Periodic Reporting for period 4 - AlgoHex (Algorithmic Hexahedral Mesh Generation)
Reporting period: 2024-08-01 to 2025-07-31
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. Automatic hexahedral meshing of general volumetric domains is a long-standing, notoriously difficult, and open problem. The development of automatized hexahedral meshing algorithms is of high practical importance since at the moment practitioners spend a substantial amount of time and resources on manual mesh generation and cleanup.
The main goal of the AlgoHex project 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.
Our research within the scope of AlgoHex lead to several breakthroughs, boosting robustness, scalability and control of integer-grid map based based hexahedral mesh generation algorithms. Most notably, (1) we developed the theory of local meshability of frame fields and devised robust practical algorithms to achieve it, (2) we introduced a conceptually novel class of robust volumetric map generation approaches called shrink-and-expand, a key component required for a robust hexmeshing algorithm, (3) we developed the first unconditionally robust and fast quantization algorithms, and (4) we collected and published HexMe, which is a challenging dataset of input domains to measure research progress and guide future developments. Moreover, we introduced the powerful odeco frame representation and developed the state-of-the-art frame field optimization techniques, and boosted performance of all stages of the hexahedral mesh generation pipeline through various novel algorithms. A strong focus of AlgoHex has been to not only research hexahedral meshing from a theoretical perspective but to develop practically relevant algorithms. Consequently, an important outcome of AlgoHex is the source code for all our novel algorithms, which can be accessed through the project webpage https://www.algohex.eu/code/(opens in new window)
The main goal of the AlgoHex project 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.
Our research within the scope of AlgoHex lead to several breakthroughs, boosting robustness, scalability and control of integer-grid map based based hexahedral mesh generation algorithms. Most notably, (1) we developed the theory of local meshability of frame fields and devised robust practical algorithms to achieve it, (2) we introduced a conceptually novel class of robust volumetric map generation approaches called shrink-and-expand, a key component required for a robust hexmeshing algorithm, (3) we developed the first unconditionally robust and fast quantization algorithms, and (4) we collected and published HexMe, which is a challenging dataset of input domains to measure research progress and guide future developments. Moreover, we introduced the powerful odeco frame representation and developed the state-of-the-art frame field optimization techniques, and boosted performance of all stages of the hexahedral mesh generation pipeline through various novel algorithms. A strong focus of AlgoHex has been to not only research hexahedral meshing from a theoretical perspective but to develop practically relevant algorithms. Consequently, an important outcome of AlgoHex is the source code for all our novel algorithms, which can be accessed through the project webpage https://www.algohex.eu/code/(opens in new window)
The 5.5 years of AlgoHex research allowed us to make significant progress toward robust algorithms for automatic hexahedral mesh generation. We are extremely happy that the highly ambitious goals of AlgoHex have been reached to a large extent. The AlgoHex project enabled a large number of publications in internationally leading journals, i.e. 10x ACM Transactions on Graphics and 5x Computer Graphics Forum, and presentations at top-tier conferences, namely ACM SIGGRAPH, ACM SIGGRAPH Asia, and EUROGRAPHICS Symposium on Geometry Processing (SGP). In the following we elaborate on a selection of main results and their dissemination.
We designed and published the HexMe dataset [Beaufort et al. 2022] (presented at SGP 2022, published in CGF), 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 methods available prior to AlgoHex (success rate less than 10%). Consequently, the HexMe dataset has been highly valuable to measure our progress within the AlgoHex project and will guide future research.
Obtaining frame-fields that are guaranteed to be meshable is the first key challenge of AlgoHex. Our novel frame-field representations and corresponding optimization algorithms [Palmer et al. 2020] (presented at ACM SIGGRAPH 2020, published in ACM TOG), enabled 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 establish it, as introduced in [Liu et al. 2023] (presented at ACM SIGGRAPH 2023, published in ACM TOG). By ensuring local meshability, we were able to increase the success rate on the HexMe dataset from below 10% to more than 50%. Local meshability is a necessary but not sufficient condition for the ultimate long-term goal of (global) meshability, however, offering promising insights and a suitable framework to tackle global meshability.
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] (presented at ACM SIGGRAPH 2022, published in ACM TOG), which turns a given seamless map into a valid combinatorial hexahedral mesh. It is a crucial step forward, effectively resolving one major robustness gap of prior hexmeshing pipelines. In [Brückler et al. 2024] (presented at SGP 2024, published in CGF) we further improved quantization by a specialized optimization scheme enabling shorter and more predictable runtimes.
Moreover, we made substantial progress regarding the task of robust volumetric map generation with fixed boundaries. In [Nigolian et al. 2023] (presented at ACM SIGGRAPH 2023, published in ACM TOG), we presented a novel shrink-and-expand framework, which improved the success rate on a highly challenging dataset from 33% to 76%. In our subsequent research [Nigolian et al. 2024] (presented at ACM SIGGRAPH Asia 2024, published in ACM TOG) we deepened the theoretical understanding of shrink-and-expand enabling strong theoretical guarantees for shellable inputs (success rate of 100% without time limit), and boosting the practical success rate (time limit 6h) from 76% to 95%.
In the pursue of scalability, in [Kohler et al. 2025] (presented at ACM SIGGRAPH 2025, published in ACM TOG) we introduced a novel parallelized hexahedral mesh extraction algorithm, enabling runtimes which are orders of magnitudes faster than the prior state of the art.
We designed and published the HexMe dataset [Beaufort et al. 2022] (presented at SGP 2022, published in CGF), 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 methods available prior to AlgoHex (success rate less than 10%). Consequently, the HexMe dataset has been highly valuable to measure our progress within the AlgoHex project and will guide future research.
Obtaining frame-fields that are guaranteed to be meshable is the first key challenge of AlgoHex. Our novel frame-field representations and corresponding optimization algorithms [Palmer et al. 2020] (presented at ACM SIGGRAPH 2020, published in ACM TOG), enabled 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 establish it, as introduced in [Liu et al. 2023] (presented at ACM SIGGRAPH 2023, published in ACM TOG). By ensuring local meshability, we were able to increase the success rate on the HexMe dataset from below 10% to more than 50%. Local meshability is a necessary but not sufficient condition for the ultimate long-term goal of (global) meshability, however, offering promising insights and a suitable framework to tackle global meshability.
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] (presented at ACM SIGGRAPH 2022, published in ACM TOG), which turns a given seamless map into a valid combinatorial hexahedral mesh. It is a crucial step forward, effectively resolving one major robustness gap of prior hexmeshing pipelines. In [Brückler et al. 2024] (presented at SGP 2024, published in CGF) we further improved quantization by a specialized optimization scheme enabling shorter and more predictable runtimes.
Moreover, we made substantial progress regarding the task of robust volumetric map generation with fixed boundaries. In [Nigolian et al. 2023] (presented at ACM SIGGRAPH 2023, published in ACM TOG), we presented a novel shrink-and-expand framework, which improved the success rate on a highly challenging dataset from 33% to 76%. In our subsequent research [Nigolian et al. 2024] (presented at ACM SIGGRAPH Asia 2024, published in ACM TOG) we deepened the theoretical understanding of shrink-and-expand enabling strong theoretical guarantees for shellable inputs (success rate of 100% without time limit), and boosting the practical success rate (time limit 6h) from 76% to 95%.
In the pursue of scalability, in [Kohler et al. 2025] (presented at ACM SIGGRAPH 2025, published in ACM TOG) we introduced a novel parallelized hexahedral mesh extraction algorithm, enabling runtimes which are orders of magnitudes faster than the prior state of the art.
AlgoHex has shaped and extended the state of the art in several research areas, significantly increasing the robustness, scalability and control of hexahedral mesh generation algorithms.
We introduced the theory of local meshability of frame fields, enabling novel algorithms enabling a breakthrough in terms of robustness. While the ultimate robustness goal of algorithms to establish global meshability has not been reached yet, we are confident that our introduced theoretical framework paves the way to fully robust methods in the near future.
Our research on volumetric map generation lead to another breakthrough in terms of robustness. Moreover, it is highly valuable as a addition to the general geometry processing toolbox and thus has the potential of an impact far beyond the specific problem of hexahedral mesh generation.
We introduced the theory of local meshability of frame fields, enabling novel algorithms enabling a breakthrough in terms of robustness. While the ultimate robustness goal of algorithms to establish global meshability has not been reached yet, we are confident that our introduced theoretical framework paves the way to fully robust methods in the near future.
Our research on volumetric map generation lead to another breakthrough in terms of robustness. Moreover, it is highly valuable as a addition to the general geometry processing toolbox and thus has the potential of an impact far beyond the specific problem of hexahedral mesh generation.