×

zbMATH — the first resource for mathematics

Iterative domain decomposition meshless method modeling of incompressible viscous flows and conjugate heat transfer. (English) Zbl 1195.76316
Summary: We develop an effective domain decomposition meshless methodology for conjugate heat transfer problems modeled by convecting fully viscous incompressible fluid interacting with conducting solids. The meshless formulation for fluid flow modeling is based on a radial basis function interpolation using Hardy inverse multiquadrics and a time-progression decoupling of the equations using a Helmholtz potential. The domain decomposition approach effectively reduces the conditioning numbers of the resulting algebraic systems, arising from convective and conduction modeling, while increasing efficiency of the solution process and decreasing memory requirements. Moreover, the domain decomposition approach is ideally suited for parallel computation. Numerical examples are presented to validate the approach by comparing the meshless solutions to finite volume method solutions provided by a commercial CFD solver.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Belytscho, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int J numer methods, 37, 229-256, (1994) · Zbl 0796.73077
[2] 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
[3] Melenk, J.M.; Babuska, I., The partition of unity finite element method: basic theory and application, Comput meth appl mech eng, 139, 289-316, (1996) · Zbl 0881.65099
[4] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics I—surface approximations and partial derivative estimates, Comput math appl, 19, 127-145, (1990) · Zbl 0692.76003
[5] 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, 19, 147-161, (1990) · Zbl 0850.76048
[6] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput math appl, 39, 123-137, (2000) · Zbl 0955.65086
[7] Franke, R., Scattered data interpolation: test of some methods, Math comput, 38, 181-200, (1982) · Zbl 0476.65005
[8] Mai-Duy, N.; Tran-Cong, T., Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equation, Eng anal bound elem, 26, 133-156, (2002) · Zbl 0996.65131
[9] Cheng, A.H.-D.; Golberg, M.A.; Kansa, E.J.; Zammito, G., Exponential convergence and H-c multiquadric collocation method for partial differential equations, Numer meth. PDE, 19, 5, 571-594, (2003) · Zbl 1031.65121
[10] Gottlieb, D.; Orzag, S.A., Numerical analysis of spectral methods: theory and applications, (1977), Society for Industrial and Applied Mathematics Bristol, England
[11] Maday, Y.; Quateroni, A., Spectral and pseudo-spectral approximations of the navier – stokes equations, SIAM J numer anal, 19, 4, 761-780, (1982) · Zbl 0503.76035
[12] Patera, A., A spectral element method of fluid dynamics: laminar flow in a channel expansion, J comput phys, 54, 468-488, (1984) · Zbl 0535.76035
[13] Macaraeg, M.; Street, C.L., Improvement in spectral collocation discretization through a multiple domain technique, Appl numer math, 2, 95-108, (1986) · Zbl 0633.76094
[14] Hwar, C.K.; Hirsch, R.; Taylor, T.; Rosenberg, A.P., A pseudo-spectral matrix element method for solution of three-dimensional incompressible flows and its parallel implementation, J comput phys, 83, 260-291, (1989) · Zbl 0672.76036
[15] Powell, M.J.D., The theory of radial basis function approximation, (), 143-167
[16] Dyn, N.; Levin, D.; Rippa, S., Numerical procedures for surface Fitting of scattered data by radial basis functions, SIAM J sci stat comput, 7, 2, 639-659, (1986) · Zbl 0631.65008
[17] Brebbia, C.A.; Partridge, P.; Wrobel, L.C., The dual reciprocity boundary element method, (1992), Computational Mechanics and Elsevier Southampton, UK · Zbl 0758.65071
[18] Golberg, M.; Chen, C.S.; Bowman, H., Some recent results and proposals for the use of radial basis functions in the BEM, Eng anal bound elem, 23, 285-296, (1999) · Zbl 0948.65132
[19] Rahaim, C.P.; Kassab, A.J., Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids, Eng anal bound elem, 18, 4, 265-272, (1996)
[20] Hardy RL. Multiquadric equations of topography and other irregular surfaces. J Geophys Res 176:1905-15.
[21] Kassab, A.; Divo, E.; Heidmann, J.; Steinthorsson, E.; Rodriguez, F., BEM/FVM conjugate heat transfer analysis of a three-dimensional film cooled turbine blade, Int J numer meth heat fluid flow, 13, 5, 581-610, (2003) · Zbl 1183.76808
[22] ()
[23] Rahaim, C.P.; Kassab, A.J.; Cavalleri, R., A coupled dual reciprocity boundary element/finite volume method for transient conjugate heat transfer, AIAA J thermophys heat transfer, 14, 1, 27-38, (2000)
[24] He, M.; Bishop, P.; Kassab, A.J.; Minardi, A., A coupled FDM/BEM solution for the conjugate heat transfer problem, Numer heat transfer part B fundam, 28, 2, 139-154, (1995)
[25] Li H, Kassab AJ. A coupled FVM/BEM solution to conjugate heat transfer in turbine blades. AIAA Paper 94-1981, presented at the AIAA/ASME sixth joint thermophysics conference, Colorado Springs, Colorado; June 20-23, 1994.
[26] Divo, E.A.; Kassab, A.J.; Rodriguez, F., Parallel domain decomposition approach for large-scale 3d boundary element models in linear and non-linear heat conduction, Numer heat transfer part B fundam, 44, 5, 417-437, (2003)
[27] Divo E, Kassab AJ, Mitteff E, Quintana L. A parallel domain decomposition technique for meshless methods applications to large-scale heat transfer problems. ASME Paper: HT-FED2004-56004.
[28] Divo, E.; Kassab, A.J., A meshless method for conjugate heat transfer problems, Eng anal, 29, 2, 136-149, (2005) · Zbl 1182.76925
[29] Harlow, F.H.; Welch, J.E., Numerical calculation of time dependent viscous incompressible flow of fluids with a free surface, Phys fluids, 8, 2182-2189, (1965) · Zbl 1180.76043
[30] Patankar, S.V., Numerical heat transfer and fluid flow, (1980), Hemisphere New York · Zbl 0521.76003
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.