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

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. 2017/4/13

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MESHLESS METHOD MESH METHOD 2017/4/13

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. 2017/4/13

Radial Basis Functions Linear: Cubic: Multiquadrics: Polyharmonic Spines: Gaussian: 2017/4/13

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. 2017/4/13

Wendland’s CS-RBFs Define For d=1, For d=2, 3, For 2017/4/13

Globally Supported RBFs 2017/4/13

Compactly Supported RBFs 2017/4/13

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 2017/4/13

Example 2017/4/13

RBF Collocation Method Meshless Method I Kansa’s Method or RBF Collocation Method 2017/4/13

RBF Collocation Method (Kansa’s Method) Consider the Poisson’s equation (1) (2) We approximate u by û by assuming (3) where 2017/4/13

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). 2017/4/13

W W 2017/4/13

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) 2017/4/13

Substituting (9) into (8) and (2), we have Notice that 2017/4/13

Example I where RBFs: MQ, c = 0.8; i.e., Grid points: 19x19 Maximum error: 8.703E-5 2017/4/13

Approximate Sol. and Maximum Norm Error by Kansa’s Method 2017/4/13

Example II (Rotating Cone) Exact solution: where t = 0.01, t [0, 2] Maximum norm error = 0.001165 with c = 0.2 (MQ). 2017/4/13

Approximate Sol. by Kansa’s Method (Rotating Cone) 2017/4/13 t=1.0

Maximum Error Norm by Kansa’s Method (Rotating Cone) 2017/4/13

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). 2017/4/13

Approximate Sol. of Kansa’s Method – Burger’s Equation 2017/4/13 t=1.0 t=1.25

Maximum Norm Error of Kansa’s Method 2017/4/13

CFD Example with the Kansa’s Method: Natural Convection with/without Phase Change 2017/4/13

Meshless Method II MFS-DRM 2017/4/13

Elliptic PDEs 2017/4/13

*MFS - METHOD OF FUNDAMENTAL SOLUTIONS WE FOCUS ON THIS *MFS - METHOD OF FUNDAMENTAL SOLUTIONS 2017/4/13

The Method of Fundamental Solutions Desingularized Method The Charge Simulation Method The Superposition Method Regular BEM 2017/4/13

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 2017/4/13

CFD Example with the MFS: Potential Flow Around Circular Cylinder 2017/4/13

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.) 2017/4/13

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. 2017/4/13

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) 2017/4/13

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. 43,1421-1435, 1998) (Requires no meshing if = circle or sphere) Others 2017/4/13

Domain Embedding Method 2017/4/13

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 2017/4/13

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 2017/4/13

For in 2D 2017/4/13

Analytic Particular Solutions L= in 3D Recall (19) Since we have 2017/4/13

Following the same integration procedure as above, we obtain (20) 2017/4/13

Numerical Example in 3D Consider the following Poisson’s problem (21) (22) Physical Domain 2017/4/13

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  2017/4/13

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) 2017/4/13

Hence, to obtain approximate particular solutions to we approximate . By linearity, (37) Where the coefficients in (10) are chosen to guarantee maximal smoothness of 2017/4/13

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) 2017/4/13

LAPLACE TRANSFORM (HEAT EQUATION) Consider the BVP (25) (26) (27) where D is a bounded domain in 2D and 3D. Let (28) (29) (30) 2017/4/13

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 2017/4/13

FINITE DIFFERENCE IN TIME (HEAT EQUATION) (36) (37) and (38) (39) (40) (41) 2017/4/13

RBF-PDE Webpages http://rbf-pde.uah.edu/ http://www.cityu.edu.hk/ma/staff/ychon.html http://www.num.math.uni-goettingen.de/schaback / http://www.usm.edu/cschen 2017/4/13

The End 2017/4/13