Development of a meshless Galerkin boundary node method for viscous fluid flows. (English) Zbl 1332.76039
Summary: In this paper, a meshless Galerkin boundary node method is developed for boundary-only analysis of the interior and exterior incompressible viscous fluid flows, governed by the Stokes equations, in biharmonic stream function formulation. This method combines scattered points and boundary integral equations. Some of the novel features of this meshless scheme are boundary conditions can be enforced directly and easily despite the meshless shape functions lack the delta function property, and system matrices are symmetric and positive definite. The error analysis and convergence study of both velocity and pressure are presented in Sobolev spaces. The performance of this approach is illustrated and assessed through some numerical examples.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
boundary integral equation; Stokes equations; biharmonic stream function; error analysis; convergenceReferences:
| [1] | Alves, C. J.S.; Silvestre, A. L., Density results using Stokeslets and a method of fundamental solutions for the Stokes equations, Eng. Anal. Bound Elem., 28, 1245-1252 (2004) · Zbl 1079.76058 |
| [2] | Belytchko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077 |
| [3] | Desimone, H.; Urouiza, S.; Arrieta, H.; Pardo, E., Solution of Stokes equations by moving least squares, Commun. Numer. Methods Eng., 14, 907-920 (1998) · Zbl 0919.76066 |
| [4] | Duarte, C. A.; Oden, J. T., H-p clouds—an h-p meshless method, Numer. Methods Partial Diff. Eq., 12, 673-705 (1996) · Zbl 0869.65069 |
| [5] | Gaspar, C., Several meshless solution techniques for the Stokes flow equations, (Ferreira, A. J.M.; Kansa, E. J.; Fasshauer, G. E.; Leitao, V. M.A., Progress on Meshless Methods (2009), Springer: Springer New York), 141-158 · Zbl 1157.76036 |
| [6] | Girault, V.; Raviart, P. A., Finite Element Approximation of the Navier-Stokes Equations (1979), Springer: Springer Berlin · Zbl 0396.65070 |
| [7] | Giroire, J.; Nedelec, J. C., Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comput., 32, 973-990 (1978) · Zbl 0405.65060 |
| [8] | Gresho, P. M.; Sani, R. L., On pressure boundary conditions for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 7, 1111-1145 (1987) · Zbl 0644.76025 |
| [9] | Gu, Y. T.; Liu, G. R., A boundary radial point interpolation method (BRPIM) for 2-D structural analyses, Struct. Eng. Mech., 15, 535-550 (2003) |
| [10] | Hwu, T. Y.; Young, D. L.; Chen, Y. Y., Chaotic advections for Stokes flows in circular cavity, J. Eng. Mech., 123, 774-782 (1997) |
| [11] | Li, X., Meshless analysis of two-dimensional Stokes flows with the Galerkin boundary node method, Eng. Anal. Bound Elem., 34, 79-91 (2010) · Zbl 1244.76079 |
| [12] | X. Li, Adaptive methodology for the meshless Galerkin boundary node method, Eng. Anal. Bound Elem. 35 (2011) 750-760.; X. Li, Adaptive methodology for the meshless Galerkin boundary node method, Eng. Anal. Bound Elem. 35 (2011) 750-760. · Zbl 1259.65170 |
| [13] | Li, X.; Zhu, J., A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math., 230, 314-328 (2009) · Zbl 1189.65291 |
| [14] | Li, X.; Zhu, J., A Galerkin boundary node method for biharmonic problems, Eng. Anal. Bound Elem., 33, 858-865 (2009) · Zbl 1244.65175 |
| [15] | Li, X.; Zhu, J., A Galerkin boundary node method for two-dimensional linear elasticity, CMES: Comput. Model. Eng. Sci., 45, 1-29 (2009) · Zbl 1357.74008 |
| [16] | Li, X.; Zhu, J., A meshless Galerkin method for Stokes problems using boundary integral equations, Comput. Methods Appl. Mech. Eng., 198, 2874-2885 (2009) · Zbl 1229.76076 |
| [17] | Li, X.; Zhu, J.; Meshless, Galerkin analysis of Stokes slip flow with boundary integral equations, Int. J. Numer. Methods Fluids, 61, 1201-1226 (2009) · Zbl 1422.76047 |
| [18] | X. Li, J. Zhu, Galerkin boundary node method for exterior Neumann problems, J. Comput. Math. 29 (2011) 243-260.; X. Li, J. Zhu, Galerkin boundary node method for exterior Neumann problems, J. Comput. Math. 29 (2011) 243-260. · Zbl 1249.65243 |
| [19] | Li, X.; Zhu, J.; Zhang, S., A hybrid radial boundary node method based on radial basis point interpolation, Eng. Anal. Bound Elem., 33, 1273-1283 (2009) · Zbl 1244.65228 |
| [20] | Liu, G. R.; Gu, Y. T., An Introduction to Meshfree Methods and Their Programming (2005), Springer: Springer Dordrecht · Zbl 1109.74371 |
| [21] | Lobovský, L.; Vimmr, J., Smoothed particle hydrodynamics and finite volume modelling of incompressible fluid flow, Math. Comput. Simul., 76, 124-131 (2007) · Zbl 1132.76043 |
| [22] | Mosov, V., Meshless RKHPU method and its applications, Math. Comput. Simul., 76, 161-165 (2007) · Zbl 1135.65399 |
| [23] | Mukherjee, Y. X.; Mukherjee, S., The boundary node method for potential problems, Int. J. Numer. Methods Eng., 40, 797-815 (1997) · Zbl 0885.65124 |
| [24] | Nedelec, J. C.; Artola, M.; Cessenat, M., Integral equations and numerical methods, (Dautray, R.; Lious, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4 (2000), Springer: Springer Berlin), 33-152 · Zbl 0944.47002 |
| [25] | Nguyena, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math. Comput. Simul., 79, 763-813 (2008) · Zbl 1152.74055 |
| [26] | Nicolazzi, L. C.; Barcellos, C. S.; Fancello, E. A.; Duarte, C. A.M., Generalized boundary element method for Galerkin boundary integrals, Eng. Anal. Bound Elem., 29, 494-510 (2005) · Zbl 1182.65182 |
| [27] | Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on Lagrangian kernels, Comput. Methods Appl. Mech. Eng., 193, 1035-1063 (2004) · Zbl 1060.74672 |
| [28] | Shrivastava, V.; Aluru, N. R., A fast boundary cloud method for 3D exterior electrostatic analysis, Int. J. Numer. Methods Eng., 59, 2019-2046 (2004) · Zbl 1170.78429 |
| [29] | Sladek, V.; Sladek, J., Regularization of hypersingular and nearly singular integrals in potential theory and elasticity, Int. J. Numer. Methods Eng., 36, 1609-1628 (1993) · Zbl 0772.73091 |
| [30] | Svacek, P., On energy conservation for finite element approximation of flow-induced airfoil vibrations, Math. Comput. Simul., 80, 1713-1724 (2010) · Zbl 1426.76060 |
| [31] | Tsai, C. C.; Young, D. L.; Cheng, A. H.D., Meshless BEM for three-dimensional Stokes flows, CMES: Comput. Model. Eng. Sci., 3, 117-128 (2002) · Zbl 1147.76595 |
| [32] | Young, D. L.; Jane, S. J.; Fan, C. M.; Murugesan, K.; Tsai, C. C., The method of fundamental solutions for 2D and 3D Stokes problems, J. Comput. Phys., 211, 1-8 (2006) · Zbl 1160.76332 |
| [33] | Young, D. L.; Jane, S. C.; Lin, C. Y.; Chiu, C. L.; Chen, K. C., Solutions of 2D and 3D Stokes laws using multiquadrics method, Eng. Anal. Bound Elem., 28, 1233-1243 (2004) · Zbl 1178.76290 |
| [34] | Young, D. L.; Wu, J. T.; Chin, C. L., Method of fundamental solutions for stokes problems by the pressure-stream function formulation, J. Mech., 24, 137-144 (2008) |
| [35] | Zhang, X. K.; Kwon, K. C.; Youn, S. K., The least-squares meshfree method for the steady incompressible viscous flow, J. Comput. Phys., 206, 182-207 (2005) · Zbl 1087.76086 |
| [36] | Zhang, J.; Tanaka, M.; Endo, M., The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems, Int. J. Numer. Methods Eng., 63, 660-680 (2005) · Zbl 1085.65115 |
| [37] | Zhu, J.; Yuan, Z., Boundary Element Analysis (2009), Science Press: Science Press Beijing |
| [38] | Zhu, J.; Zhang, S., Galerkin boundary element method for solving the boundary integral equation with hypersingularity, J. Univ. Sci. Technol. China, 37, 1357-1362 (2007) · Zbl 1174.65538 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
