On Generation of Node Distributions for Meshless PDE Discretizations
Abstract
In this paper we present an algorithm that is able to generate locally regular node layouts with spatially variable nodal density for interiors of arbitrary domains in two, three, and higher dimensions. It is demonstrated that the generated node distributions are suitable to use in the RBF-FD method, which is demonstrated by solving thermo-fluid problem in two and three dimensions. Additionally, local minimal spacing guarantees are proven for both uniform and variable nodal densities. The presented algorithm has time complexity $O(N)$ to generate $N$ nodes with constant nodal spacing and $O(N \log N)$ to generate variably spaced nodes. Comparison with existing algorithms is performed in terms of node quality, time complexity, execution time, and PDE solution accuracy.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
M. Balzer, T. Schlömer, and O. Deussen, Capacity-constrained point distributions: A variant of Lloyd's method, ACM Trans. Graphics, 28 (2009), https://doi.org/10.1145/1531326.1531392.
2.
S. Bauer, Image number K7245-1, United States Department of Agriculture, https://www.ars.usda.gov/oc/images/photos/k7245-1/.
3.
V. Bayona, N. Flyer, B. Fornberg, and G. A. Barnett, On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys., 332 (2017), pp. 257--273, https://doi.org/10.1016/j.jcp.2016.12.008.
4.
J. L. Blanco and P. K. Rai, Nanoflann: A C++ header-only Fork of FLANN, a Library for Nearest Neighbor (NN) with KD-trees, 2014, https://github.com/jlblancoc/nanoflann.
5.
R. Bridson, Fast Poisson Disk Sampling in Arbitrary Dimensions, in Proceedings of SIGGRAPH sketches, 2007, p. 22, https://doi.org/10.1145/1278780.1278807.
6.
Y. Choi and S. Kim, Node generation scheme for meshfree method by Voronoi diagram and weighted bubble packing, in Proceedings of the Fifth U.S. National Congress on Computational Mechanics, Boulder, CO, 1999.
7.
R. L. Cook, Stochastic sampling in computer graphics, ACM Trans. Graphics, 5 (1986), pp. 51--72, https://doi.org/10.1145/7529.8927.
8.
H. Couturier and S. Sadat, Performance and accuracy of a meshless method for laminar natural convection, Numer. Heat Trans. B, 37 (2000), pp. 455--467, https://doi.org/10.1080/10407790050051146.
9.
G. de Vahl Davis, Natural convection of air in a square cavity: A bench mark numerical solution, Internat. J. Numer. Methods Fluids, 3 (1983), pp. 249--264, https://doi.org/10.1002/fld.1650030305.
10.
D. Dimitrov, C. Knauer, K. Kriegel, and G. Rote, On the bounding boxes obtained by principal component analysis, in Proceedings of the 22nd European Workshop on Computational Geometry, 2006, pp. 193--196.
11.
B. Fornberg and N. Flyer, Fast generation of 2-D node distributions for mesh-free PDE discretizations, Comput. Math. Appl., 69 (2015), pp. 531--544, https://doi.org/10.1016/j.camwa.2015.01.009.
12.
B. Fornberg and N. Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences, SIAM, Philadelphia, 2015, https://doi.org/10.1137/1.9781611974041.
13.
B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), pp. 215--258, https://doi.org/10.1017/S0962492914000130.
14.
T. Fusegi, J. M. Hyun, K. Kuwahara, and B. Farouk, A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure, Int. J. Heat Mass Transf., 34 (1991), pp. 1543--1557, https://doi.org/10.1016/0017-9310(91)90295-p.
15.
A. Golbabai and E. Mohebianfar, A new method for evaluating options based on multiquadric RBF-FD method, Appl. Math. Comput., 308 (2017), pp. 130--141, https://doi.org/10.1016/j.amc.2017.03.019.
16.
G. Guennebaud, B. Jacob, et al., Eigen v3, 2010, http://eigen.tuxfamily.org.
17.
D. P. Hardin and E. B. Saff, Discretizing manifolds via minimum energy points, Not. AMS, 51 (2004), pp. 1186--1194.
18.
I. Jolliffe, Principal Component Analysis, 2nd ed., Springer Ser. Statist., Springer, New York, 2011, https://doi.org/10.1007/978-3-642-04898-2_455.
19.
G. Kosec, A local numerical solution of a fluid-flow problem on an irregular domain, Adv. Eng. Softw., 120 (2018), pp. 36--44, https://doi.org/10.1016/j.advengsoft.2016.05.010.
20.
G. Kosec and B. Šarler, Solution of thermo-fluid problems by collocation with local pressure correction, Internat. J. Numer. Methods Heat Fluid Flow, 18 (2008), pp. 868--882, https://doi.org/10.1108/09615530810898999.
21.
G. Kosec and J. Slak, RBF-FD based dynamic thermal rating of overhead power lines, in Advances in Fluid Mechanics XII, WIT Trans. Eng. Sci. 120, WIT Press, Southampton, 2018, pp. 255--262, https://doi.org/10.2495/afm180261.
22.
X.-Y. Li, S.-H. Teng, and A. Ungor, Point placement for meshless methods using sphere packing and advancing front methods, in Proceedings of ICCES'00, Los Angeles, CA, 2000.
23.
G.-R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, FL, 2002, https://doi.org/10.1201/9781420040586.
24.
Y. Liu, Y. Nie, W. Zhang, and L. Wang, Node placement method by bubble simulation and its application, CMES Comput. Model. Eng. Sci., 55 (2010), pp. 89--110, https://doi.org/10.3970/cmes.2010.055.089.
25.
R. Löhner and E. On͂ate, A general advancing front technique for filling space with arbitrary objects, Internat. J. Numer. Methods Engrg., 61 (2004), pp. 1977--1991, https://doi.org/10.1002/nme.1068.
26.
B. Mavrič and B. Šarler, Local radial basis function collocation method for linear thermo-elasticity in two dimensions, Internat. J. Numer. Methods Heat Fluid Flow, 25 (2015), pp. 1488--1510, https://doi.org/10.1108/hff-11-2014-0359.
27.
Medusa Library, http://e6.ijs.si/medusa/.
28.
S. A. Mitchell, A. Rand, M. S. Ebeida, and C. Bajaj, Variable radii Poisson-disk sampling, extended version, in Proceedings of the 24th Canadian Conference on Computational Geometry, Vol. 5, 2012.
29.
A. W. Moore, An Introductory Tutorial on KD-trees, Ph.D. thesis, University of Cambridge, Cambridge, 1991, https://www.ri.cmu.edu/pub_files/pub1/moore_andrew_1991_1/moore_andrew_1991_1.pdf.
30.
P.-O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Rev., 46 (2004), pp. 329--345, https://doi.org/10.1137/s0036144503429121.
31.
K. Reuther, B. Sarler, and M. Rettenmayr, Solving diffusion problems on an unstructured, amorphous grid by a meshless method, Int. J. Therm. Sci., 51 (2012), pp. 16--22, https://doi.org/10.1016/j.ijthermalsci.2011.08.017.
32.
V. Shankar, R. M. Kirby, and A. L. Fogelson, Robust node generation for meshfree discretizations on irregular domains and surfaces, SIAM J. Sci. Comput., 40 (2018), pp. 2584--2608, https://doi.org/10.1137/17m114090x.
33.
J. Slak and G. Kosec, Standalone Implementation of the Proposed Node Placing Algorithm, http://e6.ijs.si/medusa/static/PNP.zip.
34.
J. Slak and G. Kosec, Fast generation of variable density node distributions for mesh-free methods, in WIT Trans. Eng. Sci., 122 (2018), pp. 163--173, https://doi.org/10.2495/be410151.
35.
J. Slak and G. Kosec, Refined meshless local strong form solution of Cauchy--Navier equation on an irregular domain, Eng. Anal. Bound. Elem., 100 (2018) pp.3--13, https://doi.org/10.1016/j.enganabound.2018.01.001.
36.
J. Slak and G. Kosec, Adaptive radial basis function-generated finite differences method for contact problems, Internat. J. Numer. Methods Engrg., 2019, https://doi.org/10.1002/nme.6067.
37.
P. Wang, Y. Zhang, and Z. Guo, Numerical study of three-dimensional natural convection in a cubical cavity at high Rayleigh numbers, Int. J. Heat Mass Transf., 113 (2017), pp. 217--228, https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.057.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A3202 - A3229
DOI: 10.1137/18M1231456
ISSN (online): 1095-7197
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 10 December 2018
Accepted: 19 August 2019
Published online: 15 October 2019
Keywords
MSC codes
Authors
Funding Information
Research Foundation Flanders : G018916N
Funding Information
Young Researcher program : PR-08346
Funding Information
Javna Agencija za Raziskovalno Dejavnost RS https://doi.org/10.13039/501100004329 : P2-0095
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Guidelines for RBF-FD Discretization: Numerical Experiments on the Interplay of a Multitude of Parameter ChoicesJournal of Scientific Computing, Vol. 95, No. 1 | 15 February 2023
- Point Cloud Generation for Meshfree Methods: An OverviewArchives of Computational Methods in Engineering, Vol. 30, No. 2 | 28 September 2022
- A sharp-interface mesoscopic model for dendritic growthIOP Conference Series: Materials Science and Engineering, Vol. 1274, No. 1 | 1 Jan 2023
- Automatic mesoscopic fracture modelling of concrete based on enriched SBFEM space and quad-tree meshConstruction and Building Materials, Vol. 350 | 1 Oct 2022
- PIMesh : An automatic point cloud and unstructured mesh generation algorithm for meshless methods and finite element analysis–with applications in surgical simulationsInternational Journal for Numerical Methods in Biomedical Engineering, Vol. 38, No. 8 | 9 June 2022
- A hybrid RBF-FD and WLS mesh-free strong-form approximation method2022 7th International Conference on Smart and Sustainable Technologies (SpliTech) | 5 Jul 2022
- Implicit-Explicit Error Indicator based on Approximation Order2022 7th International Conference on Smart and Sustainable Technologies (SpliTech) | 5 Jul 2022
- Grid-free Monte Carlo for PDEs with spatially varying coefficientsACM Transactions on Graphics, Vol. 41, No. 4 | 22 July 2022
- Stability analysis of RBF-FD and WLS based local strong form meshless methods on scattered nodes2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO) | 23 May 2022
- A Meshless Solution of a Small-Strain Plasticity Problem2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO) | 23 May 2022
- Meshless method stencil evaluation with machine learning2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO) | 23 May 2022
- Parallel spatial indexing for domain discretization2022 45th Jubilee International Convention on Information, Communication and Electronic Technology (MIPRO) | 23 May 2022
- Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEsComputers & Structures, Vol. 264 | 1 May 2022
- A new radial basis integration method applied to the boundary element analysis of 2D scalar wave equationsEngineering Analysis with Boundary Elements, Vol. 136 | 1 Mar 2022
- A Strong-Form Off-Lattice Boltzmann Method for Irregular Point CloudsSymmetry, Vol. 13, No. 10 | 28 September 2021
- MedusaACM Transactions on Mathematical Software, Vol. 47, No. 3 | 26 June 2021
- Discretized Boundary Surface Reconstruction2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO) | 27 Sep 2021
- A k-d Tree Based Partitioning of Computational Domains for Efficient Parallel Computing2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO) | 27 Sep 2021
- Partition-of-Unity Based Error Indicator for Local Collocation Meshless Methods2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO) | 27 Sep 2021
- Natural Convection of Non-Newtonian Fluids in a Differentially Heated Closed Cavity2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO) | 27 Sep 2021
- p-refined RBF-FD solution of a Poisson problem2021 6th International Conference on Smart and Sustainable Technologies (SpliTech) | 8 Sep 2021
- RBF-FD analysis of 2D time-domain acoustic wave propagation in heterogeneous mediaComputers & Geosciences, Vol. 153 | 1 Aug 2021
- Node Generation for RBF-FD Methods by QR FactorizationMathematics, Vol. 9, No. 16 | 5 August 2021
- An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometriesJournal of Computational Physics, Vol. 436 | 1 Jul 2021
- Micro-combustion modelling with RBF-FD: A high-order meshfree method for reactive flows in complex geometriesApplied Mathematical Modelling, Vol. 94 | 1 Jun 2021
- Monomial Augmentation Guidelines for RBF-FD from Accuracy Versus Computational Time PerspectiveJournal of Scientific Computing, Vol. 87, No. 1 | 17 February 2021
- A divergence-free generalized moving least squares approximation with its applicationApplied Numerical Mathematics, Vol. 162 | 1 Apr 2021
- On the meshfree particle methods for fluid-structure interaction problemsEngineering Analysis with Boundary Elements, Vol. 124 | 1 Mar 2021
- A Fully Meshless Approach to the Numerical Simulation of Heat Conduction Problems over Arbitrary 3D GeometriesEnergies, Vol. 14, No. 5 | 2 March 2021
- A Least Squares Radial Basis Function Finite Difference Method with Improved Stability PropertiesSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 26 April 2021
- Fast Variable Density Node Generation on Parametric Surfaces with Application to Mesh-Free MethodsSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 15 March 2021
- Fast Variable Density 3-D Node GenerationSIAM Journal on Scientific Computing, Vol. 43, No. 1 | 13 January 2021
- A parallel implementation strategy for meshless methods based on the functional programming paradigmAdvances in Engineering Software, Vol. 151 | 1 Jan 2021
- The Kriging integration method applied to the boundary element analysis of Poisson problemsEngineering Analysis with Boundary Elements, Vol. 121 | 1 Dec 2020
- GPU accelerated RBF-FD solution of Poisson’s equation2020 43rd International Convention on Information, Communication and Electronic Technology (MIPRO) | 28 Sep 2020
- Parallel point sampling for 3D bodies2020 43rd International Convention on Information, Communication and Electronic Technology (MIPRO) | 28 Sep 2020