# Electrostatic PDE

## Governing Equations

The governing equation can easily be derived by considering Faraday's law in its stationary form (no time-derivative of the magnetic flux density) and law of Gauss. Introducing the electric potential $\mathbf{E} = -\nabla{\phi}$, we obatain $$-\nabla \cdot {\mathbf{\epsilon}\nabla{\phi}} = 0,$$ where $\epsilon$ is the permittivity. The boundary conditions can be given as $$\phi = \phi_{\textrm e} \qquad \textrm{on }\Gamma_{\textrm e},$$

$$\mathbf{D}\cdot\mathbf{n} = -\mathbf{\epsilon}\nabla{\phi} \cdot\mathbf{n} = q_{\textrm s} \qquad \textrm{on }\Gamma_{\textrm n},$$ 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 $q_{\textrm s}$ the surface charge density.

## 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>
<potential name="" value=""/>
<ground name=""/>
<charge name="" value=""/>
<chargeDensity name="" value=""/>
<fluxDensity name=""/>
<fieldParallel name=""/>

• Potential <potential name="" value=""/>: Inhomogeneous Dirichlet value $\phi_{\textrm{e}}=\textrm{value}$
• Ground <potential name=""/>: Homogeneous Dirichlet value $\phi_{\textrm{e}}=0$
• Charge <charge name="" value=""/>: Surface charge $\int_{\textrm{A}}q_s\, ds=\textrm{value}$ (Neumann-type BC), where $ds$ is the infinitesimal surface element
• Charge density <chargeDensity name="" value=""/>: Surface charge density $q_s=\textrm{value}$ (Neumann-type BC)
• Flux density <fluxDensity name="" value=""/>: Defines normal component of dielectric displacement $\mathbf{D}\cdot \mathbf{n}=\textrm{value}$ (similar to a surface charge density)
• Field parallel <fieldParallel name=""/>: $\mathbf{D}\cdot \mathbf{n}=0$ on the specified surface

## Material

In general, the permittivity is a rank-2 tensor for anisotropic materials, and the 'total' permittivity can be defined via the relative permittivity $\epsilon_{\textrm{r}}$ as $$\epsilon=\epsilon_{\textrm{0}} \epsilon_{\textrm{r}}.$$ It can be specified in the material-xml file e.g. for polyethylene either by defining an insotropic permittivity and/or a tensorial value for the anisotropic version

  <material name="polyethylene">
<electric>
<permittivity>
<linear>
<isotropic>
<real> 8.85419E-12 </real>
</isotropic>
<tensor dim1="3" dim2="3">
<real>
1.99200E-11 0.00000E+00 0.00000E+00
0.00000E+00 1.99200E-11 0.00000E+00
0.00000E+00 0.00000E+00 1.99200E-11
</real>
</tensor>
</linear>
</permittivity>
</electric>
</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 (hysteresis excluded)

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

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

• Electric Energy Density (element result) $$e = \epsilon \mathbf{E} \cdot \mathbf{E}$$
<elemResult type="elecEnergyDensity">

• Charge Density (element result) $$q_{{s}}$$
<elemResult type="elecChargeDensity">

• Electric Charge (surface region result) $$Q = \int_{{A}} q_{{s}} ds$$
<surfRegionResult type="elecCharge">

• Electric Energy (region result) $$E = \int_{{\Omega}} \mathbf{E} \cdot \mathbf{E} \, dV$$
<regionResult type="elecEnergy">