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Fast Boundary Knot Method for Solving Axisymmetric Helmholtz Problems with High Wave Number

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Fast Boundary Knot Method for Solving Axisymmetric Helmholtz Problems with High Wave Number Copyright © 2013 Tech Science Press CMES, vol.94, no.6, pp.485-505, 2013 Fast Boundary Knot Method for Solving Axisymmetric Helmholtz Problems with High Wave Number J. Lin1, W. Chen 1;2, C. S. Chen3, X. R. Jiang4 Abstract: To alleviate the difficulty of dense matrices resulting from the bound- ary knot method, the concept of the circulant matrix has been introduced to solve axi-symmetric Helmholtz problems. By placing the collocation points in a circular form on the surface of the boundary, the resulting matrix of the BKM has the block structure of a circulant matrix, which can be decomposed into a series of smaller matrices and solved efficiently. In particular, for the Helmholtz equation with high wave number, a large number of collocation points is required to achieve desired accuracy. In this paper, we present an efficient circulant boundary knot method algorithm for solving Helmholtz problems with high wave number. Keywords: boundary knot method, Helmholtz problem, circulant matrix, ax- isymmetric. 1 Introduction The boundary knot method (BKM) [Chen and Tanaka (2002); Chen (2002); Chen and Hon (2003); Wang, Ling, and Chen (2009); Wang, Chen, and Jiang (2010); Zhang and Wang (2012); Zheng, Chen, and Zhang (2013)] has been widely ap- plied for solving certain classes of boundary value problems. Instead of using the singular fundamental solution as the basis function in the method of funda- mental solutions (MFS) [Fairweather and Karageorghis (1998); Fairweather, Kara- georghis, and Martin (2003); Alves and Antunes (2005); Chen, Cho, and Golberg (2009); Drombosky, Meyer, and Ling (2009); Gu, Young, and Fan (2009); Lin, Gu, and Young (2010); Lin, Chen, and Wang (2011)], the BKM uses the non-singular general solution for the approximation of the solution. In the literature, both the MFS and BKM are classified as boundary-type meshless methods [Song and Chen 1 College of Mechanics and Materials, Hohai University, Nanjing, P.R. China, 210098 2 Corresponding to: [email protected] 3 Department of Mathematics, University of Southern Mississippi, USA 4 Nanjing Les Information Technology Co.,Ltd, Nanjing, P.R. China, 210007 486 Copyright © 2013 Tech Science Press CMES, vol.94, no.6, pp.485-505, 2013 (2009); Chen, Lin, and Wang (2011); Tan, Zhang, Wang, and Miao (2011); Liu and Sarler (2013)]. In the MFS, the determination of the location of the source points on the fictitious boundary is a challenging issue. The major difference between the MFS and BKM is that no fictitious boundary is required for the BKM in the solu- tion process. In terms of the numerical implementation, the BKM is much easier to implement than the MFS due to the use of general solutions rather than the funda- mental solutions. In general, the BKM is applicable as long as the general solution of the underlying differential operator is known. Initially, the BKM had been used exclusively for solving homogeneous problems. However, recently the BKM has been applied to solve nonhomogeneous problems through the radial basis functions and the method of particular solutions [Golberg and Chen (Computational Mechan- ics Publications); Hon and Chen (2003); Lin, Chen, and Sze (2012); Mramor, Vert- nik, and Sarler (2013)]. Since then, the BKM has been extended to solve a variety of physical problems governed by the Helmholtz, modified Helmholtz, Laplace, and the convection diffusion equations, including time-dependent problems and nonlinear problems [Jing and Zheng (2005a,b)]. Similar to the MFS, the resultant matrix of the BKM is dense and ill-conditioned [Li and Hon (2004); Liu (2008); Chen, Cho, and Golberg (2009)]. A direct solver for solving such matrices requires O(N3) operations and O(N2) memory storages. As such, there are not feasible for solving Helmholtz problems with high wave-number, where a large number of boundary collocation points is required. To our knowledge, the numerical effi- ciency of the BKM for solving these kinds of problems has not yet been published. The purpose of this paper is to introduce a more efficient BKM for solving high wave number Helmholtz problems in axi-symmetric domain. The key idea behind this approach was inspired by the matrix decomposition algorithm in the literature of the MFS [Karageorghis and Fairweather (1998, 1999, 2000); Tsangaris, Smyrlis, and Karageorghis (2004, 2006); Karageorghis, Chen, and Smyrlis (2009)]. In gen- eral, the proposed algorithm decomposes the large system of equations into small linear systems of lower order. As in the traditional matrix decomposition method, the fast fourier transform (FFT) is crucial in augmenting the speed of computation. 2 The BKM formulation d Let W be a bounded open set in R ;d = 2;3; with boundary ¶W = ¶WD [¶WN and ¶WD \ ¶WN = /0.In this paper we consider the following homogeneous Helmholtz problem (∇2 + k2)u(x) = 0; x 2 W; (1) u(x) = gd(x); x 2 ¶WD; (2) Fast Boundary Knot Method 487 ¶u(x) = g (x); x 2 ¶W ; (3) ¶n n N where k is the wave number, and gd and gn are known functions. N Let fx jg j=1 2 ¶W. The basic idea of the BKM is to approximate the solution of Eqs. (1)–(3) through a series of the general solutions y which are non-singular as follows: N u(x) ' u˜(x) = ∑ b jy(r j) (4) j=1 where r j = kx − x jk and k · k is the Euclidian norm. For the Helmholtz equation, we have [Chen (2002)] 8 J (kr); in 2D, <> 0 y(r) = sin(kr) (5) > ; in 3D. : r From Eqs. (2) and (3), by the collocation method, we have N ∑ b jy(ri j) = gd(xi); xi 2 ¶WD; j=1 (6) N ¶y(ri j) ∑ b j = gn(xi); xi 2 ¶WN: j=1 ¶n N Once the undetermined coefficient fb jg j=1 is obtained, the approximate solutions at any points can be obtained through Eq. (4). 3 Circulant matrix To solve Eq. (6) using direct solver, one requires O(N3) operations, and O(N2) memory space, which is infeasible when the number of boundary collocation points becomes large. In this section, we briefly introduce the concept of the circulant matrix and the fast fourier transform to accelerate the solution process of an axi- symmetric domain. Note that the resultant matrix from Eq. (6) is circulant if the solution domain is symmetric and the collocation points are uniformly distributed on the boundaries for two dimensional problems. The resultant matrix for the 3D case is somewhat m;n different from the 2D case. Let xi; j = f(xi; j;yi; j;zi)gi=1; j=1 be the collocation points on the boundary. They are distributed in the following circular form on the surface of the boundary i.e., xi; j = Ri cos(q j); yi; j = Ri sin(q j); 488 Copyright © 2013 Tech Science Press CMES, vol.94, no.6, pp.485-505, 2013 where 2p( j − 1) q = ; j = 1;2;··· ;n: j n Due to the axi-symmetry, the radius Ri of each concentreated circle is different for each zi. At each height zi, we have the same number of collocation points evenly distributed on a circle with radius Ri. From Eq. (6), we have Qb = h (7) where 0 1 Q11 Q12 ··· Q1m BQ21 Q22 ··· Q2m C Q = B C; (8) B . .. C @ . A Qm1 Qm2 ··· Qmm 0 1 y(kxi1 − x j1k) y(kxi1 − x j2k) ··· y(kxi1 − x jnk) By(kxi2 − x j1k) y(kxi2 − x j2k) ··· y(kxi2 − x jnk) C Q = B C; i j B . .. C @ . A y(kxin − x j1k) y(kxin − x j2k) ··· y(kxin − x jnk) and T b = [b1 b2 ··· bmn] : th th It is clear that Qi j is formulated using the circular points on the i and j circles. Due to the symmetry, the sub-matrix Qi j is circulant. Before we proceed, we would like to give a brief review of the matrix decomposi- tion of the circulant matrix. Let us consider the following n × n circulant matrix M = circ(q1;q2;··· ;qn) (9) It is well-known that M can be decomposed as follows [Li and Hon (2004)] M = U∗DU; (10) where 01 1 1 ··· 1 1 B1 w w2 ··· wn−1 C 1 B 2 4 2(n−1) C ∗ B1 w w ··· w C U = p B C; (11) n B. .. C @. A 1 wn−1 w2(n−1) ··· w(n−1)(n−1) Fast Boundary Knot Method 489 and n (k−1)(i−1) D = diag(d1;d2;··· ;dn); di = ∑ qkw ; (12) k=1 with w = e2pi=n. Note that U is a unitary matrix i.e., U∗U = UU∗ = I and I is the identity matrix Let ⊗ denote the matrix tensor product. The tensor product of a m×n matrix A and a l × k matrix B is defined as follows 0 1 a11B a12B ··· a1nB B a21B a22B ··· a2nB C A ⊗ B = B C B . .. C @ . A am1B am2B ··· amnB We also note that ∗ (Im ⊗U )(Im ⊗U) = Imn; (13) where Im is the m × m identity matrix. Pre-multiplying Eq. (7) by the block diagonal mn × mn matrix Im ⊗U and using the fact that U is unitary, we have ∗ (Im ⊗U)Q(Im ⊗U )(Im ⊗U)a = (Im ⊗U) f : (14) From the above equation, it follows that Q¯a¯ = f¯; (15) where 0 ∗ ∗ ∗ 1 UQ11U UQ12U ··· UQ1mU ∗ ∗ ∗ BUQ21U UQ22U ··· UQ2mU C Q¯ = (I ⊗U)Q(I ⊗U∗) = B C m m B . .. C @ . A ∗ ∗ ∗ UQm1U UQm2U ··· UQmmU 0 1 (16) D11 D12 ··· D1m BD21 D22 ··· D2m C = B C: B . .. C @ . A Dm1 Dm2 ··· Dmm and a¯ = (Im ⊗U)a; (17) 490 Copyright © 2013 Tech Science Press CMES, vol.94, no.6, pp.485-505, 2013 f¯ = (Im ⊗U) f : (18) In Eq.
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