Abstract: In computational fluid dynamics (CFD), mesh-smoothing methods are widely used to refine the mesh quality for achieving high-precision numerical simulations. Specifically, optimization-based smoothing is used for high-quality mesh smoothing, but it incurs significant computational overhead. Pioneer works have improved its smoothing efficiency by adopting supervised learning to learn smoothing methods from high-quality meshes. However, they pose difficulties in smoothing the mesh nodes with varying degrees and require data augmentation to address the node input sequence problem. Additionally, the required labeled high-quality meshes further limit the applicability of the proposed method. In this paper, we present graph-based smoothing mesh net (GMSNet), a lightweight neural network model for intelligent mesh smoothing. GMSNet adopts graph neural networks (GNNs) to extract features of the node’s neighbors and outputs the optimal node position. During smoothing, we also introduce a fault-tolerance mechanism to prevent GMSNet from generating negative volume elements. With a lightweight model, GMSNet can effectively smooth mesh nodes with varying degrees and remain unaffected by the order of input data. A novel loss function, MetricLoss, is developed to eliminate the need for high-quality meshes, which provides stable and rapid convergence during training. We compare GMSNet with commonly used mesh-smoothing methods on two-dimensional (2D) triangle meshes. Experimental results show that GMSNet achieves outstanding mesh-smoothing performances with 5% of the model parameters compared to the previous model, but offers a speedup of 13.56 times over the optimization-based smoothing.

基于图神经网络的网格平滑方法研究

[1]Baker TJ, 2005. Mesh generation: art or science? Prog Aerosp Sci, 41(1):29-63.

[2]Bohn J, Feischl M, 2021. Recurrent neural networks as optimal mesh refinement strategies. Comput Math Appl, 97:61-76.

[3]Bridgeman J, Jefferson B, Parsons SA, 2010. The development and application of CFD models for water treatment flocculators. Adv Eng Softw, 41(1):99-109.

[4]Cai TL, Luo SJ, Xu KYL, et al., 2021. GraphNorm: a principled approach to accelerating graph neural network training. Proc 38 th Int Conf on Machine Learning, p.1204-1215.

[5]Canann SA, Tristano JR, Staten ML, 1998. An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes. Proc 7 th Int Meshing Roundtable, p.479-494.

[6]Chen L, 2004. Mesh smoothing schemes based on optimal Delaunay triangulations. Proc IMR, p.109-120.

[7]Chen XH, Liu J, Pang YF, et al., 2020. Developing a new mesh quality evaluation method based on convolutional neural network. Eng Appl Comput Fluid Mech, 14(1):391-400.

[8]Chen XH, Liu J, Gong CY, et al., 2021. MVE-Net: an automatic 3-D structured mesh validity evaluation framework using deep neural networks. Comput Aided Des, 141:103104.

[9]Chen XH, Li TJ, Wan Q, et al., 2022. MGNet: a novel differential mesh generation method based on unsupervised neural networks. Eng Comput, 38(5):4409-4421.

[10]Chowdhary KR, 2020. Natural language processing. In: Chowdhary KR (Ed.), Fundamentals of Artificial Intelligence. Springer, New Delhi, India, p.603-649.

[11]Damjanović D, Kozak D, Živić M, et al., 2011. CFD analysis of concept car in order to improve aerodynamics. A Jövő Járműve, 1(2):67-70.

[12]Daroya R, Atienza R, Cajote R, 2020. REIN: flexible mesh generation from point clouds. Proc IEEE/CVF Conf on Computer Vision and Pattern Recognition Workshops, p.1444-1453.

[13]Du Q, Gunzburger M, 2002. Grid generation and optimization based on centroidal Voronoi tessellations. Appl Math Comput, 133(2-3):591-607.

[14]Du Q, Wang DS, 2003. Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations. Int J Numer Methods Eng, 56(9):1355-1373.

[15]Du Q, Faber V, Gunzburger M, 1999. Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev, 41(4):637-676.

[16]Durand R, Pantoja-Rosero BG, Oliveira V, 2019. A general mesh smoothing method for finite elements. Finite Elem Anal Des, 158:17-30.

[17]Escobar JM, Montenegro R, Montero G, et al., 2005. Smoothing and local refinement techniques for improving tetrahedral mesh quality. Comput Struct, 83(28-30):2423-2430.

[18]Fidkowski KJ, Chen GD, 2021. Metric-based, goal-oriented mesh adaptation using machine learning. J Comput Phys, 426:109957.

[19]Field DA, 1988. Laplacian smoothing and Delaunay triangulations. Commun Appl Numer Methods, 4(6):709-712.

[20]Freitag LA, Knupp PM, 2002. Tetrahedral mesh improvement via optimization of the element condition number. Int J Numer Methods Eng, 53(6):1377-1391.

[21]Freitag LA, Ollivier-Gooch C, 1997. Tetrahedral mesh improvement using swapping and smoothing. Int J Numer Methods Eng, 40(21):3979-4002. https://doi.org/10.1002/(SICI)1097-0207(19971115)40:21<3979::AID-NME251>3.0.CO;2-9

[22]Guo YF, Hai YQ, 2021. Adaptive surface mesh remeshing based on a sphere packing method and a node insertion/deletion method. Appl Math Model, 98:1-13.

[23]Guo YF, Wang L, Zhao K, et al., 2020. An angle-based smoothing method for triangular and tetrahedral meshes. Proc 15^th Chinese Conf on Image and Graphics Technologies and Applications, p.224-234.

[24]Guo YF, Wang CR, Ma Z, et al., 2022. A new mesh smoothing method based on a neural network. Comput Mech, 69(2):425-438.

[25]Hai YQ, Guo YF, Cheng SY, et al., 2021. Regular position-oriented method for mesh smoothing. Acta Mech Soli Sin, 34(3):437-448.

[26]Han X, Gao H, Pffaf T, et al., 2022. Predicting physics in mesh-reduced space with temporal attention. Proc Int Conf on Learning Representations.

[27]Herrmann LR, 1976. Laplacian-isoparametric grid generation scheme. J Eng Mech Div, 102(5):749-756.

[28]Kingma DP, Ba J, 2015. Adam: a method for stochastic optimization. Proc 3rd Int Conf on Learning Representations.

[29]Kipf TN, Welling M, 2016. Semi-supervised classification with graph convolutional networks.

[30]Klingner BM, Shewchuk JR, 2008. Aggressive tetrahedral mesh improvement. In: Brewer ML, Marcum D (Eds.), Proc 16th International Meshing Roundtable. Springer, Berlin, Germany, p.3-23.

[31]Knupp PM, 2001. Algebraic mesh quality metrics. SIAM J Sci Comput, 23(1):193-218.

[32]LeCun Y, Bengio Y, Hinton G, 2015. Deep learning. Nature, 521(7553):436-444.

[33]Lee DT, Schachter BJ, 1980. Two algorithms for constructing a Delaunay triangulation. Int J Comput Inform Sci, 9(3):219-242.

[34]Li GH, Müller M, Thabet A, et al., 2019. DeepGCNs: can GCNs go as deep as CNNs? Proc IEEE/CVF Int Conf on Computer Vision, p.9266-9275.

[35]Liaw R, Liang E, Nishihara R, et al., 2018. Tune: a research platform for distributed model selection and training.

[36]Lino M, Cantwell C, Bharath AA, et al., 2021. Simulating continuum mechanics with multi-scale graph neural networks.

[37]Liu Y, Wang WP, Lévy B, et al., 2009. On centroidal Voronoi tessellation—energy smoothness and fast computation. ACM Trans Graph, 28(4):101.

[38]Lloyd S, 1982. Least squares quantization in PCM. IEEE Trans Inform Theory, 28(2):129-137.

[39]Lopes NP, 2023. Torchy: a tracing JIT compiler for PyTorch. Proc 32nd ACM SIGPLAN Int Conf on Compiler Construction, p.98-109.

[40]Moukalled F, Mangani L, Darwish M, 2016. The Finite Volume Method in Computational Fluid Dynamics: an Advanced Introduction with OpenFOAM^® and Matlab. Springer, Cham, Germany, p.110-128.

[41]Pak M, Kim S, 2017. A review of deep learning in image recognition. Proc 4th Int Conf on Computer Applications and Information Processing Technology, p.1-3.

[42]Pan W, Lu XQ, Gong YH, et al., 2020. HLO: half-kernel Laplacian operator for surface smoothing. Comput Aided Des, 121:102807.

[43]Papagiannopoulos A, Clausen P, Avellan F, 2021. How to teach neural networks to mesh: application on 2-D simplicial contours. Neur Netw, 136:152-179.

[44]Parthasarathy VN, Kodiyalam S, 1991. A constrained optimization approach to finite element mesh smoothing. Finite Elem Anal Des, 9(4):309-320.

[45]Paszyński M, Grzeszczuk R, Pardo D, et al., 2021. Deep learning driven self-adaptive hp finite element method. Proc 21st Int Conf on Computational Science, p.114-121.

[46]Peng WH, Yuan ZL, Wang JC, 2022. Attention-enhanced neural network models for turbulence simulation. Phys Fluids, 34(2):025111.

[47]Pfaff T, Fortunato M, Sanchez-Gonzalez A, et al., 2021. Learning mesh-based simulation with graph networks.

[48]Prasad T, 2018. A Comparative Study of Mesh Smoothing Methods with Flipping in 2D and 3D. MS Thesis, The State University of New Jersey, New Jersey, USA.

[49]Reddy DR, 1976. Speech recognition by machine: a review. Proc IEEE, 64(4):501-531.

[50]Samstag RW, Ducoste JJ, Griborio A, et al., 2016. CFD for wastewater treatment: an overview. Water Sci Technol, 74(3):549-563.

[51]Sharp N, Crane K, 2020. A Laplacian for nonmanifold triangle meshes. Comput Graph Forum, 39(5):69-80.

[52]Song WB, Zhang MR, Wallwork JG, et al., 2022. M2N: mesh movement networks for PDE solvers. Proc 36th Conf on Neur Information Processing Systems, p.7199-7210.

[53]Spalart PR, Venkatakrishnan V, 2016. On the role and challenges of CFD in the aerospace industry. Aeronaut J, 120(1223):209-232.

[54]Ulyanov D, Vedaldi A, Lempitsky V, 2017. Instance normalization: the missing ingredient for fast stylization.

[55]Vartziotis D, Papadrakakis M, 2017. Improved GETMe by adaptive mesh smoothing. Comput Assisted Methods Eng Sci, 20(1):55-71

[56]Vartziotis D, Wipper J, 2019. The GETMe Mesh Smoothing Framework: a Geometric Way to Quality Finite Element Meshes. Taylor & Francis Group, Boca Raton, USA.

[57]Vartziotis D, Athanasiadis T, Goudas I, et al., 2008. Mesh smoothing using the geometric element transformation method. Comput Methods Appl Mech Eng, 197(45-48):3760-3767.

[58]Vollmer J, Mencl R, Müller H, 1999. Improved Laplacian smoothing of noisy surface meshes. Comput Graph Forum, 18(3):131-138.

[59]Wallwork JG, Lu JY, Zhang MR, et al., 2022. E2N: error estimation networks for goal-oriented mesh adaptation.

[60]Wang ZH, Chen XH, Li TJ, et al., 2022. Evaluating mesh quality with graph neural networks. Eng Comput, 38(5):4663-4673.

[61]Wu TF, Liu XJ, An W, et al., 2022. A mesh optimization method using machine learning technique and variational mesh adaptation. Chin J Aeronaut, 35(3):27-41.

[62]Wu ZH, Pan SR, Chen FW, et al., 2021. A comprehensive survey on graph neural networks. IEEE Trans Neur Netw Learn Syst, 32(1):4-24.

[63]Xu KJ, Gao XF, Chen GN, 2017. Hexahedral mesh quality improvement via edge-angle optimization. Comput Graphics, 70:17-27.

[64]Zhang YJ, Bajaj C, Xu GL, 2009. Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Commun Numer Methods Eng, 25(1):1-18.

[65]Zhang ZY, Wang YX, Jimack PK, et al., 2020. MeshingNet: a new mesh generation method based on deep learning. Proc 20th Int Conf on Computational Science, p.186-198.

[66]Zhang ZY, Jimack PK, Wang H, 2021. MeshingNet3D: efficient generation of adapted tetrahedral meshes for computational mechanics. Adv Eng Softw, 157-158:103021.

[67]Zhou T, Shimada K, 2000. An angle-based approach to two-dimensional mesh smoothing. Proc IMR, p.373-384.