Electric Flow PDE¶
Governing Equations¶
The governing equation can easily be derived by considering Ampére's law and using Gauss' and Ohm's law in its stationary form (no time-derivative of the free charge density). Mathematically this PDE is equivalent to the heat-conduction PDE, both of them are Poisson problems.
Introducing the electric potential \mathbf{E} = -\nabla{\phi}, we obatin \begin{equation} -\nabla \cdot {\mathbf{\gamma}\nabla{\phi}} = 0, \end{equation} where \gamma is the electric conductivity. The boundary conditions can be given as \begin{equation} \phi = \phi_{\textrm e} \qquad \textrm{on }\Gamma_{\textrm e}, \end{equation}
\begin{equation} \mathbf{J}\cdot\mathbf{n} = -\mathbf{\gamma}\nabla{\phi} \cdot\mathbf{n} = J_{\textrm n} \qquad \textrm{on }\Gamma_{\textrm n}, \end{equation} with \phi_{\textrm e} as the Dirichlet (essential) BC-value, \Gamma_{\textrm e} the essential boundary, \mathbf{n} the normal vector on Neumann (natural) boundary \Gamma_{\textrm n} and J_{\textrm n} the current density on the boundary.
Boundary conditions¶
As described above, there are two types of BC's, Neumann and Dirichlet. However, since openCFS is physics-based, there are several variations and combinations of these. They can be defined in the simulation-xml file in the BC-section
<bcsAndLoads>
<ground/>
<normalCurrentDensity name="">
<comp dof="x" value=""/>
<comp dof="y" value=""/>
</normalCurrentDensity>
<potential name="" value=""/>
</bcsAndLoads>
- Ground
<potential name=""/>
: Homogeneous Dirichlet value \phi_{\textrm{e}}=0 - Normal Current Density
<normalCurrentDensity/>
: Defines normal component of current density \mathbf{J}\cdot \mathbf{n} on a surface in the specified components<comp dof="x" ... />
- Potential
<potential name="" value=""/>
: Inhomogeneous Dirichlet value \phi_{\textrm{e}}
Material¶
In general, the conductivity is a rank-2 tensor for anisotropic materials, It can be specified in the material-xml file e.g. for copper either by defining an insotropic electric conductivity and/or a tensorial value for the anisotropic version
<material name="polyethylene">
<elecConduction>
<electricConductivity>
<linear>
<isotropic>
<real> 5.8E7 </real>
</isotropic>
<tensor dim1="3" dim2="3">
<real>
5.8E7 0.0E+00 0.0E+00
0.0E+00 5.8E7 0.0E+00
0.0E+00 0.0E+00 5.8E7
</real>
</tensor>
</linear>
</electricConductivity>
</elecConduction>
</material>
Analysis Types¶
Since in general we are dealing with a time-independent PDE, we just have a static analysis:
- Static:
Postprocessing Results¶
Node Results¶
- Electric Potential (primary solution)
Element Results¶
-
Electric Field Intensity \begin{equation} \mathbf E = -\nabla \phi \end{equation}
-
Electric Current Density \begin{equation} J = \gamma \mathbf{E} \end{equation}
-
Electric Power Density \begin{equation} e = \gamma \mathbf{E} \cdot \mathbf{E} \end{equation}
Surface Element Results¶
- Electric Normal Current Density \begin{equation} J_{\textrm{n}} = J \cdot \mathbf n \cdot \mathbf n \end{equation}
Surface Region Results¶
- Electric Current \begin{equation} J = \int_{{A}} J_{\textrm{n}}\, ds \end{equation}
Region Results¶
- Electric Power \begin{equation} E = \int_{{\Omega}} \gamma \mathbf{E} \cdot \mathbf{E} \, dV \end{equation}