Singularity-constrained octahedral fields for hexahedral meshing

Heng Liu, Paul Zhang, Edward Chien, Justin Solomon, David Bommes

Research output: Contribution to journalArticlepeer-review

71 Scopus citations

Abstract

Despite high practical demand, algorithmic hexahedral meshing with guarantees on robustness and quality remains unsolved. A promising direction follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high-quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a surfacealigned octahedral field and (2) generation of an integer-grid map that best aligns to the octahedral field. The main robustness issue stems from the fact that smooth octahedral fields frequently exhibit singularity graphs that are not appropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The first contribution of this work is an enumeration of all local configurations that exist in hex meshes with bounded edge valence, and a generalization of the Hopf-Poincaré formula to octahedral fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large nonlinear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field.

Original languageEnglish
Article numberA54
JournalACM Transactions on Graphics
Volume37
Issue number4
DOIs
StatePublished - 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 Copyright held by the owner/author(s).

Funding

D. Bommes received funding from the German Research Foundation (DFG, grant GSC 111, Aachen Institute for Advanced Study in Computational Engineering Science) and H. Liu from the Chinese Scholarship Council (CSC). J. Solomon acknowledges the generous support of Army Research Office grant W911NF-12-R-0011 (“Smooth Modeling of Flows on Graphs”), from the MIT Research Support Committee (“Structured Optimization for Geometric Problems”), and from the Skoltech–MIT Next Generation Program (“Simulation and Transfer Learning for Deep 3D Geometric Data Analysis”). We would like to thank Jan Möbius for OpenFlipper, Martin Heister-mann for help with interfacing Blender, Amir Vaxman for inspiring discussions, and the reviewers for their helpful feedback.

FundersFunder number
Chinese Scholarship Council
MIT Next Generation Program
MIT Research Support Committee
Army Research OfficeW911NF-12-R-0011
Deutsche ForschungsgemeinschaftGSC 111

    Keywords

    • Hexahedral meshing
    • Integer-grid maps
    • Octahedral fields
    • Singularity graph

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