Dear David,
Indeed, Wolfgang is correct. I gather that you’re trying to produce the sort of
displacement field that you see in the attached image. This reference
> C. Miehe. Computational micro-to-macro transitions for discretized
> micro-structures of hetero- geneous materials at finite strains based on the
> minimization of averaged incremental energy. Computer Methods in Applied
> Mechanics and Engineering, 192(5–6):559–591, January 2003. DOI:
> 10.1016/s0045- 7825(02)00564- 9.
gives a very tractable explanation on how to set up the problem in terms of a
Lagrangian periodicity frame (and sets it up in such a way that you can use
constraints to impose the periodicity).
The upshot of what’s described therein is that if you describe the microscopic
displacement as the first order expansion
u = \bar{\Grad{u}}.X + \tilde{u},
where \bar{\Grad{u}} is the applied macroscopic displacement gradient
(encompassing the shear that you wish to apply; also equal to the volume
average of the microscopic displacement gradient) and \tilde{u} is the
microscopic fluctuation field. So there is a superposition of the average field
and the fluctuation field. Then the periodicity conditions for the microscopic
displacement field are
[[u]] = \bar{\Grad{u}}.[[X]]
where [[ ... ]] is the jump of the field across the periodic interface and [[ X
]] is the difference in reference position between two points on the periodic
interfaces.
So you can see that when you’re solving for the total displacement, there is
indeed some “offset” the needs to be considered, i.e. an inhomogeneity in the
constraints that you set.
You also need to impose some additional constraints on the “corner” vertices
that describe the periodic frame in order to remove the rigid body modes.
However, if solving for the fluctuation field itself, i.e.
\tilde{u} = u - \bar{\Grad{u}}.X
then I believe (although I haven’t tried this myself) that the periodicity
condition simplifies to
[[\tilde{u}]] = 0
and, as a post processing step, you then simply superimpose the solution with
\bar{\Grad{u}}.X to get the total displacement.
I really recommend taking a look at that paper, and maybe some of the others
that Miehe has written on the topic. I found them to be quite useful and
enlightening. I do plan on adding a small function to do this in the future, as
the same sort of description of periodic constraints applies at the very least
to elasticity, magnetostatics and electrostatics.
I hope that this quick description helps compliments Wolfgang’s and David’s
replies so that you can implement what you need!
Best,
Jean-Paul
> On 27 Apr 2020, at 17:00, Wolfgang Bangerth <bange...@colostate.edu> wrote:
>
>
> David,
>
>> I'm trying to compute the effective elastic properties of a heterogeneous,
>> linear and a bi-periodic system (i.e., left-right and top-bottom periodic
>> displacement fields). To this system, I would like to apply a global
>> shearing by prescribing the displacement field of the surfaces in the form
>> of Dirichlet BCs. This seems slightly contradictory since a bi-periodic
>> system doesn't have surfaces.
>> However, we can still think of the global shearing as an average surface
>> displacement around which periodic fluctuations occur (the origin of such
>> fluctuations is due to heterogeneous elastic properties). This is
>> illustrated in the picture below.
>> bitmap.png
>> I would like to know which is the best way to do this in deal.II (I have
>> tried with make_periodicity_constraints and interpolate_boundary_values, but
>> the problem is that, as I explain before, we set apparently contradictory
>> constraints).
>
> It seems to me like the correct boundary values are of the form
>
> u(left) = u(right) + offset
>
> and similarly for the bottom/top. The point is that you encode the shearing
> in the 'offset', so you have a variation of periodic boundary conditions that
> includes this nonzero offset.
>
> I believe that the make_periodicity_constraints() function takes such an
> offset argument. Have you tried that?
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bange...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
>
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