Hexahedral Mesh Repair via Sum-of-Squares Relaxation.

The validity of trilinear hexahedral (hex) mesh elements is a prerequisite for many applications of hex meshes, such as finite element analysis. A commonly used check for hex mesh validity evaluates mesh quality on the corners of the parameter domain of each hex, an insufficient condition that neglects invalidity elsewhere in the element, but is straightforward to compute. Hex mesh quality optimizations using this validity criterion suffer by being unable to detect invalidities in a hex mesh reliably, let alone fix them. We rectify these challenges by leveraging sum-of-squares relaxations to pinpoint invalidities in a hex mesh efficiently and robustly. Furthermore, we design a hex mesh repair algorithm that can certify validity of the entire hex mesh. We demonstrate our hex mesh repair algorithm on a dataset of meshes that include hexes with both corner and face-interior invalidities and demonstrate that where naive algorithms would fail to even detect invalidities, we are able to repair them. Our novel methodology also introduces the general machinery of sum-of-squares relaxation to geometry processing, where it has the potential to solve related problems.