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A meshless geometric multigrid method for a grid with a high aspect ratio

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Abstract

In this paper, a new semi-coarsening method is proposed by extending the meshless geometric multigrid method based on full-coarsening for an anisotropic unstructured mesh. The present semi-coarsening method is based on the distance weighting for a finite volume grid and validated by solving the pressure equation of CUPID, which is a CFD code with an unstructured grid. Although the present semi-coarsening method requires more coarsening steps than the full-coarsening method to build the coarsest grid, the numbers of V-cycles required to the convergence are nearly the same for the two methods. Therefore, the proposed semi-coarsening method is found to perform much better than the conventional full-coarsening method when an unstructured grid with a high aspect ratio is involved in the V-cycle multi-grid computation. Furthermore, the present method works well with a hybrid mesh, where brick elements with high aspect ratios near the wall boundary are coupled with isotropic tetrahedrons.

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Abbreviations

α :

Volume fraction

ρ :

Density

u :

Velocity vector

P :

Pressure

Τ :

Viscous tensor

g :

Gravity vector

M i :

Interface momentum transfer rate

e :

Internal energy

E D :

Energy diffusion

Q i :

Interface energy transfer rate

Ω h :

Finest grid

v 1 :

Pre-smoothing step

v 2 :

Post-smoothing step

R :

Restriction operator

P :

Interpolation operator

A :

Stiffness matrix

k :

k-phase (gas or liquid)

References

  1. J. J. Jeong, H. Y. Yoon, I. K. Park and H. K. Cho, The CUPID code development and assessment strategy, Nuclear Engineering and Technology, 42(6) (2010) 636–655.

    Article  Google Scholar 

  2. H. K. Cho, S. J. Lee, H. K. Yoon, H. K. Kang and J. J. Jeong, Simulation of single-and two-phase natural circulation in the passive condensate cooling tank using the CUPID code, Journal of Nuclear Science and Technology, 50(7) (2013) 709–722.

    Article  Google Scholar 

  3. H. K. Yoon, J. R. Lee, H. Kim, I. K. Park, C. H. Song, H. K. Cho and J. J. Jeong, Recent improvements in the CUPID code for a multi-dimensional two-phase flow analysis of nuclear reactor components, Nuclear Engineering and Technology, 46(5) (2014) 655–666.

    Article  Google Scholar 

  4. J. R. Lee, H. Y. Yoon and H. G. Choi, Domain decomposition parallel computing for transient two-phase flow of nuclear reactors, Journal of Mechanical Science and Technology, 30(5) (2016) 2045–2057.

    Article  Google Scholar 

  5. S. T. Ha and H. G. Choi, A meshless geometric multigrid method based on a node-coarsening algorithm for the linear finite element discretization, Computers and Mathematics with Applications, 96 (2021) 31–43.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. J. Mavriplis, Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, Journal of Computational Physics, 145(1) (1998) 141–165.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Oliveira, M. A. V. Pinto, C. H. Marchi and L. K. Araki, Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids, Applied Mathematical Modelling, 36(10) (2012) 4665–4676.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Wittum, Linear iterations as smoothers in multigrid methods: theory with applications to incomplete decompositions, IMPACT of Computing in Science and Engineering, 1(2) (1989) 180–215.

    Article  MATH  Google Scholar 

  9. P. M. De Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, Journal of Computational and Applied Mathematics, 33(1) (1990) 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Wesseling and C. W. Oosterlee, Geometric multigrid with applications to computational fluid dynamics, Journal of Computational and Applied Mathematics, 128(1–2) (2001) 311–334.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. C. Lai, C. T. Wu and Y. H. Tseng, An efficient semicoarsening multigrid method for variable diffusion problems in cylindrical coordinates, Applied Numerical Mathematics, 57(5–7) (2007) 801–810.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. A. Mulder, A new multigrid approach to convection problems, 11th International Conference on Numerical Methods in Fluid Dynamics, Springer, Berlin, Heidelberg (1989).

    Google Scholar 

  13. H. B. Zubair, S. P. MacLachlan and C. W. Oosterlee, A geometric mu ltigrid method based on L-shaped coarsening for PDEs on stretched grids, Numerical Linear Algebra with Applications, 17(6) (2010) 871–894.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Francescatto and A. Dervieux, A semi-coarsening strategy for unstructured multigrid based on agglomeration, International Journal for Numerical Methods in Fluids, 26(8) (1998) 927–957.

    Article  MATH  Google Scholar 

  15. Y. Mesri, H. Guillard and T. Coupez, Automatic coarsening of three dimensional anisotropic unstructured meshes for multi-grid applications, Applied Mathematics and Computation, 218 (2012) 10500–10519.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Gruau and T. Coupez, 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multi-domain metric, Computer Methods in Applied Mechanics and Engineering, 194(48–49) (2005) 4951–4976.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. T. Ha, H. Y. Yoon and H. G. Choi, Performance comparison of interpolation operators of multigrid methods based on distance weight and area (volume) intersection approaches part I: finite volume discretization, Journal of Mechanical Science and Technology, 33(5) (2019) 2219–2224.

    Article  Google Scholar 

  18. S. T. Ha, Development of a new geometric multi-grid finite element method and a semi-implicit partitioned algorithm for fluid-structure interaction simulation, Ph.D. Dissertation, Seoul National University of Science and Technology, Republic of Korea (2019).

    Google Scholar 

  19. J. R. Lee and H. Y. Yoon, Multi-physics simulation of nuclear reactor core by coupled simulation using CUPID/MASTER, International Journal of Heat and Mass Transfer, 115 (2017) 1020–1032.

    Article  Google Scholar 

  20. J. J. Jeong, H. Y. Yoon, I. K. Park, H. K. Cho and J. Kim, A semi-implicit numerical scheme for transient two-phase flows on unstructured grids, Nuclear Engineering and Design, 238 (2008) 3403–3412.

    Article  Google Scholar 

  21. A. Chemin, T. Elguedj and A. Gravouil, Isogeometric local h-refinement strategy based on multigrids, Finite Elem. Anal. Des., 100 (2015) 77–90.

    Article  MathSciNet  Google Scholar 

  22. Y. T. Feng, D. Perić and D. R. J. Owen, A non-nested Galerkin multi-grid method for solving linear and nonlinear solid mechanics problems, Comput. Methods Appl. Mech. Eng., 144(3–4) (1997) 307–325.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. T. Ha and H. G. Choi, Performance comparison of interpolation operators of multi-grid method based on a distance weighted and area (volume) intersection approaches, part II: finite element discretization, Journal of Mechanical Science and Technology, 33(7) (2019) 3323–3331.

    Article  Google Scholar 

  24. P. R. Brune, M. G. Knepley and L. R. Scott, Unstructured geometric multigrid in two and three dimensions on complex and graded meshes, SIAM J. Sci. Comput., 35(1) (2013) A173–A191.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. A. 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. Comput., 13 (1992) 631–644.

    Article  MathSciNet  MATH  Google Scholar 

  26. I. K. Park, J. R. Lee, Y. H. Choi and D. H. Kang, A multi-scale and multi-physics approach to main steam line break accidents using coupled MASTER/CUPID/MARS code, Annals of Nuclear Energy, 135 (2020) 106972.

    Article  Google Scholar 

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Acknowledgments

This research was supported by the Nuclear Safety Research Program through the Korea Foundation of Nuclear Safety (KOFONS) under a grant from the Nuclear Safety and Security Commission (NSSC) (Grant No. 210621).

Author information

Authors and Affiliations

  1. Dept. of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Viet Nam

    Sang Truong Ha

  2. Dept. of Mechanical and Automotive Engineering, Seoul National University of Science and Technology, Seoul, Korea

    Sang Truong Ha & Hyoung Gwon Choi

  3. Korea Atomic Energy Research Institute, 989-111 Daeduk-daero, Daejeon, 34057, Korea

    Seong Ju Do & Han Young Yoon

Authors
  1. Sang Truong Ha
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  2. Hyoung Gwon Choi
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  3. Seong Ju Do
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  4. Han Young Yoon
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Corresponding author

Correspondence to Han Young Yoon.

Additional information

Sang Truong Ha is a Researcher of the Department of Mechanical Engineering at Le Quy Don technical University, Ha Noi, Viet Nam. He received his Ph.D. in Mechanical Engineering at Seoul National University of Science and Technology, Korea. His research interests include computational fluid dynamics, fluid-structure interaction and multi-grid method.

Hyoung Gwon Choi is a Professor in the Department of Mechanical/Automotive Engineering, Seoul National University of Science and Technology. He obtained a Ph.D., major in the development of CFD algorithms of finite element method, from Seoul National University, Korea.

Seong Ju Do is a Senior Researcher of the Multi-physics Computational Science Research Division, Korea Atomic Energy Research Institute, Korea. He received his Ph.D. in Mathematics from Seoul National University. His research interest includes high order numerical schemes for solving conservation laws and scientific computing.

Han Young Yoon is a Director of the Multi-physics Computational Science Research Division, Korea Atomic Energy Research Institute, Korea. He received his Ph.D. in Nuclear Engineering from University of Tokyo. His research interest includes the 2-phase flow numerical method and the multi-physics simulation of nuclear reactors.

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Ha, S.T., Choi, H.G., Do, S.J. et al. A meshless geometric multigrid method for a grid with a high aspect ratio. J Mech Sci Technol 36, 5551–5559 (2022). https://doi.org/10.1007/s12206-022-1019-4

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  • DOI: https://doi.org/10.1007/s12206-022-1019-4

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