MCHex: Marching Cubes Based Adaptive Hexahedral Mesh Generation with Guaranteed Positive Jacobian

Abstract

Grid-based methods are the most robust approach for automatic hexahedral (hex) meshing, but they struggle to achieve high boundary fidelity and element quality. Conventional pipelines remove outside elements. This yields axis-aligned surfaces that converge to the input geometry at first order. The subsequent padding and projection steps are heuristic, offering no guarantees on final boundary fidelity or mesh quality. This paper introduces MCHex, a fundamental reformulation of boundary and mesh quality handling in grid-based hex meshing. MCHex directly applies Marching Cubes (MC) inside each grid cell, ensuring that the mesh boundary is an MC surface. The key insight is that by constraining cut-cell configurations to a 3-regular polyhedron, a midpoint subdivision of these configurations produces an all-hex mesh with a guaranteed positive Jacobian for every element. MCHex provides three advantages: (1) a theoretical guarantee of positive Jacobian for all hex elements; (2) boundary convergence that matches the approximation rate of MC, together with a non-heuristic algorithm that has well-bounded time complexity and achieves the fastest wall-clock time among all existing methods; and (3) generation of manifold surfaces for arbitrary geometries and a natural padded layer. Extensive evaluation on a benchmark of 202 geometries compares MCHex against several previous state-of-the-art hex meshing methods under a slightly smaller element budget, demonstrating that MCHex consistently produces positive Jacobian meshes with similar boundary fidelity while running significantly faster. MCHex can integrate seamlessly with post-processing steps such as mesh smoothing, mesh simplification, and is suitable for simulation.