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using level-set for inclusions/non homogeneity

From: Egor Vtorushin
Subject: using level-set for inclusions/non homogeneity
Date: Wed, 17 Mar 2021 00:40:08 +0700

Dear Yves,
Could you please provide me with a hint on how to implement an inclusion with level set.
I want to implement an inversion/optimization problem with a given conductive homogeneous medium. 
There is a dipole  source with given frequency, power and location and i am modeling a field via Helmholtz equation or MaxwelL equation
Then i want to put an anomalous object (that has different non zero conductivity/k ^2) inside the media such a way so field propagation and frequency resolution is sensitive to the anomalia.
My optimization problem is to find the anomalia's position and shape to minimize a misfit with the measured field. It is close to structural_optimization.m example but i don't need holes i need an inclusion.
It still seems to me that it is very reasonable to use a LevelSet based technique to describe the anomalia and its changings.
But using the level-set raises the variable jump immediately instead of the operator coefficient jump that i need for.
I looked through some other examples(like fictitious domains) but still have no way  to come up with. 
Please share with me some hints if you have one. 
Regards, Egor Vtorushin

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