Electrostatics Model¶

Introduction¶

The electrostatics model describes a static electric field $\mathbf{E}$ in terms of the scalar electric potential $\phi$ and solves the following equation:

\[ \begin{align} \nabla \cdot \left(\varepsilon \nabla \phi\right) = -\rho, \end{align} \]

where $\varepsilon$ is the electric permittivity and $\rho$ is the density of free electric charges. With appropriate boundary conditions, this equation determines the distribution of the electric potential $\phi$ in the region of interest, and hence the corresponding distribution of the static electric field $\mathbf{E}$ via $\mathbf{E}=-\nabla\phi$. The electrostatics model is used to study electrostatic phenomena that arise due to the forces with which stationary electric charges act on each other. It can be applied to study capacitors and the energy stored in them, the forces acting on distributions of electric charges, as well as various shielding effects (Faraday cages, etc.). The electrostatics model can be used to study polarization in dielectrics and space charge distributions in semiconductors and insulators. Additionally, this model is very useful in field analysis for visualizing field lines and equipotential surfaces.

Typical industrial applications include high-voltage insulator design, capacitor and MEMS-comb-drive characterisation, electrostatic shielding and Faraday-cage studies, semiconductor space-charge analysis, and electron-optics or ion-source modelling.

Model¶

The model is created and added to the simulation with

electrostatics_model = ElectrostaticsModel(
    marker=electric_domain,
    order=2,
)

sim.get_model_manager().add_model(electrostatics_model)

where

marker specifies the domain on which the model is solved (defaults to the whole volume),
order is the polynomial order of the finite-element discretisation (default 1),
formulation selects Formulation.Continuous (H1, default) or Formulation.Discontinuous (DG); the latter is useful for problems with strong material discontinuities.

Solver¶

The model owns a solver, obtained with get_solver():

solver = electrostatics_model.get_solver()
solver.set_verbose(True)

See Solver for the available controls.

Materials¶

In the most general case, the electric permittivity $\varepsilon$ in the governing equation can be a spatially varying quantity, which can also depend on additional physical properties, such as temperature. To define $\varepsilon$ as a function of all these parameters we use a corresponding material. A material is represented by a set of functions that introduce $\varepsilon$ as a function of various parameters in the form of analytic expressions, tabulated data, or external models. For more details, please refer to Electrostatics Material.

To simplify the setup of the electric permittivity $\varepsilon$ for the most common cases, we provide a set of predefined material functions, listed below:

Name	Description
Constant	Constant relative electric permittivity $\varepsilon_\text{usr}$: $$ \varepsilon = \varepsilon_0 \varepsilon_\text{usr}. $$
Function	Spatially varying or field-dependent relative permittivity supplied through a scalar coefficient function (for inhomogeneous dielectrics or coupling with other models).

Conditions¶

To solve the governing equation in a given region, it is necessary to specify boundary conditions at the boundaries of this region. In addition, it may be necessary to specify a specific distribution of electric charges or to fix the electrical potential at a specific point within the volume of interest. For all these cases, we can use the following list of supported conditions:

Name	Supported Entities	Description
Charge Density	Volume, Boundary	This condition specifies the distribution of charge density $\rho$ in a volume, on a surface, or along a line (wire).
Displacement Field	Boundary	This condition sets a specific value of the electric displacement field $\mathbf{D}$ at the selected boundary.
Electric Potential	Volume, Boundary	This condition establishes a fixed electric potential $\phi$ in the volume or at the boundary of the region of interest.
Floating Potential	Volume	This condition imposes a constraint on the electric potential $\phi$, requiring that its value in a given region be constant.
Potential Jump	Boundary	This condition defines a discontinuous electric potential $\phi$ with a given difference in value at the boundary of two contacting regions.
Single-Point Grounded	Point	Pins $\phi = 0$ at the origin. Use to fix the gauge in pure-Neumann (charge-density-only) problems where the potential is otherwise defined only up to an additive constant.

Coefficients¶

The following functions are available in the electrostatics model for visualization or querying:

Name	Field Type	Description
Electric Potential	Scalar	Electric potential: $~ \phi$
Electric Field	Vector	Electric field vector: $~ \mathbf{E} = -\nabla \phi$
Electric Displacement Field	Vector	Electric displacement field vector: $~ \mathbf{D} = \varepsilon \mathbf{E}$
Electric Energy Density	Scalar	Electric energy density: $~ U = \frac{1}{2} \mathbf{E} \cdot \mathbf{D}$
Electric Permittivity	Scalar	Electric permittivity $\varepsilon$ as evaluated from the material.
Electrostatics Model Order	Scalar	The degree of the polynomial that represents the electric potential within each element of the mesh.