How to deal with eikonal equation '|grad(phi)| = 1'?


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16 days ago by
$\left|\nabla\phi\right|=1$|ϕ|=1  is the eikonal equation which describes the distance to enclosing walls.

How can I deal with it. It cannot be matched into the nonlinear examples because the nonlinearity comes from the derivative of trial function rather than the trial function itself.
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16 days ago by
Peter  
This isn't two bad.  First create the variational form.

$\int\left|\nabla\phi\right|\cdot\psi-\psi\mathrm{\mathrm{\mathrm{d}}}V=0$|ϕ|·ψψdV=0

From the weak form you should be able to follow one of the other nonlinear examples with phi as the function and psi as your test function.

Peter
Thank you for your reply. However, I got "Applying nonlinear operator Power to expression depending on form argument v_1." error

I was using 
F = (dot(grad(u),grad(u))**0.5-f)*v*dx​
as the form.
written 16 days ago by 迪 程  
The unknown should be a Function in nonlinear problems, not a TrialFunction.  See, e.g., the tutorial here:

http://fenics.readthedocs.io/projects/dolfin/en/2017.2.0/demos/nonlinear-poisson/python/demo_nonlinear-poisson.py.html

(Forms must be linear in TrialFunction/TestFunction arguments.)  Also, the Eikonal equation can be difficult to solve with Galerkin's method and Newton iteration.  A possible stabilized approach using SUPG and pseudo-time integration is described in this paper:

http://www.dtic.mil/dtic/tr/fulltext/u2/a582021.pdf
written 14 days ago by David Kamensky  
Sorry I missed that point when I read the document.

However, I used a functional to produce p-Poisson equation to approximate Eikonal equation when p-> infinity and it worked.
  $E\left(u\right)=\int_{\Omega}\left(\frac{1}{p}\left|\nabla u\right|^p-u\right)dV$E(u)=Ω(1p |u|pu)dV

However, it seems that there is no functional form of Eikonal equation.

written 14 days ago by 迪 程  
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