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From global to local radial basis function collocation method for transport phenomena. en. (English) Zbl 1323.65111
Leitão, V. M. A. (ed.) et al., Advances in meshfree techniques. Invited contributions based on the presentation at the ECCOMAS thematic conference on meshless methods, Lisbon, Portugal, July 11–14, 2005. Dordrecht: Springer (ISBN 978-1-4020-6094-6/hbk; 978-90-481-7533-8/pbk; 978-1-4020-6095-3/ebook). Computational Methods in Applied Sciences (Springer) 5, 257-282 (2007).
Summary: This article introduces basic concepts of meshless methods for solving partial differential equations in their strong form by collocation or least squares approximation. Global and local formulations are defined. The current achievements, based on the local form and collocation with radial basis functions are explained in detail. Heat transfer and fluid flow problems are treated. These achievements represent a simple, and at the same time more efficient version of the classical meshless radial basis function collocation (Kansa) method. Instead of global, the collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. The computational effort thus grows roughly linearly with the number of the nodes. The represented approach thus overcomes the principal large scale bottleneck of the original Kansa method and widely opens space for industrial applications of the method. The purpose of this article is to give a concentrated information on this new method, which has already been successfully applied in macroscopic and microscopic transport phenomena field, accompanied with research requirements for the future. It is devoted to practicing engineers and researchers.
For the entire collection see [Zbl 1125.74003].

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76R99 Diffusion and convection
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[1] Atluri S.N. \(The Meshless Method (MLPG) for Domain and BIE Discretizations\). Tech Science Press, Forsyth, 2004. · Zbl 1105.65107
[2] Atluri S.N. and Shen S. \(The Meshless Method\). Tech Science Press, Encino, 2002. · Zbl 1012.65116
[3] Boyd J.P. \(Chebishev and Fourier Spectral Methods\). Dover, New York, 2001.
[4] Buhmann M.D. \(Radial Basis Function: Theory and Implementations\). Cambridge University Press, Cambridge, 2003. · Zbl 1038.41001
[5] Bulinsky Z., Nowak A. and Šarler B. (2007) Numerical experiments with the local radial basi function collocation method. \(Computers, Materials, Continua\), in review, 2007.
[6] Cameron A.D., Casey J.A. and Simpson G.B. \(Benchmark Tests for Thermal Analysis\). NAFEMS National Agency for Finite Element Methods and Standards, Department of Trade and Industry, National Engineering Laboratory, Glasgow, 1986.
[7] Chen C.S., Ganesh M., Golberg M.A. and Cheng A.H.-D. Multilevel compact radial basis functions based computational scheme for some elliptic problems. \(Computers and Mathematics with Application\), 43:359-378, 2002. · Zbl 0999.65143
[8] Chen W. New RBF collocation schemes and kernel RBFs with applications. \(Lecture Notes in Computational Science and Engineering\), 26:75-86, 2002. · Zbl 1016.65094
[9] Divo E. and Kassab A.J. An efficient localized RBF meshless method for fluid flow and conjugate heat transfer. \(ASME Journal of Heat Transfer\), in print, 2006. · Zbl 1195.76316
[10] Fasshauer G.E. (1997) Solving partial differential equations by collocation with radial basis functions. In: Mehaute A.L., Rabut C. and Schumaker L.L. (Eds), \(Surface Fitting and Multiresolution Methods\), 1997, pp. 131-138. · Zbl 0938.65140
[11] 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. \(Computers and Mathematics with Application\), 19:147-161, 1990. · Zbl 0850.76048
[12] Kovačević I., Poredoš A. and Šarler B. Solving the Stefan problem by the RBFCM. \(Numer. Heat Transfer, Part B: Fundamentals\), 44:575-599, 2003.
[13] Kovačević I. and Šarler B. Solution of a phase-field model for dissolution of primary particles in binary aluminium alloys by an r-adaptive mesh-free method. \(Materials Science and Engineering\), 413/414A:423-428, 2005.
[14] Lee C.K., Liu X. and Fan S.C. Local muliquadric approximation for solving boundary value problems. \(Computational Mechanics\), 30:395-409, 2003. · Zbl 1035.65136
[15] Liu G.R. \(Mesh Free Methods\). CRC Press, Boca Raton, FL, 2003. · Zbl 1031.74001
[16] Liu G.R. and Gu Y.T. \(An Introduction to Meshfree Methods and Their Programming\). Springer, Dordrecht, 2005.
[17] Mai-Duy N. and Tran-Cong T. Numerical solution of differential equations using multiquadrics radial basis function networks. \(International Journal for Numerical Methods in Engineering\), 23:1807-1829, 2001.
[18] Mai-Duy N. and Tran-Cong T. Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks. \(Neural Networks\), 14:185-199, 2001. · Zbl 02022497
[19] 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. \(Engineering Analysis with Boundary Elements\), 26:133-156, 2002. · Zbl 0996.65131
[20] Mai-Duy N. and Tran-Cong T. Indirect RBFN method with thin plate splines for numerical solution of differential equations. \(Computer Modeling in Engineering and Sciences\), 4:85-102, 2003. · Zbl 1148.76351
[21] Nayroles B., Touzot G. and Villon P. The diffuse approximation. \(C.R. Acad. Sci. Paris\) 313-II:293-296, 1991. · Zbl 0753.65012
[22] Özisik M.N. \(Finite Difference Methods in Heat Transfer\). CRC Press, Boca Raton, FL, 1994. · Zbl 0855.65097
[23] Perko J. \(Modelling of Transport Phenomena by the Diffuse Approximate Method\). Ph.D. Thesis, School of Applied Sciences, Nova Gorica Polytechnic, Nova Gorica, 2005.
[24] Power H. and Barraco W.A. Comparison analysis between unsymmetric and symmetric RBFCMs for the numerical solution of PDE’s. \(Computers and Mathematics with Applications\), 43:551-583, 2002. · Zbl 0999.65135
[25] Sadat H. and Prax C. Application of the diffuse approximation for solving fluid flow and heat transfer problems. \(International Journal of Heat and Mass Transer\), 39:214-218, 1996. · Zbl 0979.76529
[26] 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. \(Computer Methods in Applied Mechanics and Engineering\), 192:941-954, 2003. · Zbl 1025.76036
[27] Šarler B. and Vertnik R. Meshfree explicit local radial basis function collocation method for diffusion problems. \(Computers and Mathematics with Applications\), 51:1269-1282, 2006. · Zbl 1168.41003
[28] Šarler B., Jelić N., Kovačević I., Lakner M. and Perko J. Axisymmetric multiquadrics. \(Engineering Analysis with Boundary Elements\), 30:137-142, 2006. · Zbl 1195.65190
[29] Šarler B. A radial basis function collocation approach in computational fluid dynamics. \(Computer Modeling in Engineering and Sciences\), 7:185-193, 2005. · Zbl 1189.76380
[30] Šarler B., Vertnik R. and Perko J. Application of diffuse approximate method in convective-diffusive solidification problems. \(Computers, Materials, Continua\), 2(1):77-84, 2005. · Zbl 1160.76383
[31] Šarler B., Perko J. and Chen C.S. Radial basis function collocation method solution of natural convection in porous media. \(International Journal of Numerical Methods for Heat and Fluid Flow\), 14:187-212, 2004. · Zbl 1103.76361
[32] Šarler B. Towards a mesh-free computation of transport phenomena. \(Engineering Analysis with Boundary Elements\), 26:731-738, 2002. · Zbl 1032.76628
[33] Šarler B., Kovačević I. and Chen C.S. A mesh-free solution of temperature in direct-chill cast slabs and billets. In: Mammoli A.A. and Brebbia C.A. (Eds), \(Moving Boundaries VI\). WIT Press, Southampton, 2004, pp. 271-280.
[34] Tolstykh A.I. and Shirobokov D.A. On using radial basis functions in a ”finite difference” mode with applications to elasticity problems. \(Computational Mechanics\), 33:68-79, 2003. · Zbl 1063.74104
[35] Versteeg H.K. and Malalasekera W. \(Computational Fluid Dynamics: The Finite Volume Method\). Prentice Hall, Harlow, 1995.
[36] Vertnik R. and Šarler B. Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems. \(International Journal of Numerical Methods for Heat and Fluid Flow\) (Special Issue: European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, Jyväskylä, 24-28 July 2004), 16:617-640, 2005.
[37] Vertnik R., Perko J. and Šarler B. Solution of temperature field in DC cast aluminium alloy billet by the Diffuse Approximate Method. \(Materials and Technology\), 38:257-261, 2004.
[38] Vertnik R., Založnik M. and Šarler B. Solution of transient direct chill aluminium billet casting problem with simultaneous material and interphase moving boundaries. \(Engineering Analysis with Boundary Elements\), 30: 847-855, 2006. · Zbl 1195.76325
[39] Wang J.G. and Liu G.R. On the optimal shape parameter of radial basis functions used for 2-D meshless method. \(Computer Methods in Applied Mechanics and Engineering\), 26:2611-2630, 2002. · Zbl 1065.74074
[40] Wrobel L.C. \(The Boundary Element Method -- Volume 1: Applications in Thermofluids and Acoustics\). Wiley, New York, 2001.
[41] Zienkiewicz O.C. and Taylor R.L. \(The Finite Element Method\). Butterworth-Heinemann, Oxford, 2002. · Zbl 0991.74002
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