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 \begin{equation} -\nabla \cdot {\mathbf{\epsilon}\nabla{\phi}} = 0, \end{equation} where $\epsilon$ is the permittivity. The boundary conditions can be given as \begin{equation} \phi = \phi_{\textrm e} \qquad \textrm{on }\Gamma_{\textrm e}, \end{equation}

\begin{equation} \mathbf{D}\cdot\mathbf{n} = -\mathbf{\epsilon}\nabla{\phi} \cdot\mathbf{n} = q_{\textrm s} \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 $q_{\textrm s}$ 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=""/>
          <constraint name="Electrodes" quantity="elecPotential"/>
        </bcsAndLoads>

• Potential <potential name="" value=""/>: Inhomogeneous Dirichlet value $\phi_{\textrm{e}}=\textrm{value}$
• Ground <ground 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
• Constraint <constraint name="Electrodes" quantity="elecPotential"/> : Puts a constraint on a certain area. This means the specified quantity must be the same for all nodes over the specified area.

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 \begin{equation} \epsilon=\epsilon_{\textrm{0}} \epsilon_{\textrm{r}}. \end{equation} 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) \begin{equation} \mathbf E = -\nabla \phi \end{equation}

  <elemResult type="elecFieldIntensity">
  

• Electric Energy Density (element result) \begin{equation} e = \epsilon \mathbf{E} \cdot \mathbf{E} \end{equation}

  <elemResult type="elecEnergyDensity">
  

• ** Charge Density** (element result) \begin{equation} q_{{s}} \end{equation}

  <elemResult type="elecChargeDensity">
  

• Electric Charge (surface region result) \begin{equation} Q = \int_{{A}} q_{{s}} ds \end{equation}

  <surfRegionResult type="elecCharge">
  

• Electric Energy (region result) \begin{equation} E = \int_{{\Omega}} \mathbf{E} \cdot \mathbf{E} \, dV \end{equation}

  <regionResult type="elecEnergy">
  

Tutorial

Analysis Workflow