# 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 $$-\nabla \cdot {\mathbf{\gamma}\nabla{\phi}} = 0,$$ where $\gamma$ is the electric conductivity. The boundary conditions can be given as $$\phi = \phi_{\textrm e} \qquad \textrm{on }\Gamma_{\textrm e},$$

$$\mathbf{J}\cdot\mathbf{n} = -\mathbf{\gamma}\nabla{\phi} \cdot\mathbf{n} = J_{\textrm n} \qquad \textrm{on }\Gamma_{\textrm n},$$ 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=""/>

• 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:
<analysis>
<static/>
</static>


## Postprocessing Results

#### Node Results

• Electric Potential (primary solution)
<nodeResult type="elecPotential">


#### Element Results

• Electric Field Intensity $$\mathbf E = -\nabla \phi$$
<elemResult type="elecFieldIntensity">

• Electric Current Density $$J = \gamma \mathbf{E}$$
<elemResult type="elecCurrentDensity">

• Electric Power Density $$e = \gamma \mathbf{E} \cdot \mathbf{E}$$
<elemResult type="elecPowerDensity">


#### Surface Element Results

• Electric Normal Current Density $$J_{\textrm{n}} = J \cdot \mathbf n \cdot \mathbf n$$
<surfElemeResult type="elecNormalCurrentDensity">


#### Surface Region Results

• Electric Current $$J = \int_{{A}} J_{\textrm{n}}\, ds$$
<surfRegionResult type="elecCurrent">


#### Region Results

• Electric Power $$E = \int_{{\Omega}} \gamma \mathbf{E} \cdot \mathbf{E} \, dV$$
<regionResult type="elecPower">