mesh-pde-solver#

Upload a quadrilateral 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 — but the mesh must contain quadrilateral cells (surfacefun computes on quad patches; triangle-only meshes are rejected). Two 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 quad cells, and writes a canonical Gmsh MSH 2.2 ASCII file plus preview arrays.
2. src/engine/ — a managed numbl session (createNumblSession from numbl/browser): numbl owns the worker and VFS, bootstraps the mip package manager, and runs matlab/main.m, which begins with mip load --install surfacefun. The script opens a placeholder uihtml figure that is never rendered — it is the event bridge: the host writes mesh.msh into the VFS and dispatches solve events; the script solves and sends per-patch data back.
3. matlab/solve_pde.m — parses the mesh (load_gmsh_quads.m), builds a surfacemesh from the quads, resamples it to the requested order, and solves with surfaceop.
4. src/render/SurfaceView.tsx — three.js view of the quad mesh or the per-patch solution grids with a parula colormap.

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 project 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 >= 0.4.9 (the numbl/browser managed-session entry).