Numerical homogenization strategies for extended continua

Project Leader:

Project members:

Project partners:


  • Deutsche Forschungsgemeinschaft (DFG)


There is an increasing interest in cellular structures in science and industry.
Generally, foams feature a complex material behaviour. Modelling aspects
to be considered are evolving porosity and anisotropy, size effects, non-
affine microstructures, e.g. with negative Poisson’s ratio.

If the characteristic microstructural length scale h becomes comparable to the
macroscopic length scale H, analytical homogenisation strategies fail and must
be replaced by numerical homogenisation approaches. The numerical homo-
genisation scheme to be applied is essentially based on the solution of two
nested boundary value problems (BVP), one for the macroscale and one for the
microscale. This is commonly called two-level FEM or FE2-scheme.

On the macroscale, the macro-deformation gradient and the micro-deformations
are calculated and projected to the microscale. The stress and hyperstress ans-
wer of the microstructure are homogenised and transferred back to the macro-

Outline of the project:

The aim of the project is to model the macroscopic behaviour of cellular materials
based on the inherent microscopical properties. This results in generalised con-
tinua of micromorphic type. To solve this highly nonlinear problem, a combination
of two FE-codes is applied:

  • AdhoC4: a higher-order finite element code for solid mechanics.

PANDAS is used on the macroscale. For this it is continously enhanced by extended macro theories during this project. AdhoC4 will serve as a solver for the microscopic problem. It is based on standard continuum higher-order finite elements allowing for an anisotropic Ansatz where each component of the displacement field can be chosen individually. With these elements the beam-like structures on the micro-level can be discretised efficiently, i.e. by using a high polynomial degree p in longitudinal direction while the polynomial degree q in thickness direction can be lower.
Both programs work in a partitioned solution approach (i.e. exchanging strain- and stress-like quantities) via Message Passing Interface (MPI).


  • Düster, A.; Rank, E.; Diebels, S.; Ebinger, T.; Steeb. H.:
    Second order homogenization method based on higher order finite elements.

    In: PAMM, hrsg. von GAMM, Bd. 5, 2005, pp. 391-392
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    Numerical homogenization of foam-like structures based on the FE2- approach.
    In: PAMM, hrsg. von GAMM, Bd. 8 (1), 2008, pp. 10549-10550
  • Jänicke, R.; Steeb, H.; Diebels, S.; Sehlhorst, H.-G.; Düster, A.; Rank, E.:
    Investigations on an elastic micropolar continuum model for large deformations.
    In: PAMM, hrsg. von GAMM, Bd. 8 (1), 2008, pp. 10573-10574
  • Sehlhorst, H.-G.; Jänicke, R.; Düster, A.; Rank, E.; Steeb, H.; Diebels, S.:
    Numerical investigations of foam-like materials by nested high-order
    finite element methods
    . In: Computational Mechanics, hrsg. von Peter Wriggers, Volume 45, Number 1 / Dezember 2009, Springer Verlag
  • Jänicke, R.; Diebels, S.; Sehlhorst, H.-G.; Düster, A.:
    Two-scale modelling of micromorphic continua.
    In: Continuum Mechanics and Thermodynamics, hrsg. von Stefan Seelecke und Lev Truskinovsky, Volume 21 Number 4 / September 2009, Springer Verlag
  • Sehlhorst, H.-G.; Düster, A.; Jänicke, R.; Diebels, S.:
    On Dirichlet boundary conditions in second-order FE2-schemes.
    In: PAMM, hrsg. von GAMM, Bd. 10 (1), 2010, pp. 423-424
  • Sehlhorst, H.-G.:
    Numerical homogenization stategies for cellular materials with applications in structural mechanics. Fortschritt-Berichte VDI, Reihe 18 Mechanik/Bruchmechanik Nr. 333, VDI Verlag GmbH, Düsseldorf 2012