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Finite Element Embedded Library and Language in C++
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2D Maxwell Dimulation in a Diode
Author
Thomas Strub
Philippe Helluy
Christophe Prud'homme
Date
2011-06-01




Description

The Maxwell equations read:

\begin{eqnarray*} \frac{-1}{c^{2}}\frac{\partial {{\bm E}}}{\partial t}+\nabla\times {{\bm B}}& = & \mu_{0} {{\bm J}}\\ {{\bm B}}{t}+\nabla\times {{\bm E}}& = & 0\\ \nabla \cdot {{\bm B}}& = & 0\\ \nabla \cdot {{\bm E}}& = & \frac{\rho}{\epsilon_{o}} \end{eqnarray*}

where \({{\bm E}}\) is the electric field, \({{\bm B}}\) the magnetic field, \({{\bm J}}\) the current density, \( c \) the speed of light, \( \rho \) density of electric charge, \( \mu_ {0} \) the vacuum permeability and \( \epsilon_ {0} \) the vacuum permittivity.
In the midst industrial notament in aeronautics, systems Products must verify certain standards such as the receipt an electromagnetic wave emitted by a radar does not cause the inefficassité of part or all of the hardware in the system.
Thus, the simulation of such situations can develop when or during the certification of a new product to test its reaction to such attacks.
Also note that the last two equations are actually initial conditions, since if we assume they are true at the moment \( t = 0\) then it can be deduced from the first two.
At \(t=0s\), we suppose that

\begin{eqnarray} \nabla \cdot {{\bm B}}& = & 0\\ \nabla \cdot {{\bm E}}& = & \frac{\rho}{\epsilon_{o}} \end{eqnarray}

Suppose that \({{\bm B}} = (B_x, B_y, B_z )^T\) and \({{\bm E}}=(E_x,E_y,E_z)^T\)

i.e.

\begin{eqnarray} \frac{\partial B_{x}}{\partial x}(t=0)+\frac{\partial B_{y}}{\partial y}(t=0)+\frac{\partial B_{z}}{\partial z} & (t=0)= & 0\\ \frac{\partial E_{x}}{\partial x}(t=0)+\frac{\partial E_{y}}{\partial y}(t=0)+\frac{\partial E_{z}}{\partial z} & (t=0)= & \frac{\rho}{\epsilon_{o}} \end{eqnarray}

Differentiating the first of these two equations with respect to time, we get:

\begin{multline} \label{eq:6} \frac{\partial}{\partial t}\frac{\partial}{\partial x}B_{x}+\frac{\partial}{\partial t}\frac{\partial}{\partial y}B_{y}+\frac{\partial}{\partial t}\frac{\partial}{\partial z}B_{z} = \\ \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}E_{z}-\frac{\partial}{\partial z}E_{y}\right)+\frac{\partial}{\partial y}\left(\frac{\partial}{\partial z}E_{x}-\frac{\partial}{\partial x}E_{z}\right)+\frac{\partial}{\partial z}\left(\frac{\partial}{\partial x}E_{y}-\frac{\partial}{\partial y}E_{x}\right)\\ = 0 \end{multline}

thanks to

\begin{equation} \label{eq:3} {{\bm B}}_{t}+\nabla\times {{\bm E}} =0 \end{equation}

So, for all \(t\geq0\),

\begin{equation} \label{eq:4} \nabla \cdot {{\bm B}}(t)=\nabla\cdot(0)=0 \end{equation}

We deduce the same way the second equation, using the charge conservation equation :

\begin{equation} \label{eq:2} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho {{\bm J}}) = 0 \end{equation}

Theory

Implementation

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