Calculator
Example data table
Use these cases to sanity-check wavelength and skin-depth guidance.
| Case | Material | εr | μr | σ (S/m) | f (Hz) | L (m) | Suggested Ne |
|---|---|---|---|---|---|---|---|
| 1 | FR-4 (low-loss approx.) | 4.3 | 1 | 0 | 1.0e9 | 0.05 | 150 |
| 2 | Copper conductor region | 1 | 1 | 5.8e7 | 1.0e6 | 0.01 | 200 |
| 3 | Ferrite-like medium | 12 | 200 | 1 | 5.0e6 | 0.02 | 120 |
Formula used
This calculator solves a 1D Poisson-type equation in weak form.
Electrostatics
-d/dx( ε dV/dx ) = ρ
E = -dV/dx
Energy ≈ ∫ (ε E² / 2) dVMagnetostatics (vector potential)
-d/dx( (1/μ) dA/dx ) = J
B = dA/dx
Energy ≈ ∫ (B² / (2μ)) dVFor linear elements, each element contributes (k/h) [[1,-1],[-1,1]] and a constant-source load of (s·h/2)[1,1]. Quadratic elements use the standard 3-node stiffness and load weights.
How to use this calculator
- Choose a mode: electrostatics, magnetostatics, or estimator.
- Enter geometry (L, area) and material (εr, μr, σ).
- Set frequency to get wavelength and skin-depth recommendations.
- Select element order and Ne; start with the suggested Ne.
- Pick boundary type and set values at x=0 (and x=L if needed).
- Click Compute to view results above this form.
- Download CSV/PDF after a 1D solve to archive outputs.
Professional article
1) What this tool estimates
Finite element electromagnetics replaces continuous fields with nodal unknowns. This calculator reports wavelength, skin depth, and a recommended element size to resolve field variation. It also estimates degrees of freedom and matrix memory, helping you choose a practical mesh before running multi-dimensional models in practical engineering workflows safely.
2) Governing 1D field problems
For electrostatics, potential follows a Poisson form with charge density as the source. For magnetostatics, a single component of vector potential can represent current-driven fields in a simplified geometry. Both reduce to second-order operators with material coefficients, making them ideal for demonstrating assembly, boundary conditions, and postprocessing.
3) Weak form and element contributions
The weak form multiplies the governing equation by a test function and integrates over the domain. After integration by parts, derivatives move onto the test function and boundary terms appear naturally. Each element contributes a local stiffness matrix scaled by coefficient k and length h, plus a load vector from constant sources.
4) Material parameters and frequency metrics
Relative permittivity and permeability set wave speed through vp = c/√(εrμr). When conductivity is nonzero, skin depth δ ≈ √(2/(ωμσ)) estimates how quickly fields decay inside conductors. A conservative mesh rule resolves the smallest relevant length scale by comparing wavelength and skin depth targets.
5) Meshing guidance with numbers
As a baseline, use at least ten elements per wavelength in dielectric regions. In conducting regions, capture skin depth with at least five elements across δ. The calculator outputs an element size recommendation h and a suggested element count over L. Increase Ne when sharp sources, discontinuities, or steep gradients are present.
6) Boundary conditions and physical meaning
Dirichlet boundaries fix the primary variable, such as V(0) or A(0), and are useful for references and known potentials. Neumann boundaries specify flux-like quantities: displacement Dn for electrostatics or magnetic field Hn for magnetostatics. The weak form converts these into a simple right-hand-side contribution at the boundary node.
7) Solver cost and memory planning
Even 1D systems illustrate how FEM matrices grow. Linear elements create a tridiagonal structure; quadratic elements expand the bandwidth. Sparse storage is dramatically smaller than dense storage, especially as nodes increase. The estimator reports approximate sparse and dense memory so you can decide whether direct factorizations or iterative methods are appropriate.
8) Interpreting results for engineering use
Nodal solutions provide potential or vector potential profiles, while element midpoints provide field estimates such as E or B. Energy integrals quantify stored energy and support convergence checks. For reliable studies, refine Ne until field and energy changes are small. Use the CSV and PDF exports to compare cases consistently.
FAQs
1) Is this a full 3D solver?
No. It provides a 1D FEM solve for two common field prototypes plus a mesh and resource estimator. Use it to plan inputs and sanity-check scales before larger simulations.
2) How should I choose Ne?
Start from the suggested Ne based on wavelength or skin depth. Then refine Ne until the field profile and energy change only slightly between runs. Regions with sharp sources may need extra refinement.
3) What does the Neumann input represent?
For electrostatics it represents normal displacement Dn at x=L, in C/m². For magnetostatics it represents normal magnetic field Hn at x=L, in A/m. It is applied as a boundary load.
4) Why does conductivity affect the recommendation?
Conductivity introduces skin depth in time-harmonic behavior. When δ is small, the field changes rapidly near surfaces. The estimator recommends smaller elements to resolve that length scale.
5) When should I use quadratic elements?
Quadratic elements can reduce error for smooth solutions at similar element counts. They increase bandwidth and cost per element, so they are most helpful when you want higher accuracy without extreme refinement.
6) Why is the energy labeled “per meter depth”?
The model is 1D with a cross-sectional area. If you interpret the system as extruded uniformly in the out-of-plane direction, the computed energy corresponds to one meter of that depth.
7) Can I trust the memory numbers exactly?
They are approximate. Real solvers store additional data, preconditioners, and factorization fill-in. Treat the estimator as a planning guide, then validate against your solver’s actual reports.