My Version:
2016.1.0
Hi all,
I am using the linear elasticity equation with my stress tensor
$$\sigma(u) = \lambda \cdot \mathrm{tr}(\epsilon(u))I + 2\mu\epsilon(u)$$
and my strain tensor
$$\epsilon(u) = \frac{1}{2} (\nabla u + \nabla u^{T}).$$
I got the weak formulation as
$$\int_{\Omega} \sigma(u) : \epsilon(v) dx = \int_{\Omega} fv dx$$
I want to optimize the shape of my grid with a volume constraint, ie
$$c(\Omega) = 0$$.
The Volume should be constant, i.e. Endvolume = Initialvolume
For this approach I want to use an augmented lagrangian method, my steepest descent prototype is working great (I use the volume formulation for the shape gradient of my functional), however my augmented lagrangian method isn't working at all.
I think my problem is in the normal vector of the shape derivative of the volume integral $\int_{\Omega} 1 dx$ which is $\int_{\Gamma}\langle n, V \rangle ds$. I assemble the shape gradient as a neumann condition in the RHS of my equation with my lagrange Multipliers (which don't intersect with $\lambda$, $\mu$ stated in elasticity equation).
I tried to use
~~~python
n = FacetNormal(Omega)
~~~
but my steepest descent in my augmented lagrangian method does not converge. I've read that the normal vector return by this function has length 1, which might not be appropriate for the shape derivative of my volume integral as stated as above.
