Dear Dr. Bangerth,
Thank you for the explanation. I am trying to study fluid-structure
interaction and use Dealii as the solid solver for the problem. However, I
only have a traction vector at individual points. This is because the
traction vector is obtained from a discrete fluid solver (Lattice Boltzmann
solver). The traction vector is discontinuous and I do not have a function
to describe it.
Thanks,
Qin
在 2018年10月9日星期二 UTC+8上午3:46:36，Wolfgang Bangerth写道：
>
> On 10/07/2018 09:29 AM, Lucky Qin wrote:
> >
> > I would like to apply a traction boundary condition on the surface of a
> > cantilever beam. Suppose I have a vector named
> > /traction[dof_handler.n_dofs()/*], *which gives the various traction on
> > each dof.
>
> To augment the answer Jean-Paul has already given:
>
> The way you see things when you define such a traction vector is not the
> way you should be thinking of tractions in the finite element context.
> There, you should think of a traction boundary condition as a function
> g(x) so that
>
> n . sigma(u(x)) = g(x)
>
> where sigma(u) gives you the stress at a point u. In linear elasticity,
> for example, this would be something like
>
> sigma(u) = lambda eps(u) + 2*mu*div(u)*I
>
> if I recall correctly. The point I want to make is that the traction is
> defined *for every point of the boundary*. This then allows you to
> compute contributions to the right hand side of the discrete problem by
> approximating the boundary integral
>
> T_i = \int_{\partial \Omega} g(x) phi_i(x) dx
>
> via quadrature. For this to work, you have to have the traction at every
> point of the boundary.
>
> On the other hand, you only provide the traction at individual points
> (assuming that's how you have defined your 'traction' array), which is
> not good enough to actually do the integration.
>
> The reason for all of this is that you will want to work on a sequence
> of successively finer meshes to explore convergence of your solution.
> For this, it is not enough to just know the traction at individual
> points -- you need to be able to evaluate it everywhere.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
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