mesh-pde-solver#

Upload a triangle or quad surface mesh, pick a PDE, tweak its right-hand side and coefficients, and solve it on the surface — entirely in your browser. The mesh is converted to Gmsh format with meshio (via Pyodide), and the PDE is solved by surfacefun running on numbl, a MATLAB-compatible runtime, in a web worker. The solution renders in a rotatable 3D view (drag to rotate, scroll to zoom).

PDEs#

• Poisson (Laplace–Beltrami) — Δu = f. On a closed surface the problem is rank-deficient: f is projected to mean zero and the mean-zero solution is returned. On an open surface, zero Dirichlet data is imposed.
• Helmholtz (variable coefficient) — (Δ + c)u = f with c(x, y, z) an arbitrary expression.

The right-hand side f and coefficient c are MATLAB expressions in the surface coordinates x, y, z (elementwise operators: .*, .^, …), with presets to start from. The polynomial order per patch is adjustable (accuracy vs. time).

Meshes#

Uploads go through meshio, so any of .msh .vtk .vtu .obj .off .ply .inp .mesh .bdf .avs works — the mesh must contain triangle or quadrilateral cells (surfacefun computes on either patch type, but not both at once, so a mixed mesh has its quads split into triangles). Three sample meshes are bundled. The converted Gmsh file can be downloaded. Whether the surface is closed or open is detected from the edge connectivity.

How it works#

1. src/mesh/ — meshio in Pyodide parses the upload, keeps the triangle and quad cells, and writes a canonical Gmsh MSH 4.1 ASCII file plus preview arrays.
2. matlab/ — the whole solve is one MATLAB script, generated by filling the parameters into matlab/solve_template.m: mip load --install surfacefun, load the mesh with surfacemesh.import, resample to the requested order, solve with surfaceop, write result.json.
3. src/engine/ — each solve boots a fresh managed numbl session (createNumblSession from numbl/browser): numbl owns the worker and VFS and bootstraps the mip package manager. The host stages the generated script and the converted mesh, runs the script standalone, and reads result.json back before disposing the worker.
4. src/render/SurfaceView.tsx — three.js view of the mesh or the per-patch solution data with a parula colormap.

The exact script the Solve button runs can be downloaded from the UI (even before solving). It also runs in desktop MATLAB with the downloaded converted .msh next to it: install surfacefun via mip, or comment out the mip load line and put surfacefun and its dependencies on your path.

The first visit downloads the Python runtime (~15 MB, browser-cached) and the surfacefun/chebfun packages (~28 MB). Installed MATLAB packages persist in IndexedDB across page loads (numbl wipes them after 24 h of inactivity), so later visits skip the package downloads.

Development#

npm install
npm run dev          # local dev server
npm run build        # static build in dist/
npm run engine-test  # headless solver check in Node (no browser)
python3 scripts/make_samples.py   # regenerate public/samples/

The engine test runs the exact MATLAB script the worker runs, shimming numbl's synchronous-XHR websave/webread with curl (responses cached in .cache/), and checks a Poisson solve against an exact spherical-harmonic solution.

Requires numbl >= 0.4.12 — NumblSession.readFile (0.4.10), enumeration-class support and the 1×1-tensor broadcast-assignment fix (0.4.11), which surfacefun's surfacemesh.import / patchtype depend on, and the 1×1-tensor gather-orientation fix (0.4.12), which surfacefun's trianglepts depends on.