Laplace Equation (Electrostatic Field)
Theory and Physics
Laplace Equation
Professor, how is the Laplace equation different from the Poisson equation?
In regions without charge ($\rho_v = 0$), the Poisson equation becomes the Laplace equation:
The electric potential distribution in spaces without free charge, such as inside insulators between electrodes or inside dielectrics, is determined by this.
So, high-voltage equipment insulation design mostly uses the Laplace equation?
Yes. The potential distribution is determined solely by the electrode potentials (Dirichlet BC). Calculating the electric field distribution in insulators and equipotential lines falls under the domain of the Laplace equation.
Maximum Principle
An important property of the Laplace equation: The maximum and minimum values of the potential always lie on the boundary (not inside the domain).
So, if there are no extreme values inside, it would be strange if an FEM result showed a potential peak inside the domain, right?
Correct. This can be used for validating result plausibility. If there is a peak inside, it's likely a mesh issue or a mistake in setting up the charge source.
Summary
- $\nabla^2 \phi = 0$ — Charge-free region
- Maximum Principle — Extreme values exist only on the boundary
- Fundamental equation for insulation design — Electric field distribution determined solely by electrode potentials
"No maximum/minimum values in Laplace equation solutions" — Impact on practical work
The "maximum principle" possessed by solutions of the Laplace equation is a subtly important property in CAE practice. In spaces where no charge exists (regions where $\nabla^2\phi = 0$ holds), the maximum and minimum values of the potential always appear on the boundary. This means a "local maximum point of potential" cannot suddenly appear inside the domain. This is useful for insulation design, as it confirms that "the location of strongest electric field = on the boundary (electrode or dielectric interface)". Therefore, you don't need to check the entire interior in detail; just increase accuracy near the boundary. In high-voltage equipment design, techniques leveraging this principle to concentrate mesh refinement in specific areas are commonly used in the field.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. Time-varying magnetic flux density generates 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 heating (IH) cookers 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 law. 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 $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that charge is the divergence source of electric flux. 【Everyday example】Rubbing hair with a plastic sheet creates static electricity, making hair stand up—charged sheet (charge) radiates electric field lines outward, exerting force on light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the absence 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, the formulation using 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 do not depend on 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
- Isotropy 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: 0.2–1.4T |
| Magnetic field strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS Oe (Oersted): 1 Oe = 79.577 A/m |
| Current density $J$ | A/m² | Calculated from conductor cross-section 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 Solution
The FEM for the Laplace equation is the Poisson equation with its right-hand side set to zero. $[K]\{\phi\} = \{0\}$ (no charge) + boundary conditions.
The solution method is exactly the same. The difference is that the right-hand side vector contains only terms originating from the Dirichlet conditions of the electrodes.
Comparison with Analytical Solutions
| Problem | Analytical Solution | Application |
|---|---|---|
| Parallel plates | $\phi = V_0 (1 - x/d)$ | Simplest verification |
| Coaxial cylinders | $\phi = V_0 \ln(r/b)/\ln(a/b)$ | Cable verification |
| Concentric spheres | $\phi = V_0 (1/r - 1/b)/(1/a - 1/b)$ | 3D spherical symmetry verification |
Summary
- Solution method identical to Poisson equation — Right-hand side zero
- Analytical solutions effective for FEM verification — Parallel plates, coaxial cylinders
"Solving Laplace equation in fluids too" — Universality of harmonic functions
The Laplace equation $\nabla^2\phi = 0$ governs not only electrostatic potential but also a surprisingly wide variety of physical phenomena: velocity potential in incompressible potential flow, temperature distribution in steady-state heat conduction, pressure field in porous media, etc. This means "even with different boundary conditions, it can be solved with the same solver code." In practice, people who have written electrostatic field analysis code sometimes experience that "replacing permittivity with thermal conductivity turned it into a thermal analysis." This mathematical commonality is called "analogy," and in the past, experiments were conducted to measure fluid pressure fields using electrical analog circuits. The Laplace equation is a "common language" of nature.
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: Handles nonlinear materials and inhomogeneous media. BEM: Naturally handles infinite domains (open region problems). Hybrid FEM-BEM is also effective.
Nonlinear convergence (Magnetic saturation)
Nonlinearity of B-H curve handled by Newton-Raphson method. Residual criterion: $||R||/||R_0|| < 10^{-4}$ is common.
Frequency domain analysis
Reduced to a steady-state problem via time-harmonic assumption. Requires complex number operations.
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