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Algorithmic Hexahedral Mesh Generation

Periodic Reporting for period 2 - AlgoHex (Algorithmic Hexahedral Mesh Generation)

Reporting period: 2021-08-01 to 2023-01-31

Digital geometry representations are nowadays a fundamental ingredient of many applications, as for instance CAD/CAM, fabrication, shape optimization, biomedical engineering, and numerical simulation. Among volumetric discretizations, the preferred choice is often a hexahedral mesh, i.e. a decomposition of the domain into conforming cube-like elements. For simulations, a hexahedral mesh offers accuracy and efficiency that cannot be obtained with alternatives like tetrahedral meshes, specifically when dealing with higher-order PDEs. So far, automatic hexahedral meshing of general volumetric domains is a long-standing, notoriously difficult, and open problem.

Our main goal is to develop algorithms for automatic hexahedral meshing of general volumetric domains that are (i) robust, (ii) scalable, and (iii) offer precise control on regularity, approximation error, and element sizing/anisotropy. Our approach is designed to replicate the success story of recent integer-grid map based algorithms for 2D quadrilateral meshing. The underlying methodology offers the essential global perspective on the problem that was lacking in previous attempts, mostly failing since local considerations cannot prevent global inconsistencies. Preliminary results of integer-grid map based hexahedral meshing are encouraging and a breakthrough is in reach.
The first 30 months of the project allowed us to make significant progress on our path toward robust algorithms for automatic hexahedral mesh generation.

We designed and published the HexMe dataset [Beaufort et al. 2022], which contains 189 input domains that are challenging for hexahedral mesh generation algorithms. The focus has been on collecting problematic geometries including complicated feature curve and feature surface arrangements. Such domains are of high practical importance but not well handled by available methods. Our evaluation shows that state-of-the-art frame-field-based hex meshing algorithms succeed only on less than 10% of these inputs. Consequently, the HexMe dataset will be highly valuable to measure our progress within the AlgoHex project and will furthermore guide all future research on hex meshing. The HexMe dataset has been designed with evolution in mind. Due to its automated workflow, it is easy to adapt the selection of models whenever the state of the art progresses.

Obtaining frame-fields that are guaranteed to be meshable is the first key challenge that we target in AlgoHex. Our initial step in this direction has been a novel class of frame-field representations and corresponding optimization algorithms [Palmer et al. 2020], enabling better singularity graphs and thus improved meshability.
Subsequently, we made substantial progress by developing the theory of local meshability of frame-fields, and a novel algorithm to turn an arbitrary given frame-field into a (similar) locally meshable one. Local meshability is a necessary but not sufficient condition for (global) meshability, our ultimate goal. Nevertheless, by ensuring local meshability, we were able to increase the success rate on the HexMe dataset from below 10% to more than 50% with our prototype implementation. The corresponding publication is currently in preparation.

The second key challenge of AlgoHex consists in robustly constructing a locally injective integer-grid map for a given meshable frame field. We developed a provably robust construction and quantization of the motorcycle complex (MC) [Brückler et al. 2022], which turns a given seamless map into a valid combinatorial hexahedral mesh. Our novel algorithm is a crucial step forward since previous techniques based on greedy rounding regularly generate invalid quantizations and thus have been a major source of failures of the hex meshing pipeline.
Our research and developments on the local meshability of frame-fields and the robust construction and quantization of the motorcycle complex already substantially extend the state of the art, and significantly increase the success rate of the corresponding hex meshing algorithms.

In the second stage of AlgoHex, we target the resolution of all remaining robustness issues and will specifically focus on (i) the global meshability of frame-fields and (ii) local injectivity guarantees of the quantized integer-grid map. Both components are essential for automatic hexahedral meshing algorithms and seem to be in reach based on our deepened understanding of the involved mathematical objects.

We are confident that AlgoHex will enable a level of robustness that is sufficient for industrial applications. Until the end of the project, we expect algorithms for automatic hexahedral meshing of general volumetric domains that are (i) robust, (ii) scalable, and (iii) offer precise control on regularity, approximation error, and element sizing/anisotropy.
AlgoHex has the goal of automatically decomposing volumetric domains into conforming hexahedra