Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are a ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this article, we study in detail the dual approach, and propose four main contributions to it: (i) We enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) We show that schemes are internally asymmetric, therefore not only their construction is ambiguous, but different implementative choices lead to hexahedral meshes with different singular structure; (iii) We explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) We enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing the tight topological requirements imposed by The work of Marco Livesu was partly supported by EU ERC Advanced Grant CHANGE No. 694515. Gianmarco Cherchi gratefully acknowledges the support to his research by PON R&I 2014-2020 AIM1895943-1. Authors' addresses: M. Livesu, CNR IMATI, via De Marini 6, 16149 Genoa, Italy; email: marco.livesu@gmail.com; L. Pitzalis, University of Cagliari and CRS4, via Ospedale 72, 09124 Cagliari, Italy; email: luca.pitzalis94@gmail.com; G. Cherchi, University of Cagliari, via Ospedale 72, 09124 Cagliari, Italy; email: g.cherchi@unica.it. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2021 Association for Computing Machinery. 0730-0301/2021/12-ART15 $15.00 https://doi.org/10.1145/3494456 previous approaches. Our extensive experiments show that our transition schemes consistently outperform prior art in terms of ability to converge to a valid solution, amount and distribution of singular mesh edges, and element count. Last but not least, we publicly release our code and reveal a conspicuous amount of technical details that were overlooked in previous literature, lowering an entry barrier that was hard to overcome for practitioners in the field.