Validating Hexahedra through their Boundaries
Abstract
Trilinear hexahedral mesh elements are used in finite element analysis to simulate volumetric phenomena. Realism of the simulation mandates that mesh elements through the trilinear map maintain positive volume, or mathematically, that the trilinear map maintains positive Jacobian determinant (jacdet). While jacdet positivity generally needs to be verified in the full element volume, we prove Knupp's conjecture which posits that for trilinear hexahedra, jacdet positivity on the boundary of the element is sufficient to guarantee positivity in the entire element. We further prove that for any valid hexahedron, its globally minimal jacdet must reside on the boundary. Lastly, we prove that the globally minimal jacdet on the boundary of a hexahedron, valid or not, must be achieved within a finite set of candidate points that can be determined by quartic root finding. Combining these results, we can algorithmically validate a hexahedron through evaluation of its jacdet on a finite set of boundary points.
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BibTeX
@misc{zhang2026validating,
title = {{Validating Hexahedra through their Boundaries}},
author = {Paul Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19926},
eprint = {2609.19926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19926},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Validating Hexahedra through their Boundaries
AU - Paul Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19926
UR - https://arxiv.org/abs/2609.19926
AB - Trilinear hexahedral mesh elements are used in finite element analysis to simulate volumetric phenomena. Realism of the simulation mandates that mesh elements through the trilinear map maintain positive volume, or mathematically, that the trilinear map maintains positive Jacobian determinant (jacdet). While jacdet positivity generally needs to be verified in the full element volume, we prove Knupp's conjecture which posits that for trilinear hexahedra, jacdet positivity on the boundary of the element is sufficient to guarantee positivity in the entire element. We further prove that for any valid hexahedron, its globally minimal jacdet must reside on the boundary. Lastly, we prove that the globally minimal jacdet on the boundary of a hexahedron, valid or not, must be achieved within a finite set of candidate points that can be determined by quartic root finding. Combining these results, we can algorithmically validate a hexahedron through evaluation of its jacdet on a finite set of boundary points.
ER -