A Quadric-Based Framework for Local Curved-Volume Computation on Tetrahedral Meshes

Fast, accurate, and conservative geometry near moving fluid interfaces remains critical for multiphase simulation on unstructured meshes. At coarse resolution, maintaining low drift and geometric fidelity often requires substantial refinement or specialized reconstruction. We present a local quadric-patch curved-volume framework on tetrahedral meshes: crease-aware patches are fitted, canonicalized, classified, and evaluated by subtracting matched analytic sub-volumes to recover the missing curved patch volume for each piecewise-linear (PL) surface triangle, then split conservatively into vertex dual volumes. On analytic quadrics (e.g., spheres and ellipsoids) we achieve near-round-off accuracy at sub-millisecond per-element cost (single thread). In interface-only transients, cube-to-sphere relaxation reduces final enclosure error from 7.1% to about 0.04%, and Rayleigh l=2 oscillations reduce long-time volume drift by more than two orders at coarse resolution. The method provides a practical drop-in geometry module for curvature-corrected volume integration in multiphase solvers.