 Greens function methods 
Definition:  Computational method
for simulating
microstructure
evolution and
solving linear
problems. 
Explanation:  Greens functions
methods are employed
in the
SFB with different
aspects:
firstly for solving
linear elastic
problems in
multiphase
materials,
and secondly for
predicting diffusion
limited
microstructure
evolution during
solidification and
melting problems.
For the first
application, the
misfit
stresses at coherent
interfaces are
expressed in terms
of localized force
terms at the
interfaces. They act
as point forces, for
which the solution
of the elastic
problem in an
infinite sample is
known via Greens
function. Therefore,
the total
deformation is a
superposition of the
forces from all
interfaces. Similar
considerations are
possible for
specific finite
geometries, which
typically lead to
image forces at
free surfaces. For
arbitrary
geometries, the
description results
in
boundary integral
methods.
For the second
aspect of
predicting
microstructure
evolution, the
Greens function
methods are a
complementary
approach to phase
field modeling.
For an example of
solidification, the
partitioning at the
moving interfaces
under local
equilibrium
effectively leads to
source terms for the
diffusion problem in
the bulk, in analogy
to the mechanical
problems discussed
above. The
additional challenge
here is that the
location of the
interfaces is not
known in advance and
it needs to be
solved as the part
of the problem. The
entire problem would
then be reformulated
as
integrodifferential
equations, which
have to be solved
selfconsistently
within
the
sharp interface
concept. This is
a particularly
useful approach for
the steady state
regime, where it can
often lead to
more efficient
computation and
precise descriptions
than phase
field modeling
and
allows to extract
analytical scaling
relations. In the
combination with the
diffuse interface
description of pphase
field modeling
which
can decide about the
stability of the
steady state
solutions, it
becomes a
powerful way to
simulate
microstructure
evolution. 
Picture / Figure / Diagram: 

Cellular structures during isothermal solidification in peritectic systems. Depending on the lamella spacing of the primary solidification, either dendritic or cellular microstructures are kinetically preferred.


SFBLink:  In the project A9, Greens function methods are used for modeling peritectic solidification, grain boundary melting and hydride formation near surfaces. 
References:  G. Boussinot, C. Hüter, R. Spatschek, E. A. Brener, Acta Mat. 75 (2014), 212. C. Hüter, et al., Phys. Rev. B, 89 (2014), 224104. 

