Electrostatic Potential Model

In electrostatic applications, Simcenter STAR-CCM+ calculates the electric field induced by a distribution of electric charges from the electric potential.

Electrostatic Potential Equation

Simcenter STAR-CCM+ computes the electric potential from:

Figure 1. EQUATION_DISPLAY

- \oint_{A} ε \nabla ϕ \cdot d a = \int_{V} ρ d V

(4272)

which is obtained by integrating Eqn. (4237) over the cell domain. Eqn. (4272) states that the total electric flux through a closed surface $A$ is equal to the electric charge that is contained in the volume $V$ bounded by the surface $A$ .

Simcenter STAR-CCM+ discretizes and solves Eqn. (4272) for the electric potential using the finite volume method. The electric field and the electric flux density are calculated from the potential using Eqn. (4236) and Eqn. (4219).

Source Term

The source term for Eqn. (4272) is the electric charge density $ρ$ . The term on the right-hand side of Eqn. (4272) represents the total electric charge contained in the volume $V$ .

Boundary and Interface Conditions

At the domain boundaries, the solution must satisfy either Dirichlet boundary conditions, which define the electric potential $ϕ$ , or Neumann boundary conditions, which define the component of the electric potential gradient $\nabla ϕ$ normal to the boundary [848].

At a boundary

Γ

, Neumann boundary conditions can be defined by specifying either the:

electric flux density $D$ (see Eqn. (4219))
specific electric flux $D_{n}$ , that is, the component of $D$ normal to $Γ$ :
Figure 2. EQUATION_DISPLAY

$D_{n} = - ε \nabla ϕ \cdot n$
(4273)

where $n$ is the boundary surface normal.
total electric flux through $Γ$ , that is:
Figure 3. EQUATION_DISPLAY

$- \int_{Γ} (ε \nabla ϕ) \cdot d a$
(4274)

The total and specific electric flux can also be prescribed at contact interfaces.

Electrostatic Force Density

In electrostatics, the Maxwell stress tensor can be defined as:

Figure 4. EQUATION_DISPLAY

σ_{E S} (D) = \frac{1}{ε} D \otimes D - \frac{ε}{2} (E \cdot E) I

(4275)

At an interface between two materials, Simcenter STAR-CCM+ calculates the electrostatic force density at the interface as:

Figure 5. EQUATION_DISPLAY

f_{E S} = (σ_{{E S}_{1}} - σ_{{E S}_{0}}) \cdot n_{01}

(4276)

where $σ_{{E S}_{1}}$ and $σ_{{E S}_{0}}$ represent the electrostatic stress tensor at each side of the interface, and $n_{01}$ is the surface normal pointing from side 0 to side 1.