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Simulation of the incompressible Navier–Stokes via integrated radial basis function based on finite difference scheme

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Abstract

Integrated radial basis function based on finite difference (IRBF–FD) method is presented in this paper for the solution of incompressible Navier–Stokes equations. A semi-implicit temporal scheme is first used to discretize the time variable of the incompressible Navier–Stokes (NS) equations. We consider the same discrete scheme for the time variable for both pressure–Poisson equation and vorticity stream function formulation. For solving lid-driven cavity flow and backward-facing step flow, we used the vorticity-stream formulation. The proposed method approximates the function derivatives at a knot in terms of the function values on a collection of nodes existing in the support domain of the node. We also utilize an algorithm for finding the optimal shape parameter for each stencil based on the range of condition numbers. It can be seen that no special treatment is needed to impose the essential boundary conditions. The efficiency, accuracy and robustness of the presented method are demonstrated by comparing the current method with existing methods.

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References

  1. Abbaszadeh M, Dehghan M (2020) Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method. Comput Meth Appl Mech Engin 364:112914

    Article  MathSciNet  MATH  Google Scholar 

  2. Abide S, Viazzo S (2005) A 2D compact fourth-order projection decomposition method. J Comput Phys 206(1):252–276

    Article  MathSciNet  MATH  Google Scholar 

  3. Armaly BF, Durst F, Pereira JCF, Schonung B (1983) Experimental and theoretical investigation of backward-facing step flow. J Fluid Mech 127:473–496

    Article  Google Scholar 

  4. Bardos C (2002) Navier–Stokes Equations and Turbulence

  5. Cheng YM, Bai FN, Peng MJ (2014) A novel interpolating element free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl Math Model 38:5187–5197

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng YM, Li J (2005) A meshless method with complex variables for elasticity. Acta Physica Sinica 54:4463–4471

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng YM, Peng M (2005) Boundary element free method for elastodynamics. Sci China G 48:641–657

    Article  Google Scholar 

  8. Chung D, Pullin DI (2009) Large-eddy simulation and wall modelling of turbulent channel flow. J Fluid Mech 631:281–309

    Article  MathSciNet  MATH  Google Scholar 

  9. Dehghan M, Abbaszadeh M (2016) Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier-Stokes equation. Comput Meth Appl Mech Engin 311:856–888

    Article  MathSciNet  MATH  Google Scholar 

  10. Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71(1):16–30

    Article  MathSciNet  MATH  Google Scholar 

  11. Dehghan M, Shokri A (2008) A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math Comput Simul 79(3):700–715

    Article  MathSciNet  MATH  Google Scholar 

  12. Ebrahimijahan A, Dehghan M, Abbaszadeh M (2021) Numerical simulation of shallow water waves based on generalized equal width (GEW) equation by compact local integrated radial basis function method combined with adaptive residual subsampling technique. Nonlinear Dyn 105(4):3359–3391

    Article  Google Scholar 

  13. Fasshauer G. E (1997) Solving partial differential equations by collocation with radial basis functions. In Surface Fitting and Multiresolution Methods, Mehaute AL, Rabut C, Schumaker LL (eds), 131-138

  14. Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes Equations And Turbulence. Cambridge University Press

    Book  MATH  Google Scholar 

  15. Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230(6):2270–2285

    Article  MathSciNet  MATH  Google Scholar 

  16. Fornberg B, Lehto E, Powell C (2013) Stable calculation of Gaussian-based RBF-FD stencils. Comput Math Appl 65(4):627–637

    Article  MathSciNet  MATH  Google Scholar 

  17. Gargari SF, Kolahdoozan M, Afshar MH (2018) Mixed discrete least squares meshfree method for solving the incompressible Navier-Stokes equations, Engin. Anal. Bound Elem 88:64–79

    Article  MathSciNet  MATH  Google Scholar 

  18. Gartling DK (1990) A test problem for outflow boundary conditions-flow over a backward-facing step. Inter J Numer Meth Fluid 11:953–967

    Article  Google Scholar 

  19. Ghia U, Ghia KN, Shin CT (1982) High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys 48(3):387–411

    Article  MATH  Google Scholar 

  20. Gonzalez-Rodriguez P, Bayona V, Moscoso M, Kindelan M (2015) Laurent series based RBF-FD method to avoid ill-conditioning. Eng Anal Bound Elem 52:24–31

    Article  MathSciNet  MATH  Google Scholar 

  21. Griffith BE (2009) An accurate and efficient method for the incompressible Navier-Stokes equations using the projection method as a preconditioner. J Comput Phys 228(20):7565–7595

    Article  MathSciNet  MATH  Google Scholar 

  22. Gu L (2003) Moving Kriging interpolation and element-free Galerkin method. Int J Numer Meth Eng 56:1–11

    Article  MATH  Google Scholar 

  23. Guo X, Li W, Iorio F (2016) Convolutional neural networks for steady flow approximation. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining 481-490

  24. Hirsch C (1988) Numerical Computation of Internal and External Flows. John Wiley & Sons

    MATH  Google Scholar 

  25. Hoffmann KA, Chiang ST (2000) Computational Fluid Dynamics, vol I. Engineering Education System, Wichita, KS

    Google Scholar 

  26. Kansa ET (1990) Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics: surface approximations and partial derivative estimates. Comput Math Appl 19(6–8):127–145

    Article  MathSciNet  MATH  Google Scholar 

  27. Kashefi A, Staples AE (2018) A finite-element coarse-grid projection method for incompressible flow simulations. Adv Comput Math 44(4):1063–1090

    Article  MathSciNet  MATH  Google Scholar 

  28. Kashefi A (2020) A coarse grid projection method for accelerating free and forced convection heat transfer computations. Results Math 75(1):1–24

    Article  MathSciNet  MATH  Google Scholar 

  29. Kashefi A (2021) A coarse-grid projection method for accelerating incompressible MHD flow simulations. In Press, Engin. Comput., pp 1–15

  30. Kashefi A (2020) Coarse grid projection methodology: a partial mesh refinement tool for incompressible flow simulations. Bull Iranian Math Soc 46(1):177–181

    Article  MathSciNet  MATH  Google Scholar 

  31. Kashefi A, Rempe D, Guibas LJ (2021) A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries. Phys Fluids 33(2):027104

    Article  Google Scholar 

  32. Kelly JM, Divo EA, Kassab AJ (2014) Numerical solution of the two-phase incompressible Navier–Stokes equations using a GPU-accelerated meshless method, Engin. Anal. Bound Elem 40:36–49

    Article  MathSciNet  MATH  Google Scholar 

  33. Kim P, Kim D, Piao X, Bak S (2020) A completely explicit scheme of Cauchy problem in BSLM for solving the Navier-Stokes equations. J Comput Phys 401:109028

    Article  MathSciNet  MATH  Google Scholar 

  34. Kim P, Bak S (2021) Algorithm for a cost-reducing time-integration scheme for solving incompressible Navier-Stokes equations. Comput Meth Appl Mech Eng 373:113546

    Article  MathSciNet  MATH  Google Scholar 

  35. Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier-Stokes equations. J Comput Phys 59(2):308–323

    Article  MathSciNet  MATH  Google Scholar 

  36. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158

    Article  MathSciNet  MATH  Google Scholar 

  37. Lentine M, Zheng W, Fedkiw R (2010) A novel algorithm for incompressible flow using only a coarse grid projection. ACM Trans Graphics (TOG) 29(4):1–9

    Article  Google Scholar 

  38. Liew KM, Wang WQ, Zhang LX, He XQ (2007) A computational approach for predicting the hydroelasticity of flexible structures based on the pressure Poisson equation. Int J Nume Methods Eng 72:1560–1583

    Article  MathSciNet  MATH  Google Scholar 

  39. Liew KM, Zhao X, Ferreira AJM (2011) A review of meshless methods for laminated and functionally graded plates and shells. Composite Structures 93:2031–2041

    Article  Google Scholar 

  40. Loukopoulos VC, Messaris GT, Bourantas GC (2013) Numerical solution of the incompressible Navier-Stokes equations in primitive variables and velocity-vorticity formulation. Appl Math Comput 222:575–588

    MathSciNet  MATH  Google Scholar 

  41. Mai-Duy N, Tran-Cong T (2005) An efficient indirect RBFN-based method for numerical solution of PDEs. Numer Meth PDE 21(4):770–790

    Article  MathSciNet  MATH  Google Scholar 

  42. Mai-Duy N, Tran-Cong T (2013) A compact five-point stencil based on integrated RBFs for 2D second-order differential problems. J Comput Phys 235:302–321

    Article  MathSciNet  Google Scholar 

  43. Mramor K, Vertnik R, Šarler B (2013) Low and intermediate Re solution of lid driven cavity problem by local radial basis function collocation method. CMC-Comput Mater Con 1:1–21

    MATH  Google Scholar 

  44. Muratova G, Martynova T, Andreeva E, Bavin V, Wang ZQ (2020) Numerical Solution of the Navier-Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers. Symmetry 12(2):233

    Article  MATH  Google Scholar 

  45. 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

    Article  MathSciNet  MATH  Google Scholar 

  46. John V, Matthies G, Rang J (2006) A comparison of time-discretization/linearization approaches for the incompressible Navier-Stokes equations. Comput Meth Appl Mech Eng 195(44–47):5995–6010

    Article  MathSciNet  MATH  Google Scholar 

  47. Reis GA, Tasso IVM, Souza LF, Cuminato JA (2015) A compact finite differences exact projection method for the Navier-Stokes equations on a staggered grid with fourth-order spatial precision. Comput Fluids 118:19–31

    Article  MathSciNet  MATH  Google Scholar 

  48. Roberts NV, Demkowicz L, Moser R (2015) A discontinuous Petrov-Galerkin methodology for adaptive solutions to the incompressible Navier-Stokes equations. J Comput Phys 301:456–483

    Article  MathSciNet  MATH  Google Scholar 

  49. Sarra SA (2006) Integrated multiquadric radial basis function approximation methods. Comput Math Appl 51(8):1283–1296

    Article  MathSciNet  MATH  Google Scholar 

  50. Sarra SA (2012) A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains. Appl Math Comput 218(19):9853–9865

    MathSciNet  MATH  Google Scholar 

  51. Sedaghatjoo Z, Dehghan M, Hosseinzadeh H (2018) Numerical solution of 2D Navier-Stokes equation discretized via boundary elements method and finite difference approximation. Eng Anal Boundary Elements 96:64–77

    Article  MathSciNet  MATH  Google Scholar 

  52. Shu C, Wu YL (2007) Integrated radial basis functions-based differential quadrature method and its performance. Int J Numerical Methods Fluids 53(6):969–984

    Article  MATH  Google Scholar 

  53. Spalart PR, Watmuff JH (1993) Experimental and numerical study of a turbulent boundary layer with pressure gradients. J Fluid Mech 249:337–371

    Article  Google Scholar 

  54. Tai CH, Zhao Y, Liew KM (2005) Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme. Comput Methods Appl Mech Eng 194:3949–3983

    Article  MATH  Google Scholar 

  55. Tu J, Yeoh GH, Liu C (2018) Computational Fluid Dynamics: a Practical Approach. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  56. Thamareerat N, Luadsong A, Aschariyaphotha N (2016) The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations. SpringerPlus 5(1):417

    Article  Google Scholar 

  57. Tabbakh Z, Seaid M, Ellaia R, Ouazar D, Benkhaldoun F (2019) A local radial basis function projection method for incompressible flows in water eutrophication, Engin. Anal. Bound. Elem. 106:528–540

    Article  MathSciNet  MATH  Google Scholar 

  58. Yanwen M, Dexun F, Kobayashi T, Taniguchi N (1999) Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme. Int J Numer Methods Fluids 30(5):509–521

    Article  MathSciNet  MATH  Google Scholar 

  59. Yun-Xin Z, Yong-Ji T (2006) Meshless schemes for unsteady Navier-Stokes equations in vorticity formulation using radial basis functions. J Comput Appl Math 192(2):328–338

    Article  MathSciNet  Google Scholar 

  60. Zhang X, Song K. Z, Lu M. W, Liu X (2000) Meshless methods based on collocation with radial basis functions, Computational Mechanics, 26 (4) 333-343

    Article  MATH  Google Scholar 

  61. Zhang Z, Liew KM, Cheng YM, Lee YY (2008) Analyzing 2D fracture problems with the improved element free Galerkin method. Eng Anal Bound Elem 32:241–250

    Article  MATH  Google Scholar 

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Acknowledgements

Authors thank the reviewers for their comments and suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Ali Ebrahimijahan, Mehdi Dehghan & Mostafa Abbaszadeh

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  1. Ali Ebrahimijahan
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  2. Mehdi Dehghan
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  3. Mostafa Abbaszadeh
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Correspondence to Mehdi Dehghan.

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Ebrahimijahan, A., Dehghan, M. & Abbaszadeh, M. Simulation of the incompressible Navier–Stokes via integrated radial basis function based on finite difference scheme. Engineering with Computers 38, 5069–5090 (2022). https://doi.org/10.1007/s00366-021-01543-z

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