Difference between revisions of "Magnetodynamics with cohomology conditions"

Revision as of 14:58, 15 January 2013

\(\renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\Grad}[1]{\mathbf{\text{grad}}\,{#1}} \newcommand{\Curl}[1]{\mathbf{\text{curl}}\,{#1}} \newcommand{\Div}[1]{\text{div}\,{#1}} \newcommand{\Real}[1]{\text{Re}({#1})} \newcommand{\Imag}[1]{\text{Im}({#1})} \newcommand{\pvec}[2]{{#1}\times{#2}} \newcommand{\psca}[2]{{#1}\cdot{#2}} \newcommand{\E}[1]{\,10^{#1}} \newcommand{\Ethree}{{\mathbb{E}^3}} \newcommand{\Etwo}{{\mathbb{E}^2}} \newcommand{\Units}[1]{[\mathrm{#1}]} \)Here we represent an 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 $M_c \cap M_a = \partial M_c \cap \partial M_a$ hold. We assume that $M$ is connected and has no holes nor voids, i.e. its Betti numbers are $b_0(M)$ = 1 and $b_1(M) = b_2(M) = 0$.

The conducting subdomain $M_c$ consists of two parts: The workpiece being heated is a hollow cylinder around which the inductor coil wraps. The terminals of the inductor coil extend to the boundary $\partial M$ of the domain.

The boundary value problem

We solve time harmonic magnetoquasistatic approximation to the Maxwell's equations in $M$:

\begin{align} \Curl{\vec{h}} &= \vec{j}, \\ \Curl{\vec{e}} + i \omega \vec{b} &= 0 , \\ \Div{\vec{b}} &= 0 , \\ \end{align} with constitutive equations \begin{align} \vec{b} &= \mu \vec{h} , \\ \vec{e} &= \rho \vec{j} . \end{align}

In order to treat the domain $M$ as a circuit element, we apply magnetic isolation at its boundary $\partial M$. This is achieved by the boundary condition \begin{align} \vec{b} \cdot \vec{n} = 0 \textrm{ on } \partial M. \end{align}

In the non-conducting domain $M_a$ we have no currents: \begin{align} \Curl{\vec{h}} = 0 \textrm{ in } M_a, \end{align} and the current cannot pass from the conducting subdomain $M_c$ to the non-conducting subdomain $M_a$. Also, the current cannot pass through the boundary $\partial M$ to the non-conducting subdomain $M_a$. Thus, the conditions \begin{align} \vec{j} \cdot \vec{n} = 0 \textrm{ on } \partial M_a \cap \partial M_c \textrm{ and on } M_a \cap \partial M = S_2 \end{align} are required to hold at the interfaces. The terminals on $M_c \cap \partial M$ which connect the inductor coil to an external circuit are modeled as equipotential surfaces, i.e. as perfectly conducting surfaces. Thus, we apply a boundary condition \begin{align} \vec{e} \times \vec{n} = 0 \textrm{ on } M_c \cap \partial M = S_1 \end{align} on them.

Note that the boundary $\partial M$ is now decomposed on two parts according to the boundary condition applied on it: $\partial M = S_1 \cap S_2$. On $S_1 = M_c \cap \partial M$ we have the condition $\vec{e} \times \vec{n} = 0$ and on $S_2 = M_a \cap \partial M$ we have the condition $\vec{j} \cdot \vec{n} = 0$. We can either drive the problem by the net current $I_s$ through the terminals, or by the net voltage difference $V_s$ between the terminals: \begin{align} \int_{\zeta} \vec{e} \cdot \mathrm{d} \vec{l} &= V_s, \\ \int_{\Sigma} \vec{j} \cdot \vec{n} \, \mathrm{d} a = \int_{\Sigma} \Curl{\vec{h}} \cdot \vec{n} \, \mathrm{d} a = \int_{\partial \Sigma} \vec{h} \cdot \mathrm{d} \vec{l} &= I_s, \\ \end{align} where $\zeta$ is a curve on $S_2$ between the terminals on $S_1$ and $\Sigma \subset M$ is a surface that isolates the terminals. The boundary $\partial \Sigma$ also lies on the part $S_2$ of the boundary of the domain. In addition, the fields $\vec{e}$ and $\vec{j}$ are linked by the constitutive equation $\vec{e} = \rho \vec{j}$. All these considerations hint a duality between the electric field $\vec{e}$ and the current density $\vec{j}$ and between the voltage and the current conditions.

A class $[\zeta]$ of curves on $S_2$ between the terminals on $S_1$ belong to so-called relative homology space $H_1(S_2, \partial S_2)$, while the class $[\partial \Sigma]$ of curves belong to a homology space $H_1(S_2)$. The integral conditions on fields $\vec{e}$ and $\vec{j}$ (or $\vec{h}$) fixing $V_s$ and $I_s$ are called cohomology conditions, since they consider integrals over an element of a homology space.

$T-\Omega$ potential formulation

In this formulation we solve for the magnetic field $\vec{h}$ to obtain the induced eddy currents in the as $\vec{j} = \Curl{\vec{h}}$. Let us denote by $N(M_a)$ the set of nodes in the mesh of $M_a$ and by $E(M_c / \partial M_c)$ the set of edges in $M_c$ that are not $\partial M_c$.

Denote by $\mathsf{n}_i$ nodal shape functions associated with the nodes of the mesh and by $\mathsf{e}_i$ edge elements associated with the edges of the mesh. By $\mathsf{E}_i$ we denote so-called 1-cohomology basis function, which is an integer combination of edge elements on $M$: $$ \mathsf{E}_i = \sum_j z^j_i \mathsf{e}_j $$ where the integer coefficient vector $\vec{z}_i$ is produced by the cohomology solver of Gmsh.

Now, the unknown magnetic field $\vec{h}$ is of the form $$ \vec{h} = \sum_{i \in N(M_a)} \phi_i \Grad{\mathsf{n}_i} + \sum_{i \in E(M_c / \partial M_c)} t_i \mathsf{e}_i + I_1 \mathsf{E}_1 + I_2 \mathsf{E}_2, $$ where the cohomology basis functions $\mathsf{E}_1$ and $\mathsf{E}_2$ are produced by the computation of the cohomology space $H^1(M_a)$ in Gmsh. That is, the function space from which we look for the approximate solution is spanned by nodal shape functions $\mathsf{n}_i$ of $M_a$, edge elements $\mathsf{e}_i$ of $M_c / \partial M_c$ and by the cohomology basis functions $\mathsf{E}_i$ of the cohomology space $H^1(M_a)$. Those functions shall also serve as the test functions in the following weak formulation.

TBD.

\begin{align} -\int_M \mu \vec{h} \cdot \vec{h}' \, \mathrm{d} V &= 0 \quad \forall \, \vec{h}' \\ \int_{M_c} \rho \Curl{\vec{h}} \cdot \Curl{\vec{h}} \, \mathrm{d} V + i \omega \int_M \mu \vec{h} \cdot \vec{h}' \, \mathrm{d} V &= - \int_{S_2} \vec{e} \times \vec{h}' \cdot \vec{n} \, \mathrm{d} a \quad \forall \, \vec{h}' \end{align}

$A-V$ potential formulation

\begin{align} \int_M \mu^{-1} \Curl{\vec{a}} \cdot \Curl{\vec{a}}' \, \mathrm{d} V + \int_{M_c} \rho^{-1} \vec{e} \cdot \vec{a}' \, \mathrm{d} V + i \omega \int_{M_c} \rho^{-1} \vec{a} \cdot \vec{a}' \, \mathrm{d} V &= 0 \quad \forall \, \vec{a}' \\ \int_{M_c} \rho^{-1} \vec{e}' \cdot \vec{e}' \, \mathrm{d} V + i \omega \int_{M_c} \rho^{-1} \vec{a} \cdot \vec{e}' \, \mathrm{d} V &= - \int_{S_1} \vec{j} \varphi' \, \mathrm{d} a \quad \forall \, \vec{e}' \end{align}

Implementation

Indheat.geo: problem geometry and cohomology computation in Gmsh

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

Indheat.pro: weak formulation in GetDP

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

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.