Electrostatics (v-formulation)

This page derives the weak formulation for electrostatics using the electric scalar potential $v$ and describes the available interactions and couplings in Allsolve.

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 \tag{1} \\[10pt] \frac{\partial \boldsymbol{E}}{\partial t} &= 0. \tag{2} \end{align*}

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

\begin{align*} \nabla \cdot \varepsilon \boldsymbol{E} &= \rho \tag{3} \\[5pt] \nabla \times \boldsymbol{E} &= 0, \tag{4} \end{align*}

where $\text{(3)}$ is Gauss’s law and $\text{(4)}$ 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, \tag{5} \end{align*}

where $v$ is a scalar potential of the vector field $\boldsymbol{E}$ . By substituting $\text{(5)}$ into Gauss’s law we obtain a Poisson’s equation

\begin{align*} \nabla \cdot (-\varepsilon \nabla v) &= \rho \tag{6} \\[5pt] \nabla \cdot (\varepsilon \nabla v) + \rho &= 0. \tag{7} \end{align*}

Weak formulation

To obtain the weak form, multiply by the test function $v^{\prime}$ and integrate over the domain $\Omega$ :

\begin{align*} \int_{\Omega}\; (\nabla \cdot (\varepsilon \nabla v))\; v^{\prime}\; d \Omega + \int_{\Omega}\; \rho\; v^{\prime}\; d \Omega = 0. \tag{8} \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. \tag{9} \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. \tag{10} \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. \tag{11} \end{align*}

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 an electric potential value in point or region in Volts
Example	$v = 100$ applies an 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 follows the same principles, regardless of which physics module it belongs to. See how periodicity is used in magnetism as a reference: Periodicity in electric motors

Couplings to Other Physics

This formulation supports the following couplings:

Piezoelectricity
- Solid mechanics
- Elastic waves
Large displacement
- Mesh deformation