Find a 3D function knowing the numerical gradient in discrete point (mesh)

by estragiotti   Last Updated October 09, 2019 14:20 PM

I have a rectangular mesh of dimension $n,m$ and a scalar field $\theta(x,y)$ defined over the different nodes of the mesh. I want to link $\theta(x,y)$ to an initial condition level-set function $ \phi $ using this relation:

$$ \theta(x,y) = \frac{\pi}{2} - \arctan \Bigg( \frac{\frac{\partial \phi}{\partial y}}{\frac{\partial \phi}{\partial x}} \Bigg) $$

Since there are infinite functions that verify the above relation I've added another dummy constraint:

$$ \| \nabla \phi \| = 1 \implies \sqrt{(\frac{\partial \phi}{\partial x})^2 + (\frac{\partial \phi}{\partial y})^2} = 1$$ $$ \frac{\partial \phi}{\partial x} = \pm \sqrt{1-(\frac{\partial \phi}{\partial y})^2}$$

I know the scalar field $\theta(x,y)$ everywhere, and with the two relation written above, I also know the numerical value of the gradient of the function.

So right now, I would integrate the first relation to find $\phi$, but I'm completely blocked. I don't know exactly how to deal with such a discrete problem. I need the value of the function $\phi$ over the nodes of the mesh. I was thinking to integrate

$$ \int d\phi = \int \sqrt{\frac{\tan^2(\frac{\pi}{2} - \theta(x,y))}{\tan^2(\frac{\pi}{2} - \theta(x,y))+1}} \:\:dy$$

and I've tried to implement it in MATLAB without results.

Have you any idea on how to deal with such a problem?

PS: I can if needed query my MATLAB script to get more points for $\theta(x,y)$.

Thanks in advance! Enrico

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