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国际计算热科学期刊
ESCI SJR: 0.249 SNIP: 0.434 CiteScore™: 1.4

ISSN 打印: 1940-2503
ISSN 在线: 1940-2554

国际计算热科学期刊

DOI: 10.1615/ComputThermalScien.2019025846
pages 401-422

LOCALIZED RADIAL BASIS FUNCTIONS AND DIFFERENTIAL QUADRATURE-MESHLESS METHOD FOR SIMULATING COMPRESSIBLE FLOWS

Ebrahim Nabizadeh
Department of Mechanical Engineering, Rice University, Houston, Texas 77005, USA
Darrell W. Pepper
NCACM, Department of Mechanical Engineering, University of Nevada Las Vegas, Las Vegas, NV 89154, USA

ABSTRACT

A numerical approach based on the meshless method is used to simulate compressible flow. The meshless, or mesh-free, method circumvents the need to generate a mesh. Since there is no connectivity among the nodes, the method can be easily implemented for any geometry. However, one of the most fundamental issues in numerically simulating compressible flow is the lack of conservation, which can be a source of unpredictable errors in the solution process. This problem is particularly evident in the presence of steep gradient regions and shocks that frequently occur in highspeed compressible flow problems. To resolve this issue, a conservative localized meshless method based on radial basis functions and differential quadrature (RBF-DQ) has been developed. An upwinding scheme, based on the Roe method, is added to capture steep gradients and shocks. In addition, a blended RBF is used to decrease the dissipation ensuing from the use of low shape parameters. A set of test problems are used to confirm the accuracy and reliability of the algorithm, and the method applied to the solution of Euler's equation.

REFERENCES

  1. Afiatdoust, F. and Esmaeilbeigi, M., Optimal Variable Shape Parameters Using Genetic Algorithm for Radial Basis Function Approximation, Ain Shams Eng. J., vol. 6, no. 2, pp. 639-647,2015.

  2. Atluri, S.N. and Zhu, T.L., A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics, Comput. Mech., vol. 22, pp. 117-127,1998.

  3. Atluri, S.N. and Zhu, T.L., New Concepts in Meshless Methods, Int. J. Numer. Methods Eng., vol. 47, pp. 537-556,2000.

  4. Batina, J.T., A Gridless Euler/Navier-Stokes Solution Algorithm for Complex Aircraft Applications, AIAA Paper, vol. 93-0333, 1993.

  5. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., and Krysl, P., Meshless Methods: An Overview and Recent Developments, Comput. Methods Appl. Mech. Eng., vol. 139, nos. 1-4, pp. 3-47,2003.

  6. Belytschko, T., Lu, Y., and Gu, L., Element-Free Galerkin Methods, Int. J. Numer. Methods Eng., vol. 37, pp. 229-256,1994.

  7. Chinchapatnam, P.P., Djidjeli, K., and Nair, P.B., Radial Basis Function Meshless Method for the Steady Incompressible Navier-Stokes Equations, Int. J. Comput. Math., vol. 84, pp. 1509-1521,2007.

  8. Chiu, K.-Y.E., Wang, Q.Q., Hu, R., and Jameson, A., A Conservative Mesh-Free Scheme and Generalized Framework for Conservation Laws, SIAM. J. Sci. Comput, vol. 34, no. 6, pp. 2896-2916,2012.

  9. Divo, E., and Kassab, A.J., An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer, J. Heat Transf., vol. 129, pp. 124-136,2007.

  10. Duarte, C.A. and Oden, J.T., An H-P Adaptive Method Using Clouds, Comput. Methods Appl. Mech. Eng., vol. 139, pp. 237-262, 1996.

  11. Fantuzzi, N., Tornabene, F., Viola, E., and Ferreira, A.J.M., A Strong Formulation Finite Element Method (SFEM) based on RBF and GDQ Techniques for the Static and Dynamics Analysis of Laminated Plates of Arbitrary Shape, Meccanica, vol. 49, pp. 2503-2542,2014.

  12. Franke, C. and Schaback, R., Solving Partial Differential Equations by Collocation Using Radial Basis Functions, Appl. Math. Comput., vol. 93, no. 1, pp. 73-82,1998.

  13. Franke, R., Scattered Data Interpolation: Tests of Some Methods, Math. Comput., vol. 38, no. 157, pp. 181-200,1982.

  14. Gingold, R.A. and Monaghan, J., Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars, Mon. Not. R. Astron. Soc., vol. 181, pp. 375-389,1977.

  15. Godunov, S.K., A Finite-Difference Method for the Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics, Mat. Sb., vol. 47, pp. 271-306,1959.

  16. Hardy, R.L., Multiquadric Equations of Topography and Other Irregular Surfaces, J. Geo. Res., vol. 76, no. 8, pp. 1905-1915, 1971.

  17. Harris, M., Kassab, A., and Divo, E., An RBF Interpolation Blending Scheme for Effective Shock-Capturing, Int. J. Comput. Methods Exp. Meas., vol. 5, no. 3, pp. 281-292,2017.

  18. Hou, T. Y. and LeFloch, P.G., Why Nonconservative Schemes Converge to Wrong Solutions: Error Analysis, Math. Comput., vol. 62, no. 206, pp. 497-530,1994.

  19. Kansa, E.J., 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.

  20. Larsson, E. and Fornberg, B., 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.

  21. Lax, P. and Wendroff, B., Systems of Conservation Laws, Commun. Pure Appl. Math, vol. 13, no. 2, pp. 217-237,1960.

  22. LeVeque, R.J., Numerical Methods for Conservation Laws, New York: Springer Science and Business Media, 1992.

  23. Ling, L. and Kansa, E.J., A Least-Squares Preconditioner for Radial Basis Functions Collocation Methods, Adv. Comput. Math., vol. 23, no. 1,pp. 31-54,2005.

  24. Liszka, T.J., Duarte, C.A.M., and Tworzydlo, W.W., HP-Meshless Cloud Method, Comput. Methods Appl. Mech. Eng., vol. 139, pp. 263-288,1996.

  25. Liu, G.R., Meshfree Methods: Moving Beyond the Finite Element Method, Boca Raton, FL: CRC Press, 2003.

  26. Liu, M.B. and Liu, G.R., Smoothed Particle Hydrodynamics (SPH): An Overview and Recent Developments, Arch. Comput. Meth. Eng., vol. 17, pp. 25-76,2010.

  27. Liu, W.K., Jun, S., Li, S., Adee, J., and Belytschko, T., Reproducing Kernel Particle Methods for Structural Dynamics, Int. J. Numer. Meth. Eng., vol. 38, pp. 1655-1679,1985.

  28. Lu, Y. Y., Belytschko, T., and Gu, L., A New Implementation of the Element Free Galerkin Method, Comput. Methods Appl. Mech. Eng., vol. 113, pp. 397-414,1994.

  29. Lucy, L.B., A Numerical Approach to the Testing of the Fission Hypothesis, Astron. J, vol. 82, pp. 1013-1024,1977.

  30. Mai-Duy, N. and Tran-Cong, T., Mesh-Free Radial Basis Function Network Methods with Domain Decomposition for Approximation of Functions and Numerical Solution of Poisson's Equations, Eng. Anal. Bound. Elem, vol. 26, no. 2, pp. 133-156, 2002.

  31. Monaghan, J., An Introduction to SPH, Comput. Phys. Commun., vol. 48, pp. 89-96,1988.

  32. Morinishi, K., A Gridless Type Solution for High Reynolds Number Multielement Flow Fields, AIAA Paper, 95-1856, 1995.

  33. Mukherjee, Y.X. and Mukherjee, S., The Boundary Node Method for Potential Problems, Int. J. Numer. Methods Eng., vol. 40.5, pp. 797-815,1997.

  34. Nayroles, B., Touzot, G., and Villon, P., Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Comput. Mech, vol. 10, pp. 307-318,1992.

  35. Noguchi, H., Kawashima, T., and Miyamura, T., Element Free Analyses of Shell and Spatial Structures, Int. J. Numer. Methods Eng., vol. 47, pp. 1215-1240,2000.

  36. Onate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L., and Sacco, C., A Stabilized Finite Point Method for Analysis of Fluid Mechanics Problems, Comput. Methods Appl. Mech. Eng., vol. 139, pp. 315-346,1996.

  37. Randles, P.W. and Libersky, L.D., Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications, Comput. Methods Appl. Mech. Eng., vol. 139, pp. 375-408,1996.

  38. Roe, P.L., Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, J. Comput. Phys, vol. 43, no. 2, pp. 357-372,1981.

  39. Sarler, B. and Vertnik, R., Meshfree Explicit Local Radial Basis Function Collocation Method for Diffusion Problems, Comput. Math. Appl., vol. 51, no. 8, pp. 1269-1282,2006.

  40. Shu, C., Ding, H., Chen, H.Q., and Wang, T.G., An Upwind Local RBF-DQ Method for Simulation of Inviscid Compressible Flows, Comput. Methods Appl. Mech. Eng., vol. 194, no. 18, pp. 2001-2017,2005.

  41. Shu, C., Ding, H., and Yeo, K.S., Local Radial Basis Function-Based Differential Quadrature Method and its Application to Solve Two-Dimensional Incompressible Navier-Stokes Equations, Comput. Methods Appl. Mech. Eng., vol. 192, no. 7, pp. 941-954, 2003.

  42. Sweby, P.K., High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM J. Numer. Anal., vol. 21, no. 5, pp. 995-1011,1984.

  43. Tornabene, F., Fantuzzi, N., Ubertini, F., and Viola, E., Strong Formulation Finite Element Method based on Differential Quadrature: A Survey, Appl. Mech. Rev., vol. 67, no. 2, p. 020801,2015.

  44. Tota, P. V. and Wang, Z.J., Meshfree Euler Solver Using Local Radial Basis Functions for Inviscid Compressible Flows, 18th AIAA Comput. Fluid Dynam. Conf, Miami, FL, 2007.

  45. Wang, J.G. and Liu, G.R., On the Optimal Shape Parameters of Radial Basis Functions Used for 2-D Meshless Methods, Comput. Methods Appl. Mech. Eng., vol. 191, no. 23, pp. 2611-2630,2002.

  46. Wang, J.C.T. and Widhopf, G.F., A High-Resolution TVD Finite Volume Scheme for the Euler Equations in Conservation Form, J. Comput. Phys, vol. 84, no. 1, pp. 145-173,1989.

  47. Waters, J. and Pepper, D.W., Global versus Localized RBF Meshless Methods for Solving Incompressible Fluid Flow with Heat Transfer, Numer. Heat Transf., Part B, vol. 68, no. 3, pp. 185-203,2015.

  48. Wu, Y.L. and Shu, C., Development of RBF-DQ Method for Derivative Approximation and Its Application to Simulate Natural Convection in Concentric Annuli, Comput. Mech, vol. 29, no. 6, pp. 477-485,2002.

  49. Yao, G., Islam, S.-U., and Sarler, B., A Comparative Study of Global and Local Meshless Methods for Diffusion-Reaction Equation, CMES, vol. 59, no. 2, pp. 127-154,2010.

  50. Zhou, X., Hon, Y.C., and Li, J., Overlapping Domain Decomposition Method by Radial Basis Functions, Appl. Numer. Math., vol. 44, nos. 1-2, pp. 241-255,2003.


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