A meshless IRBFN-based method for transient problems

CMES - Computer Modeling in Engineering and Sciences

Volumn 7, Issue 2, 2005, Pages 149-171

  Mai Cao, L ^a   Tran Cong, T ^a  

^a UNIVERSITY OF SOUTHERN QUEENSLAND   (Australia)

Author keywords

IRBFN method; Meshless method; Radial basis functions; Time dependent PDEs

Indexed keywords

APPROXIMATION THEORY; BOUNDARY ELEMENT METHOD; FINITE DIFFERENCE METHOD; FINITE ELEMENT METHOD; INTEGRATION; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS; PROBLEM SOLVING;

INDIRECT RADIAL BASIS FUNCTION NETWORK (IRBFN) METHOD; MESHLESS METHODS; RADIAL BASIS FUNCTIONS; THIN PLATE SPLINES; TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS;

RADIAL BASIS FUNCTION NETWORKS;

EID: 15544389485     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. Atluri, S.; Han, Z.; Rajendran, A. (2004): A new implementation of the meshless finite volume method, through the mlpg "mixed" approach. CMES: Computer Modeling in Engineering & Sciences, vol. 6, no. 6, pp. 491-513.
2. Atluri, S.; Shen, S. (2002): The Meshless Local Petrov-Galerkin (MLPG) Method. Tech Science Press, Encino, USA.
3. Atluri, S.; Shen, S. (2002): The meshless local Petrov-Galerkin (MLPG) method: A simple and less-costly alternative to the finite element and boundary element methods. CMES: Computer Modeling in Engineering & Sciences, vol. 3, no. 1, pp. 11-52.
4. Atluri, S.; Zhu, T. (1998): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, vol. 22, pp. 117-127.
5. Atluri, S.; Zhu, T. (1998): A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computational modeling and simulation. Modeling Simulation in Engineering, vol. 3, pp. 187-196.
6. Beatson, R.; Powell, M. (1990): Univariate multiquadric approximation: Quasi-interpolation to scattered data. Technical Report DAMTP 1990/NA7, University of Cambridge, 1990.
7. 2 in multiquadric interpolation. Computers and Mathematics with Applications, vol. 21, no. 9, pp. 29-42.
8. Chen, C.; Golberg, M.; Hashed, Y. (1998): A meshfree method for linear diffusion equations. Numerical Heat Transfer, Part B, vol. 33, pp. 469-486.
9. Fasshauer, G. (1996): Soving partial differential equations by collocation with radial basis functions. In Le Mehaute, A.; Rabut, C.; Schumaker, L.(Eds): Proceedings of Chamonix, pp. 1-8, Nashville. Vanderbilt University Press.
10. Franke, R. (1982): Scattered data interpolation: Test of some methods. Mathematics of Computation, vol. 38, no. 157, pp. 181-200.
11. Golberg, M.; Chen, C. (1999): The Method of Fundamental Solutions for Potential, Helmholtz and Diffusion Problems, chapter 4, Boundary Integral Methods: Numerical and Mathematical Aspects, pp. 105-176. WIT Press and Computational Mechanics Publications, Boston, Southampton, 1999.
12. Golberg, M.; Chen, C. (2001): An efficient mesh-free method for nonlinear reaction-diffusion equations. CMES: Computer Modeling in Engineering and Sciences, vol. 2, no. 1, pp. 87-95.
13. Han, Z.; Atluri, S. (2004): A meshless local petrov-galerkin (mlpg) approach for 3-dimensional elasto-dynamics. CMC:Computer Materials and Continua, vol. 1, no. 2, pp. 129-140.
14. Hon, Y. (2002): A quasi-radial basis functions method for American options pricing. Computer and Mathematics with Applications, vol. 43, no. 3/5, pp. 513-524.
15. Hon, Y.; Cheung, K.; Mao, X.; Kansa, E. (1999): Multiquadric solution for shallow water equations. ASCE Journal of Hydraulic Engineering, vol. 125, no. 5, pp. 524-533.
16. Ingber, M.; Phan-Thien, N. (1992): A boundary element approach for parabolic differential equations using a class of particular solutions. Applied Mathematical Modelling, vol. 16, pp. 124-132.
17. Kansa, E. (1990): Multiquadrics - A scattered data approximation scheme with applications to computational fluid-dynamics-I. Surface approximations and partial derivative estimates. Computers and Mathematics with Applications, vol. 19, no. 8/9, pp. 127-145.
18. Kansa, E. (1990): 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 Applications, vol. 19, no. 8/9, pp. 147-161.
19. Kovacevic, I.; Poredos, A.; Sarler, B. (2003): Solving the Stefan problem with the radial basis function collocation method. Numerical Heat Transfer, Part B: Fundamentals, vol. 44, pp. 1-24.
20. Mai-Cao, L.; Tran-Cong, T. (2003): A meshless level set approach to interface capturing. In Atluri, S.; Beskos, D.; Polyzos, D.(Eds): Proceedings of the 2003 International Conference on Computational & Experimental Engineering & Sciences, Encino. Tech Science Press.
21. Mai-Duy, N.; Tran-Cong, T. (2001): Numerical solution of differential equations using multiquadric radial basis function networks. Neural Networks, vol. 14, pp. 185-199.
22. Mai-Duy, N.; Tran-Cong, T. (2001): Numerical solution of navier-stokes equations using multiquadric radial basis function networks. International Journal for Numerical Methods in Fluids, vol. 37, pp. 65-86.
23. Mai-Duy, N.; Tran-Cong, T. (2003): Approximation of function and its derivatives using radial basis function networks. Applied Mathematical Modelling, vol. 27, pp. 197-220.
24. Moody, J.; Darken, C. (1989): Fast learning in networks of locally-tuned processing units. Neural Computations, vol. 1, pp. 281-294.
25. Moridis, G.; Kansa, E. (1994): The Laplace transform multiquadradric method: A highly accurate scheme for the numerical solution of linear partial differential equations. Journal of Applied Science and Computations, vol. 1, no. 2, pp. 375-407.
26. Perko, J.; Chen, C.; Sarler, B. (2001): A polygon-free numerical solution of steady natural convection in solid-liquid systems. In Sarler, B.; Brebbia, C.(Eds): Moving Boundaries VI: Computational Modelling of Free and Moving Boundary Problems, (Computational and Experimental Methods, Vol. 4), pp. 111-122, Boston. WIT Press.
27. Quarteroni, A.; Sacco, R.; Saleri, F. (2000): Numerical Mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, New York, USA.
28. Rippa, S. (1999): An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, vol. 11, pp. 193-210.
29. Sarler, B.; Perko, J.; Chen, C. (2004): Radial basis function collocation method solution of natural convection in porous media. International journal of numerical methods for heat and fluid flow, vol. 14, no. 2, pp. 187-212.
30. Sarler, B.; Perko, J.; Chen, C.; Kuhn, G. (2001): A meshless approach to natural convection. In Atluri, S.; Nishioka, T.; Kikuchi, M.(Eds): Advances in Computational Engineering & Sciences, ICES'01: [Proceedings of the International Conference on Computational Engineering & Sciences, 19-25 August 2001, Puerto Vallarta, Mexico], Palmdale, CA. Tech Science Press.
31. Zebib, A. (1984): A Chebyshev method for the solution of boundary value problems. Journal of Computational Physics, vol. 53, pp. 443-455.
32. Zerroukat, M.; Djidjeli, K.; Charafi, A. (2000): Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations. International Journal for Numerical Methods in Engineering, vol. 48, pp. 19-35.
33. Zerroukat, M.; Power, H.; Chen, C. (1998): A numerical method for heat transfer problems using collocation and radial basis functions. International Journal for Numerical Methods in Engineering, pp. 1263-1278.