Poisson Equation (Electrostatic Field)
Theory and Physics
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.
Physical Meaning of Each Term
- Electric Field Term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a time-varying magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic Field Term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Electric current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ becomes non-negligible, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet on hair makes hair stand up due to static electricity—electric field lines radiate outward from the charged sheet (electric charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis derived from Gauss's law.
- Magnetic Flux Conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the non-existence of magnetic monopoles. 【Everyday Example】Cutting a bar magnet in half does not create a magnet with only a N pole or only a S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, a formulation using the vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and edge effects can be ignored
- Isotropic assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive laws are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
Variable SI Unit Notes / Conversion Memo Magnetic Flux Density $B$ T (Tesla) 1T = 1 Wb/m². Permanent magnets: 0.2 to 1.4T Magnetic Field Strength $H$ A/m Horizontal axis of B-H curve. Conversion from CGS unit Oe (Oersted): 1 Oe = 79.577 A/m Current Density $J$ A/m² Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect Permeability $\mu$ H/m $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m Electrical Conductivity $\sigma$ S/m Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise
| 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.
Physical Meaning of Each Term
- Electric Field Term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a time-varying magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic Field Term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Electric current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ becomes non-negligible, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet on hair makes hair stand up due to static electricity—electric field lines radiate outward from the charged sheet (electric charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis derived from Gauss's law.
- Magnetic Flux Conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the non-existence of magnetic monopoles. 【Everyday Example】Cutting a bar magnet in half does not create a magnet with only a N pole or only a S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, a formulation using the vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and edge effects can be ignored
- Isotropic assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive laws are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2 to 1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS unit Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
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."
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulations. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs BEM (Boundary Element Method)
FEM discretizes the entire domain, suitable for complex geometries and nonlinear materials. BEM transforms the governing equation into a boundary integral equation, solving it on the boundary. It reduces dimensionality and is effective for infinite domain problems (e.g., electromagnetic wave radiation).
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