Poisson Equation (Electrostatic Field)
Poisson Equation (Electrostatic Field): Theoretical Foundations
Poisson Equation
Professor, what is the Poisson equation for electrostatic fields?
It's the result of combining Gauss's Law with the relationship between electric field and electric potential:
For a general dielectric material:
This is the governing equation for electrostatic field FEM. It finds the electric potential $\phi$ when the charge distribution $\rho_v$ is given.
It resembles the equilibrium equation for structures, $\nabla \cdot \sigma + f = 0$.
Mathematically, they are the same elliptic partial differential equation. Just substitute: Young's modulus in structures → permittivity $\varepsilon$, external force → charge density $\rho_v$, displacement → electric potential $\phi$.
Boundary Conditions
Type Mathematical Form Physical Meaning Dirichlet $\phi = \phi_0$ Electrode (fixed potential) Neumann $\partial\phi/\partial n = 0$ Symmetry plane, insulating surface Mixed $\varepsilon \partial\phi/\partial n = \sigma_s$ Surface charge density
Summary
- $\nabla \cdot (\varepsilon \nabla \phi) = -\rho_v$ — Governing equation for electrostatic field FEM
- Elliptic PDE — Same mathematical structure as structural FEM
- Dirichlet = Electrode, Neumann = Symmetry/Insulation
Coffee Break Casual Talk
Why the Poisson Equation is Active in Lithium-Ion Battery Design
In lithium-ion battery charge/discharge simulations, it's necessary to simultaneously solve for the lithium-ion concentration distribution and the electric potential distribution within the electrolyte. The electric potential distribution is precisely governed by the Poisson equation $\nabla^2 \phi = -\rho/\varepsilon$. Inside the battery, the permittivity differs greatly between the positive electrode, negative electrode, and electrolyte, and the setting of boundary conditions critically affects accuracy. In practice, when battery manufacturers like Panasonic use CAE for "optimizing the porous structure of electrodes," they estimate local current density from the potential distribution solved by the Poisson equation to identify locations prone to degradation. This humble second-order differential equation quietly supports cutting-edge battery technology.
| Type | Mathematical Form | Physical Meaning |
|---|---|---|
| Dirichlet | $\phi = \phi_0$ | Electrode (fixed potential) |
| Neumann | $\partial\phi/\partial n = 0$ | Symmetry plane, insulating surface |
| Mixed | $\varepsilon \partial\phi/\partial n = \sigma_s$ | Surface charge density |
- $\nabla \cdot (\varepsilon \nabla \phi) = -\rho_v$ — Governing equation for electrostatic field FEM
- Elliptic PDE — Same mathematical structure as structural FEM
- Dirichlet = Electrode, Neumann = Symmetry/Insulation
Why the Poisson Equation is Active in Lithium-Ion Battery Design
In lithium-ion battery charge/discharge simulations, it's necessary to simultaneously solve for the lithium-ion concentration distribution and the electric potential distribution within the electrolyte. The electric potential distribution is precisely governed by the Poisson equation $\nabla^2 \phi = -\rho/\varepsilon$. Inside the battery, the permittivity differs greatly between the positive electrode, negative electrode, and electrolyte, and the setting of boundary conditions critically affects accuracy. In practice, when battery manufacturers like Panasonic use CAE for "optimizing the porous structure of electrodes," they estimate local current density from the potential distribution solved by the Poisson equation to identify locations prone to degradation. This humble second-order differential equation quietly supports cutting-edge battery technology.
Computational Methods for Poisson Equation (Electrostatic Field)
FEM Discretization
Weak form via Galerkin method, discretized per element:
$[B]$ is the gradient matrix of shape functions. This form replaces $[D]$ in structural analysis's $[B]^T [D] [B]$ with $\varepsilon [I]$.
For 2D triangular elements, $B$ is constant (linear element), so it can be calculated by hand, right?
Exactly. The electrostatic version of the CST triangular element is perfect for textbook exercises. The element stiffness matrix is $\varepsilon \cdot A_e / (4A_e) \cdot [b_i b_j + c_i c_j]$.
Summary
- Stiffness Matrix $[K] = \int \varepsilon [B]^T[B] d\Omega$ — Same form as structures
- Can be calculated by hand for linear triangular elements — Ideal for FEM education
Solving the Poisson Equation with FDM—Intuition of the 5-Point Star Difference
When discretizing the Poisson equation with the Finite Difference Method (FDM) in 2D, it takes the form "potential at the target grid point = average of the potentials at the four surrounding points + charge density term." This is the "5-point star" difference formula, intuitively meaning "if there is no charge, the potential at that point becomes the average of its neighbors." This property is directly linked to the speed of iterative solvers like SOR (Successive Over-Relaxation), where the choice of the over-relaxation factor $\omega$ alone can double the convergence speed or conversely cause divergence. Even with FEM being mainstream today, many engineers implement the Poisson equation with FDM for educational purposes or concept verification, making it an excellent entry point for numerical techniques leveraging matrix "sparsity."