Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method
Abstract
:1. Introduction
2. Lattice Boltzmann Method
3. Radial Basis Function Method
3.1. RBF for Solving the Streaming Step
3.2. Basis Function
4. Results and Discussions
4.1. One-Dimensional Problems
4.1.1. 1D Diffusion
4.1.2. 1D Advection–Diffusion
4.2. 2D Lid-Driven Cavity
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Bawazeer, S.; Mohamad, A.; Oclon, P. Natural convection in a differentially heated enclosure filled with low prandtl number fluids with modified lattice boltzmann method. Int. J. Heat Mass Transf. 2019, 143, 118562. [Google Scholar] [CrossRef]
- Mohamad, A.A. Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes, 2nd ed.; Springer: New York, NY, USA, 2019. [Google Scholar]
- Mohamad, A.; Tao, Q.; He, Y.; Bawazeer, S. Treatment of transport at the interface between multilayers via the lattice boltzmann method. Numer. Heat Transf. Part B Fundam. 2015, 67, 124–134. [Google Scholar] [CrossRef]
- Bawazeer, S. Lattice Boltzmann Method with Improved Radial Basis Function Method; University of Calgary: Calgary, AB, Canada, 2019. [Google Scholar]
- Filippova, O.; Hänel, D. Grid refinement for lattice-bgk models. J. Comput. Phys. 1998, 147, 219–228. [Google Scholar] [CrossRef]
- Kandhai, D.; Soll, W.; Chen, S.; Hoekstra, A.; Sloot, P. Finite-difference lattice-bgk methods on nested grids. Comput. Phys. Commun. 2000, 129, 100–109. [Google Scholar] [CrossRef]
- Yu, D.; Mei, R.; Shyy, W. A multi-block lattice boltzmann method for viscous fluid flows. Int. J. Numer. Methods Fluids 2002, 39, 99–120. [Google Scholar] [CrossRef]
- Yu, Z.; Fan, L.-S. An interaction potential based lattice boltzmann method with adaptive mesh refinement (amr) for two-phase flow simulation. J. Comput. Phys. 2009, 228, 6456–6478. [Google Scholar] [CrossRef]
- Crouse, B.; Rank, E.; Krafczyk, M.; Tölke, J. A lb-based approach for adaptive flow simulations. Int. J. Mod. Phys. B 2003, 17, 109–112. [Google Scholar] [CrossRef]
- Wu, J.; Shu, C. A solution-adaptive lattice boltzmann method for two-dimensional incompressible viscous flows. J. Comput. Phys. 2011, 230, 2246–2269. [Google Scholar] [CrossRef]
- Chen, Y.; Kang, Q.; Cai, Q.; Zhang, D. Lattice boltzmann method on quadtree grids. Phys. Rev. E 2011, 83, 026707. [Google Scholar] [CrossRef]
- Lagrava, D.; Malaspinas, O.; Latt, J.; Chopard, B. Advances in multi-domain lattice boltzmann grid refinement. J. Comput. Phys. 2012, 231, 4808–4822. [Google Scholar] [CrossRef] [Green Version]
- Eitel-Amor, G.; Meinke, M.; Schröder, W. A lattice-boltzmann method with hierarchically refined meshes. Comput. Fluids 2013, 75, 127–139. [Google Scholar] [CrossRef]
- Fakhari, A.; Lee, T. Numerics of the lattice boltzmann method on nonuniform grids: Standard lbm and finite-difference lbm. Comput. Fluids 2015, 107, 205–213. [Google Scholar] [CrossRef]
- Fakhari, A.; Lee, T. Finite-difference lattice boltzmann method with a block-structured adaptive-mesh-refinement technique. Phys. Rev. E 2014, 89, 033310. [Google Scholar] [CrossRef] [PubMed]
- Guzik, S.M.; Weisgraber, T.H.; Colella, P.; Alder, B.J. Interpolation methods and the accuracy of lattice-boltzmann mesh refinement. J. Comput. Phys. 2014, 259, 461–487. [Google Scholar] [CrossRef]
- Succi, S.; Amati, G.; Benzi, R. Challenges in lattice boltzmann computing. J. Stat. Phys. 1995, 81, 5–16. [Google Scholar] [CrossRef]
- Nannelli, F.; Succi, S. The lattice boltzmann equation on irregular lattices. J. Stat. Phys. 1992, 68, 401–407. [Google Scholar] [CrossRef]
- Xi, H.; Peng, G.; Chou, S.-H. Finite-volume lattice boltzmann schemes in two and three dimensions. Phys. Rev. E 1999, 60, 3380. [Google Scholar] [CrossRef] [Green Version]
- Peng, G.; Xi, H.; Duncan, C.; Chou, S.-H. Finite volume scheme for the lattice boltzmann method on unstructured meshes. Phys. Rev. E 1999, 59, 4675. [Google Scholar] [CrossRef]
- Shrestha, K.; Mompean, G.; Calzavarini, E. Finite-volume versus streaming-based lattice boltzmann algorithm for fluid-dynamics simulations: A one-to-one accuracy and performance study. Phys. Rev. E 2016, 93, 023306. [Google Scholar] [CrossRef] [Green Version]
- Cevik, F.; Albayrak, K. A fully implicit finite volume lattice boltzmann method for turbulent flows. Commun. Comput. Phys. 2017, 22, 393–421. [Google Scholar] [CrossRef]
- Cao, N.; Chen, S.; Jin, S.; Martinez, D. Physical symmetry and lattice symmetry in the lattice boltzmann method. Phys. Rev. E 1997, 55, R21. [Google Scholar] [CrossRef]
- Mei, R.; Shyy, W. On the finite difference-based lattice boltzmann method in curvilinear coordinates. J. Comput. Phys. 1998, 143, 426–448. [Google Scholar] [CrossRef]
- Guo, Z.; Zhao, T.-S. Explicit finite-difference lattice boltzmann method for curvilinear coordinates. Phys. Rev. E 2003, 67, 066709. [Google Scholar] [CrossRef] [PubMed]
- Sofonea, V.; Sekerka, R.F. Viscosity of finite difference lattice boltzmann models. J. Comput. Phys. 2003, 184, 422–434. [Google Scholar] [CrossRef]
- Sofonea, V.; Lamura, A.; Gonnella, G.; Cristea, A. Finite-difference lattice boltzmann model with flux limiters for liquid-vapor systems. Phys. Rev. E 2004, 70, 046702. [Google Scholar] [CrossRef] [Green Version]
- El-Amin, M.; Sun, S.; Salama, A. On the stability of the finite difference based lattice boltzmann method. Procedia Comput. Sci. 2013, 18, 2101–2108. [Google Scholar] [CrossRef] [Green Version]
- Hejranfar, K.; Ezzatneshan, E. Implementation of a high-order compact finite-difference lattice boltzmann method in generalized curvilinear coordinates. J. Comput. Phys. 2014, 267, 28–49. [Google Scholar] [CrossRef]
- Polasanapalli, S.R.G.; Anupindi, K. A high-order compact finite-difference lattice boltzmann method for simulation of natural convection. Comput. Fluids 2019, 181, 259–282. [Google Scholar] [CrossRef]
- Yoshida, H.; Nagaoka, M. Lattice boltzmann method for the convection–diffusion equation in curvilinear coordinate systems. J. Comput. Phys. 2014, 257, 884–900. [Google Scholar] [CrossRef]
- Rao, P.R.; Schaefer, L.A. Numerical stability of explicit off-lattice boltzmann schemes: A comparative study. J. Comput. Phys. 2015, 285, 251–264. [Google Scholar] [CrossRef] [Green Version]
- Hejranfar, K.; Saadat, M.H. Preconditioned weno finite-difference lattice boltzmann method for simulation of incompressible turbulent flows. Comput. Math. Appl. 2018, 76, 1427–1446. [Google Scholar] [CrossRef]
- Krivovichev, G.V.; Mikheev, S.A. On the stability of multi-step finite-difference-based lattice boltzmann schemes. Int. J. Comput. Methods 2019, 16, 1850087. [Google Scholar] [CrossRef]
- Lee, T.; Lin, C.-L. A characteristic galerkin method for discrete boltzmann equation. J. Comput. Phys. 2001, 171, 336–356. [Google Scholar] [CrossRef]
- Li, Y.; LeBoeuf, E.J.; Basu, P. Least-squares finite-element scheme for the lattice boltzmann method on an unstructured mesh. Phys. Rev. E 2005, 72, 046711. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Li, Y.; LeBoeuf, E.J.; Basu, P. Least-squares finite-element lattice boltzmann method. Phys. Rev. E 2004, 69, 065701. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Bardow, A.; Karlin, I.V.; Gusev, A.A. General characteristic-based algorithm for off-lattice boltzmann simulations. EPL 2006, 75, 434. [Google Scholar] [CrossRef]
- Jo, J.C.; Roh, K.W.; Kwon, Y.W. Finite element based formulation of the lattice boltzmann equation. Methods 2008, 6, 7. [Google Scholar] [CrossRef] [Green Version]
- Krivovichev, G. On the finite-element-based lattice boltzmann scheme. Appl. Math. Sci. 2014, 8, 1605–1620. [Google Scholar] [CrossRef]
- Patel, S.; Lee, T. A new splitting scheme to the discrete boltzmann equation for non-ideal gases on non-uniform meshes. J. Comput. Phys. 2016, 327, 799–809. [Google Scholar] [CrossRef] [Green Version]
- Shu, C.; Peng, Y.; Chew, Y. Simulation of natural convection in a square cavity by taylor series expansion-and least squares-based lattice boltzmann method. Int. J. Mod. Phys. C 2002, 13, 1399–1414. [Google Scholar] [CrossRef] [Green Version]
- Shu, C.; Niu, X.; Chew, Y. Taylor series expansion and least squares-based lattice boltzmann method: Three-dimensional formulation and its applications. Int. J. Mod. Phys. C 2003, 14, 925–944. [Google Scholar] [CrossRef]
- Shu, C.; Niu, X.; Chew, Y. Taylor-series expansion and least-squares-based lattice boltzmann method: Two-dimensional formulation and its applications. Phys. Rev. E 2002, 65, 036708. [Google Scholar] [CrossRef] [Green Version]
- Shu, C.; Chew, Y.; Niu, X. Least-squares-based lattice boltzmann method: A meshless approach for simulation of flows with complex geometry. Phys. Rev. E 2001, 64, 045701. [Google Scholar] [CrossRef]
- Lin, X.; Wu, J.; Zhang, T. A mesh-free radial basis function–based semi-Lagrangian lattice Boltzmann method for incompressible flows. Int. J. Numer. Methods Fluids 2019, 91, 198–211. [Google Scholar] [CrossRef]
- Musavi, S.H.; Ashrafizaadeh, M. Meshless lattice boltzmann method for the simulation of fluid flows. Phys. Rev. E 2015, 91, 023310. [Google Scholar] [CrossRef]
- He, X.; Luo, L.-S.; Dembo, M. Some progress in lattice boltzmann method. Part i. Nonuniform mesh grids. J. Comput. Phys. 1996, 129, 357–363. [Google Scholar] [CrossRef] [Green Version]
- He, X.; Doolen, G.D. Lattice boltzmann method on a curvilinear coordinate system: Vortex shedding behind a circular cylinder. Phys. Rev. E 1997, 56, 434. [Google Scholar] [CrossRef]
- He, X.; Doolen, G. Lattice boltzmann method on curvilinear coordinates system: Flow around a circular cylinder. J. Comput. Phys. 1997, 134, 306–315. [Google Scholar] [CrossRef]
- Chen, H. Volumetric formulation of the lattice boltzmann method for fluid dynamics: Basic concept. Phys. Rev. E 1998, 58, 3955. [Google Scholar] [CrossRef]
- Guo, P.; Qian, F.; Zhang, W.; Yan, H.; Wang, Q.; Zhao, C. Radial basis function interpolation supplemented lattice Boltzmann method for electroosmotic flows in microchannel. Electrophoresis 2021, 42, 2171–2181. [Google Scholar] [CrossRef]
- Bawazeer, S.A.; Baakeem, S.S.; Mohamad, A. A new radial basis function approach based on hermite expansion with respect to the shape parameter. Mathematics 2019, 7, 979. [Google Scholar] [CrossRef] [Green Version]
- Bawazeer, S. Stability and Accuracy of Lattice Boltzmann Method. Master’s Thesis, University of Calgary, Calgary, AB, Canada, 2013. [Google Scholar]
- Lee, H.; Bawazeer, S.; Mohamad, A. Boundary conditions for lattice boltzmann method with multispeed lattices. Comput. Fluids 2018, 162, 152–159. [Google Scholar] [CrossRef]
- Baakeem, S.S.; Bawazeer, S.A.; Mohamad, A. Comparison and evaluation of shan-chen model and most commonly used equations of state in multiphase lattice boltzmann method. Int. J. Multiph. Flow 2020, 128, 103290. [Google Scholar] [CrossRef]
- Bawazeer, S.A.; Baakeem, S.S.; Mohamad, A. A critical review of forcing schemes in lattice boltzmann method: 1993–2019. Arch. Comput. Methods Eng. 2021, 28, 4405–4423. [Google Scholar] [CrossRef]
- Lee, T.; Lin, C.-L. An eulerian description of the streaming process in the lattice boltzmann equation. J. Comput. Phys. 2003, 185, 445–471. [Google Scholar] [CrossRef]
- Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids; Clarendon Press: Oxford, UK, 1992. [Google Scholar]
- Ghia, U.; Ghia, K.N.; Shin, C. High resolutions for incompressible flow using the navier-stokes equations and a multigrid method. J. Comput. Phys. 1982, 48, 387–411. [Google Scholar] [CrossRef]
Model | Mesh | Error | Time per Iteration (s) | ||
---|---|---|---|---|---|
Distribution | Size | Norm | Norm per Node | ||
Streaming | Uniform | 100 | 0.004746 | 4.75 × 10−5 | 1.58 × 10−4 |
Interpolation | Uniform | 100 | 0.734430 | 7.34 × 10−3 | 2.26 × 10−4 |
Interpolation | Stretched | 100 | 2.186103 | 2.19 × 10−2 | 2.28 × 10−4 |
Interpolation | Uniform | 200 | 0.197118 | 9.81 × 10−4 | 2.70 × 10−4 |
Interpolation | Stretched | 200 | 0.074056 | 3.68 × 10−4 | 2.50 × 10−4 |
Interpolation | Uniform | 300 | 0.108697 | 3.56 × 10−4 | 2.92 × 10−4 |
Interpolation | Stretched | 300 | 0.046173 | 1.53 × 10−4 | 2.79 × 10−4 |
Model | Mesh | Error | Time per Iteration (s) | ||
---|---|---|---|---|---|
Distribution | Size | Norm | Norm per Node | ||
Streaming | Uniform | 100 | 0.018873 | 1.89 × 10−4 | 1.66 × 10−4 |
Interpolation | Uniform | 100 | 0.567023 | 5.67 × 10−3 | 2.31 × 10−4 |
Interpolation | Stretched | 100 | 1.317779 | 1.32 × 10−2 | 2.43 × 10−4 |
Interpolation | Uniform | 200 | 0.179421 | 8.93 × 10−4 | 2.54 × 10−4 |
Interpolation | Stretched | 200 | 0.087822 | 4.37 × 10−4 | 2.59 × 10−4 |
Interpolation | Uniform | 300 | 0.117243 | 3.88 × 10−4 | 2.79 × 10−4 |
Interpolation | Stretched | 300 | 0.071195 | 2.36 × 10−4 | 2.92 × 10−4 |
Model | Mesh | No. of Steps | Time | Error w.r.t Ghai et al.’s Results | |||
---|---|---|---|---|---|---|---|
Distribution | Size | Total (s) | Per Iteration (s) | ||||
Streaming | Uniform | 40 | 7227 | 173.9970 | 0.0241 | 0.0410 | 0.0367 |
Interpolation | Uniform | 40 | 8332 | 256.7160 | 0.0308 | 0.0353 | 0.0368 |
Interpolation | Stretched | 40 | 7620 | 237.4643 | 0.0312 | 0.1529 | 0.1028 |
Interpolation | Uniform | 60 | 8395 | 564.0292 | 0.0672 | 0.1214 | 0.0806 |
Interpolation | Stretched | 60 | 8629 | 616.4658 | 0.0714 | 0.0453 | 0.0448 |
Interpolation | Uniform | 80 | 9000 | 1285.9117 | 0.1429 | 0.0372 | 0.0315 |
Interpolation | Stretched | 80 | 8906 | 1302.6666 | 0.1463 | 0.0668 | 0.0532 |
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Bawazeer, S.A.; Baakeem, S.S.; Mohamad, A.A. Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method. Mathematics 2022, 10, 501. https://doi.org/10.3390/math10030501
Bawazeer SA, Baakeem SS, Mohamad AA. Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method. Mathematics. 2022; 10(3):501. https://doi.org/10.3390/math10030501
Chicago/Turabian StyleBawazeer, Saleh A., Saleh S. Baakeem, and Abdulmajeed A. Mohamad. 2022. "Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method" Mathematics 10, no. 3: 501. https://doi.org/10.3390/math10030501
APA StyleBawazeer, S. A., Baakeem, S. S., & Mohamad, A. A. (2022). Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method. Mathematics, 10(3), 501. https://doi.org/10.3390/math10030501