99 and deals with the investigation of generalized Finite Element Method (gFEM) for the numerical solution of boundary value problems with oscillating coefficients or geometries. Under the project, we develop a systematic approach to analysis and implementation of high order gFEM for elliptic problems with oscillating coefficients and domains. In particular, we aim at generating families of Finite Elements which simultaneously discretize the macroscopic and the microscopic variables. The key to our approach are representation formulas for the full homogenization problems on unbounded domains. These representations are based on an assumption of scale separation and generalized Fourier inversion integrals (continous analogues of the Bloch series expansions). The Fourier representation formulas were generalized to perforated domains and also to the diffusions in networks which are 1-dimensional domains embedded in two or three dimensional space.

gFEM is based on two-scale FE spaces obtained by augmenting the standard piecewise polynomial FE spaces with problem-dependent, non-polynomial micro shape functions that reflect the oscillatory fine-scale behavior of the exact solution at the small length scale. Generalized Finite Element Methods do not require scale resolution since the appropriate small-scale behavior is built into the micro shape functions. The crucial question in this context is, however, how to choose the shapefunctions. Here the Fourier representations come into play: in the gFEM we propose resolve the fine-scale effects of the solution with the Fourier kernels which are precomputed as solution of a unit-cell problem which depends on suitable wave parameters.

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 done together with Dr. Markus Melenk from the Max Planck Institut Leipzig within the german priority program ("Schwerpunkt") Mehrskalenprobleme. New tools, namely parameter dependent regularity analysis for homogenization problems where the unit-cell problem does not admit full elliptic regularity are developed.

Contacts

Prof. Christoph Schwab, Dr. Ana-Maria Matache and Andreas W. Rüegg.