Re: [Getfem-users] Periodic BC

[Yves Renard]

On dimanche 19 septembre 2010, Artur A. Poźniak wrote:
> 2010/9/16 Yves Renard <address@hidden>
> 
> > On mercredi 15 septembre 2010, Artur A. Poźniak wrote:
> > > Hello All,
> > > 
> > > I would like to impose periodic BC to my linear elastic system (2D). I
> > 
> > need
> > 
> > > the *derivative* of one component of the displacement field to be equal
> > 
> > on
> > 
> > > respective edges, i.e. {u_y},x on left = {u_y},x on right. ,x denotes
> > > partial derivative along x and u_y is the y-component of the
> > > displacement vector u.
> > > I know that there is no brick responsible for this in GetFEM, so I
> > > kindly ask for some help. I would expect it to be done with the use of
> > 
> > additional
> > 
> > > system of Lagrange multipliers. Is that correct?
> > > 
> > > Thanks in advance
> > > Artur Pozniak
> > 
> > You are right, there is no specific brick to prescribe a periodic
> > boundary condition (this would be possible to build one).
> > You do not have to prescribe the equality of a derivative (or a stress in
> > elasticity), the continuity of the unknown is sufficient.
> > If the degrees of freedom correspond from a boundary to the other (in the
> > case
> > of a regular mesh for instance) then you can easily bouild the constraint
> > matrix prescribing the eaqulity of these degrees of freedom (and add a
> > constraint brick). If the mesh is not the same, you have to prescribe the
> > equality in a weak form with a multiplier which is more difficult.
> > 
> > Yves.
> 
> Dear Yves,
> 
> first of all I would like to say thank you for the answer. I would also to
> ask you about the possibility of imposing such a boundary conditions
> (periodicity of a derivative). My domain is a repeating unit cell (RUC). I
> imposed the boundary conditions mentioned by you at the beginning but I did
> not get the expected results since I know what is the behavior of the
> model. It is also imposible to build structured mesh. Could you please
> point out some papers dealing with the idea of weak formulation of
> periodic boundary conditions? I failed to find anything useful.
> 
> Sorry for bothering you
> and many thanks in advance!
> Artur Pozniak


A weak bonding condition is something like

\int_{\Gamma} (u^h_+ - u^h_-) w_h = 0  for all w_h in W_h

where \int_{\Gamma} is the integral on the considered boundary, u^h_+, u^h_-
 are the value of the unknown on both sides and W_h is a correct space of 
multipliers. For non matching meshes, this is called a mortar technique. there 
is a wide litterature on that (see 
http://en.wikipedia.org/wiki/Mortar_methods).

But, what is sure is that for a 2D elastic problem (so an order two problem) 
this is not correct to prescribe the equality of the derivative. First of all 
because the derivative on the boundary has not a clear sense. What has to be 
equal is the corresponding Neumann condition. But this is automatically 
ensured when you prescribe the equality of the displacements (action-reaction 
law).


Yves.




-- 

  Yves Renard (address@hidden)       tel : (33) 04.72.43.87.08
  Pole de Mathematiques, INSA-Lyon             fax : (33) 04.72.43.85.29
  20, rue Albert Einstein
  69621 Villeurbanne Cedex, FRANCE
  http://math.univ-lyon1.fr/~renard

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