id_X looks like this:
template<int spacedim>
class Identity: public Function<spacedim>
{
public:
Identity() : Function<spacedim>() {}
virtual void vector_value(const Point<spacedim> &p, Vector<double> &
value) const;
virtual double value(const Point<spacedim> &p, const unsigned int
component = 0) const;
};
Identity<spacedim> id_X();
Do I need to implement nabla_X id_X from W.'s expression (v_h,
k_{bar,h})_X = - (nabla_X v_h, nabla_X id_X)_X above, or is there a way
to convince dealii to do this for me?
At the moment, the Identity class doesn't know anything about the Manifold
object that I have attached to my Triangulation, so at a minimum I figure I
would either have to tell Identity about that or attach the Identity
Function to something that knows about the manifold, but somehow it doesn't
seem right to attach it to another finite element space.
On Wednesday, August 17, 2016 at 12:48:05 PM UTC-4, Wolfgang Bangerth wrote:
>
> On 08/17/2016 10:22 AM, thomas stephens wrote:
> > *On to the question: *It is a fact that the vector mean curvature k_bar
> > is equal to the surface laplacian of the identity function id_X on the
> > manifold X, k_bar = laplace_beltrami id_X. Equation (4.4) in [1] is
> > the weak form of this identity: (k_bar, eta_bar) = - (surface gradient
> > of id_X, surface gradient of eta_bar) for all eta_bar in some
> > vector-valued finite element space. (For clarity, k_bar = k*nu, where
> > nu is a unit normal vector, and k = k_1 + k_2 is the 'mean' curvature).
> >
> > I am trying to use dealii to compute k_bar from this expression, but it
> > is not clear to me how set up this problem. It does not look like a PDE
> > to me since the unknown is k_bar, not id_X.
>
> It isn't a PDE because (on an analytically known surface X) you can
> evaluate the curvate at each point in isolation without having to know
> anything about the curvate anywhere else.
>
> But it's an equation nonetheless that is best solved in variational
> form. Let's say, you approximate k_bar by k_{bar,h} from some finite
> element space, then it satisfies the variational equality
>
> (v_h, k_{bar,h})_X = - (nabla_X v_h, nabla_X id_X)_X
>
> On the left hand side, this results in a mass matrix that you will have
> to invert to obtain the mean curvature.
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
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