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.
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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