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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

    <normalCurrentDensity name="">
        <comp dof="x" value=""/>
        <comp dof="y" value=""/>
    <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}}


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">
           <real> 5.8E7 </real>
        <tensor dim1="3" dim2="3">
            5.8E7 0.0E+00 0.0E+00
            0.0E+00 5.8E7 0.0E+00
            0.0E+00 0.0E+00 5.8E7

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)
<nodeResult type="elecPotential">

Element Results

  • Electric Field Intensity \begin{equation} \mathbf E = -\nabla \phi \end{equation}
<elemResult type="elecFieldIntensity">
  • Electric Current Density \begin{equation} J = \gamma \mathbf{E} \end{equation}
<elemResult type="elecCurrentDensity">
  • Electric Power Density \begin{equation} e = \gamma \mathbf{E} \cdot \mathbf{E} \end{equation}
<elemResult type="elecPowerDensity">

Surface Element Results

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

Surface Region Results

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

Region Results

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