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Scale transitions in space and time for materials
One of the goals of the workshop, i.e. to bring together researchers of different fields working on
scale transitions, was well achieved. The research disciplines represented by the speakers and
participants were mathematics, physics, bio- and chemical engineering, theoretical and applied
mechanics as well as computational and material sciences. The workshop lasted one full week
with 50 participants from
within and outside the
One of the key challenges in scale transitions is to retain the relevant physical parameters by
either coarse-graining or homogenizing the fine scale variations. Different methodological
approaches and terminologies are used in various fields, and it was obvious throughout the
workshop that the respective communities are often not aware of the work done in neighboring
areas. Among the originally scheduled issues were: nonlinear behaviour, defects, discreteness,
interfaces and surfaces, non-affinity, and time-dependent behavior.
The program was scheduled through a daily focus on a different theme:
1. spatial scale transitions,
2. temporal scale transitions,
3. discrete to continuous transitions,
4. scale transitions with space-time interactions, and
5. complex fluids, soft matter or granular matter.
Within these topics, special sessions were organized for (i) flash-poster presentations, followed
by a poster session; (ii) definition of the goal and presentation of open questions beforehand; (iii)
an interactive session in small groups on the definition of homogenization versus coarse-graining;
(iv) an interactive session in small groups on particular difficulties characterizing soft-matter and
the importance of handling inhomogeneities in scale-transitions, and (v) the concluding session,
revealing the “Lessons Learned”.
Scale transitions in space and time are typically presented in a space-time diagram, as shown
below for the particular case of metals:
Among the Lessons Learned are the results of the interactive workshop 1:
W1-A: The terminology used in various fields and their characteristic differences evolved into the
summary in the Table below.
Homogenization (mostly in space) leads to the same type of eqs. (possibly with a reduced set of
dofs [degrees of freedom]). In contrast, CG [=coarse-graining] generally applies to timedependent
phenomena and is based on ensemble averaging and can lead to different equations,
reduced dofs. It relies on the correlation between fine scale fluctuations, which should NOT be
averaged out but, instead, lead to irreversibility and emergent behavior. Other terms and methods
were discussed – Hierarchical Multiscale Modeling (HMM), up-scaling, micro-macro transitions –,
but they can be classified in one of these two main groups.
W1-B Q2: The second main question was: “What is the methodological complexity that
discriminates upscaling in time from upscaling in space?” Discussions among participants
indicated that length-scale jumps are easier to identify; thinking in terms of length-scales is more
intuitive for the majority of researchers working on solids. Multi-scale in time invokes the
correlation of fine, rapidly varying phenomena; the community working on small oscillatory and
wave phenomena is likely to be more familiar in this respect. Furthermore, the role of
discreteness for time-scaling vs. discrete phenomena in space was discussed. Discreteness
seems to bring a coupling between space and time. A general problem is the infinite propagation
speed in some models, which assume quasi-static deformations. Discretization is an
approximation that makes propagation-speeds finite. The participants seemed to indicate that the
history-dependent behaviour is the main complexity to handle in time-scale transitions in solids.
However, within GENERIC (see below), basic ingredients to do so seem to be well in place if time
scale separation holds.
W2-A: Among the results from Workshop 2 are the different interpretations of soft-matter (versus
solids) and what makes soft materials special. The following issues can be special for soft-matter:
fluid- and solid-like behavior is important at the same time; different time- and length-scales are
involved; the systems are highly heterogeneous/ disordered; energy/kBT is around unity; they
display long time tails; small changes can lead to big effects.
W2-B: The role of inhomogeneities for scale transitions needed first a definition of non-affinity vs.
inhomogeneity (what comes first and leads to the other?). Then the size/scale of inhomogeneities
is linked the scale-jump and possibly determines its upper-limit. The question what one can do
when the length-scale of inhomogeneities changes (sometimes rapidly) during a process
remained unclear. In any case one has to be careful with smoothing out inhomogeneities, since
they make the system behavior change with their presence and state. The length scale set by
inhomogeneities can lead to new state-variables and parameters like porosity, a structure tensor
or, e.g., pair-correlation functions.
The outcome of all discussions during the lectures and the various discussion sessions were
summarized in the Lessons Learned session, which can be summarized as follows:
· Mathematical tools exist that justify the limit passage from micro to macro, for
homogenization of energy-driven systems. A "smart" Γ-limit (‘smart’ indicating the "right"
variables were chosen) but there is no general recipe (except for special/trivial cases). There
seems to be, however, a (partially unexplored) connection between GENERIC (see below)
and the mathematical tools at hand.
· GENERIC received extensive attention in the programme and was profiled as a very powerful
methodology, enabling systematic scale transitions relying on coarse graining and correlation
of fluctuations. The scale separation in time is thereby essential (Hütter). Understanding the
link between the handling of the entropy terms versus some proposed stochastic methods
(with particular probability distribution functions) remains open but seems feasible.
· The notion of history was often mentioned as a really complicating factor in time scale
transitions, but this does not seem to be justified completely. Handling history dependent
behaviour can be done by incorporating the related micro-information into a state variable
and thereby avoiding (as for example in Maxwell fluids) the memory kernel. The choice of the
"right" state variable is essential and requires a good physical insight in the system studied.
· There is no essential contradiction between kinetics (evolution) controlled solution methods
vs. equilibrium-driven methods. The equilibrium-driven methods usually only provide a
necessary condition and hence a lower-bound for the real evolution path. The kinetics
controlled methods are more rich and detailed and incorporate the physics of the evolution.
Phase field models are well developed methods to resolve several energy driven-kineticslimited
microstructure evolution processes. They are nowadays mature and reach out now to
coupling to other scales and more complex mechanical behavior (kinetics in mechanics and
materials science do not refer to inertia!).
· Dissipation is inherently coupled to fine scale fluctuations. The precise definition of dissipative
events can be scale-dependent, e.g. where systems may be either considered as open or
closed. The issue of a fluctuation-dissipation theorem and related emergence of entropy was
not worked out to the end.
· Biophysical systems are characterized by complexity. To handle this, use is made of an
efficient scale separation map. This map classifies each of the processes at different spatial
and temporal scales. Model for each of the processes being available, most emphasis is
given to the coupling between different processes (= resolving multiple processes at different
scales). This is not a true scale transition (= resolving a single process across the scales),
and the methods used therefore more focus on solving the materiomics loop.
· Peridynamics was presented as a promising method, based on solving integral equations
rather than PDEs. The method seems to show great promise for problems that present
discontinuities, but there are still many challenges ahead before its potential can be fully
exploited. It was applied to non-local elasticity, but how to go beyond (finding the kernel for
nonlinear material behaviour) is an open issue. The numerical implementation and solution
seems very convenient, relying on a discrete (MD-like) solution strategy. Problems to be
resolved are the proper handling of various types of boundary conditions and interface
conditions (resolving heterogeneities). At this stage, its relation to large scale transitions and
to SPH (smooth particle hydrodynamics) was not yet fully clear.
· GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling): For
this promising method, most important aspect is the choice of internal variables, for which a
healthy insight and intuition is required. For a scale transition in time, time-scale separation is
· Spelling checkers: Some mathematical tools exist now that allows for a proof of the limit
passage. In a primitive form, GENERIC can be used as a spelling checker, but this
methodology seems more powerful than just that.
· An inherent space-time coupling seems to exist upon studying the same process at different
scales. This does not exclude the existence of fast-large and slow-small processes (as
exemplified for bio-systems). A true scale transition (for the same process) perpendicular to
the diagonal in the x-t-diagram does not seem trivial (if it exists at all). More importantly, a
space-time coupling may abruptly change due to the occurrence of instabilities (e.g.
microstructure evolution, strain bursts, dislocation avalanches, etc., where a rapid evolution
occurs at a larger length scale). Such instabilities may seriously compromise the scale
separation assumed up front and hence may render several methods inapplicable (e.g.
homogenization for localization of deformation). This implies that, even when one succeeds
to develop a macro-model based on a micro-model, the occurrence of an instability may call
for the explicit incorporation of the finer scale that controls the physics of the instability.
Many of the original questions that were posed by organizers and participants (see sheets on the
website) were discussed and partially answered. In summary, a better focus and improved insight
was gained, but also many new questions were raised and challenges for further work and future
collaborations could be identified.
Some pending issues discussed that were unresolved at this stage:
1. Does time-scale separation hold in metal plasticity?
2. How to extend and handle interfacial problems and heterogeneities in GENERIC (seems
3. feasible though)?
4. Does it make sense at all to treat scale transition in time separate from those in space?
5. What are the conceptual differences for different CG methods, in particular since some CG
6. methods only resolve a small iterative step, which is insufficient.
We thank all participants for their contributions, input and active participation. Financial support is
acknowledged from the
organizers strongly acknowledge the hospitality at the Lorentz-Center and the efficient and
competent help of the local staff.
K. Bertoldi (
H. Steeb H. Steeb (Enschede/Bochum, Netherlands/Germany)
S. Luding S. Luding (Enschede, Netherlands)
M. Geers M. Geers (Eindhoven, Netherlands)
E. van der Giessen E. van der Giessen (Groningen, Netherlands)