Skip to main content

Advertisement

SpringerLink
Log in

Evaluating mesh quality with graph neural networks

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

The quality of the finite element mesh has a considerable effect on the efficiency and accuracy of computational fluid dynamics (CFD) simulations. To ensure the generated mesh is of good quality, many quality metrics have been proposed to assess the generated mesh, such as aspect ratio, skewness, Jacobian ratio, etc. Such metrics, however, are primarily employed to detect locally distorted mesh elements. There are still no justifiable thresholds for determining whether the generated mesh is of sufficient quality for simulation. Consequently, it is necessary for the professionals to assess the generated mesh afterward which is time-consuming and labor-intensive work. With the ability to learn features on the graph, the graph neural networks have been successfully applied in many application areas to reduce human-computer interaction. In this paper, we define mesh quality evaluation as a graph classification problem. We first propose a novel and sparse-implemented algorithm to transform the mesh data into graph data. We then introduce a deep graph neural network, GMeshNet, to evaluate the mesh quality. Experimental results on the NACA-Market and NACA6510 mesh datasets demonstrate the effectiveness of our proposed network.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Spalart P, Venkatakrishnan V (2016) On the role and challenges of CFD in the aerospace industry. Aeronaut J 120(1223):209–232. https://doi.org/10.1017/aer.2015.10

    Article  Google Scholar 

  2. Watanabe N, Miyamoto S, Kuba M et al (2003) The CFD application for efficient designing in the automotive engineering. SAE Trans 1476–1482. https://doi.org/10.4271/2003-01-1335

  3. Das S, Paul S, Doloi B (2019) Application of cfd and vapor bubble dynamics for an efficient electro-thermal simulation of edm: an integrated approach. Int J Adv Manuf Technol 102. https://doi.org/10.1007/s00170-018-3144-x

  4. Wan S, Ang Y, Sato T et al (2013) Process modeling and cfd simulation of two-way abrasive flow machining. Int J Adv Manuf Technol 71:1077–1086. https://doi.org/10.1007/s00170-013-5550-4

  5. Ho-Le K (1988) Finite element mesh generation methods: a review and classification. Comput Aided Des 20(1):27–38. https://doi.org/10.1016/0010-4485(88)90138-8

    Article  MATH  Google Scholar 

  6. Gammon M (2018) A review of common geometry issues affecting mesh generation. In: 2018 AIAA Aerospace Sciences Meeting, p 1402. https://doi.org/10.2514/6.2018-1402

  7. Li H et al (2012) Finite element mesh generation and decision criteria of mesh quality. China Mech Eng 23(3):368

    Google Scholar 

  8. Knupp P (2007) Remarks on mesh quality. Tech. rep., Sandia National Lab.(SNL-NM), Albuquerque, NM (United States)

  9. Guo Y, Liu Y, Oerlemans A et al (2016) Deep learning for visual understanding: a review. Neurocomputing 187:27–48. https://doi.org/10.1016/j.neucom.2015.09.116

    Article  Google Scholar 

  10. Wu Z, Pan S, Chen F et al (2020) A comprehensive survey on graph neural networks. IEEE Trans Neural Netw Learn Syst 32(1):4–24. https://doi.org/10.1109/TNNLS.2020.2978386

    Article  MathSciNet  Google Scholar 

  11. Chen X, Liu J, Pang Y et al (2020) Developing a new mesh quality evaluation method based on convolutional neural network. Eng Appl Comput Fluid Mech 14(1):391–400. https://doi.org/10.1080/19942060.2020.1720820

    Article  Google Scholar 

  12. Sarrate J, Palau J, Huerta A (2003) Numerical representation of the quality measures of triangles and triangular meshes. Commun Numer Methods Eng 19(7):551–561. https://doi.org/10.1002/cnm.585

    Article  MathSciNet  MATH  Google Scholar 

  13. Nie C, Liu J, Sun S (2003) Study on quality measures for tetrahedral mesh. Chin J Comput Mech 20(5):579–582

    Google Scholar 

  14. Kwok W, Chen Z (2000) A simple and effective mesh quality metric for hexahedral and wedge elements. In: IMR, pp 325–333

  15. Knupp PM (2001) Algebraic mesh quality metrics. SIAM J Sci Comput 23(1):193–218. https://doi.org/10.1137/S1064827500371499

    Article  MathSciNet  MATH  Google Scholar 

  16. Knupp PM (2003) Algebraic mesh quality metrics for unstructured initial meshes. Finite Elem Anal Des 39(3):217–241. https://doi.org/10.1016/S0168-874X(02)00070-7

    Article  MATH  Google Scholar 

  17. Chauhan VK, Dahiya K, Sharma A (2019) Problem formulations and solvers in linear SVM: a review. Artif Intell Rev 52(2):803–855. https://doi.org/10.1007/s10462-018-9614-6

    Article  Google Scholar 

  18. Chetouani A (2017) A 3D mesh quality metric based on features fusion. Electron Imaging 2017(20):4–8. https://doi.org/10.2352/ISSN.2470-1173.2017.20.3DIPM-001

    Article  Google Scholar 

  19. Sprave J, Drescher C (2021) Evaluating the quality of finite element meshes with machine learning. arXiv:2107.10507

  20. Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. Adv Neural Inf Process Syst 25:1097–1105. https://doi.org/10.1145/3065386

    Article  Google Scholar 

  21. Wang J, Zheng T, Lei P et al (2019) A hierarchical convolution neural network CNN-based ship target detection method in spaceborne sar imagery. Remote Sens 11(6):620. https://doi.org/10.3390/rs11060620

    Article  Google Scholar 

  22. Jalilian E, Uhl A, Kwitt R (2017) Domain adaptation for cnn based iris segmentation. In: 2017 International Conference of the Biometrics Special Interest Group (BIOSIG), pp 1–6. https://doi.org/10.23919/BIOSIG.2017.8053502

  23. Chen X, Liu J, Gong C et al (2021) MVE-Net: An automatic 3-d structured mesh validity evaluation framework using deep neural networks. Comput Aided Des 141(103):104. https://doi.org/10.1016/j.cad.2021.103104

    Article  MathSciNet  Google Scholar 

  24. Yu B, Yin H, Zhu Z (2018) Spatio-temporal graph convolutional networks: a deep learning framework for traffic forecasting. In: Proceedings of the 27th International Joint Conference on Artificial Intelligence, pp 3634–3640. https://doi.org/10.24963/ijcai.2018/505

  25. Monti F, Bronstein MM, Bresson X (2017) Geometric matrix completion with recurrent multi-graph neural networks. In: Proceedings of the 31st International Conference on Neural Information Processing Systems, pp 3700–3710

  26. Gilmer J, Schoenholz SS, Riley PF et al (2017) Neural message passing for quantum chemistry. In: International conference on machine learning, PMLR, pp 1263–1272

  27. Qiu J, Tang J, Ma H et al (2018) Deepinf: Social influence prediction with deep learning. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp 2110–2119. https://doi.org/10.1145/3219819.3220077

  28. Kipf TN, Welling M (2016) Semi-supervised classification with graph convolutional networks. https://doi.org/10.1145/3459637.3482477. arXiv preprint arXiv:1609.02907

  29. Veličković P, Cucurull G, Casanova A et al (2017) Graph attention networks. arXiv preprint arXiv:1710.10903

  30. Ying R, You J, Morris C et al (2018) Hierarchical graph representation learning with differentiable pooling. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp 4805–4815

  31. Cangea C, Veličković P, Jovanović N, et al (2018) Towards sparse hierarchical graph classifiers. arXiv preprint arXiv:1811.01287

  32. Li Q, Han Z, Wu XM (2018) Deeper insights into graph convolutional networks for semi-supervised learning. In: Thirty-Second AAAI conference on artificial intelligence

  33. Li G, Muller M, Thabet A, et al (2019) DeepGCNs: Can GCNs go as deep as CNNs? In: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp 9267–9276, https://doi.org/10.1109/ICCV.2019.00936

  34. Li G, Xiong C, Thabet A et al (2020) Deepergcn: All you need to train deeper gcns. arXiv:2006.07739

  35. Ba JL, Kiros JR, Hinton GE (2016) Layer normalization. arXiv:1607.06450

  36. Lee J, Lee I, Kang J (2019) Self-attention graph pooling. In: International Conference on Machine Learning, PMLR, pp 3734–3743

  37. Xu K, Li C, Tian Y et al (2018) Representation learning on graphs with jumping knowledge networks. In: International Conference on Machine Learning, PMLR, pp 5453–5462

  38. Ioffe S, Szegedy C (2015) Batch normalization: Accelerating deep network training by reducing internal covariate shift. In: Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML’15, pp 448–456

  39. Reddi SJ, Kale S, Kumar S (2019) On the convergence of adam and beyond. arXiv:1904.09237

  40. Hamilton WL, Ying R, Leskovec J (2017) Inductive representation learning on large graphs. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. Curran Associates Inc., Red Hook, NY, USA, NIPS’17, pp 1025-1035

Download references

Funding

This research work was supported in part by the National Numerical Windtunnel project (NNW2019ZT5-A10) and National Key Research and Development Program of China.

Author information

Author notes

  1. Zhichao Wang and Xinhai Chen contributed equally to this work.

Authors and Affiliations

  1. Science and Technology on Parallel and Distributed Processing Laboratory, National University of Defense Technology, Changsha, 410073, China

    Zhichao Wang, Xinhai Chen, Chunye Gong & Jie Liu

  2. Laboratory of Software Engineering for Complex System, National University of Defense Technology, Changsha, 410073, China

    Zhichao Wang, Xinhai Chen, Tieju Li, Chunye Gong & Jie Liu

  3. China Aerodynamics Research and Development Center, Mianyang, 621000, China

    Yufei Pang

Authors
  1. Zhichao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinhai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tieju Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chunye Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yufei Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jie Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Data collection and analysis, network design, experiment and result analysis were performed by ZW and XC. The first draft of the manuscript was written by ZW and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jie Liu.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Chen, X., Li, T. et al. Evaluating mesh quality with graph neural networks. Engineering with Computers 38, 4663–4673 (2022). https://doi.org/10.1007/s00366-022-01720-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-022-01720-8

Keywords

Search

Navigation