HIGH ORDER B-SPLINE COLLOCATION METHOD AND ITS APPLICATION FOR HEAT TRANSFER PROBLEMS

Infimum D. Yulamda, Mas Irfan P. Hidayat

Abstract


High order B-spline collocation for solving boundary value problem is presented in this paper. The approach employs high order B-spline basis functions with high approximation and continuity properties to handle problem domain with scattered or random distribution of knot points.  Using appropriate B-spline basis function construction, the new approach introduces no difficulties in imposing both Dirichlet and Neumann boundary conditions in the problem domain. Several numerical examples in arbitrary domains, both regular and irregular shaped domains, are considered in the present study. In addition, simulation results concerning with heat transfer applications are further presented and discussed.

Keywords


B-splines; high order; approximation; meshless; arbitrary domains; heat transfer

Full Text:

PDF

References


Atluri, S.N. and Zhu, T. (1998), "A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics", Comput. Mech. Vol. 22, pp. 117–127.

Bathe, K.J. (1996), "Finite Element Procedures", Prentice Hall, Inc., New Jersey, USA.

Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng. Vol. 37, pp. 229–256.

Boroomand, B., Najjar, M. and Onate, E. (2009), "The generalized finite point method", Comput. Mech. Vol. 44, pp. 173–190.

Botella, O. (2002), "On a collocation B-spline method for the solution of the Navier–Stokes equations", Comp. Fluids. Vol. 31, pp. 397-420.

Burla, R.K. and Kumar, A.V. (2008), "Implicit boundary method for analysis using uniform B-spline basis and structured grid", Int. J. Numer. Meth. Eng. Vol. 76, pp. 1993-2028.

Chaniotis, A.K. and Poulikakos, D. (2004), “High order interpolation and differentiation using B-splines,” Journal of Computational Physics, Vol. 197, pp. 253–274.

Chui, C.K. (1988), "Multivariate Splines", second ed., SIAM, Philadelphia, USA.

de Boor, C. (2001), "A Practical Guide to Splines", revised ed., Springer, New York, USA.

Dehghan, M. and Ghesmati, A. (2010), "Combination of meshless local weak and strong (MLWS) forms to solve the two-dimensional hyperbolic telegraph equation", Eng. Anal. Bound. Elem. Vol. 34, pp. 324–336.

Farin, G. (2002), "Curves and Surfaces for Computer Aided Geometric Design", fifth ed., Academic Press, San Diego, CA.

Hoggar, S.G. (2006), "Mathematics of Digital Images", first ed., Cambridge University Press, Cambridge.

Hollig, K. (2003), "Finite Element Methods with B-splines", SIAM, Philadelphia, USA.

Jator, S. and Sinkala, Z. (2007), "A high order B-spline collocation method for linear boundary value problems", App. Math. Comp., Vol. 191, pp. 100–116.

Johnson, R.W. (2005), "Higher order B-spline collocation at the Greville abscissae", App. Num. Math., Vol. 52, pp. 63–75.

Leitao, V.M.A., Alves, C.J.S. and Duarte, C.A. (Eds.) (2007), "Advances in Meshfree Techniques", Springer, Netherlands.

Liu, G.R. and Gu, Y.T. (2003), "A meshfree method: meshfree weak–strong (MWS) form method, for 2-D solids", Comp. Mech., Vol. 33, pp. 2-14.

Liu, X. and Tai, K. (2006), "Point interpolation collocation method for the solution of partial differential equations", Eng. Anal. Bound. Elem., Vol. 30, pp. 598–609.

Liu, X., Liu, G.R., Tai, K. and Lam, K.Y. (2005), "Radial point interpolation collocation method (RPICM) for partial differential equations", Comp. Math. App., Vol. 50, pp. 1425-1442.

Ohayon, R. (2004), Fluid–Structure Interaction Problem, in: E. Stein, R. de Borst, T.J.R. Hugues (Eds.), Encyclopedia of Computational Mechanics, Vol. 2, John Wiley, pp. 683–694.

Ramsak, M. and Skerget, L. (2004), "A subdomain boundary element method for high-Reynolds laminar flow using stream function-vorticity formulation", Int. J. Num. Meth. Fluids., Vol. 46, pp. 815–847.

Salomon, D. (2006), "Curves and Surfaces for Computer Graphics", Springer Science + Business Media, Inc. New York, USA.

Wang, H. and Qin, Q.H. (2006), "A meshless method for generalized linear or nonlinear Poisson-type problems", Eng. Anal. Bound. Elem., Vol. 30, pp. 515–521.

Yu, K.H., Kadarman, A.H. and Djojodihardjo, H. (2010), "Development and implementation of some BEM variants-A critical review", Eng. Anal. Bound. Elem., Vol. 34, pp. 884–899.

Zahiri, S., Daneshmand, F. and Akbari, M.H. (2009), "Using meshfree weak-strong form method for 2-D heat transfer problem", Proceedings of ASME 2009 International Mechanical Engineering Congress and Exposition, IMECE2009-12525, Lake Buena Vista, Florida, USA, November 13-19, 2009.

Zhang, X., Song, K.Z., Lu, M.W. and Liu, X. (2000), "Meshless methods based on collocation with radial basis functions", Comput. Mech., Vol. 26, pp. 333–343.




DOI: http://dx.doi.org/10.12962/j2746279X.v1i2.16945

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.

 
View My Stats