Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach

References

• G. Liu, Meshfree Methods: Moving beyond the Finite Element Method. Boca Raton, FL: CRC Press, 2009.
   Google Scholar
• H. Li and S. Mulay, Meshless Methods and Their Numerical Properties. Boca Raton, FL: CRC Press, 2013.
   Google Scholar
• G. Fasshauer, “Solving partial differential equations by collocation with radial basis functions,” presented at the Proceedings of the 3rd International Conference on Curves and Surfaces, Chamonix-Mont-Blanc, 1996.
   Google Scholar
• C. Franke and R. Schaback, “Solving partial differential equations by collocation using radial basis functions,” Appl. Math. Comput., vol. 93, no. 1, pp. 73–82, 1998. DOI: 10.1016/S0096-3003(97)10104-7.
   Web of Science ®Google Scholar
• M. Golberg, C. Chen, and H. Bowman, “Some recent results and proposals for the use of radial basis functions in the BEM,” Eng. Anal. Bound. Elem., vol. 23, no. 4, pp. 285–296, 1999. DOI: 10.1016/S0955-7997(98)00087-3.
   Web of Science ®Google Scholar
• E. Larsson and B. Fornberg, “A numerical study of some radial basis function based solution methods for elliptic PDEs,” Comput. Math. Appl., vol. 46, no. 5, pp. 891–902, 2003. DOI: 10.1016/S0898-1221(03)90151-9.
   Web of Science ®Google Scholar
• S. Sarra and E. Kansa, “Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations,” Adv. Comput. Mech., vol. 2, pp. 1–206, 2009.
   Google Scholar
• B. Fornberg and N. Flyer, “Solving PDEs with radial basis functions,” Acta Numer., vol. 24, pp. 215–258, 2015. DOI: 10.1017/S0962492914000130.
   Web of Science ®Google Scholar
• M. Li, C. S. Chen, and C. H. Tsai, “Meshless method based on radial basis functions for solving parabolic partial differential equations with variable coefficients,” Numer. Heat Transfer B Fund., vol. 57, no. 5, pp. 333–347, 2010. DOI: 10.1080/10407790.2010.481489.
   Web of Science ®Google Scholar
• E. Kansa, “Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations,” Comput. Math. Appl., vol. 19, no. 8, pp. 147–161, 1990. DOI: 10.1016/0898-1221(90)90271-K.
   Web of Science ®Google Scholar
• C. Lee, X. Liu, and S. Fan, “Local multiquadric approximation for solving boundary value problems,” Comput. Mech., vol. 30, no. 5, pp. 396–409, 2003. DOI: 10.1007/s00466-003-0416-5.
   Web of Science ®Google Scholar
• B. Šarler and R. Vertnik, “Meshfree explicit local radial basis function collocation method for diffusion problems,” Comput. Math. Appl., vol. 51, no. 8, pp. 1269–1282, 2006. DOI: 10.1016/j.camwa.2006.04.013.
   Web of Science ®Google Scholar
• G. Chandhini and Y. Sanyasiraju, “Local RBF-FD solutions for steady convection-diffusion problems,” Int. J. Numer. Meth. Eng., vol. 72, no. 3, pp. 352–378, 2007. DOI: 10.1002/nme.2024.
   Web of Science ®Google Scholar
• E. Divo and A. J. Kassab, “Localized meshless modeling of Natural-Convective viscous flows,” Numer. Heat Transfer B Fund., vol. 53, no. 6, pp. 487–509, 2008. DOI: 10.1080/10407790802083190.
   Web of Science ®Google Scholar
• J. Sun, H.-L. Yi, and H.-P. Tan, “Local RBF meshless scheme for coupled radiative and conductive heat transfer,” Numer. Heat Transfer A Appl., vol. 69, no. 12, pp. 1390–1404, 2016. DOI: 10.1080/10407782.2016.1139959.
   Web of Science ®Google Scholar
• T. T. V. Le, N. Mai-Duy, K. Le-Cao, and T. Tran-Cong, “A time discretization scheme based on integrated radial basis functions for heat transfer and fluid flow problems,” Numer. Heat Transfer B Fund., vol. 74, no. 2, pp. 498–518, 2018. DOI: 10.1080/10407790.2018.1515329.
   Web of Science ®Google Scholar
• K. Le-Cao, N. Mai-Duy, and T. Tran-Cong, “An effective integrated-RBFN Cartesian-grid discretization for the stream functionvorticitytemperature formulation in nonrectangular domains,” Numer. Heat Transfer B Fund., vol. 55, no. 6, pp. 480–502, 2009. DOI: 10.1080/10407790902827470.
   Web of Science ®Google Scholar
• M. Q. Wang, C. X. Wang, M. Li, and C. S. Chen, “A numerical scheme based on FD-RBF to solve Fractional-Diffusion inverse heat conduction problems,” Numer. Heat Transfer A Appl., vol. 68, no. 9, pp. 978–992, 2015. DOI: 10.1080/10407782.2014.986376.
   Web of Science ®Google Scholar
• N. Li, Z. Tan, and X. Feng, “Novel two-level discretization method for high dimensional semilinear elliptic problems base on RBF-FD scheme,” Numer Heat Transfer B Fund., vol. 72, no. 5, pp. 349–360, 2017.
   Web of Science ®Google Scholar
• G. Wright and B. Fornberg, “Scattered node compact finite difference-type formulas generated from radial basis functions,” J. Comput. Phys., vol. 212, no. 1, pp. 99–123, 2006. DOI: 10.1016/j.jcp.2005.05.030.
   Web of Science ®Google Scholar
• B. Šarler, “From global to local radial basis function collocation method for transport phenomena,” in Computational Methods in Applied Sciences, V. Leitão, C. Alves, and C. Duarte, Eds. Springer: Dordrecht, The Netherlands, 2007. Adv. Meshfree Tech., vol. 5, pp. 257–282, 2007.
   Google Scholar
• G. Yao, Siraj-Ul-Islam, and B. Šarler, “Assessment of global and local meshless methods based on collocation with radial basis functions for parabolic partial differential equations in three dimensions,” Eng. Anal. Bound. Elem, vol. 36, no. 11, pp. 1640–1648, 2015.
   Google Scholar
• J. Waters, and D. Pepper, “Global versus localized RBF meshless methods for solving incompressible fluid flow with heat transfer,” Numer. Heat Transfer B Fund., vol. 68, no. 3, pp. 185–203, 2015. DOI: 10.1080/10407790.2015.1021590.
   Web of Science ®Google Scholar
• E. Divo and A. Kassab, “An efficient localized radial basis function meshless method for fluid flow and conjugate heat transfer,” J. Heat Transfer, vol. 129, no. 2, pp. 124–136, 2007. DOI: 10.1115/1.2402181.
   Web of Science ®Google Scholar
• G. Kosec and B. Šarler, “Solution of thermo-fluid problems by collocation with local pressure correction,” Int. J. Numer. Methods Heat Fluid Flow, vol. 18, no. 7/8, pp. 868–882, 2008. DOI: 10.1108/09615530810898999.
   Web of Science ®Google Scholar
• P. Chinchapatnam, K. Djidjeli, P. Nair, and M. Tan, “A compact RBF-FD based meshless method for the incompressible Navier-Stokes equations,” presented at the Proceedings of the IMechE, Part M. J. Eng. Maritime Environ., vol. 223, no. 3, pp. 275–290, 2009. DOI: 10.1243/14750902JEME151.
   Google Scholar
• G. Kosec, and B. Šarler, “Solution of a low prandtl number natural convection benchmark by a local meshless method,” Int. J. Numer. Methods Heat Fluid Flow, vol. 23, no. 1, pp. 189–204, 2013. DOI: 10.1108/09615531311289187.
   Web of Science ®Google Scholar
• Z. Wang, Z. Huang, W. Zhang, and G. Xi, “A meshless local radial basis function method for two-dimensional incompressible Navier-Stokes equations,” Numer. Heat Transfer B Fund., vol. 67, no. 4, pp. 320–337, 2015. DOI: 10.1080/10407790.2014.955779.
   Web of Science ®Google Scholar
• A. Chorin, “Numerical solution of the Navier-Stokes equations,” Math. Comput., vol. 22, no. 104, pp. 745–762, 1968. DOI: 10.1090/S0025-5718-1968-0242392-2.
   Web of Science ®Google Scholar
• R. Zamolo, and E. Nobile, “Two algorithms for fast 2D node generation: Application to RBF meshless discretization of diffusion problems and image halftoning,” Comput. Math. Appl., vol. 75, no. 12, pp. 4305–4321, 2018. DOI: 10.1016/j.camwa.2018.03.031.
   Web of Science ®Google Scholar
• R. Finkel and J. Bentley, “Quad trees a data structure for retrieval on composite keys,” Acta Inform., vol. 4, no. 1, pp. 1–9, 1974. DOI: 10.1007/BF00288933.
   Google Scholar
• R. Floyd and L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inform. Display, vol. 17, no. 2, pp. 75–77, 1976.
   Google Scholar
• V. Bayona, N. Flyer, B. Fornberg, and G. Barnett, “On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs,” J. Comput. Phys., vol. 332, pp. 257–273, 2017. DOI: 10.1016/j.jcp.2016.12.008.
   Web of Science ®Google Scholar
• R. Hardy, “Multiquadric equations of topography and other irregular surfaces,” J. Geophys. Res., vol. 76, no. 8, pp. 1905–1915, 1971. DOI: 10.1029/JB076i008p01905.
   Web of Science ®Google Scholar
• R. Franke, “A Critical Comparison of Some Methods for Interpolation of Scattered Data (Technical Report),” Monterey, CA: Naval Postgraduate School, Technical Report NPS-53-79-003, 1980.
   Google Scholar
• R. Franke, “Scattered data interpolation: tests of some methods,” Math. Comput., vol. 38, no. 157, pp. 181–200, 1982. DOI: 10.1090/S0025-5718-1982-0637296-4.
   Web of Science ®Google Scholar
• N. Flyer, B. Fornberg, V. Bayona, and G. Barnett, “On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy,” J. Comput. Phys., vol. 321, pp. 21–38, 2016. DOI: 10.1016/j.jcp.2016.05.026.
   Web of Science ®Google Scholar
• G. Fasshauer, “Meshfree approximation methods with Matlab,” in Interdisciplinary Mathematical Sciences, vol. 6, ch. 9: Conditionally Positive Definite Radial Functions. Singapore: World Scientific, 2007.
   Google Scholar
• R. Horn, and C. Johnson, Matrix Analysis. Cambridge, UK: Cambridge University Press, 1990.
   Google Scholar
• S. Armfield, “Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids,” Comput. Fluids, vol. 20, no. 1, pp. 1–17, 1991. DOI: 10.1016/0045-7930(91)90023-B.
   Web of Science ®Google Scholar
• H. van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp., vol. 13, no. 2, pp. 631–644, 1992. DOI: 10.1137/0913035.
   Web of Science ®Google Scholar
• Y. Saad, “Iterative methods for sparse linear systems,” in Preconditioning Techniques, 2nd ed., ch. 10. Philadelphia, PA: SIAM, 2003.
   Google Scholar
• E. Cuthill, and J. McKee, “Reducing the bandwidth of sparse symmetric matrices,” in Proceedings of the 1969 24th National Conference, ACM ’69. New York, NY: ACM, pp. 157–172, 1969.
   Google Scholar
• R. J. A. Janssen, and R. A. W. M. Henkes, “Accuracy of finite-volume discretizations for the bifurcating natural-convection flow in a square cavity,” Numer Heat Transfer B Fund., vol. 24, no. 2, pp. 191–207, 1993. DOI: 10.1080/10407799308955889.
   Web of Science ®Google Scholar
• E. Nobile, “Simulation of time-dependent flow in cavities with the additive-correction multigrid method, part II: applications,” Numer Heat Transfer B Fund., vol. 30, no. 3, pp. 351–370, 1996. DOI: 10.1080/10407799608915087.
   Web of Science ®Google Scholar
• A. Fortin, M. Jardak, J. Gervais, and R. Pierre, “Localization of hopf bifurcations in fluid flow problems,” Int. J. Numer. Methods Fluids, vol. 24, no. 11, pp. 1185–1210, 1997. DOI: 10.1002/(SICI)1097-0363(19970615)24:11<1185::AID-FLD535>3.3.CO;2-O.
   Web of Science ®Google Scholar
• C.-H. Bruneau and M. Saad, “The 2D lid-driven cavity problem revisited,” Comput. Fluids, vol. 35, no. 3, pp. 326–348, 2006. DOI: 10.1016/j.compfluid.2004.12.004.
   Web of Science ®Google Scholar
• S. Paolucci and D. R. Chenoweth, “Transition to chaos in a differentially heated vertical cavity,” J. Fluid Mech., vol. 201, no. 1, pp. 379, 1989. DOI: 10.1017/S0022112089000984.
   Google Scholar
• D. Contrino, P. Lallemand, P. Asinari, and L.-S. Luo, “Lattice-Boltzmann simulations of the thermally driven 2D square cavity at high Rayleigh numbers,” J. Comput. Phys., vol. 275, pp. 257–272, 2014. DOI: 10.1016/j.jcp.2014.06.047.
   Web of Science ®Google Scholar
• P. Le Quéré, “Accurate solutions to the square thermally driven cavity at high rayleigh number,” Comput. Fluids, vol. 20, no. 1, pp. 29–41, 1991. DOI: 10.1016/0045-7930(91)90025-D.
   Web of Science ®Google Scholar
• J.-H. Chen, W. G. Pritchard, and S. J. Tavener, “Bifurcation for flow past a cylinder between parallel planes,” J. Fluid Mech., vol. 284, no. 1, pp. 23–41, 1995. DOI: 10.1017/S0022112095000255.
   Google Scholar
• L. Zovatto and G. Pedrizzetti, “Flow about a circular cylinder between parallel walls,” J. Fluid Mech., vol. 440, pp. 1–25, 2001.
   Web of Science ®Google Scholar