Adaptive Shape Parameter (ASP) Technique for Local Radial Basis Functions (RBFs) and Their Application for Solution of Navier Strokes Equations
The concept of adaptive shape parameters (ASP) has been presented for solution of incompressible Navier Strokes equations using mesh-free local Radial Basis Functions (RBF). The aim is to avoid ill-conditioning of coefficient matrices of RBF weights and inaccuracies in RBF interpolation resulting from non-optimized shape of basis functions for the cases where data points (or nodes) are not distributed uniformly throughout the domain. Unlike conventional approaches which assume globally similar values of RBF shape parameters, the presented ASP technique suggests that shape parameter be calculated exclusively for each data point (or node) based on the distribution of data points within its own influence domain. This will ensure interpolation accuracy while still maintaining well conditioned system of equations for RBF weights. Performance and accuracy of ASP technique has been tested by evaluating derivatives and laplacian of a known function using RBF in Finite difference mode (RBFFD), with and without the use of adaptivity in shape parameters. Application of adaptive shape parameters (ASP) for solution of incompressible Navier Strokes equations has been presented by solving lid driven cavity flow problem on mesh-free domain using RBF-FD. The results have been compared for fixed and adaptive shape parameters. Improved accuracy has been achieved with the use of ASP in RBF-FD especially at regions where larger gradients of field variables exist.
<p>[1] E. J. Kansa, “Multiquadrics - a scattered data approximation scheme
with applications to computational fluid-dynamics .2. solutions to
parabolic, hyperbolic and elliptic partial-differential equations,” Computers
and Mathematics with Applications, vol. 19, no. 8-9, pp. 147–161,
1990.
[2] W. Chen and M. Tanaka, “A meshless, integration-free, and boundaryonly
rbf technique,” Computers and Mathematics with Applications,
vol. 43, no. 3-5, pp. 379–391, 2002.
[3] M. C. V. Bayona, M. Moscoso and M. Kindelan, “Rbf-fd formulas and
convergence properties,” Journal of Computational Physics, vol. 229,
no. 22, pp. 8281–8295, 2010.
[4] N. Mai-Duy and T. Tran-Cong, “Numerical solution of differential
equations using multiquadric radial basis function networks,” Neural
Networks, vol. 14, no. 2, pp. 185–199, 2001.
[5] J. G. Wang and G. R. Liu, “On the optimal shape parameters of radial
basis functions used for 2-d meshless methods,” Computer Methods in
Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611–
2630, 2002.
[6] K. D. P. Phani Chinchapatnam and P. B. Nair, “Radial basis function
meshless method for the steady incompressible navierstokes equations,”
International Journal of Computer Mathematics, vol. 84, no. 10, pp.
1509–1521, 2007.
[7] C. Franke and R. Schaback, “Solving partial differential equations
by collocation using radial basis functions,” Applied Mathematics and
Computation, vol. 93, no. 1, pp. 73–82, 1998.
[8] H. D. C. Shu and K. S. Yeo, “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, vol. 192, no. 7-8, pp. 941–954, 2003.
[9] A. I. Tolstykh and D. A. Shirobokov, “On using radial basis functions
in a ”finite difference mode” with applications to elasticity problems,”
Computational Mechanics, vol. 33, no. 1, pp. 68–79, 2003.
[10] Y. Sanyasiraju and G. Chandhini, “Local radial basis function based
gridfree scheme for unsteady incompressible viscous flows,” Journal of
Computational Physics, vol. 227, no. 20, pp. 8922–8948, 2008.
[11] P. B. N. P. Phani Chinchapatnam, K. Djidjeli and M. Tan, “A compact
rbf-fd based meshless method for the incompressible navier-stokes
equations,” Proceedings of the Institution of Mechanical Engineers Part
M-Journal of Engineering for the Maritime Environment, vol. 223,
no. M3, pp. 275–290, 2009.
[12] G. B. Wright and B. Fornberg, “Scattered node compact finite differencetype
formulas generated from radial basis functions,” Journal of Computational
Physics, vol. 212, no. 1, pp. 99–123, 2006.
[13] C. F. L. C. S. Huang and A. H. D. Cheng, “Error estimate, optimal
shape factor, and high precision computation of multiquadric collocation
method,” Engineering Analysis with Boundary Elements, vol. 31, no. 7,
pp. 614–623, 2007.
[14] L. I. M. Gherlone and M. D. Sciuva, “A novel algorithm for shape
parameter selection in radial basis functions collocation method,” Composite
Structures, vol. 94, no. 2, pp. 453–461, 2012.
[15] S. Rippa, “An algorithm for selecting a good value for the parameter
c in radial basis function interpolation,” Advances in Computational
Mathematics, vol. 11, no. 2-3, pp. 193–210, 1999.
[16] A. G. S. Hamed Meraji and P. Malekzadeh, “An efficient algorithm
based on the differential quadrature method for solving navier-stokes
equations,” International Journal for Numerical Methods in Fluids, pp.
n/a–n/a, 2012.
[17] R. Franke, “Scattered data interpolation: Tests of some method,” Mathematics
of Computation, vol. 38, no. 157, pp. 181–200, 1982.
[18] U. K. N. G. Ghia and C. Shin, “High-re solutions for incompressible
flow using the navier-stokes equations and a multigrid method,” Journal
of Computational Physics, vol. 48, no. 3, pp. 387 – 411, 1982.
[19] W. F. Spotz and G. F. Carey, “High-order compact scheme for the
steady stream-function vorticity equations,” International Journal for
Numerical Methods in Engineering, vol. 38, no. 20, pp. 3497–3512,
1995.</p>
<p>[1] E. J. Kansa, “Multiquadrics - a scattered data approximation scheme
with applications to computational fluid-dynamics .2. solutions to
parabolic, hyperbolic and elliptic partial-differential equations,” Computers
and Mathematics with Applications, vol. 19, no. 8-9, pp. 147–161,
1990.
[2] W. Chen and M. Tanaka, “A meshless, integration-free, and boundaryonly
rbf technique,” Computers and Mathematics with Applications,
vol. 43, no. 3-5, pp. 379–391, 2002.
[3] M. C. V. Bayona, M. Moscoso and M. Kindelan, “Rbf-fd formulas and
convergence properties,” Journal of Computational Physics, vol. 229,
no. 22, pp. 8281–8295, 2010.
[4] N. Mai-Duy and T. Tran-Cong, “Numerical solution of differential
equations using multiquadric radial basis function networks,” Neural
Networks, vol. 14, no. 2, pp. 185–199, 2001.
[5] J. G. Wang and G. R. Liu, “On the optimal shape parameters of radial
basis functions used for 2-d meshless methods,” Computer Methods in
Applied Mechanics and Engineering, vol. 191, no. 23-24, pp. 2611–
2630, 2002.
[6] K. D. P. Phani Chinchapatnam and P. B. Nair, “Radial basis function
meshless method for the steady incompressible navierstokes equations,”
International Journal of Computer Mathematics, vol. 84, no. 10, pp.
1509–1521, 2007.
[7] C. Franke and R. Schaback, “Solving partial differential equations
by collocation using radial basis functions,” Applied Mathematics and
Computation, vol. 93, no. 1, pp. 73–82, 1998.
[8] H. D. C. Shu and K. S. Yeo, “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, vol. 192, no. 7-8, pp. 941–954, 2003.
[9] A. I. Tolstykh and D. A. Shirobokov, “On using radial basis functions
in a ”finite difference mode” with applications to elasticity problems,”
Computational Mechanics, vol. 33, no. 1, pp. 68–79, 2003.
[10] Y. Sanyasiraju and G. Chandhini, “Local radial basis function based
gridfree scheme for unsteady incompressible viscous flows,” Journal of
Computational Physics, vol. 227, no. 20, pp. 8922–8948, 2008.
[11] P. B. N. P. Phani Chinchapatnam, K. Djidjeli and M. Tan, “A compact
rbf-fd based meshless method for the incompressible navier-stokes
equations,” Proceedings of the Institution of Mechanical Engineers Part
M-Journal of Engineering for the Maritime Environment, vol. 223,
no. M3, pp. 275–290, 2009.
[12] G. B. Wright and B. Fornberg, “Scattered node compact finite differencetype
formulas generated from radial basis functions,” Journal of Computational
Physics, vol. 212, no. 1, pp. 99–123, 2006.
[13] C. F. L. C. S. Huang and A. H. D. Cheng, “Error estimate, optimal
shape factor, and high precision computation of multiquadric collocation
method,” Engineering Analysis with Boundary Elements, vol. 31, no. 7,
pp. 614–623, 2007.
[14] L. I. M. Gherlone and M. D. Sciuva, “A novel algorithm for shape
parameter selection in radial basis functions collocation method,” Composite
Structures, vol. 94, no. 2, pp. 453–461, 2012.
[15] S. Rippa, “An algorithm for selecting a good value for the parameter
c in radial basis function interpolation,” Advances in Computational
Mathematics, vol. 11, no. 2-3, pp. 193–210, 1999.
[16] A. G. S. Hamed Meraji and P. Malekzadeh, “An efficient algorithm
based on the differential quadrature method for solving navier-stokes
equations,” International Journal for Numerical Methods in Fluids, pp.
n/a–n/a, 2012.
[17] R. Franke, “Scattered data interpolation: Tests of some method,” Mathematics
of Computation, vol. 38, no. 157, pp. 181–200, 1982.
[18] U. K. N. G. Ghia and C. Shin, “High-re solutions for incompressible
flow using the navier-stokes equations and a multigrid method,” Journal
of Computational Physics, vol. 48, no. 3, pp. 387 – 411, 1982.
[19] W. F. Spotz and G. F. Carey, “High-order compact scheme for the
steady stream-function vorticity equations,” International Journal for
Numerical Methods in Engineering, vol. 38, no. 20, pp. 3497–3512,
1995.</p>
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:65163", author = "A. Javed and K. Djidjeli and J. T. Xing", title = "Adaptive Shape Parameter (ASP) Technique for Local Radial Basis Functions (RBFs) and Their Application for Solution of Navier Strokes Equations", abstract = "The concept of adaptive shape parameters (ASP) has been presented for solution of incompressible Navier Strokes equations using mesh-free local Radial Basis Functions (RBF). The aim is to avoid ill-conditioning of coefficient matrices of RBF weights and inaccuracies in RBF interpolation resulting from non-optimized shape of basis functions for the cases where data points (or nodes) are not distributed uniformly throughout the domain. Unlike conventional approaches which assume globally similar values of RBF shape parameters, the presented ASP technique suggests that shape parameter be calculated exclusively for each data point (or node) based on the distribution of data points within its own influence domain. This will ensure interpolation accuracy while still maintaining well conditioned system of equations for RBF weights. Performance and accuracy of ASP technique has been tested by evaluating derivatives and laplacian of a known function using RBF in Finite difference mode (RBFFD), with and without the use of adaptivity in shape parameters. Application of adaptive shape parameters (ASP) for solution of incompressible Navier Strokes equations has been presented by solving lid driven cavity flow problem on mesh-free domain using RBF-FD. The results have been compared for fixed and adaptive shape parameters. Improved accuracy has been achieved with the use of ASP in RBF-FD especially at regions where larger gradients of field variables exist.
", keywords = "CFD, Meshless Particle Method, Radial Basis Functions, Shape Parameters", volume = "7", number = "9", pages = "1846-10", }
