1. Scientific Computing with Radial Basis Functions
C.S. Chen
Department of Mathematics
University of Southern Mississippi
U.S.A.

2. PURPOSE OF THE LECTURE
TO SHOW HOW RADIAL BASIS FUNCTIONS (RBFs) CAN BE USED TO PROVIDE 'MESH-FREE' METHODS FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS.
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3. 2017/4/13

4. MESHLESS METHOD
MESH METHOD
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5. Advantages of Meshless Methods
It requires neither domain nor surface discretization.
The formulation is similar for 2D and 3D problems.
It does not involve numerical integration.
Ease of learning.
Ease of coding.
Cost effectiveness due to the man-power reduction involved for the meshing.
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6. Radial Basis Functions
Linear:
Cubic:
Multiquadrics:
Polyharmonic Spines:
Gaussian:
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7. Compactly Supported RBFs
References
Z. Wu, Multivariate compactly supported positive definite radial
functions, Adv. Comput. Math., 4, pp , 1995.
R. Schaback, Creating surfaces from scattered data using radial basis
functions, in Mathematical Methods for Curves and Surface,
eds. M. Dahlen, T. Lyche and L. Schumaker, Vanderbilt Univ. Press,
Nashville, pp , 1995
W. Wendland, Piecewise polynomial, positive definite and compactly
supported RBFs of minumal degree, Adv. Comput. Math., 4, pp 389-
396, 1995.
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8. Wendland’s CS-RBFs
Define
For d=1,
For d=2, 3,
For
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9. Globally Supported RBFs
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10. Compactly Supported RBFs
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11. Surface Reconstruction Scheme
Assume that
To approximate f by
we usually require fitting the given
of pairwise distinct centres with the imposed
data set
conditions
The linear system
is well-posed if the interpolation matrix is non-singular
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12. Example
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13. RBF Collocation Method
Meshless Method I
Kansa’s Method or
RBF Collocation Method
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14. RBF Collocation Method (Kansa’s Method)
Consider the Poisson’s equation
(1)
(2)
We approximate u by û by assuming
(3)
where
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15. By substituting (3), (5) into (1)-(2), we have
(4)
(5)
By substituting (3), (5) into (1)-(2), we have
(6)
(7)
can be obtained by solving NN system (6)-(7).
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16. W W
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17. For parabolic problems such as heat equation, we have
(8)
where t is the time step, and un and un+1 are the solutions
at time tn=n t and tn+1=(n+1) t.
Similar to elliptic problems, we assume
(9)
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18. Substituting (9) into (8) and (2), we have
Notice that
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19. Example I where RBFs: MQ, c = 0.8; i.e., Grid points: 19x19
Maximum error: 8.703E-5
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20. Approximate Sol. and Maximum Norm Error by Kansa’s Method
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21. Example II (Rotating Cone)
Exact solution:
where
t = 0.01, t [0, 2]
Maximum norm error = with c = 0.2 (MQ).
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22. Approximate Sol. by Kansa’s Method (Rotating Cone)
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t=1.0

23. Maximum Error Norm by Kansa’s Method (Rotating Cone)
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24. Example III (Burgers’ Equation)
Exact Solution
t = 0.01, t [0, 1.25],  = 0.05
Maximum norm error = with c = 0.2 (MQ).
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25. Approximate Sol. of Kansa’s Method – Burger’s Equation
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t=1.0
t=1.25

26. Maximum Norm Error of Kansa’s Method
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27. CFD Example with the Kansa’s Method:
Natural Convection with/without Phase Change
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28. Meshless Method II
MFS-DRM
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29. Elliptic PDEs
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30. *MFS - METHOD OF FUNDAMENTAL SOLUTIONS
WE FOCUS ON THIS
*MFS - METHOD OF FUNDAMENTAL SOLUTIONS
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31. The Method of Fundamental Solutions
Desingularized Method
The Charge Simulation Method
The Superposition Method
Regular BEM
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32. THE MFS To solve BVPs for
Let G(P,Q) satisfy be a fundamental solution for L.
Choose a surface S Rd containing D in its interior and m points {Qk}1m on S.
Approximate u by
n points on the boundary (MESHLESS)
m points-sources on the source-set
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33. CFD Example with the MFS:
Potential Flow Around Circular Cylinder
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34. MFS - SATISFYING BOUNDARY CONDITON
To satisfy the boundary conditions one can use a variety of techniques
Galerkin’s Method (N. Limic, Galerkin-Petrov method for Helmholtz equation on exterior problems, Glasnik Mathematicki, 36, , 1981)
Least Squares (G. Fairweather & A. Kargeorghis, The MFS for elliptic BVPs, Adv. Comp. Math., 9, 69-95, 1998)
Collocation (M.A. Golberg & C.S. Chen, The MFS for potential, Helmholtz and diffusion problems, Chapter 4, Boundary Integral Methods: Numerical and Mathematical Aspects, ed. M.A. Golberg, WIT Press and Computational Mechanics Publ. Boston, Southampton, , 1999.)
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35. COLLOCATION Choose n points on and set
If B is linear, satisfy the linear equations
A major concern with this method is the ill-conditioning of
However, this generally does not affect accuracy.
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36. The Splitting Method Consider the following equation (10) (11) Where
is a bounded open nonempty domain
with sufficiently regular boundary
Let
where
satisfying
(12)
but does not necessary satisfy the boundary condition in (11).
v satisfies
(13)
(14)
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37. Particular Solution Domain integral Domain integral
Atkinson’s method (C.S. Chen, M.A. Golberg & Y.C. Hon, The MFS and quasi-Monte Carlo method for diffusion equations, Int. J. Num. Meth. Eng. 43, , 1998)
(Requires no meshing if = circle or sphere)
Others
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38. Domain Embedding Method
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39. Dual Reciprocity Method (DRM)
Assume that
and that we can obtain an analytical solution to
(15)
Then
To approximate f by
we usually require fitting the given
data set
of pairwise distinct centres with the imposed
conditions
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40. is well-posed if the interpolation matrix is non-singular
The linear system
(16)
is well-posed if the interpolation matrix is non-singular
(17)
where
(18)
and
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41. For
in 2D
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42. Analytic Particular Solutions L= in 3D
Recall
(19)
Since
we have
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43. Following the same integration procedure as above, we obtain
(20)
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44. Numerical Example in 3D Consider the following Poisson’s problem (21)
(22)
Physical Domain
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45. to approximate the forcing term.
We choose
to approximate the forcing term.
To evaluate particular solutions, we choose 300 quasi-random points
in a box [-1.5,1.5]x[-0.5,0.5]x[-0.5,0.5]. The numerical results are
compute along the x-axis with y=z=0.
The effect of various scaling factor 
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46. RADIAL BASIS FUNCTION :
POLYHARMONIC SPLINES
M.A. Golberg, C.S. Chen & Y.F. Rashed, The annihilator method for Computing particular solutions to PDEs, Eng. Anal. Bound. Elem., 23(3), , 1999.
A. Muleshkov, M.A. Golberg & C.S. Chen, Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Computational Mechanics, 2000.
(32)
(33)
where
(34)
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47. Hence, to obtain approximate particular solutions to
we approximate
. By linearity,
(37)
Where the coefficients in (10) are chosen to guarantee maximal
smoothness of
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48. Time-Dependent Problems
For time-dependent problems, we consider two approaches to convert problems to Helmholtz equation
LAPLACE TRANSFORM
FINITE DIFFERENCES IN TIME
ALSO POSSIBLE
OPERATOR SPLITTING (RAMACHANDRAN AND BALANKRISHMAN)
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49. LAPLACE TRANSFORM (HEAT EQUATION) Consider the BVP (25) (26) (27)
where D is a bounded domain in 2D and 3D.
Let
(28)
(29)
(30)
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50. LAPLACE TRANSFORM (WAVE EQUATION) Consider BVP (31) (32) (33) (34)
(35)
Similar approach can be applied to hyperbolic-heat
equation and heat equations with memory
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51. FINITE DIFFERENCE IN TIME
(HEAT EQUATION)
(36)
(37)
and
(38)
(39)
(40)
(41)
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52. RBF-PDE Webpages http://rbf-pde.uah.edu/ /
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53. The End
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