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## Re: [Getfem-users] boundary problem with elastostatic

 From: radames42 Subject: Re: [Getfem-users] boundary problem with elastostatic Date: Wed, 23 Nov 2011 00:25:41 +0100

Dear Yves,

many thanks for your reply. I agree with you, that I have to specify a Neuman boundary condition. AFAIK, it is normaly given by:

u_i,j * n_j = p1 * n_i on S_N  (Notation with Einstein summation convention)

or in weak form

int_S_N u_i,j * n_j * v_i= int_S_N p1 * n_i * v_i   for all test functions v

In my elastostatic problem I have

sigma_ij * n_j = (lambda * delta_ij * u_k,k + 2 * mu * (u_i,j + u_j,i)/2 ) * n_j = p1 * n_i on S_N1

where the unknown is the displacement vector field u. My problem is how to specify this boundary condition (in 3D)?

E.g. how do I get the normal of the face at given point of the mesh m?

If I use the (normal) source term brick, how does it know to use sigma_ij and not u_i,j?

I figured out, that I cannot use the coordinates x,y,z but need to use x[0], x[1], x[2] instead in m.eval("[...]").

I use the demo_tripod.py demo as my template.

As the equations are unreadable in my first email I repeat them here:

• e_ij = (u_i,j + u_j,i)/2              in V
• sigma_ij = lambda delta_ij epsilon_kk + 2 mu epsilon_ij       in V    with lambda, mu Lamé coefficient
• sigma_ij,j + b_j = 0                in V with b data
• u_i = w_i                              on S_D with w data
• sigma_ij n_j = p1 n_i             on S_N1 with p1 data and n the unit normal of the surface (face)
• sigma_ij n_j = p2 n_i             on S_N2 with p2 data and n the unit normal of the surface (face)

Somehow my mailer dropped the references I've read but probably not understand completly. Here they are

> -----Ursprüngliche Nachricht-----

> Von: Yves Renard
> Gesendet: Tue. 22.11.2011 17:21
> Betreff: Re: [Getfem-users] boundary problem with elastostatic
>
>
>
> Neumann conditions are taken into account by the source term brick.
> In weak
> formulation, a Neumann condition is a source term on the
> corresponding
> boundary. See examples and
>
>
> Yves.
>
>
> On mardi 22 novembre 2011, address@hidden wrote:
> > Dear all,
> >
> > I'm a newbie to getfem++.
> >
> > I want to solve an isotropic homogenous elastostatic problem. The
> (first)
> > mesh is ready but I cannot see how to include my boundary
> conditions.
> >
> > Here are the equations of my problem:
> >
> > e_ij = (u_i,j + u_j,i)/2 in V
> > sigma_ij =lambda delta_ij epsilon_kk + 2 mu epsilon_ij in V with
> lambda, mu
> > Lamé coefficient sigma_ij,j + b_j = 0 in V with b data
> > u_i = w_i on S_D with w data
> > sigma_ij n_j = p1 n_i on S_N1 with p1 data and n the unit normal of
> the
> > surface (face) sigma_ij n_j = p2 n_i on S_N2 with p2 data and n the
> unit
> > normal of the surface (face)
> >
> > p1 and p2 are gas pressures. Therfore there is no tangential
> traction at
> > the surfaces S_Ni, but only normal pressure. S_N1 + S_N2 + S_D =
> partial
> > V.
> >
> >
> > For the first three equations I can use the linear elasticity brick
> [1]
> > AFAICS. For the Dirichlet condition I can use the Dirichlet brick
> [2]. But
> > with the two Neumann boundaries I cannot see how to specify them.
> The
> > unknown of the FEM is the displacement field u. The boundary
> condition is
> > given in terms of sigma but not (directly) in terms of u. I expect
> that
> > the Neumann brick [3] implements u_i,j n_j = v_i, but that's not
> what I
> > need. How can I proceed?
> > At the moment I use the python interface, but I can move to the c++
> > interface if it can solve my problem easier.
> >
> > [1]
> > [2]
> > [3]
> >
> >
> > Many thanks for any hints.
> >
> > Kind regards,
> >
> >
> >
> >
> > ---
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> > _______________________________________________
> > Getfem-users mailing list
> > https://mail.gna.org/listinfo/getfem-users
>
>
> --
>
> 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
>
> ---------
>
>
> -----Ursprüngliche Nachricht Ende-----

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