Periodicity with simultaneous displacement controlled BCs

[David F]

Hello everyone,

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.

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).

Thank you in advance.

P.S.: suggestions of alternative procedures to achieve the same goal are welcome.

[Wolfgang Bangerth]

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: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[David Wells]

Hi David,

There are a lot of different ways to solve this problem, but I think the relevant solution is to consider a more Lagrangian perspective - I recommend that you do not modify the Triangulation directly (i.e., don't move vertices) but instead represent the initial deformation as a finite element field. You can then combine the initial deformation with MappingFEField to compute quantities in the deformed configuration - this should enable you to correctly set periodic boundary conditions without having to worry about wobbly edges.

Does this work for you?

Thanks,

David

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[Jean-Paul Pelteret]

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

 

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[David F]

Thank you all, I found the way to do it. For future users trying to solve the same problem, I also found extremely helpful this paper:

Effective properties of composite materials with periodic microstructure: a computational approach, J.C. Michel, H. Moulinec, P. Suquet, Comput. Methods Appl. Mech. Engrg. 172 (1999) 109-143

David.