Homogenization methods and multiscale modeling:
Marc G.D. Geers1, Varvara G. Kouznetsova1,
K. Matous2 and Julien Yvonnet3
1Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven, The Netherlands.
2Department of Aerospace and Mechanical
University of Notre Dame
Notre Dame, IN, 46556, USA.
3Laboratoire Modelisation et Simulation Multi
Universite Paris-Est Marne-la-Vallee,
This chapter focuses on
computational multiscale methods for the mechanical
response of nonlinear heterogeneous materials. After a
short historical note, a brief overview is given of
some recent activities in the field, with a particular
focus on nonlinear homogenization methods. The
two-scale nonlinear computational homogenization
scheme for mechanics is presented, along with details
on representative unit cell aspects and statistics.
Model performance is advocated through a decoupled
implementation and multiscale schemes based on the
nonuniform transformation field analysis.
High-performance parallel multi-scale implementations
of the computational homogenization scheme are
addressed in more detail.
spite of the progress made over the past decades, a lot
of work still remains to be done in the broad field of
multi-scale computational engineering. Among the ongoing
and expected trends and challenges, the following ones
- 3D, the third dimension (see section 7):
Most applications focus on
two-dimensional descriptions of the considered materials
with their governing deformation mechanisms. Without any
doubt, more realistic threedimensional computations and
experimental analyses will be necessary, e.g. Shan and
Gokhale (2001); Mosby and Matous (2015). The progress to
be made here is evidently coupled to the increase in
computational power in the coming decade(s).
The high computational cost of nonlinear
multiscale solution methods calls for novel efficient
approaches that adequately balance computational speed
and accuracy. The further development of model reduction
techniques in combination with nonlinear multiscale
schemes will be a necessity to make further progress.
- Interaction with materials science, physics
The various cross-sections presented in this
overview have clearly illustrated the growing
interaction with materials science. The need for more
accurate microstructural deformation models goes hand in
hand with the need for more physics in the applied
models. At the micron scale and even more at smaller
scales, interaction with other physical phenomena is of
major importance. It is expected that this trend will
become more pronounced in the near future. As mentioned
in the introduction, multi-scale methods are of general
importance and attract attention from all fields in
science and engineering. In particular, the developments
in the physics community and the computational
mathematics community are quite relevant for the
nonlinear mechanics of materials. Increasing the
interaction with these neighbouring fields, may speed up
the developments considerably.
- Multi-scale versus multiple scales:
There is a growing interest in establishing
correct scale transitions for various nonlinear problems
in mechanics of materials, where in the mean time some
problems are probably best tackled by considering
multiple scales in a single domain. Challenges remain in
both, where the most challenging example is probably to
transition from damage to fracture across all length
- Temporal scale transitions:
Besides spatial scales, a lot more attention
has to be given to temporal scales. Engineering
approaches easily resort to accelerated tests to assess
the lifetime of a material in a particular application.
Many small-scale deformation mechanisms are
characterized by typical time scales, which cannot be
altered. Accelerated testing is therefore not always an
adequate tool, since it may inhibit certain small-scale
deformation processes. The proper incorporation of
various (extreme) time scales in multi-scale models
therefore remains a challenge. Methods like the GENERIC
scheme (Ottinger, 2005; Hutter and Tervoort, 2008b),
offer clear opportunities in this sense.
KM was supported in part by the U.S.
Department of Energy, National Nuclear Security
Administration under the Predictive Science Academic
Alliance Program, under contract no. DE-NA0002377. KM
would like to thank two of his graduate research
assistants, M. Mosby and A. Gillman, for their numerous
contributions to the multi-scale modeling work presented
in this chapter.
the chapter here.
(c) 2015 Notre
Dame and Dr. Karel Matous