Magnetodynamics with cohomology conditions

Here we represent and induction heating eddy current problem that utilizes the homology and cohomology solver of Gmsh.

Problem definition

The domain

Let $M \subset \mathbb{R}^3$ and let $\partial M = S_1 \cup S_2$ so that $\partial S_1 = \partial S_2 = S_1 \cap S_2$ denote the 3D modeling domain and its 2D boundary that is decomposed in two parts. Furthermore, the domain $M$ is decomposed in a conducting subdomain $M_c$ and a non-conducting subdomain $M_a$ so that $M = M_c \cup M_a$ and $M_c \cap M_a = \partial M_c \cap \partial M_a$. We assume that $M$ is connected and has no holes nor voids, i.e. its Betti numbers are $b_0(M)$ = 1 and $b_1(M) = b_2(M) = 0$.

Topology of the modeling domain.

Implementation

Indheat.geo: problem geometry and cohomology computation in Gmsh

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Direct link to file `Magnetodynamics/GMSH_GETDP/indheat.geo'

Indheat.pro: weak formulation in GetDP

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How to use

All the files (.geo and .pro) must be located in the same directory.

Meshing the domain and computing the cohomology

Go to the directory and then type:

gmsh indheat.geo -3

After the mesh is built, a file "indheat.msh" should have been created in the directory.

Solving the problem with GetDP

In a Terminal, type (in the right directory)

getdp indheat.pro -solve MagDynTOComplex -pos MagDynTO

to solve with $T-\Omega$ formulation, or

getdp indheat.pro -solve MagDynAVComplex -pos MagDynAV

to solve with $A-V$ formulation.

Showing the result

Open the file "jTO.pos" or "jAV.pos" with Gmsh by typing "gmsh jTO.pos" or "gmsh jAV.pos" in a terminal in the right directory.

Result

Boundary mesh of the conducting regions. Current density in the conducting regions.