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MGNet: a novel differential mesh generation method based on unsupervised neural networks

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Abstract

Mesh generation accounts for a large number of workloads in the numerical analysis. In this paper, we introduce a novel differential method MGNet for structured mesh generation. The proposed method poses the meshing task as an optimization problem. It takes boundary curves as input, employs a well-designed neural network to study the potential meshing (mapping) rules, and finally outputs the mesh with a desired number of cells. The whole process is unsupervised and does not require a priori knowledge or measured datasets. We evaluate the performance of MGNet in terms of mesh quality, network designs, robustness, and overhead on different geometries and governing equations (elliptic and hyperbolic). The experimental results prove that, in all cases, the proposed method is capable of generating acceptable meshes and achieving comparable or superior meshing performance to the traditional algebraic and differential methods. The proposed MGNet also outperforms other neural network-based solvers and enables fast mesh generation using feedforward prediction techniques.

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References

  1. Chang-Hoi A, Sang-Soo L, Hyuek-Jae L, Soo-Young L (1991) A self-organizing neural network approach for automatic mesh generation. IEEE Trans Magn 27(5):4201–4204. https://doi.org/10.1109/20.105028

    Article  Google Scholar 

  2. Chen X, Liu J, Li S, Xie P, Chi L, Wang Q (2018) Tamm: a new topology-aware mapping method for parallel applications on the tianhe-2a supercomputer. In: Vaidya J, Li J (eds) Algorithms and architectures for parallel processing. Springer, Cham, pp 242–256

    Chapter  Google Scholar 

  3. Karman SL, Anderson WK, Sahasrabudhe M (2006) Mesh generation using unstructured computational meshes and elliptic partial differential equation smoothing. AIAA J 44(6):1277–1286. https://doi.org/10.2514/1.15929

    Article  Google Scholar 

  4. Pochet A, Celes W, Lopes H, Gattass M (2017) A new quadtree-based approach for automatic quadrilateral mesh generation. Eng Comput 33(2):275–292. https://doi.org/10.1007/s00366-016-0471-0

    Article  Google Scholar 

  5. Liseikin VD (2004) A computational differential geometry approach to grid generation. Springer, Berlin

    Book  MATH  Google Scholar 

  6. Lu F, Qi L, Jiang X, Liu G, Liu Y, Chen B, Pang Y, Hu X (2020) Nnw-gridstar: interactive structured mesh generation software for aircrafts. Adv Eng Softw 145:102803. https://doi.org/10.1016/j.advengsoft.2020.102803

    Article  Google Scholar 

  7. Shragge J (2006) Differential mesh generation. SEG Tech Progr Expand Abstr 10(1190/1):2369975

    Google Scholar 

  8. Haynes RD, Howse AJM (2015) Alternating Schwarz methods for partial differential equation-based mesh generation. Int J Comput Math 92(2):349–376. https://doi.org/10.1080/00207160.2014.891733

    Article  MathSciNet  MATH  Google Scholar 

  9. Brink AR, Najera-Flores DA, Martinez C (2020) The neural network collocation method for solving partial differential equations. Neural Comput Appl. https://doi.org/10.1016/j.jcp.2021.110364

    Article  Google Scholar 

  10. Papagiannopoulos A, Clausen P, Avellan F (2021) How to teach neural networks to mesh: application on 2-d simplicial contours. Neural Netw 136:152–179. https://doi.org/10.1016/j.neunet.2020.12.019

    Article  Google Scholar 

  11. Lu X, Tian B (2010) Research on mesh generation in the finite element numerical analysis based on radial basis function neural network. In: 2010 international conference on computer application and system modeling (ICCASM ), vol 11, pp 157–160. 10.1109/ICCASM.2010.5623240

  12. Chen X, Liu J, Gong C, Li S, Pang Y, Chen B (2021) Mve-net: an automatic 3-d structured mesh validity evaluation framework using deep neural networks. Comput Aided Des 141:103104. https://doi.org/10.1016/j.cad.2021.103104

    Article  MathSciNet  Google Scholar 

  13. Chen X, Liu J, Pang Y, Chen J, Chi L, Gong C (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 

  14. Chen X, Chen R, Wan Q, Xu R, Liu J (2021) An improved data-free surrogate model for solving partial differential equations using deep neural networks. Sci Rep. https://doi.org/10.1038/s41598-021-99037-x

    Article  Google Scholar 

  15. Jin X, Cai S, Li H, Karniadakis GE (2021) NSFnets (Navier–Stokes flow nets): physics-informed neural networks for the incompressible Navier–Stokes equations. J Comput Phys 426:109951. https://doi.org/10.1016/j.jcp.2020.109951

    Article  MathSciNet  MATH  Google Scholar 

  16. Liang D, Wang Y, Chen C, Liu Y, Wang F (2020) A clustering-based approach to vortex extraction. J Vis. https://doi.org/10.1007/s12650-020-00636-z

    Article  Google Scholar 

  17. Alfonzetti S, Coco S, Cavalieri S, Malgeri M (1996) Automatic mesh generation by the let-it-grow neural network. IEEE Trans Magn 32(3):1349–1352. https://doi.org/10.1109/20.497496

    Article  Google Scholar 

  18. Ahmet Ç, Ahmet A (2002) Neural networks based mesh generation method in 2-d. In: Shafazand H, Tjoa AM (eds) EurAsia-ICT 2002: information and communication technology. Springer, Berlin, Heidelberg, pp 395–401

  19. Zhang Z, Wang Y, Jimack PK, Wang H (2020) Meshingnet: A new mesh generation method based on deep learning. In: Krzhizhanovskaya VV, Závodszky G, Lees MH, Dongarra JJ, Sloot PMA, Brissos S, Teixeira J (eds) Computational Science—ICCS 2020. Springer, Cham, pp 186–198

  20. Huang K, Krügener M, Brown A, Menhorn F, Bungartz H-J, Hartmann D (2021) Machine learning-based optimal mesh generation in computational fluid dynamics .https://arxiv.org/abs/2102.12923

  21. Geuzaine C, Remacle J.-F(2009) Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities.International Journal for Numerical Methods in Engineering 79(11), 1309–1331

  22. Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707. https://doi.org/10.1016/j.jcp.2018.10.045

    Article  MathSciNet  MATH  Google Scholar 

  23. Dwivedi V, Srinivasan B (2020) Physics informed extreme learning machine (PIELM)—a rapid method for the numerical solution of partial differential equations. Neurocomputing 391:96–118. https://doi.org/10.1016/j.neucom.2019.12.099

    Article  Google Scholar 

  24. Arthurs CJ, King AP (2021) Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations. J Comput Phys. https://doi.org/10.1016/j.jcp.2021.110364

    Article  MathSciNet  MATH  Google Scholar 

  25. Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay E, Louppe G (2012) Scikit-learn: machine learning in python. J Mach Learn Res 12:2825–2830

    MathSciNet  MATH  Google Scholar 

  26. Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M, Kudlur M, Levenberg J, Monga R, Moore S, Murray D, Steiner B, Tucker P, Vasudevan V, Warden P, Zhang X (2016) TensorFlow: a system for large-scale machine learning. In: Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation.OSDI’16, pp. 265–283. USENIX Association, USA

  27. Morales J, Nocedal J (2011) Remark on algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. ACM Trans Math Softw 38:7. https://doi.org/10.1145/2049662.2049669

    Article  MATH  Google Scholar 

  28. Raissi M, Yazdani A, Karniadakis GE (2020) Hidden fluid mechanics: learning velocity and pressure fields from flow visualizations. Science 367(6481):1026–1030. https://doi.org/10.1126/science.aaw4741

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang S, Teng Y, Perdikaris P(2020) Understanding and mitigating gradient pathologies in physics-informed neural networks.https://arxiv.org/abs/2001.04536

  30. Ramachandran P, Zoph B, Le QV (2017) Searching for activation functions. https://arxiv.org/abs/1710.05941

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Acknowledgements

This research work was supported in part by the National Key Research and Development Program of China (2017YFB0202104, 2018YFB0204301).

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Author notes

  1. Xinhai Chen and Qian Wan have contributed equally to this work.

Authors and Affiliations

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

    Xinhai Chen, Xiaoyu He & Yufei Pang

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

    Xinhai Chen, Tiejun Li, Qian Wan, Chunye Gong & Jie Liu

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  1. Xinhai Chen
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  2. Tiejun Li
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  3. Qian Wan
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  4. Xiaoyu He
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Correspondence to Tiejun Li.

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Chen, X., Li, T., Wan, Q. et al. MGNet: a novel differential mesh generation method based on unsupervised neural networks. Engineering with Computers 38, 4409–4421 (2022). https://doi.org/10.1007/s00366-022-01632-7

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