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Scientific Computing with Radial Basis Functions

Scientific Computing with Radial Basis Functions. C.S. Chen Department of Mathematics University of Southern Mississippi U.S.A. PURPOSE OF THE LECTURE.

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Scientific Computing with Radial Basis Functions

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  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.

  3. MESHLESS METHOD MESH METHOD

  4. 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.

  5. Radial Basis Functions Linear: Cubic: Multiquadrics: Polyharmonic Spines: Gaussian:

  6. Compactly Supported RBFs • References • Z. Wu, Multivariate compactly supported positive definite radial • functions, Adv. Comput. Math., 4, pp. 283-292, 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. 477-496, 1995 • W. Wendland, Piecewise polynomial, positive definite and compactly • supported RBFs of minumal degree, Adv. Comput. Math., 4, pp 389- • 396, 1995.

  7. Wendland’s CS-RBFs Define For d=1, For d=2, 3, For

  8. Globally Supported RBFs  =1+r

  9. Compactly Supported RBFs

  10. 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

  11. Example

  12. Meshless Method I Kansa’s Method or RBF Collocation Method

  13. RBF Collocation Method (Kansa’s Method) Consider the Poisson’s equation (1) (2) We approximate u by û by assuming (3) where

  14. (4) (5) By substituting (3), (5) into (1)-(2), we have (6) (7) can be obtained by solving NN system (6)-(7).

  15. W W

  16. 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)

  17. Substituting (9) into (8) and (2), we have Notice that

  18. Example I where RBFs: MQ, c = 0.8; i.e., Grid points: 19x19 Maximum error: 8.703E-5

  19. Approximate Sol. and Maximum Norm Error by Kansa’s Method

  20. Example II (Rotating Cone) Exact solution: where t = 0.01, t [0, 2] Maximum norm error = 0.001165 with c = 0.2 (MQ).

  21. Approximate Sol. by Kansa’s Method (Rotating Cone) t=0.5 t=0.75 t=1.25 t=1.0

  22. Maximum Error Norm by Kansa’s Method (Rotating Cone) t=0.75 t=0.5 t=1.0 t=1.25

  23. Example III (Burgers’ Equation) Exact Solution t = 0.01, t [0, 1.25],  = 0.05 Maximum norm error = 0.0719 with c = 0.2 (MQ).

  24. Approximate Sol. of Kansa’s Method – Burger’s Equation t=0.75 t=0.5 t=1.0 t=1.25

  25. Maximum Norm Error of Kansa’s Method t=0.75 t=0.5 t=1.0 t=1.25

  26. CFD Example with the Kansa’s Method: Natural Convection with/without Phase Change

  27. Meshless Method II MFS-DRM

  28. Elliptic PDEs

  29. WE FOCUS ON THIS *MFS - METHOD OF FUNDAMENTAL SOLUTIONS

  30. The Method of Fundamental Solutions Desingularized Method The Charge Simulation Method The Superposition Method Regular BEM

  31. 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 THE MFS To solve BVPs for n points on the boundary (MESHLESS) m points-sources on the source-set

  32. CFD Example with the MFS: Potential Flow Around Circular Cylinder

  33. 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, 245-260, 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, 105-176, 1999.)

  34. COLLOCATION Choose n points on and set If B is linear, satisfy the linear equations A major concern with this method is theill-conditioning of However, this generally does not affect accuracy.

  35. 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)

  36. Particular Solution • Domainintegral 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. 43,1421-1435, 1998) • (Requires no meshing if = circle or sphere) • Others

  37. Domain Embedding Method D

  38. 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

  39. The linear system (16) is well-posed if the interpolation matrix is non-singular (17) where (18) and

  40. For in 2D

  41. Analytic Particular Solutions L= in 3D Recall (19) Since we have

  42. Following the same integration procedure as above, we obtain (20)

  43. Numerical Example in 3D Consider the following Poisson’s problem (21) (22) Physical Domain

  44. to approximate the forcing term. We choose 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 

  45. 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), 267-274, 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)

  46. Hence, to obtain approximate particular solutions to we approximate . By linearity, (37) Where the coefficients in (10) are chosen to guarantee maximal smoothness of

  47. For time-dependent problems, we consider two approaches to convert problems to Helmholtz equation Time-Dependent Problems • LAPLACE TRANSFORM • FINITE DIFFERENCES IN TIME ALSO POSSIBLE • OPERATOR SPLITTING (RAMACHANDRAN AND BALANKRISHMAN)

  48. LAPLACE TRANSFORM (HEAT EQUATION) Consider the BVP (25) (26) (27) where D is a bounded domain in2D and 3D. Let (28) (29) (30)

  49. 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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