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

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2015/4/302 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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2015/4/303

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4 MESH METHOD MESHLESS METHOD

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2015/4/305 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. Advantages of Meshless Methods

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2015/4/306 Radial Basis Functions Linear: Cubic: Multiquadrics: Polyharmonic Spines: Gaussian:

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2015/4/307 Compactly Supported RBFs References Z. Wu, Multivariate compactly supported positive definite radial functions, Adv. Comput. Math., 4, pp , 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 , 1995.

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2015/4/308 Wendland’s CS-RBFs Define For d=1, For For For d=2, 3,

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2015/4/309 Globally Supported RBFs =1+r

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2015/4/3010 Compactly Supported RBFs

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2015/4/3011 Assume that To approximate f bywe usually require fitting the given data set of pairwise distinct centres with the imposed conditions The linear system is well-posed if the interpolation matrix is non-singular Surface Reconstruction Scheme

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2015/4/3012 Example

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2015/4/3013 Kansa’s Method or RBF Collocation Method Meshless Method I

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2015/4/3014 RBF Collocation Method (Kansa’s Method) Consider the Poisson’s equation We approximate u by û by assuming where (1) (2) (3)

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2015/4/3015 (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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2015/4/3016

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2015/4/3017 For parabolic problems such as heat equation, we have where t is the time step, and u n and u n+1 are the solutions at time t n =n t and t n+1 =(n+1) t. Similar to elliptic problems, we assume (8) (9)

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2015/4/3018 Substituting (9) into (8) and (2), we have Notice that

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2015/4/3019 Example I where RBFs: MQ, c = 0.8; i.e., Grid points: 19x19 Maximum error: 8.703E-5

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2015/4/3020 Approximate Sol. and Maximum Norm Error by Kansa’s Method

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2015/4/3021 Example II (Rotating Cone) where Exact solution: t = 0.01, t [0, 2] Maximum norm error = with c = 0.2 (MQ).

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2015/4/3022 Approximate Sol. by Kansa’s Method (Rotating Cone) t=0.5 t=0.75 t=1.0 t=1.25

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2015/4/3023 Maximum Error Norm by Kansa’s Method (Rotating Cone) t=0.5 t=0.75 t=1.0t=1.25

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2015/4/3024 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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2015/4/3025 Approximate Sol. of Kansa’s Method – Burger’s Equation t=0.5 t=0.75 t=1.0t=1.25

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2015/4/3026 Maximum Norm Error of Kansa’s Method t=0.5 t=0.75 t=1.0 t=1.25

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2015/4/3027 CFD Example with the Kansa’s Method: Natural Convection with/without Phase Change

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2015/4/3028 MFS-DRM Meshless Method II

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2015/4/3029 Elliptic PDEs

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2015/4/3030 WE FOCUS ON THIS *MFS - METHOD OF FUNDAMENTAL SOLUTIONS

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2015/4/3031 The Superposition Method The Method of Fundamental Solutions Desingularized Method The Charge Simulation Method Regular BEM

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2015/4/3032 Let G(P,Q) satisfy be a fundamental solution for L. Choose a surface S R d containing D in its interior and m points {Q k } 1 m on S. Approximate u by To solve BVPs for n points on the boundary (MESHLESS) m points-sources on the source-set THE MFS

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2015/4/3033 CFD Example with the MFS: Potential Flow Around Circular Cylinder

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2015/4/3034 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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2015/4/3035 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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2015/4/3036 The Splitting Method Consider the following equation Whereis a bounded open nonempty domain with sufficiently regular boundary Letwheresatisfying but does not necessary satisfy the boundary condition in (11). (10) (11) v satisfies (12) (13) (14)

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2015/4/3037 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 Particular Solution

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2015/4/3038 D Domain Embedding Method

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2015/4/3039 Assume that and that we can obtain an analytical solutionto Then To approximate f bywe usually require fitting the given data set of pairwise distinct centres with the imposed conditions Dual Reciprocity Method (DRM) (15)

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2015/4/3040 The linear system is well-posed if the interpolation matrix is non-singular (16) (17) where and (18)

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2015/4/3041 For in 2D

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2015/4/3042 Analytic Particular Solutions L= in 3D Recall Since we have (19)

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2015/4/3043 Following the same integration procedure as above, we obtain (20)

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2015/4/3044 Numerical Example in 3D Consider the following Poisson’s problem Physical Domain (21) (22)

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2015/4/3045 The effect of various scaling factor 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.

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2015/4/3046 M.A. Golberg, C.S. Chen & Y.F. Rashed, The annihilator method for Computing particular solutions to PDEs, Eng. Anal. Bound. Elem., 23(3), , A. Muleshkov, M.A. Golberg & C.S. Chen, Particular solutions of Helmholtz- type operators using higher order polyharmonic splines, Computational Mechanics, where (32) (33) (34) RADIAL BASIS FUNCTION : POLYHARMONIC SPLINES

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2015/4/3047 Hence, to obtain approximate particular solutions to we approximate. By linearity, Where the coefficients in (10) are chosen to guarantee maximal smoothness of (37)

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2015/4/3048 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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2015/4/3049 (HEAT EQUATION) Consider the BVP where D is a bounded domain in 2D and 3D. Let (28) (25) (26) (27) (29) (30)

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2015/4/3050 (WAVE EQUATION) Consider BVP Similar approach can be applied to hyperbolic-heat equation and heat equations with memory (31) (32) (33) (34) (35)

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2015/4/3051 (HEAT EQUATION) and (36) (37) (38) (39) (40) (41)

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2015/4/ /

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2015/4/3053 The End

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