×

A single domain velocity-vorticity fast multipole boundary domain element method for three dimensional incompressible fluid flow problems. II. (English) Zbl 1465.76065

Summary: In the present work, the Fast Multipole Boundary Element Method (FMM/BEM) for solving three-dimensional incompressible fluid flow problems governed by the Navier-Stokes equations is proposed. With the velocity-vorticity formulation the pressure gradient is eliminated from the equations. The kinematics equation, related to the velocity field satisfies continuity and provides a direct boundary condition for the vorticity equation. The single-domain approach is used for the discretization of the entire computational volume. The system of equations is compressed into two vectors and a preconditioner matrix, which is negligible in size. The involved unknowns are velocities, vorticities, and boundary vorticity fluxes. The two governing equations are coupled together in a convergent Newton-Raphson iteration scheme, successfully used for the solution of 3D fluid flow problems on a 32 GB memory computer. The degrees of freedom of the benchmark problems are above to 300000, which is an unreachable limit for the conventional single-domain BEM.
For Part I, see [the author, Eng. Anal. Bound. Elem. 106, 359–370 (2019; Zbl 1464.74319)].

MSC:

76M15 Boundary element methods applied to problems in fluid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids

Citations:

Zbl 1464.74319

Software:

OpenFOAM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ISBN 978-0471720393
[2] ISBN 978-0080441078 · Zbl 1051.74052
[3] Gao, X.-W.; Guo, L.; Zhang, C., Three-step multi-domain BEM solver for nonhomogeneous material problems, Eng Anal Bound Elem, 31, 12, 965-973 (2007) · Zbl 1259.74046
[4] Peng, H.-F.; Bai, Y.-G.; Yang, K.; Gao, X.-W., Three-step multi-domain BEM for solving transient multi-media heat conduction problems, Eng Anal Bound Elem, 37, 11, 1545-1555 (2007) · Zbl 1287.80009
[5] Florez, W. F.; Power, H., Comparison between continuous and discontinuous boundary elements in the multidomain dual reciprocity method for the solution of the two-dimensional Navier-Stokes equations, Eng Anal Bound Elem, 25, 1, 57-69 (2001) · Zbl 1023.76029
[6] Ravnik, J.; Škerget, L.; Žunic, Z., Combined single domain and subdomain BEM for 3d laminar viscous flow, Eng Anal Bound Elem, 33, 03, 420-424 (2009) · Zbl 1244.76057
[7] ISBN 978-0-387-34042-5 · Zbl 1119.65119
[8] Gortsas, T. V.; Tsinopoulos, S. V.; Polyzos, D., An advanced ACA/BEM for solving 2d large-scale problems with multiconnected domains, Comput Methods Eng Sci, 107, 4, 321-343 (2015)
[9] Rodopoulos, D. C.; Gortsas, T. V.; Tsinopoulos, S. V.; Polyzos, D., ACA/BEM for solving large-scale cathodic protection problems, Eng Anal Bound Elem, 106, 139-148 (2019) · Zbl 1464.78023
[10] Bucher, H. F.; Wrobel, L. C.; Mansur, W. J.; Magluta, C., Fast solution of problems with multiple load cases by using wavelet-compressed boundary element matrices, Commun Numer Methods Eng, 19, 387-399 (2003) · Zbl 1018.65134
[11] Ntalaperas D., Tsinopoulos S.V., Polyzos D.. A fast wavelet/BEM for wave scattering problems. Advanced Topics in Scattering and Biomedical Engineering, A Charalampopoulos, D I Fotiadis, D Polyzos Eds, World Scientific2009;:414-421.
[12] Xiao, J.; Ye, W.; Cai, Y.; Zhang, J., Precorrected FFT accelerated BEM for large-scale transient elastodynamic analysis using frequency-domain approach, Commun Numer Methods Eng, 90, 1, 116-134 (2011) · Zbl 1242.74186
[13] ISBN 978-0-511-60504-8 (paperback)
[14] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J Comput Phys, 73, 2, 325-348 (1987) · Zbl 0629.65005
[15] Wang, H.; Lei, T.; Li, J.; Huang, J.; Yao, Z., A parallel fast multipole accelerated integral equation scheme for 3D stokes equations, Int J Numer Methods Eng, 70, 7, 812-839 (2007) · Zbl 1194.76221
[16] Young, D. L.; Liu, Y. H.; Eldho, T. I., A combined BEM-FEM model for the velocity-vorticity formulation of the Navier-Stokes equations in three dimensions, Eng Anal Bound Elem, 24, 4, 307-316 (2000) · Zbl 0970.76060
[17] Žunič, Z.; Hriberšek, M.; Škerget, L.; Ravnik, J., 3-D boundary element-finite element method for velocity-vorticity formulation of the Navier-Stokes equations, Eng Anal Bound Elem, 31, 3, 191-288 (2007)
[18] Sellountos, E.; Tiago, J.; Sequeira, A., Meshless velocity-vorticity local boundary integral equation (lbie) method for two dimensional incompressible Navier-Stokes equations, Int J Numer Methods Heat Fluid Flow, 29, 11, 4034-4073 (2019)
[19] Ravnik, J.; Škerget, L.; Žunic, Z., Fast single domain-subdomain BEM algorithm for 3d incompressible fluid flow and heat transfer, Int J Numer Methods Eng, 77, 12, 1627-1645 (2009) · Zbl 1158.76374
[20] Sellountos, E. J., A single domain velocity-vorticity fast multipole boundary domain element method for two dimensional incompressible fluid flow problems, Eng Anal Bound Elem, 106, 359-370 (2019) · Zbl 1464.74319
[21] Škerget, L.; Hriberšek, M.; Žunic, Z., Natural convection flows in complex cavities by BEM, Int J Numer Methods Heat Fluid Flow, 13, 6, 720-735 (2003) · Zbl 1042.76055
[22] Wong, K. L.; Baker, A. J., A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm, Int J Numer Methods Fluids, 38, 2, 99-123 (2001) · Zbl 1009.76059
[23] Škerget, L.; Hriberšek, M.; Kuhn, G., Computational fluid dynamics by boundary-domain integral method, Int J Numer Methods Eng, 46, 8, 1291-1311 (1999) · Zbl 0951.76053
[24] Hriberšek, M.; Škerget, L., Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries, Comput Methods Appl Mech Eng, 194, 39-41, 4196-4220 (2005) · Zbl 1151.76537
[25] Tibaut, J.; Škerget, L.; Ravnik, J., Acceleration of a BEM based solution of the velocity-vorticity formulation of the Navier-Stokes equations by the cross approximation method, Eng Anal Bound Elem, 82, 1, 17-26 (2017) · Zbl 1403.76114
[26] Bourantas, G.; Skouras, E.; Loukopoulos, V.; Nikiforidis, G., Meshfree point collocation schemes for 2D steady state incompressible Navier-Stokes equations in velocity-vorticity formulation for high values of Reynolds number, Comput Methods Eng Sci, 59, 1, 31-63 (2010) · Zbl 1231.76069
[27] Liu, C. H., Numerical solution of three-dimensional Navier-Stokes equations by a velocity-vorticity method, Int J Numer Methods Fluids, 35, 5, 533-557 (2001) · Zbl 1013.76059
[28] Lo, D. C.; Murugesan, K.; Young, D. L., Numerical solution of three-dimensional velocity-vorticity Navier-Stokes equations by finite difference method, Int J Numer Methods Fluids, 47, 12, 1469-1487 (2004) · Zbl 1155.76369
[29] Shen, L.; Liu, Y. J., An adaptive fast multipole boundary element method for three-dimensional potential problems, Comput Mech, 39, 6, 681-691 (2007) · Zbl 1198.74113
[30] ISBN 978-0521880688 · Zbl 1132.65001
[31] Chaillat, S.; Bonnet, M.; Semblat, J.-F., A multi-level fast multipole BEM for 3-d elastodynamics in the frequency domain, Comput Methods Appl Mech Eng, 197, 49-50, 4233-4249 (2008) · Zbl 1194.74109
[32] Yoshida, K.-I., Applications of fast multipole method to boundary integral equation method (2001), Department of Global Environment Engineering: Department of Global Environment Engineering Kyoto University of Japan
[33] https://openfoam.org
[34] ISBN 978-11-1900-299-4 (paperback)
[35] Ding, H.; Shu, C.; Yeo, K.; Xu, D., Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method, Comput Methods Appl Mech Eng, 195, 7-8, 481-798 (2006) · Zbl 1222.76072
[36] Lo, D. C.; Murugesan, K.; Young, D. L., Numerical solution of three-dimensional velocity-vorticity Navier-Stokes equations by finite difference method, Int J Numer Methods Fluids, 47, 12, 1469-1487 (2005) · Zbl 1155.76369
[37] Guiggiani, M.; Krishnasamy, G.; Rudolphi, T. J.; Rizzo, F., A general algorithm for the numerical solution of hypersingular boundary integral equations., ASME J Appl Mech, 59, 3, 604-614 (1992) · Zbl 0765.73072
[38] Ma, H.; Kamiya, N., Distance tranformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Eng Anal Bound Elem, 26, 4, 329-339 (2002) · Zbl 1003.65133
[39] Armaly, B. F.; Durst, F.; Pereira, J. C.F.; Schönung, B., Experimental and theoretical investigation of backward-facing step flow, J Fluid Mech, 127, 473-496 (1983)
[40] Chiang, T. P.; Sheu, W. H., A numerical revisit of backward-facing step flow problem, Phys Fluids, 11, 4, 862-874 (1999) · Zbl 1147.76361
[41] Biswas, G.; Breuer, M.; Durst, F., Backward-facing step flows for various expansion ratios at low and moderate Reynolds numbers, J Fluids Eng (ASME), 126, 3, 362-374 (2004)
[42] Malamataris, N. A., A numerical investigation of wall effects in three-dimensional, laminar flow over a backward facing step with a constant aspect and expansion ratio, Int J Numer Methods Fluids, 71, 9, 1073-1102 (2013) · Zbl 1430.76122
[43] Liakos, A.; Malamataris, N. A., Topological study of steady state, three dimensional flow over a backward facing step, Comput Fluids, 118, 1-18 (2015)
[44] Williams, P. T.; Baker, A. J., Numerical simulations of laminar flow over a 3d backward-facing step, Int J Numer Methods Fluids, 24, 11, 1159-1183 (1999) · Zbl 0886.76048
[45] Erturk, E., Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions, Comput Fluids, 37, 6, 633-655 (2008) · Zbl 1237.76102
[46] ISBN 007-124493-X
[47] ISBN 978-0123875822
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.