zbMATH — the first resource for mathematics

The least-squares meshfree method for the steady incompressible viscous flow. (English) Zbl 1087.76086
Summary: A least-squares meshfree method (LSMFM) based on the first-order velocity-pressure-vorticity formulation for two-dimensional steady incompressible viscous flow is presented. The discretization of all governing equations is implemented by the least-squares method. The equal-order moving least-squares (MLS) approximation is employed. Gauss quadrature is used in the background cells constructed by the quadtree algorithm and the boundary conditions are enforced by the penalty method. The matrix-free element-by-element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. A numerical example with analytical solution for the Stokes problem is devised to analyze the error estimates of the LSMFM. Also, cavity-driven flow for the Stokes problem and the flow past a circular cylinder at low Reynolds numbers for the steady incompressible viscous flow are solved. Through the comparisons of the LSMFM’s results with other experimental and numerical results, the numerical features of the presented LSMFM are investigated and discussed.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Oden, J.T.; Jacquotte, O.P., Stability of some mixed finite element methods for Stokesian flows, Comput. methods. appl. mech. engrg., 43, 231-248, (1984) · Zbl 0598.76033
[2] Ferziger, J.H.; Peric, M., Computational methods for fluid dynamics, (1996), Springer Berlin · Zbl 0869.76003
[3] Hughes, T.J.R.; Liu, W.K.; Brooks, A., Finite element analysis of incompressible viscous flows by the penalty function formulation, J. comput. phys., 30, 1-75, (1979)
[4] Hughes, T.J.R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Comput. methods. appl. mech. engrg., 7, 1261-1295, (1987) · Zbl 0638.76080
[5] Jiang, B.N., The least-squares finite element method-theory and applications in computational fluid dynamics and electromagnetics, (1998), Springer Berlin
[6] Lucy, L.B., A numerical approach to the testing the fission hypothesis, Astron. J., 8, 1013-1024, (1977)
[7] Gingold, R.A.; Monaghan, J.J., Kernal estimates as a basis for general particle methods in hydrodynamics, J. comput. phys., 46, 429-453, (1982) · Zbl 0487.76010
[8] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. struct., 11, 83-95, (1980) · Zbl 0427.73077
[9] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[10] Belytschko, T.; Krongauz, Y.; Fleming, M.; Organ, D.; Liu, W.K., Smoothing and accelerated computation in the element free Galerkin method, J. comput. appl. math., 74, 111-126, (1996) · Zbl 0862.73058
[11] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. methods engrg., 38, 1655-1679, (1995) · Zbl 0840.73078
[12] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[13] Babuska, I.; Melenk, J.M., The partition of unity method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[14] C.A. Duarte, J.T. Oden, Hp clouds - a meshless method to solve boundary-value problems, Technical Report 95-05, Texas Institute for Computational and Applied Mathematics, Austin, 1995
[15] Duarte, C.A.; Oden, J.T., An h-p adaptive method using clouds, Comput. methods. appl. mech. engrg., 139, 237-262, (1996) · Zbl 0918.73328
[16] Atluri, S.N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 117-127, (1998) · Zbl 0932.76067
[17] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse element approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[18] Liu, W.K.; Jun, S.; Sihling, D.T.; Chen, Y.J.; Hao, W., Multiresolution reproducing kernel particle method for computational fluid dynamics, Int. J. numer. methods fluids, 24, 1391-1415, (1997) · Zbl 0880.76057
[19] Sadat, H.; Couturier, S., Performance and accuracy of a meshless method for laminar natural convection, Numer heat tranf. B, 37, 455-467, (2000)
[20] Yagawa, G.; Shirazaki, M., Parallel computing for incompressible flow using a nodal-based method, Comput. mech., 23, 209-217, (1999) · Zbl 0962.76553
[21] Cheng, M.; Liu, G.R., A novel finite point method for flow simulation, Int. J. numer. methods fluids, 39, 1161-1178, (2002) · Zbl 1053.76056
[22] Kim, D.W.; Kim, Y.S., Point collocation methods using the fasting moving least square reproducing kernel approximation, Int. J. numer. methods engrg., 56, 1445-1464, (2003) · Zbl 1054.76066
[23] Park, S.H.; Youn, S.K., The least-squares meshfree method, Int. J. numer. methods engrg., 52, 997-1012, (2001) · Zbl 0992.65123
[24] Park, S.H.; Kwon, K.C.; Youn, S.K., A study on the convergence of least-squares meshfree method under inaccurate integration, Int. J. numer. methods engrg., 56, 1397-1419, (2003) · Zbl 1027.65141
[25] Park, S.H.; Kwon, K.C.; Youn, S.K., A posterior error estimates and an adaptive scheme of least-squares meshfree method, Int. J. numer. methods engrg., 58, 1213-1250, (2003) · Zbl 1032.76655
[26] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods. appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075
[27] Krysl, P.; Belytschko, T., ESFLIB: A library to compute the element free Galerkin shape functions, Comput. methods. appl. mech. engrg., 190, 2181-2205, (2001) · Zbl 1013.74080
[28] Tang, L.Q.; Tsang, T.T.H., An efficient least-squares finite element method for incompressible flows and transport processes, Int. J. comput. fluid dyn., 4, 21-39, (1995)
[29] Gorge, P.L., Automatic mesh generation - application to finite element methods, (1991), Wiley Masson
[30] Berg, M.D.; Kreveld, M.V.; Overmars, M.; Schwwarzkopf, O., Computational geometry - algorithms and applications, (1997), Springer Berlin
[31] Jiang, B.N., On the least-squares method, Comput. methods. appl. mech. engrg., 152, 239-257, (1998) · Zbl 0968.76040
[32] S.H. Park, A study on error estimates and an adaptive scheme of least-squares meshless method, Ph.D. Dissertation, Korea Advanced Institute of Science and Technology, 2001
[33] Liu, W.K.; Li, S.; Belytschko, T., Reproducing least square kernel Galerkin method, (I) methodology and convergence, Comput. methods. appl. mech. engrg., 143, 113-154, (1997) · Zbl 0883.65088
[34] Jiang, B.N.; Chang, C.L., Least-squares finite elements for the Stokes problems, Comput. methods. appl. mech. engrg., 78, 297-311, (1990) · Zbl 0706.76033
[35] Pan, F.; Acrivos, A., Steady flows in rectangular cavities, J. fluid. mech., 28, 643-655, (1967)
[36] Shankar, P.N., The eddy structure in Stokes flow in a cavity, J. fluid. mech., 250, 371-383, (1993) · Zbl 0775.76038
[37] Thom, A., The flow past circular cylinders at low speeds, Proc. roy. soc. A, 141, 651-669, (1933) · JFM 59.0765.01
[38] F. Homann, Der Einfluss grosser Zähigkeit bei der strömung um den Zylinder und um die Kugel, Z. Angew. Math. Mech. 16 (1936) 153. Translation: The effect of high viscosity on the flow around a cylinder and around a sphere, NACA TM 1334 (1952)
[39] Taneda, S., Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers, J. phys. soc. jpn., 11, 302-306, (1956)
[40] Tritton, D.J., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. fluid. mech., 6, 547-567, (1959) · Zbl 0092.19502
[41] Grove, A.S.; Shair, F.H.; Petersen, E.E.; Acrivos, A., An experimental investigation of the steady separated flow past a circular cylinder, J. fluid. mech., 19, 60-80, (1964) · Zbl 0117.42506
[42] Acrivos, A.; Leal, L.G.; Snowden, D.D.; Pan, F., Further experiments on steady separated flows past bluff objects, J. fluid. mech., 34, 25-48, (1968)
[43] Takami, H.; Keller, H.B., Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder, Phys. fluids, 12, Suppl. II., (1969), II-51 · Zbl 0206.55004
[44] Dennis, S.C.R; Chang, G.Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. fluid. mech., 42, 471-489, (1970) · Zbl 0193.26202
[45] Nieuwstadt, F.; Keller, H.B., Viscous flow past circular cylinders, Comput. fluids, 1, 59, (1973) · Zbl 0328.76022
[46] D. Sucker, H Brauer, Fluiddynamik bei der angeströmten Zylindern, Wärme- und Stoffübertragung. 8 (1975) 149
[47] Fornberg, B., A numerical study of steady viscous flow past a circular cylinder, J. fluid. mech., 98, 819-855, (1980) · Zbl 0428.76032
[48] Braza, M.; Chassaing, P.; Minh, H.Ha., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. fluid mech., 165, 79-130, (1986) · Zbl 0596.76047
[49] Shklyar, A.; Arbel, A., Numerical method for calculation of the incompressible flow in general curvilinear co-ordinates with double staggered grid, Int. J. numer. meth. fluids, 41, 1273-1294, (2003) · Zbl 1047.76082
[50] Silva, A.L.F Lima E; Silveira-Neto, A.; Damasceno, J.J.R., Numerical simulation of two-dimensional flow over a circular cylinder using the immersed boundary method, J. comput. phys., 189, 351-370, (2003) · Zbl 1061.76046
[51] Ding, H.; Shu, C.; Yeo, K.S.; Xu, D., Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method, Comput. methods. appl. mech. engrg., 193, 727-744, (2004) · Zbl 1068.76062
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.