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Meshless techniques for convection dominated problems

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Abstract

In this paper, the stability problem in the analysis of the convection dominated problems using meshfree methods is first discussed through an example problem of steady state convection-diffusion. Several techniques are then developed to overcome the instability issues in convection dominated phenomenon simulated using meshfree collocation methods. These techniques include: the enlargement of the local support domain, the upwind support domain, the adaptive upwind support domain, the biased support domain, the nodal refinement, and the adaptive analysis. These techniques are then demonstrated in one- and two-dimensional problems. Numerical results for example problems demonstrate the techniques developed in this paper are effective to solve convection dominated problems, and in these techniques, using adaptive local support domain is the most effective method. Comparing with the conventional finite difference method (FDM) and the finite element method (FEM), the meshfree method has found some attractive advantages in solving the convection dominated problems, because it easily overcomes the instability issues.

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References

  1. Atluri SN, Shen SP (2002) The Meshless Local Petrov-Galerkin (MLPG) method. Tech Science Press, Encino USA

  2. Atluri SN, Kim HG, Cho JY (1999) A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Comput Mech 24:348–372

    Article  MATH  MathSciNet  Google Scholar 

  3. Behrens J, Iske A, Käser M (2002) Adaptive meshfree method of backward characteristics for nonlinear transport equations, Meshfree Methods for Partial Differential Equations. Gribel, M. Schweitzer, MA (eds) Springer-Verlag, Heidelberg, pp 21–36

  4. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Engrg 37:229–256

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng M, Liu GR (2002) A novel finite point method for flow simulation. Int J Numer Meth Fl 39(12):1161–1178

    Article  MATH  Google Scholar 

  6. De S, Bathe KJ (2000) The method of finite spheres. Comput Mech 25:329–345

    Article  MATH  MathSciNet  Google Scholar 

  7. Gu YT, Liu GR (2001a) A meshless Local Petrov-Galerkin (MLPG) formulation for static and free vibration analysis of thin plates. Comput Model Eng Sci 2(4):463–476

    MATH  Google Scholar 

  8. Gu YT, Liu GR (2001b) A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput Mech 27:188–198

    Article  MATH  Google Scholar 

  9. Gu YT, Liu GR (2001c) A local point interpolation method for static and dynamic analysis of thin beams. Computer Methods Appl Mech Engrg 190:5515–5528

    Article  MATH  Google Scholar 

  10. Kansa EJ (1990), Multiquadrics-A Scattered Data Approximation Scheme with Applications to Computational Fluid dynamics. Computers Math Applic 19(8/9):127–145

  11. Lam KY, Wang QX, Hua Li (2004) A novel meshless approach – Local Kriging (LoKriging) method with two-dimensional structural analysis. Comput Mech 33:235–244

    Article  MATH  MathSciNet  Google Scholar 

  12. Liszka TJ, Duarte CA, Tworzydlo WW (1996) Hp-Meshless cloud method. Comput. Methods Appl Mech Engrg 139:263–288

    Article  MATH  Google Scholar 

  13. Lin H, Atluri SN (2000) Meshless Local Petrov-Galerkin (MLPG) method for conection-difusion problems. Comput Model Engrg Sci 1(2):45–60

    MATH  Google Scholar 

  14. Liu GR (2002) Mesh Free Methods: Moving Beyond the Finite Element Method. CRC press, Boca Raton, USA

    Google Scholar 

  15. Liu GR, Gu YT (2001) A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. J Sound Vibr 246(1):29–46

    Article  Google Scholar 

  16. Liu GR, Gu YT (2005) An introduction to MFree methods and their programming. Springer Press, Berlin

  17. Liu Xin, Liu GR, Tai Kang, Lam KY (2002) Radial Basis Point Interpolation Collocation Method For 2-d Solid Problem, Advances in Meshfree and X-FEM Methods. In: Liu GR eds. Proceedings of the 1st Asian Workshop on Meshfree methods World Scientific, 35–40

  18. Oñate E, Idelsohn S, Zienkiewicz OZ, Taylor RL (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Engrg 39:3839–3867

    Article  Google Scholar 

  19. Oñate E (1998), Derivation of stabilized equations for numerical solution of sdvictive-diffusive transport and fluid flow problems. Comp Meth Appl Mech Engrg 151:233–265

    Article  Google Scholar 

  20. Wang JG and Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Method Eng 54:1623–1648

    Article  MATH  Google Scholar 

  21. Wang QX, Hua L, Lam KY et al. (2004) Analysis of microelectromechanical systems (MEMS) by meshless local kriging (Lokriging) method. J Chinese Inst Eng 27(4):573–583

    MATH  Google Scholar 

  22. Wang QX, Li H, Lam KY (2005) Development of a new meshless - point weighted least-squares (PWLS) method for computational mechanics. Comput Mech 35(3):170–181

    Article  MATH  MathSciNet  Google Scholar 

  23. Wu Z (1992) Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx Theory Appl 8:1–10

    Google Scholar 

  24. Wu Z and Schaback R (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J Numer Anal 13:13–27

    MathSciNet  Google Scholar 

  25. Xiao JR, McCarthy MA (2003a) Meshless analysis of the obstacle problem for beams by the MLPG method and subdomain variational formulations. Eur J Mech A-Solid 22(3):385–399

    Google Scholar 

  26. Xiao JR, McCarthy MA (2003b) On the use of radial basis functions in a local weighted meshless method. Second M.I.T. Conference on Computational Fluid and Solid Mechanics. June 17–20

  27. Zienkiewicz OC, Taylor RL (2000) The Finite Element Method, 5th ed. B-H, Oxford, UK

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Authors and Affiliations

  1. Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260

    Y. T. Gu & G. R. Liu

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  1. Y. T. Gu
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  2. G. R. Liu
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Correspondence to Y. T. Gu.

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Gu, Y., Liu, G. Meshless techniques for convection dominated problems. Comput Mech 38, 171–182 (2006). https://doi.org/10.1007/s00466-005-0736-8

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  • DOI: https://doi.org/10.1007/s00466-005-0736-8

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