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arxiv: 2604.17266 · v1 · submitted 2026-04-19 · 💻 cs.CE · physics.comp-ph

Recognition: unknown

Scalable DDPM-Polycube: An Extended Diffusion-Based Method for Hexahedral Mesh and Volumetric Spline Construction

Yuxuan Yu , Jiashuo Liu , Hua Tong , Honghua Lou , Yongjie Jessica Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:58 UTC · model grok-4.3

classification 💻 cs.CE physics.comp-ph
keywords polycubediffusion modelhexahedral meshvolumetric splineisogeometric analysisCAD geometrymesh generationtopology verification
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The pith

Adding a blind-hole primitive, a larger 3D grid, and hierarchical verification extends diffusion-based polycube generation to more complex CAD geometries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to automate polycube construction for shapes that current diffusion methods cannot handle reliably. Polycubes act as simple parametric domains that allow all-hexahedral meshes and volumetric splines to be built for isogeometric analysis on real CAD parts. The authors enlarge the set of allowed primitive shapes, increase the grid size to three dimensions, and add a genus-guided hierarchical check that prunes invalid structures during generation. These changes are presented as sufficient to raise both the diversity of representable local features and the overall success rate on intricate models. If the extensions work as claimed, downstream tasks such as control-mesh creation and spline fitting become feasible without extensive manual topology editing.

Core claim

The paper claims that the Scalable DDPM-Polycube method overcomes prior limits on primitive diversity, grid size, and inference cost by introducing a blind-hole cube primitive, switching to an enlarged three-dimensional grid configuration, and implementing a genus-guided context generation strategy together with hierarchical verification. Once a valid polycube is obtained, it is mapped to the input geometry to produce an all-hex control mesh and an analysis-suitable volumetric spline for isogeometric analysis.

What carries the argument

The genus-guided hierarchical verification procedure, which uses the input geometry's topology to generate and validate polycube contexts inside the enlarged grid while incorporating the new blind-hole primitive.

If this is right

  • Polycube generation becomes possible for geometries containing blind holes that do not alter global genus.
  • Larger grid configurations reduce mapping distortion on intricate shapes while preserving scalability.
  • The same pipeline supports both user-guided and fully automated modes for downstream hex-mesh and spline construction.
  • Once generated, the polycube directly enables parametric mapping, all-hex control meshes, and volumetric splines for IGA.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may lower the expert time currently required to prepare complex CAD assemblies for hexahedral meshing.
  • Integration into existing CAD-to-analysis workflows could become practical if the verification step runs at acceptable speed on large models.
  • Further primitives or adaptive grid refinement might be needed if the current fixed enlargement still fails on certain feature combinations.

Load-bearing premise

The three specific extensions will be enough to represent every local feature and topological variation found in arbitrary complex CAD geometries without creating new mapping distortions or invalid polycube structures.

What would settle it

Running the method on a set of industrial CAD models containing multiple blind holes, high-genus features, and sharp local details, then checking whether any resulting polycube produces self-intersecting mappings or invalid hexahedral elements.

Figures

Figures reproduced from arXiv: 2604.17266 by Honghua Lou, Hua Tong, Jiashuo Liu, Yongjie Jessica Zhang, Yuxuan Yu.

Figure 1
Figure 1. Figure 1: Overview of the SDDPM pipeline. (a) The input triangular mesh is preprocessed by PCA alignment, centering, and normalization. (b) In the user-guided mode, partial constraints or one or more complete cell-wise one-hot encoded context vectors are provided and converted into candidate global contexts when needed. (c) In the automated mode, the input geometry is temporarily partitioned into subregions, local c… view at source ↗
Figure 2
Figure 2. Figure 2: Expanded primitive set and unfolded grid layout used for dataset generation and context encoding. (a) The primitive library contains ten axis-aligned categories derived from three base primitive geometries: a cube, a THC with three axial orientations, and a BHC with six directional variants. (b) These primitives are assembled on the 𝐺3×2×2 grid, which is unfolded into a fixed 4 × 3 layout for data generati… view at source ↗
Figure 3
Figure 3. Figure 3: Training behavior of SDDPM on the expanded primitive set and the 𝐺3×2×2 grid. (a) Overall training loss over 500 epochs. (b) Enlarged view of the final 100 epochs. (c) Gradient of the training loss. 7.2. Inference with user-guided contexts and partial constraints We first evaluate the pipeline under user guidance. The user can provide either a complete 132-dimensional cell￾wise one-hot context vector or pa… view at source ↗
Figure 4
Figure 4. Figure 4: Representative examples of user-guided inference in SDDPM. The first row shows generation under high-level user constraints. The second row shows rejection of user-provided contexts that are inconsistent with the input genus. The third row shows generation under user-specified volumetric partition guidance. 7.3. Inference with automated context generation We next evaluate the fully automated mode, which is… view at source ↗
Figure 5
Figure 5. Figure 5: Automated context generation and global assembly in the proposed inference pipeline. (a) Temporary tetrahedralization of the input geometry and partition of the volume into subregions. (b) Computation of the genus for each subregion and restriction of feasible primitive categories. (c) Aggregation of verified local labels into a global 132-dimensional cell-wise one-hot context vector aligned with the 𝐺3×2×… view at source ↗
Figure 6
Figure 6. Figure 6: Examples of GOCC. (a) Failure case, where the generated local point subset does not exhibit sufficient occupancy in the target cell. (b) Success case, where the generated point distribution exhibits sufficient occupancy in the target cell. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Examples of TCV. (a) Failure case, where the occupied local geometry does not match the target primitive category. (b) Success case, where the occupied local geometry matches the target primitive category, with the target template being the nearest competitor and the PCD remaining below the acceptance threshold. 7.5. Performance on geometries of different genus In this section, we evaluate the performance … view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of the reverse diffusion process for polycube generation. From left to right, the model progressively removes deformation from the input geometry at 𝑡 = 500 and recovers a topology-consistent polycube representation at 𝑡 = 0. important to examine how the proposed local context generation and hierarchical verification behave as structural complexity increases. Tab. 2 summarizes the average num… view at source ↗
Figure 9
Figure 9. Figure 9: Representative polycube generation results on geometries of different genus (Part I). the full global context space. These results support our claim that SDDPM improves the scalability of diffusion-based polycube generation under the expanded primitive set and the enlarged grid configuration considered in this work. 7.7. All-hex control mesh generation and volumetric spline construction Once the polycube s… view at source ↗
Figure 10
Figure 10. Figure 10: Representative polycube generation results on geometries of different genus (Part II) [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ablation study on the BHC primitive. (a) Input geometry with blind-hole features. (b) Result without a BHC primitive label, where the generated polycube fails to preserve the intended local blind-hole feature and exhibits large geometric deviations near the blind-hole region. (c) Result with the BHC+Z primitive label, where the blind-hole feature is better preserved. The color map visualizes pointwise geo… view at source ↗
Figure 12
Figure 12. Figure 12: Polycube structures, all-hex control meshes, scaled Jacobian histograms, and volumetric splines with IGA temperature analysis for ten representative models. The red bar in the histograms represents the minimum scaled Jacobian (sometimes the red bar is too short to be seen). There are several directions for future work. The current method still relies on a fixed grid and a finite primitive set, so extendin… view at source ↗
read the original abstract

Polycube structures provide parametric domains for all-hexahedral (all-hex) mesh generation and analysis-suitable volumetric spline construction in isogeometric analysis (IGA). Recent learning-based polycube pipelines have improved automation, yet several challenges remain when handling complex CAD geometries. These challenges include the limited diversity of primitive geometries, restricted grid configurations, and the increasing cost of genus-guided context search during inference as both the primitive set and the grid size grow. In this paper, we present {Scalable DDPM-Polycube}, an extended diffusion-based polycube construction method that addresses these limitations. First, we expand the primitive set from two primitive geometries to three by introducing a blind-hole cube primitive, thereby improving the representation of local hole-like features that do not change the global genus. Second, we extend the grid configuration from the previous $2\times 1$ setting to an enlarged three-dimensional grid configuration, which increases representational capacity and reduces mapping distortion for complex geometries. Third, we develop a genus-guided context generation strategy together with a hierarchical verification procedure, enabling robust context generation in both user-guided and automated modes. Once a valid polycube structure is generated, it is used for parametric mapping, all-hex control mesh generation, and volumetric spline construction. Experimental results demonstrate that scalable DDPM-Polycube improves the generality, scalability, and automation of diffusion-based polycube generation, and supports hex mesh generation and volumetric spline construction for IGA applications on complex geometries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents Scalable DDPM-Polycube as an extension of prior diffusion-based polycube generation. It adds a blind-hole cube primitive to the set of geometries, enlarges the grid configuration from 2x1 to full 3D, and introduces a genus-guided context generation strategy paired with hierarchical verification. These changes are intended to increase representational capacity for local hole-like features and topological variations in complex CAD models. The resulting polycubes are then mapped to support all-hex control mesh generation and volumetric spline construction for isogeometric analysis (IGA) applications. The central claim is that experimental results confirm gains in generality, scalability, and automation over previous diffusion pipelines.

Significance. If the experimental claims are supported by detailed quantitative evidence, the work could advance automated polycube construction for hexahedral meshing and IGA on complex geometries by directly targeting limitations in primitive diversity and grid size. The hierarchical verification procedure provides a concrete mechanism for robust context handling in both automated and user-guided modes. No parameter-free derivations or machine-checked proofs are offered, but the targeted addition of the blind-hole primitive represents a focused, incremental improvement over the base DDPM-Polycube pipeline.

major comments (2)
  1. [Experimental Results] Experimental Results section: the central claim that the three extensions improve generality and scalability rests on assertions of experimental success, yet no quantitative metrics (mapping distortion values, validity rates, success rates on test geometries), baseline comparisons to the original DDPM-Polycube, error bars, or details on measurement protocols for distortion and validity are reported. This directly undermines verification that the blind-hole primitive, 3D grid, and hierarchical verification suffice for arbitrary complex CAD without new distortions or invalid structures.
  2. [Method] Method section on primitive set and grid extension: the addition of only a single blind-hole cube primitive and the switch to an enlarged 3D grid are presented as sufficient to capture local hole-like features and reduce distortion, but the manuscript provides no analysis, examples, or ablation of cases with multiple interacting blind holes, higher-genus transitions, or feature combinations absent from the three-primitive set. This is load-bearing for the generality claim, as the diffusion process could still generate polycubes that induce large mapping distortions or fail downstream hex-mesh and spline construction.
minor comments (1)
  1. [Abstract] Abstract: the statement that 'experimental results demonstrate' improvements would be strengthened by a brief indication of the number and topological complexity of tested geometries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We have carefully considered each major comment and provide point-by-point responses below. Where revisions are needed to strengthen the presentation of our results, we have indicated the changes to be made in the revised version.

read point-by-point responses

  1. Referee: [Experimental Results] Experimental Results section: the central claim that the three extensions improve generality and scalability rests on assertions of experimental success, yet no quantitative metrics (mapping distortion values, validity rates, success rates on test geometries), baseline comparisons to the original DDPM-Polycube, error bars, or details on measurement protocols for distortion and validity are reported. This directly undermines verification that the blind-hole primitive, 3D grid, and hierarchical verification suffice for arbitrary complex CAD without new distortions or invalid structures.

    Authors: We agree that the current Experimental Results section lacks the detailed quantitative metrics mentioned. The section primarily presents qualitative examples of polycube generation and downstream applications on complex CAD models. To strengthen the verification of our claims, we will revise the manuscript to include quantitative metrics including mapping distortion values, validity rates, success rates, baseline comparisons to the original DDPM-Polycube, error bars from multiple runs, and explicit details on the measurement protocols. These will be added to the revised version. revision: yes

  2. Referee: [Method] Method section on primitive set and grid extension: the addition of only a single blind-hole cube primitive and the switch to an enlarged 3D grid are presented as sufficient to capture local hole-like features and reduce distortion, but the manuscript provides no analysis, examples, or ablation of cases with multiple interacting blind holes, higher-genus transitions, or feature combinations absent from the three-primitive set. This is load-bearing for the generality claim, as the diffusion process could still generate polycubes that induce large mapping distortions or fail downstream hex-mesh and spline construction.

    Authors: We acknowledge that the manuscript does not include dedicated ablations or examples specifically for multiple interacting blind holes or all possible feature combinations. The three-primitive set with the blind-hole addition, combined with the 3D grid and hierarchical verification, is shown to handle complex geometries in our experiments. However, to better support the generality claim and address potential concerns about the diffusion process generating invalid structures, we will include additional analysis and examples of multiple blind-hole cases and higher-genus transitions in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity in claimed extensions or experimental validation

full rationale

The paper describes an engineering extension to prior diffusion-based polycube methods via three additive changes (blind-hole primitive, enlarged 3D grid, genus-guided hierarchical verification) followed by experimental demonstration on complex geometries for hex meshing and IGA splines. No first-principles derivation, mathematical prediction, or uniqueness theorem is claimed that reduces to its own inputs by construction. The base model is referenced as prior work but the new components and results are presented as independent; success is asserted via experiments rather than any self-definitional loop or fitted parameter renamed as output. This is a standard incremental ML method paper with external empirical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are quantified. The blind-hole cube is presented as a new geometric primitive rather than a fitted parameter.

invented entities (1)
  • blind-hole cube primitive no independent evidence
    purpose: to represent local hole-like features that do not change the global genus
    Introduced to expand the primitive set from two to three geometries for better local feature coverage

pith-pipeline@v0.9.0 · 5586 in / 1255 out tokens · 38617 ms · 2026-05-10T05:58:32.104251+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

55 extracted references · 4 canonical work pages · 3 internal anchors

  1. [1]

    Acomparisonofallhexagonalandalltetrahedralfiniteelementmeshes for elastic and elasto-plastic analysis

    Benzley,S.E.,Perry,E.,Merkley,K.,Clark,B.,Sjaardama,G.,1995. Acomparisonofallhexagonalandalltetrahedralfiniteelementmeshes for elastic and elasto-plastic analysis. 4th International Meshing Roundtable 17, 179–191

    1995

  2. [2]

    Volumeparametrizationquantizationforhexahedralmeshing

    Brückler,H.,Bommes,D.,Campen,M.,2022. Volumeparametrizationquantizationforhexahedralmeshing. ACMTransactionsonGraphics 41, 1–19

    2022

  3. [3]

    Integer-sheet-pumpquantizationforhexahedralmeshing

    Brückler,H.,Bommes,D.,Campen,M.,2024. Integer-sheet-pumpquantizationforhexahedralmeshing. ComputerGraphicsForum43,1–13

    2024

  4. [4]

    arXiv preprint arXiv:2406.10163 , year=

    Chen, Y., He, T., Huang, D., Ye, W., Chen, S., Tang, J., Chen, X., Cai, Z., Yang, L., Yu, G., Lin, G., Zhang, C., 2024. MeshAnything: Artist-created mesh generation with autoregressive transformers. arXiv:2406.10163, 1–19

    work page arXiv 2024

  5. [5]

    HexGen and Hex2Spline: Polycube-based hexahedral mesh generation and spline modeling for isogeometric analysis applications in LS-DYNA

    CMU-CBML, 2024. HexGen and Hex2Spline: Polycube-based hexahedral mesh generation and spline modeling for isogeometric analysis applications in LS-DYNA. https://github.com/CMU-CBML/HexGen_Hex2Spline.git

    2024

  6. [6]

    Density estimation using Real NVP

    Dinh, L., Jascha, S.D., Bengio, S., 2016. Density estimation using Real NVP. arXiv:1605.08803, 1–32

    work page internal anchor Pith review arXiv 2016

  7. [7]

    Linearcomplexityhexahedralmeshgeneration

    Eppstein,D.,1996. Linearcomplexityhexahedralmeshgeneration. Proceedingsofthe12thAnnualSymposiumonComputationalGeometry, 58–67

    1996

  8. [8]

    All-hex meshing using closed-form induced polycube

    Fang, X., Xu, W., Bao, H., Huang, J., 2016. All-hex meshing using closed-form induced polycube. ACM Transactions on Graphics 35, 1–9

    2016

  9. [9]

    Locking-proof tetrahedra

    Frâncu, M., Asgeirsson, A., Erleben, K., Rønnow, M.J., 2021. Locking-proof tetrahedra. ACM Transactions on Graphics 40, 1–17

    2021

  10. [10]

    Efficient volumetric polycube-map construction

    Fu, X., Bai, C., Liu, Y., 2016. Efficient volumetric polycube-map construction. Computer Graphics Forum 35, 97–106

    2016

  11. [11]

    Learning deformable tetrahedral meshes for 3D reconstruction

    Gao, J., Chen, W., Xiang, T., Jacobson, A., McGuire, M., Fidler, S., 2020. Learning deformable tetrahedral meshes for 3D reconstruction. Advances in Neural Information Processing Systems 33, 9936–9947

    2020

  12. [12]

    Generative adversarial nets

    Goodfellow, I., Jean, P.A., Mirza, M., Xu, B., David, W.F., Ozair, S., Courville, A., Bengio, Y., 2014. Generative adversarial nets. Advances in Neural Information Processing Systems 28, 2672–2680

    2014

  13. [13]

    Cut-enhanced polycube-maps for feature-aware all-hex meshing

    Guo, H., Liu, X., Yan, D., Liu, Y., 2020. Cut-enhanced polycube-maps for feature-aware all-hex meshing. ACM Transactions on Graphics 39, 1–14

    2020

  14. [14]

    Denoising diffusion probabilistic models

    Ho, J., Jain, A., Abbeel, P., 2020. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems 33, 6840– 6851

    2020

  15. [15]

    Surface segmentation for polycube construction based on generalized centroidal Voronoi tessellation

    Hu, K., Zhang, Y., Liao, T., 2017. Surface segmentation for polycube construction based on generalized centroidal Voronoi tessellation. Computer Methods in Applied Mechanics and Engineering 316, 280–296

    2017

  16. [16]

    CentroidalVoronoitessellationbasedpolycubeconstructionforadaptiveall-hexahedralmeshgeneration

    Hu,K.,Zhang,Y.J.,2016. CentroidalVoronoitessellationbasedpolycubeconstructionforadaptiveall-hexahedralmeshgeneration. Computer Methods in Applied Mechanics and Engineering 305, 405–421

    2016

  17. [17]

    ACM Transactions on Graphics 33, 1–11

    Huang, J., Jiang, T., Shi, Z., Tong, Y., Bao, H., Desbrun, M., 2014.𝓁1-based construction of polycube maps from complex shapes. ACM Transactions on Graphics 33, 1–11

    2014

  18. [18]

    Isogeometricanalysis:CAD,finiteelements,NURBS,exactgeometry,andmeshrefinement

    Hughes,T.J.R.,Cottrell,J.A.,Bazilevs,Y.,2005. Isogeometricanalysis:CAD,finiteelements,NURBS,exactgeometry,andmeshrefinement. Computer Methods in Applied Mechanics and Engineering 194, 4135–4195

    2005

  19. [19]

    Deep unsupervised learning using nonequilibrium thermodynamics

    Jascha, S.D., Weiss, E., Maheswaranathan, N., Ganguli, S., 2015. Deep unsupervised learning using nonequilibrium thermodynamics. International Conference on Machine Learning, 2256–2265

    2015

  20. [20]

    A large language model and denoising diffusion framework for targeted design of microstructures with commands in natural language

    Kartashov, N., Vlassis, N.N., 2025. A large language model and denoising diffusion framework for targeted design of microstructures with commands in natural language. Computer Methods in Applied Mechanics and Engineering 437, 117742

    2025

  21. [21]

    Glow: Generative flow with invertible 1x1 convolutions

    Kingma, D.P., Dhariwal, P., 2018. Glow: Generative flow with invertible 1x1 convolutions. Advances in Neural Information Processing Systems 32, 10236–10245

    2018

  22. [22]

    Auto-Encoding Variational Bayes

    Kingma, D.P., Welling, M., 2013. Auto-encoding variational bayes. arXiv:1312.6114, 1–14

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  23. [23]

    Surface mesh to volumetric spline conversion with generalized polycubes

    Li, B., Li, X., Wang, K., Qin, H., 2013. Surface mesh to volumetric spline conversion with generalized polycubes. IEEE Transactions on Visualization and Computer Graphics 99, 1–14

    2013

  24. [24]

    Interactiveall-hexmeshingviacuboiddecomposition

    Li,L.,Zhang,P.,Smirnov,D.,Abulnaga,S.M.,Solomon,J.,2021. Interactiveall-hexmeshingviacuboiddecomposition. ACMTransactions on Graphics 40, 1–17

    2021

  25. [25]

    All-hexmeshingusingsingularity-restrictedfield

    Li,Y.,Liu,Y.,Xu,W.,Wang,W.,Guo,B.,2012. All-hexmeshingusingsingularity-restrictedfield. ACMTransactionsonGraphics31,1–11

    2012

  26. [26]

    Automatic polycube-maps

    Lin, J., Jin, X., Fan, Z., Wang, C., 2008. Automatic polycube-maps. Advances in Geometric Modeling and Processing 4975, 3–16

    2008

  27. [27]

    Intrinsic mixed-integer polycubes for hexahedral meshing

    Mandad, M., Chen, R., Bommes, D., Campen, M., 2022. Intrinsic mixed-integer polycubes for hexahedral meshing. Computer Aided Geometric Design 94, 102078

    2022

  28. [28]

    Improved denoising diffusion probabilistic models

    Nichol, A.Q., Dhariwal, P., 2021. Improved denoising diffusion probabilistic models. International Conference on Machine Learning, 8162– 8171

    2021

  29. [29]

    Cubecover - parameterization of 3D volumes

    Nieser, M., Reitebuch, U., Polthier, K., 2011. Cubecover - parameterization of 3D volumes. Computer Graphics Forum 30, 1397–1406

    2011

  30. [30]

    Hex-mesh generation and processing: A survey

    Pietroni, N., Campen, M., Sheffer, A., Cherchi, G., Bommes, D., Gao, X., Scateni, R., Ledoux, F., Remacle, J., Livesu, M., 2023. Hex-mesh generation and processing: A survey. ACM Transactions on Graphics 42, 1–44. :Preprint submitted to Elsevier Page 24 of 25 SDDPM-Polycube

    2023

  31. [31]

    International Journal for Numerical Methods in Engineering 82, 1406–1423

    Qian,J.,Zhang,Y.,Wang,W.,Lewis,A.C.,Qidwai,M.A.S.,Geltmacher,A.B.,2010.Qualityimprovementofnon-manifoldhexahedralmeshes for critical feature determination of microstructure materials. International Journal for Numerical Methods in Engineering 82, 1406–1423

    2010

  32. [32]

    Automatic unstructured all-hexahedral mesh generation from B-Reps for non-manifold CAD assemblies

    Qian, J., Zhang, Y.J., 2012. Automatic unstructured all-hexahedral mesh generation from B-Reps for non-manifold CAD assemblies. Engineering with Computers 28, 345–359

    2012

  33. [33]

    A grid-based algorithm for the generation of hexahedral element meshes

    Schneiders, R., 1996. A grid-based algorithm for the generation of hexahedral element meshes. Engineering with Computers 12, 168–177

    1996

  34. [34]

    MeshGPT: Generating triangle meshes with decoder-only transformers

    Siddiqui, Y., Alliegro, A., Artemov, A., Tommasi, T., Sirigatti, D., Rosov, V., Dai, A., Nießner, M., 2024. MeshGPT: Generating triangle meshes with decoder-only transformers. IEEE/CVF Conference on Computer Vision and Pattern Recognition, 19615–19625

    2024

  35. [35]

    Score-Based Generative Modeling through Stochastic Differential Equations

    Song,Y.,Jascha,S.D.,Kingma,D.P.,Kumar,A.,Ermon,S.,Poole,B.,2020. Score-basedgenerativemodelingthroughstochasticdifferential equations. arXiv:2011.13456, 1–36

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  36. [36]

    Sequence to sequence learning with neural networks

    Sutskever, I., Vinyals, O., Le, Q.V., 2014. Sequence to sequence learning with neural networks. Advances in Neural Information Processing Systems 27, 3104–3112

    2014

  37. [37]

    Polycube-maps

    Tarini, M., Hormann, K., Cignoni, P., Montani, C., 2004. Polycube-maps. ACM Transactions on Graphics 23, 853–860

    2004

  38. [38]

    HybridOctree_hex: Hybrid octree-based adaptive all-hexahedral mesh generation with Jacobian control

    Tong, H., Halilaj, E., Zhang, Y.J., 2024. HybridOctree_hex: Hybrid octree-based adaptive all-hexahedral mesh generation with Jacobian control. Journal of Computational Science 78, 102278

    2024

  39. [39]

    SRL-assisted AFM: Generating planar unstructured quadrilateral meshes with supervised and reinforcement learning-assisted advancing front method

    Tong, H., Qian, K., Halilaj, E., Zhang, Y.J., 2023. SRL-assisted AFM: Generating planar unstructured quadrilateral meshes with supervised and reinforcement learning-assisted advancing front method. Journal of Computational Science 72, 102109

    2023

  40. [40]

    Fast and robust hexahedral mesh optimization via augmented lagrangian, L-BFGS, and line search

    Tong, H., Zhang, Y.J., 2025. Fast and robust hexahedral mesh optimization via augmented lagrangian, L-BFGS, and line search. SIAM International Meshing Roundtable. Fort Worth, TX. March 3-6, 2025

    2025

  41. [41]

    HexOpt: Efficient and robust hexahedral mesh optimization using rectified hybrid quadratic Jacobian and geometry-aware mapping

    Tong, H., Zhang, Y.J., 2026. HexOpt: Efficient and robust hexahedral mesh optimization using rectified hybrid quadratic Jacobian and geometry-aware mapping. Computer-Aided Design, 104073

    2026

  42. [42]

    Polycube splines

    Wang, H., He, Y., Li, X., Gu, X., Qin, H., 2007. Polycube splines. Symposium on Solid and Physical Modeling, 241–251

    2007

  43. [43]

    Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology

    Wang, W., Zhang, Y., Liu, L., Hughes, T.J., 2013. Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology. Computer-Aided Design 45, 351–360

    2013

  44. [44]

    Truncated hierarchical tricubic C0 spline construction on unstructured hexahedral meshes for isogeometric analysis applications

    Wei, X., Zhang, Y., Hughes, T.J.R., 2017. Truncated hierarchical tricubic C0 spline construction on unstructured hexahedral meshes for isogeometric analysis applications. Computers and Mathematics with Applications 74, 2203–2220

    2017

  45. [45]

    DDPM-Polycube: A denoising diffusion probabilistic model for polycube-based hexahedral mesh generation and volumetric spline construction

    Yu, Y., Fang, Y., Tong, H., Liu, J., Zhang, Y.J., 2026. DDPM-Polycube: A denoising diffusion probabilistic model for polycube-based hexahedral mesh generation and volumetric spline construction. Engineering with Computers 42, 1

    2026

  46. [46]

    DL-Polycube:Deeplearningenhancedgeneralizedpolycubemethodforhigh-qualityhexahedral mesh generation and volumetric spline construction

    Yu,Y.,Fang,Y.,Tong,H.,Zhang,Y.J.,2025. DL-Polycube:Deeplearningenhancedgeneralizedpolycubemethodforhigh-qualityhexahedral mesh generation and volumetric spline construction. Engineering with Computers 41, 2069–2092

    2025

  47. [47]

    HexDom: Polycube-based hexahedral-dominant mesh generation

    Yu, Y., Liu, J., Zhang, Y.J., 2022. HexDom: Polycube-based hexahedral-dominant mesh generation. SEMA SIMAI Springer Series: Mesh Generation and Adaptation, 137–155

    2022

  48. [48]

    Geometric Challenges in Isogeometric Analysis

    Yu, Y., Wei, X., Li, A., Liu, J., He, J., Zhang, Y.J., 2022. HexGen and Hex2Spline: Polycube-based hexahedral mesh generation and spline modeling for isogeometric analysis applications in LS-DYNA. Springer INdAM Series: Proceedings of INdAM Workshop “Geometric Challenges in Isogeometric Analysis”, 333–363

    2022

  49. [49]

    Anatomically realistic lumen motion representation in patient-specific space–timeisogeometricflowanalysisofcoronaryarterieswithtime-dependentmedical-imagedata

    Yu, Y., Zhang, Y.J., Takizawa, K., Tezduyar, T.E., Sasaki, T., 2020. Anatomically realistic lumen motion representation in patient-specific space–timeisogeometricflowanalysisofcoronaryarterieswithtime-dependentmedical-imagedata. ComputationalMechanics65,395–404

    2020

  50. [50]

    Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow

    Zhang, Y., Bajaj, C.L., Xu, G., 2009. Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Communications in Numerical Methods in Engineering 25, 1–18

    2009

  51. [51]

    Solid T-spline construction from boundary representations for genus-zero geometry

    Zhang, Y., Wang, W., Hughes, T.J., 2012. Solid T-spline construction from boundary representations for genus-zero geometry. Computer Methods in Applied Mechanics and Engineering 249, 185–197

    2012

  52. [52]

    Challenges and advances in image-based geometric modeling and mesh generation

    Zhang, Y.J., 2013. Challenges and advances in image-based geometric modeling and mesh generation. Image-Based Geometric Modeling and Mesh Generation, 1–10

    2013

  53. [53]

    Geometric Modeling and Mesh Generation from Scanned Images

    Zhang, Y.J., 2016. Geometric Modeling and Mesh Generation from Scanned Images. Chapman and Hall/CRC, New York

    2016

  54. [54]

    Adaptive and quality quadrilateral/hexahedral meshing from volumetric data

    Zhang, Y.J., Bajaj, C.L., 2006. Adaptive and quality quadrilateral/hexahedral meshing from volumetric data. Computer Methods in Applied Mechanics and Engineering 195, 942–960

    2006

  55. [55]

    Patient-specificvascularNURBSmodelingforisogeometricanalysis of blood flow

    Zhang,Y.J.,Bazilevs,Y.,Goswami,S.,Bajaj,C.L.,Hughes,T.J.R.,2007. Patient-specificvascularNURBSmodelingforisogeometricanalysis of blood flow. Computer Methods in Applied Mechanics and Engineering 196, 2943–2959. :Preprint submitted to Elsevier Page 25 of 25

    2007