Difference between revisions of "Magnetodynamics with cohomology conditions"
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== Problem definition == | == 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 [http://en.wikipedia.org/wiki/Betti_number Betti numbers] are $b_0(M)$ = 1 and $b_1(M) = b_2(M) = 0$. | ||
| + | |||
| + | [[File:indheatdomain.jpg|400px|frameless|alt=Domain|Topology of the modeling domain.]] | ||
| + | |||
| + | === Partial differential equations and boundary and cohomology conditions === | ||
=== $T-\Omega$ potential formulation === | === $T-\Omega$ potential formulation === | ||
| Line 29: | Line 37: | ||
; Solving the problem with GetDP: | ; Solving the problem with GetDP: | ||
In a Terminal, type (in the right directory) | In a Terminal, type (in the right directory) | ||
| − | :getdp indheat.pro -solve | + | :getdp indheat.pro -solve MagDynTOComplex -pos MagDynTO |
; Showing the result | ; Showing the result | ||
Revision as of 13:40, 7 January 2013
Here we represent and induction heating eddy current problem that utilizes the homology and cohomology solver of Gmsh.
Contents
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.
Partial differential equations and boundary and cohomology conditions
$T-\Omega$ potential formulation
$A-V$ potential formulation
Implementation
Indheat.geo: problem geometry and cohomology computation in Gmsh
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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 -algo frontal3d -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
- Showing the result
Open the file "jTO.pos" with Gmsh by typing "gmsh jTO.pos" in a terminal in the right directory.
Result
Boundary mesh of the conducting regions. Current density in the conducting regions.
