Electrostatics toolbox

This toolbox solves a quasi-static conduction model for the electric potential \(V\) in conducting materials. The starting point is the Maxwell equations.

1. Fields and parameters (names and units)

1.1. Fields

Symbol Physical name Toolbox field name SI unit Role

\(V\)

electric potential

electric-potential

\(V\)

primary unknown

\(\mathbf{E}\)

electric field

electric-field

\(V.m^{-1}\)

derived from \(-\nabla V\)

\(\mathbf{J}\)

current density

current-density

\(A.m^{-2}\)

derived from \(-\sigma \nabla V\)

\(q_J\)

Joule losses density

joules-losses

\(W.m^{-3}\)

derived from \(\sigma \lVert \nabla V \rVert^2\)

\(\mathbf{D}\)

electric displacement

not exported

\(C.m^{-2}\)

used in transient formulation

\(\rho\)

electric charge density

not exported

\(C.m^{-3}\)

used in transient formulation

1.2. Required material properties and units

Property Symbol SI unit Required in steady model Required in transient model Possible dependency

electric conductivity

\(\sigma\)

\(S.m^{-1}\)

yes

yes

may depend on \(\mathbf{E}\) (e.g. \(\sigma=\sigma(\mathbf{E})\)), and more generally on model symbols

electric permittivity

\(\epsilon\)

\(F.m^{-1}\)

no

yes

may depend on \(\mathbf{E}\) (e.g. \(\epsilon=\epsilon(\mathbf{E})\)) for nonlinear dielectrics
In the current toolbox implementation, \(\sigma\) is available as material property while \(\epsilon\) is not yet assembled in the electric solver.

1.3. Parameters and boundary data

Symbol Name JSON name/key SI unit Comment

\(\sigma\)

electric conductivity

material property sigma (electric-conductivity)

\(S.m^{-1}\)

required in current implementation, can be nonlinear (e.g. \(\sigma(\mathbf{E})\))

\(\epsilon\)

electric permittivity

planned transient material property

\(F.m^{-1}\)

transient target model property, can be nonlinear (e.g. \(\epsilon(\mathbf{E})\)); not yet assembled in current implementation

\(V_D\)

imposed potential

BC electric_potential / electric_potential_imposed

\(V\)

Dirichlet boundary data

\(g\)

imposed normal current density

BC surface_charge_density / charge_density

\(A.m^{-2}\)

Neumann-type boundary data

\(I\)

imposed total current

integral-current condition

\(A\)

currently unavailable in this toolbox

\(t\)

time

model parameter t

\(s\)

used in transient expressions

\(\mathbf{n}\)

outward unit normal

geometric quantity

dimensionless

boundary orientation

2. Quasi-static electric model

In quasi-static conditions, the charge conservation equation is

\[\nabla \cdot \mathbf{J} = 0\]

with Ohm’s law

\[\mathbf{J} = -\sigma \nabla V\]

which gives the strong form solved in this toolbox:

\[\nabla \cdot \left( \sigma \nabla V \right) = 0 \quad \text{in } \Omega.\]

The unknown is the scalar potential \(V\). The electric field and current density are then recovered from \(V\). From a physical viewpoint, this is a conduction model: currents are driven by potential differences and weighted locally by the conductivity.

For multiple materials \(\Omega_i\), the conductivity \(\sigma_i\) is piecewise defined. The implementation enforces continuity through the conforming finite-element space across interfaces.

A volumetric electric source term (Poisson right-hand side) is currently not assembled in this toolbox implementation.

3. Derived quantities

After solving for \(V\), the toolbox computes

\[\mathbf{E} = -\nabla V, \qquad \mathbf{J} = -\sigma \nabla V, \qquad q_J = \sigma \lVert \nabla V \rVert^2.\]

where \(q_J\) is the Joule losses density.

3.1. Boundary flux and power quantities

For boundary interpretations, it is useful to name explicitly the normal current flux and its integrated value on a surface \(\Gamma\):

\[j_n = \mathbf{J}\cdot\mathbf{n} \quad [A.m^{-2}], \qquad I_\Gamma = \int_\Gamma \mathbf{J}\cdot\mathbf{n}\,d\Gamma \quad [A].\]

The electrical power flux density through a boundary is

\[p_e = V\,\mathbf{J}\cdot\mathbf{n} \quad [W.m^{-2}]\]

and the associated terminal power is \(P_\Gamma=\int_\Gamma p_e\,d\Gamma\) with unit \(W\).

4. Boundary conditions

Boundary conditions encode how the device is electrically connected to its environment. In practice, one usually combines potential constraints (to drive current and fix the reference level) with natural/flux conditions on other boundaries.

4.1. Electric potential imposed (Dirichlet)

On \(\Gamma_D\):

\[V = V_D.\]

This is exposed in JSON with electric_potential (or alias electric_potential_imposed). This boundary condition is typically used on terminals where voltage is controlled.

4.2. Ground

Ground is a specialized Dirichlet condition:

\[V = 0 \quad \text{on } \Gamma_g.\]

This is exposed with the JSON key ground. It provides a potential reference and avoids the constant-offset indeterminacy of pure Neumann problems.

4.3. Surface charge density / normal current density (Neumann)

On \(\Gamma_N\):

\[\mathbf{J} \cdot \mathbf{n} = g \quad \Leftrightarrow \quad -\sigma \nabla V \cdot \mathbf{n} = g.\]

This is exposed with surface_charge_density (alias charge_density). It models injected/extracted normal current density on a boundary. If no such condition is set on a boundary part, the natural homogeneous condition applies:

\[\mathbf{J}\cdot\mathbf{n}=0.\]

4.4. Integral current condition

The integral-current boundary condition

\[\int_{\Gamma_I}\mathbf{J}\cdot\mathbf{n}\,d\Gamma = I\]

is not available in this standard electric toolbox implementation (it is an HDG-oriented feature). Conceptually, this condition is useful for terminal-driven models where total current is prescribed instead of pointwise flux.

5. Time dependence

For time-dependent electric evolution in conducting media, a transient electro-quasistatic model can be considered. Compared with the steady model, it introduces capacitive effects through the displacement field.

Starting from charge conservation:

\[\partial_t \rho + \nabla\cdot\mathbf{J}=0\]

with

\[\mathbf{E}=-\nabla V,\qquad \mathbf{D}=\epsilon\mathbf{E}=-\epsilon\nabla V,\qquad \rho=\nabla\cdot\mathbf{D},\]

and conductive current

\[\mathbf{J}=-\sigma\nabla V,\]

the transient electro-quasistatic PDE reads

\[\nabla\cdot\left(\sigma\nabla V + \epsilon\,\partial_t\nabla V\right)=0.\]

When \(\epsilon\to 0\), this model reduces to the steady conduction equation.

5.1. Initial condition (transient case)

A transient solve requires an initial state, for example:

\[V(\mathbf{x},0)=V_0(\mathbf{x}).\]

5.2. Boundary-condition interpretation in transient mode

Potential (Dirichlet) conditions keep the same form as in steady mode. For Neumann-like boundaries, one must choose whether the imposed normal quantity is:

  • conductive current only: \(\mathbf{J}\cdot\mathbf{n}\)

  • or total current including displacement: \((\mathbf{J}+\partial_t\mathbf{D})\cdot\mathbf{n}\)

This choice must be fixed consistently in model equations, weak form, and user-facing boundary-condition semantics.

The current electric toolbox implementation remains steady-state only. The transient formulation above is a target model for future implementation.