Electrostatics (v-formulation)

Strong formulation

The Electrostatics v-formulation is derived based on the electrostatic approximation. As a starting point, we have the conditions

\begin{align} \frac{\partial \boldsymbol{B}}{\partial t} &= 0 \\[10pt] \frac{\partial \boldsymbol{E}}{\partial t} &= 0. \end{align}

Now with these conditions and the Maxwell’s equations, we have

\begin{align} \nabla \cdot \varepsilon \boldsymbol{E} &= \rho \\[5pt] \nabla \times \boldsymbol{E} &= 0, \end{align}

where (3) is Gauss’s law and (4) is Faraday’s law. From Faraday’s law we see that curl of $\boldsymbol{E}$ is zero. Hence $\boldsymbol{E}$ is a conservative field, which means that there exists a scalar function $v$ such that

\begin{align} \boldsymbol{E} = -\nabla v, \end{align}

where $v$ is a scalar potential of the vector field $\boldsymbol{E}$ . By substituting (6) into Gauss’s law we obtain a Poisson’s equation

\begin{align} \nabla \cdot (-\varepsilon \nabla v) &= \rho \\[5pt] \nabla \cdot (\varepsilon \nabla v) + \rho &= 0. \end{align}

Weak formulation

The partial differential equation is multiplied by the test function $v^{\prime}$ and integrated over the whole domain $\Omega$ to get

\begin{align} \int_{\Omega}\; (\nabla \cdot (\varepsilon \nabla v))\; v^{\prime}\; d \Omega + \int_{\Omega}\; \rho\; v^{\prime}\; d \Omega = 0. \end{align}

Applying the Leibniz rule for nabla operator we get

\begin{align} \int_{\Omega}\; \nabla \cdot (v^{\prime}\; \varepsilon \nabla v)\; d \Omega - \int_{\Omega}\; (\nabla v^{\prime} \cdot \varepsilon \nabla v)\; d \Omega + \int_{\Omega}\; \rho\; v^{\prime}\; d \Omega = 0. \end{align}

For the first term we can use the Divergence theorem

\begin{align} \int_{\Gamma}\; \boldsymbol{n} \cdot (v^{\prime}\; \varepsilon \nabla v)\; d \Gamma - \int_{\Omega}\; (\nabla v^{\prime} \cdot \varepsilon \nabla v)\; d \Omega + \int_{\Omega}\; \rho\; v^{\prime}\; d \Omega = 0. \end{align}

Rearranging the terms and using relation $\boldsymbol{E} = - \nabla v$ on the Neumann term, we obtain

\begin{align} \int_{\Omega}\; (-\varepsilon \nabla v \cdot \nabla v^{\prime})\; d \Omega + \int_{\Gamma}\; (-\varepsilon \boldsymbol{E} \cdot \boldsymbol{n})\; v^{\prime} \; d \Gamma + \int_{\Omega}\; \rho\; v^{\prime}\; d \Omega. \end{align}

Available Interactions

Constraint

Applies a fixed value to the electric scalar potential $v$ . Use this when you need to fix the electric potential at a node or within a region. This can be used to drive a potential difference between two capacitor plates.

How to use:

Provide a electric potential value in point or region in Volts

Example:

$v = 100$ applies a electric scalar potential of $100\:V$ to the specified region.

Unit: Electric potential in Volts (V)

Lump V/Q

Applies a lumped voltage $V$ or electric charge $Q$ to a specific region or node. Used to model simplified circuit elements or charge distributions where the detailed field distribution is not explicitly resolved, but replaced with an equivalent lumped voltage or charge.

How to use:

Specify the target curve. From Actuation mode, select either voltage, charge or circuit coupling. Fill in the value.

Example: $Q = 1*10^{-9}$ applies a charge of $1\:nC$ to the specified curve.

Unit: Voltage in Volts (V) or electric charge in Coulombs (C)

Periodicity

Imposes periodic boundary conditions on the electric potential $v$ between two boundaries. Reduces the computational domain size for geometrically symmetric or antisymmetric problems, avoiding the need to model the full geometry.

Example:

Periodicity is similar in every physics section. This example is from $\varphi$ -formulation, but workflows are identical:

The periodicity of an electric motor allows modeling only a fraction of the full geometry, such as one pole pair or one quarter, while still capturing the complete field behavior.

Periodicity in electric motor

Couplings to Other Physics

This formulation supports the following couplings:

Piezoelectricity (Solid mechanics)

Large displacement (Mesh deformation)

Piezoelectricity (Elastic waves)