Pure NumPy/SciPy 2D Electrostatic FEM Solver (Dependency-Free & Platform-Independent)

[ratwolfzero]

Hi everyone! 👋

I wanted to share capacitor-fem, a self-contained 2D finite-element electrostatics solver designed for real-world capacitor geometries (parallel plates, coaxial cables, and arbitrary shapes built from simple primitives).

🛠️ Why another solver?

Most FEM tools require native dependencies like gmsh or compiled C++ extensions. This project is built entirely on pure NumPy, SciPy, and Matplotlib. It's just a single file (capacitor_fem_universal.py) that runs anywhere—from high-performance desktops to Jupyter notebooks and even Pydroid 3 on Android!

🔬 Key Features & Numerical Architecture

• Vectorized Sparse Assembly: Element stiffness matrices are computed for all triangles simultaneously using NumPy broadcasting, pushed straight into SciPy’s csr_matrix for a fast sparse LU solve.
• Energy-Based Capacitance: Instead of integrating numerically noisy boundary charges ($\mathbf{D}\cdot\mathbf{n}$), it computes capacitance via total electrostatic stored energy ($C = \frac{2W}{(\Delta V)^2}$), providing much higher stability on structured grids.
• Grid Alignment Handling: Features a strict snap_to_grid validation mechanism to prevent geometric rounding errors from changing simulated gaps.
• Advanced Geometries: Supports graded Cartesian refinement, independent plate widths, and optional rounded (filleted) plate edges to remove artificial corner field singularities.

📊 Validation & Limitations

The core engine is verified against exact analytical 1D solutions to within machine precision ($~10^{-15}$ relative error). It handles realistic challenges like material-interface straddling and domain truncation effects, all of which are deeply documented in the repository.

Check out the repo here: Capacitor_FEM

I would love to hear your feedback, especially on potential future work like cut-cell boundary treatments or nonlinear dielectrics!