Challenging variational formulation

variational formulation
1

Does anyone have a clue as to how to implement the following variational form in FEniCS?

\dots + \dfrac{\partial}{\partial x} \left[ |\nabla \phi|^2 A(\mathbf{n}) \dfrac{\partial A(\mathbf{n})}{\partial \left( \dfrac{\partial \phi}{\partial x}\right)} \right] + \dfrac{\partial}{\partial y} \left[ |\nabla \phi|^2 A(\mathbf{n}) \dfrac{\partial A(\mathbf{n})}{\partial \left( \dfrac{\partial \phi}{\partial y}\right)} \right]

where \phi(\mathbf{x},t) is a scalar field and A(\mathbf{n}) is a nonlinear function of the approximate normal:

\mathbf{n} \approx \dfrac{\nabla \phi}{|\nabla \phi|}

In particular, I am not sure how to deal with the absolute value operator and the differentiation with respect to the directional derivatives of \phi.

Thanks!

2

So, first, Iā€™m guessing that these are terms from the strong form of a conservation law

\cdots + \nabla\cdot\mathbf{F}(\nabla\phi)\text{ ,}

where the flux \mathbf{F} is

\mathbf{F}(\nabla\phi) = \vert\nabla\phi\vert^2 A(\mathbf{n})\frac{\partial A(\mathbf{n})}{\partial\nabla\phi}\text{ ,}

so a typical corresponding term from the weak form would be

\cdots - \int_\Omega\mathbf{F}(\nabla\phi)\cdot\nabla\psi\,d\mathbf{x}\text{ ,}

where \psi is a test function. Partial UFL code for \mathbf{F} might look something like

gp = variable(grad(phi)) # (needed for diff to work)
gp2 = dot(gp,gp)
def safeSqrt(x):
    # Regularized sqrt; more robust for use with derivative(), in 
    # case sqrt(x) ends up in the denominator)
    return sqrt(x + DOLFIN_EPS)
n = gp/safeSqrt(gp2)
A = # (something depending on n)
dA_dgp = diff(A,gp)
F = gp2*A*dA_dgp
1 Like