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Axisymmetric multiquadrics
Engineering Analysis with Boundary Elements 30 (2006) 137–142 www.elsevier.com/locate/enganabound
Research note
Axisymmetric multiquadrics Bozˇidar Sˇarler a,*, Nikola Jelic´ b, Igor Kovacˇevic´ a, Mitja Lakner c, Janez Perko a b
a Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Vipavska 13, Nova Gorica 5000, Slovenia Department of Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria and LECAD Group, University of Ljubljana, SI-1000 Ljubljana, Slovenia c Faculty of Civil Engineering and Geodesy, University of Ljubljana, Ljubljana, Slovenia
Received 1 June 2005; received in revised form 7 October 2005; accepted 25 October 2005 Available online 15 December 2005
Abstract This paper reviews the previous axisymmetric global interpolation functions used in the context of the dual reciprocity boundary element method and dual reciprocity method of fundamental solutions connected to axisymmetric Laplace operator. It complements our axisymmetric thin plate splines [1] with the axisymmetric form of the Hardy’s multiquadrics ðr2 C r02 Þm=2 ; mZG1. This new functions can be used in the improved Golberg–Chen–Karur [2] type of approximations. The basic equations are accompanied by a set of related expressions that permit straightforward use of the developed global interpolation functions in a broad spectrum of dual reciprocity boundary element method and method of fundamental solutions, and meshless direct collocation like discrete approximate procedures. q 2005 Elsevier Ltd. All rights reserved. Keywords: Boundary element method; Method of fundamental solutions; Collocation method; Radial basis functions; Dual reciprocity method; Multiquadrics; Axisymmetry
1. Introduction Axisymmetric geometry and field problems occur very frequently in science and engineering. The discrete approximate solutions of the different governing equations in such situations are of pronounced importance. The fusion of the boundary element method and global interpolation emerges in a variety of dual reciprocity (DR) boundary element method (BEM) discrete approximative procedures [3,4] that give reasonable evaluations of the governing equations. Two very comprehensive overviews have been published [5,6] regarding the use of the different global approximation functions in the BEM context. However, the mathematical properties of such methods are nowadays far from being sufficiently understood. Because of the unresolved theoretical answers to related existence, uniqueness, convergence, and stability issues, many numerical experiments and comparisons have been traditionally made in an ad-hoc manner in the DRBEM literature. The problem of global interpolation outside of the BEM context has been much more closely investigated
mathematically [7]. Corresponding analyses show that the use of the radial basis class of functions [8] represents a proper choice for multidimensional global interpolation. Most of the related advances focus on the augmented thin plate splines (ATPS) and multiquadrics (MQ). The ATPS are known to give the minimized curvature of the interpolation and the MQ could, depending on the choice of the free parameter, converge very rapidly. Karur and Ramachadran [9] first gave DRBEM numerical examples with ATPS and claim a superior solution to the heuristic ‘one-plus-r’ global approximation functions in two-dimensional planar problems. Golberg et al. [2] used MQ in the method of fundamental solutions (DRMFS) variant with global interpolation. They demonstrate up to three orders of magnitude of improvement in accuracy over ATPS and ‘oneplus-r’ functions provided that the free parameter is properly chosen. Surprisingly, not many DRBEM solutions structured with the fundamental solution of the Laplace equation deal with axisymmetric problems. In the pioneering work concerning this DRBEM aspect, Wrobel and Telles [10] heuristically use the global approximation functions of the form
* Corresponding author. Fax: C1470 1919. E-mail addresses: [email protected] (B. Sˇarler), [email protected]. at (N. Jelic´), [email protected] (I. Kovacˇevic´), [email protected] (M. Lakner), [email protected] (J. Perko).
1 pr 2 2 1=2 j Z ðp Kp Þ C ðp Kp Þ 1K ; r nr z nz A n 4 pnr
0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2005.10.003
with the notation elaborated in the next chapter. Massee´ and Marcouiller [11] found this function inadequate and after
(1)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
138
several numerical experiments proposed 2 2 p0 ðpr Kpnr Þ2 A jn Z pnr ðpr Kpnr Þ C ðpz Kpnz Þ Cðpz Kpnz Þ
2 1=2
3
C pr ;
(2)
with p0 representing a small positive constant which was set to 0.01. The axisymmetric form of the scaled augmented thin plate splines has been developed in [1]. Its successful implementation and testing in the classical DRBEM is demonstrated in [12,13] where they appear in the context of solving the convective–diffusive problems with non-linear boundary conditions, material properties, and phase-change. These functions have been in addition used in DRBEM solving of the temperature field in DC casting of alluminium alloy billets [14] where they appear in a full-scale industrial context. Chen et al. [15] developed the solution of the Poisson equation based on the DRMFS. To make use of the MFS, it is necessary to calculate a particular solution, which can be subtracted off, so that the MFS can be used to solve the resulting Laplace problem. This presents a novel problem, since the axisymmetric Poisson operator does not have constant coefficients, so previous methods based on radial basis functions cannot be used. To overcome this, the source term is approximated by a two-dimensional polynomial in r and z as in Goldberg et al. [16]. One can then obtain polynomial particular solutions by the method of undetermined coefficients. The principal incitements for this paper are two. The first is that the axisymmetric ATPS cannot be used in the context of transport phenomena that extend with one coordinate to infinity, because they do not decay with growing distance from the collocation point. Such arrangements are of extreme importance for example in environmental transport phenomena. The second fact is the fact that the axisymmetric form of MQ have not been deduced yet and can be applied instead of polynomials (for example in [15]). The present paper thus focuses on a relatively complex derivation of the axisymmetic MQ and related expressions for use in the spectrum of DRBEM, DRMFS, and Kansa [17,18] like discrete approximate procedures. 2. Derivation The interpolation of the scalar function F 2R3 with the three dimensional MQ 3jn could be represented [19,20] in the following form FðpÞ z 3 jn ðpÞzn ;
n Z 1; 2; .; N C 1;
The MQ of interest are qffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Z rn2 C r02 ; 3 n;1
(6)
3 jn;3
Z ðrn2 C r02 Þ3=2 ;
(7)
3 jn;3
1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 rn C r02
(8)
3 jn;K3
Z
1 ; ðrn2 C r02 Þ3=2
with rn Z jrn j; rn Z pKpn ;
3 jNC1
Z 1:
1 ^ ð2r 2 C 5r02 Þ 3 jn;1 Z 24 n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r4 rn2 C r02 C 0 log Rðrn Þ; 8rn
1 ^ ð8r 4 C 26rn2 r02 C 33r02 Þ 3 jn;3 Z 240 n C 1 ^ 3 jn;K1 Z 2
(13)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rn2 C r02
r06 log Rðrn Þ; 16rn
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 rn2 C r02 C 0 log Rðrn Þ; 2rn
(15)
1 log Rðrn Þ; (16) rn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Rðrn Þ h rn C rn2 C r02 . However, instead of working with above functions we prefer to introduce linear combinations 3jn,A and 3jn,B as follows ^
3 jn;K3
3 jn;A
ZK
Z 3 jn;K1 C
r02 j 2 3 n;K3
1 r02 1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C 2 2 2 2 ðrn C r02 Þ3=2 rn C r0
(3)
(4)
3 jn;A
(5)
3 jn;B
r02 j Z 4 3 n;K1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 1 rn2 C r02 K 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 rn2 C r02
(17)
(18)
with corresponding solutions 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rn2 C r02 ;
^
r 1 Z 3 j^ n;K1 C 0 3 j^ n;K3 Z 2 2
^
r2 1 Z 3 j^ n;1 K 0 3 j^ n;K1 Z ðrn2 C r02 Þ3=2 : 12 4
and from the constraint i Z 1; 2; .; N:
(11)
are, respectively
Fðpi Þ Z 3 jn ðpi Þzn ; Z 0;
(10)
The corresponding solutions of Poisson equation in spherical coordinates ^ 1 v 2 v3 j n r (12) Z 3 jn vr r 2 vr
3 jn;B Z 3 jn;1 K
3 jNC1 ðpi Þzi
n Z 1; 2; .; N;
where Pn stands for the position vector of collocation point n. The augmentation function is
where p stands for the position vector and N stands for the number of collocation points. The Einstein summation convention is used. The NC1 coefficients zn are determined from the N collocation equations i Z 1; 2; .; N
(9)
(19)
(20)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
The linear combinations have been introduced because only they provide the closed form solution for axisymmetric problems. It turns out that 1 ^ (21) 3 jn;A Z 3 jn;1 2 and 1 ^ j : (22) 3 jn;B Z 12 3 n;3 The axisymmetry of the function F implies the invariance of the coordinate Pf Fðpðpr ; p4 ; pz ÞÞ Z Fðpðpr ; pz ÞÞ;
(23)
with pr, pf, and pz representing the cylindrical coordinates. For such functions, the interpolation (3) could be written as 0 1 ð 1 Fðpðpr ; pz ÞÞ z @ j ðpðpr ; p4 ; pz ÞÞdp4 Azn ; 2p 3 n (24) 2p n Z 1; 2; .; N C 1: The axisymmetric MQ jn are thus derived by the representation of the three dimensional MQ in cylindrical coordinates and subsequent integration over the symmetry coordinate pr ð 1 2p j ðpðp ; p ÞÞ Z j ðpðpr ; p4 ; pz ÞÞdp4 : (25) r z A n 2p 0 3 n It can be quickly shown that for arbitrary function f(jrj) relation qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 2p f rn2 C r02 dp4 2p 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 p=2 Z f ln 1Kkn2 sin2 a da (26) p 0 is valid, with the axial distances ln and kn defined as pr pnr l2n Z ðpr C pnr Þ2 C ðpz Kpnz Þ2 C r02 ; kn2 Z 4 2 : ln
(27)
Henceforth, the axisymmetric MQ in Cases A and B are Z
2 r2 Kðkn Þ C 03 Dðkn Þ; pln pln
n Z 1; 2; .; N;
(28)
A jn;B Z
2 r2 ln Eðkn ÞK 0 Kðkn Þ; p 2pln
n Z 1; 2; .; N;
(29)
A jn;A
where Kðkn Þ Z and Eðkn Þ Z
ð p=2 0
ð p=2 0
ð1Kkn2 sin2 4ÞK1=2 d4
(30)
ð1Kkn2 sin2 4Þ1=2 d4
(31)
stand for the complete elliptic integral of the first and second kind, respectively, and abbreviation 1 Dðkn Þ Z Eðkn Þ (32) 1Kkn2
139
has been introduced. The adjacent augmentation function is A jNC1 Z 1. Accordingly, the axisymmetric MQ interpolation reads Fðpðpr ; pz ÞÞ z A jn ðpðpr ; pz ÞÞzn ;
n Z 1; 2; .; N C 1:
(33)
The axisymmetric harmonic MQ’s A j^ n are derived by the integration of the three-dimensional harmonic MQ 3 j^ n over the symmetry coordinate pr ð 1 2p ^ ^ j ðpðpr ; p4 ; pz ÞÞdp4 ; (34) A jn ðpðpr ; pz ÞÞ Z 2p 0 3 n which yields in Case A 1 ^ l Eðk Þ; n Z 1; 2; .; N; A jn;A Z p n n and in Case B l3n Cðkn Þ; n Z 1; 2; .; N; 18p with abbreviation Cðkn Þ Z 2 2Kkn2 Eðkn ÞK 1Kkn2 Kðkn Þ: ^
A jn;B
Z
(35)
(36)
(37)
The augmented harmonic function is A j^ NC1 Z 1=6ðp2r C p2z Þ.
3. Related expressions Case A Partial derivatives of the axisymmetric MQ in Case A are given by
vA jn;A 2 k r2 vDðkn Þ vkn Z Dðkn ÞKKðkn Þ C n 20 pln kn vpı 2 ln vkn vpı (38)
2 3r02 vln K 2 Kðkn Þ C 2 Dðkn Þ ; vpı 2ln pln v2 A jn;A vpı vpj
2 2 vDðkn Þ kn r02 v2 Dðkn Þ ZK ðDðkn ÞKKðkn ÞÞK K pln kn kn vkn 2 l2n vkn2
vk vk 2 3k r 2 vDðkn Þ Dðkn ÞKKðkn Þ C n 20 ! n nK 2 vpı vpj pln kn 2 ln vkn
vkn vln vkn vln 4 r2 ! C C 3 Kðkn Þ C3 20 Dðkn Þ vpı vpj vpj vpı pln ln
vl vl 2 k r 2 vDðkn Þ v2 kn ! n nC Dðkn ÞKKðkn Þ C n 20 vpı vpj pln kn 2 ln vkn vpı vpj
2 3 r2 v2 l n K 2 Kðkn Þ C 20 Dðkn Þ ; 2 ln vpı vpj pln (39)
vA j^ n;A l vk 1 vl Z n ½Eðkn ÞKKðkn Þ n C Eðkn Þ n ; vpı pkn vpı p vpı
(40)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
140
v2 A j^ n;A l vk vk Z n 2 ½Kðkn ÞKDðkn Þ n n vpı vpj vpı vpj pkn 1 vkn vln vkn vln C ½Eðkn ÞKKðkn Þ C pkn vpı vpj vpj vpı 2
C
2
ln v kn 1 v ln ½Eðkn ÞKKðkn Þ C Eðkn Þ pkn vpı vpj p vpı vpj
v2 ^ 1 A jNC1 Z ; 3 vp2ı
(47)
v2 ^ jZ0: vpr pz
(48)
(41)
with indexes , j taking values r and z with the appropriate explicit relations given in Appendix A. Case B. Partial derivatives of the axisymmetric MQ in Case B are given by
vk vA jn;B 2l 1 r2 n Z n Eðkn ÞKKðkn ÞK 20 Dðkn ÞKKðkn Þ 4 ln vpı pkn vpı
2 1 r2 vln Eðkn Þ C 20 Kðkn Þ C ; ð42Þ p 4 ln vpı
4. Conclusions The present paper develops all necessary expressions for using the axisymmetric MQ in connection with the axisymmetric Laplace operator in DRBEM, DRMFS, and Kansa-like procedures. The corresponding double precision Fortran-77 subroutines can be obtained from the first author upon the request. The developed functions are currently being used in mesh-reduced solutions of various diffusion problems. A related publication follows. Acknowledgements
2
v A jn;B vpı vpj Z
2ln 1 r02 kn r02 vDðkn Þ Kðk ÞKDðk ÞC ðDðk ÞKKðk ÞÞK n n n n 2 l2n 4 l2n vkn pkn2
vkn vkn 2 1 r02 C Eðkn ÞKKðkn ÞC 2 ðDðkn ÞKKðkn ÞÞ ! 4 ln vpı vpj pkn (43) vkn vln vkn vln 1 r02 vln vln ! C K 3 Kðkn Þ p ln vpı vpj vpj vpı vpı vpj
2 2ln 1 r02 v kn C Eðkn ÞKKðkn ÞK 2 ðDðkn ÞKKðkn ÞÞ 4 ln pkn vpı vpj
2 2 1 r02 v ln C Eðkn ÞC 2 Kðkn Þ ; p 4 ln vpı vpj
vA j^ n;B 1 3 vCðkn Þ vkn 1 2 vl l l Cðkn Þ n ; Z C 18p n vkn vpı 6p n vpı vpı
v2 A j^ n;B l3 v2 Cðkn Þ vkn vkn l2 vCðkn Þ Z n C n 2 vpı vpj 18p vkn vpı vpj 6p vkn vkn vln vkn vln l vl vl ! C C n Cðkn Þ n n vpı vpj vpj vpı 3p vpı vpj l3n
2
l2n
(44)
(45)
2
vCðkn Þ v kn v ln C C Cðkn Þ : 18p vkn vpı vpj 6p vpı vpj The related partial derivatives of the augmented harmonic function are v ^ 1 (46) A jNC1 Z pı ; vpı 3
A part of the present research has been performed in the framework of the project J2-6403-1540-04: Modelling and Simulation of Solid-Liquid Systems, sponsored by the Slovenian Ministry of High Education, Science and Technology. Appendix A The partial derivatives of kn used are pr C pnr vkn 1 Z kn K ; 2pr vpr l2n
(A1)
vkn p Kp ZKkn z 2 nz ; vpz ln
(A2)
ðpr C pnr Þ2 1 pr C pnr v2 k n 1 Z kn 3 K 2K 2K ; ln 4pr vp2r l4n pr l2n
(A3)
v2 k n ðpz Kpnz Þ2 1 Z k 3 K ; n l2n vp2z l4n
(A4)
pr C pnr v2 k n pz Kpnz 1 Z kn 3 K : 2pr vpz pr l2n l2n
(A5)
The partial derivatives of ln used are pr C pnr vln Z ; vpr ln
(A6)
vln p Kpnz Z z ; vpz ln
(A7)
v2 ln 1 ðpr C pnr Þ2 ðpz Kpnz Þ2 C r02 Z K Z ; ln vp2r l3n l3n
(A8)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
ðpr C pnr Þ2 C r02 v2 l n 1 ðpz Kpnz Þ2 Z K Z ; ln vp2z l3n l3n ðpr C pnr Þðpz Kpnz Þ v2 l n ZK : vpr pz l3n
(A9)
(A10)
Appendix B The partial derivatives of C(kn) used are vCðkn Þ vCðkn Þ vkn Z ; vpı vkn vpı
(B1)
v2 Cðkn Þ v2 Cðkn Þ vkn 2 vCðkn Þ v2 kn Z C ; vkn vp2ı vpı vp2ı vkn2
(B2)
v2 Cðkn Þ v2 Cðkn Þ vkn vkn vCðkn Þ v2 kn Z C ; vpr pz vkn vpr vpz vkn2 vpr vpz
(B3)
vCðkn Þ 3 Z ð1K2kn2 ÞEðkn ÞKð1Kkn2 ÞKðkn Þ ; vkn kn
(B4)
v Cðkn Þ 3 ZK 2 ð1 C 4kn2 ÞEðkn ÞKð1 C 2k2 ÞKðkn Þ ; 2 vkn kn
(B5)
EðkÞ Z
141
2
p 1 1 3 2 k4 1 3 5 2 k6 1K K K/ : k2 K 2 2 2 4 2 4 6 3 5 (C2)
Expansions used when lim k/1 LK1 9 7 ð1Kk2 Þ C LK ð1Kk2 Þ2 KðkÞ Z L C 4 64 6 25 37 LK C ð1Kk2 Þ3 C/; 256 30
1 1 3 13 2 LK ð1Kk Þ C LK EðkÞ Z1 C ð1Kk2 Þ2 2 2 16 12 15 6 LK ð1Kk2 Þ3 C/ C ðC4Þ 128 5 with LZ log 4K1=2 logð1Kk2 Þ.
Appendix D Suitable approximations [24] for K(k) and E(k) are in the form
2
GðkÞ Z a0 C a1 k 0 C a2 k 0 2 C a3 k 0 3 C a4 k 0 4 C ½b0
based on [21–24] vKðkn Þ 1 1 Z Eðkn ÞK Kðkn Þ; 2 vkn kn kn ð1Kkn Þ
(B6)
v2 Kðkn Þ 1K3kn2 1K2kn2 ZK 2 Eðkn Þ C 2 Kðkn Þ; 2 2 2 kn ð1Kkn Þ kn ð1Kkn2 Þ vkn
(B7)
vEðkn Þ 1 1 Z Eðkn ÞK Kðkn Þ; vkn kn kn
(B8)
v2 Eðkn Þ 1 1 ZK 2 Eðkn Þ C 2 Kðkn Þ: vkn2 kn ð1Kkn2 Þ kn
(B9)
(B10)
v2 Dðkn Þ 1 Z 2 ½ðK1 C 7kn2 C 2kn4 ÞEðkn Þ 2 vkn kn ð1Kkn2 Þ3 C ð1K6kn2 C 5kn4 ÞKðkn Þ:
C b1 k 0 C b2 k 0 2 C b3 k 0 3 C b4 k 0 4log
1 C eðkÞ; k0
(D1)
with k 0 Z1Kk2, je(k)j!2!10K8, and coefficients in Table D1.
Similarly vDðkn Þ 1 C kn2 1 Z Eðkn ÞK Kðkn Þ; 2 2 vkn kn ð1Kkn Þ kn ð1Kkn2 Þ
(C3)
Table D1 Coefficients used in the approximation (74) of the complete elliptic integrals K(k) and E(k) G
K
E
a0 a1 a2 a3 a4 b0 b1 b2 b3 b4
1.38629 436112 0.09666 344259 0.03590 092383 0.03742 563713 0.01451 196212 0.50000 000000 0.12498 593597 0.06880 248576 0.03328 355346 0.00441 787012
1.00000 000000 0.44325 141463 0.06260 601220 0.04757 383546 0.01736 506451 0.00000 000000 0.24998 368310 0.09200 180037 0.04069 697526 0.00526 449639
Appendix E ðB11Þ
Appendix C The most important limiting expressions [23] are Expansions used when lim k/0
2 p 1 1 3 2 4 1 3 5 2 6 2 1C KðkÞ Z k C k C k C/ ; 2 2 2 4 2 4 6 (C1)
Limiting expressions in case prZ0 Case A. Limiting expression of the axisymmetric MQ RBF is given by A jn;A Z
1 r2 C 02 : ln 2ln
(E1)
Partial derivatives of the axisymmetric MQ are given by vA jn;A 1 vl 3 r 2 vl ZK 2 n K 40 n ; vpı ln vpı 2 ln vpı
(E2)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
142
v2 A jn;A 2 3r 2 vln vln 1 3 r 2 v2 ln Z 3 1 C 20 K 2 1 C 20 : 2 ln vpı vpj vpı vpj ln ln vpı vpj ln (E3) Limiting expression, and partial derivatives for harmonic axisymmetric MQ RBF are given by ln vA j^ n;A 1 vln v2 A j^ n;A 1 v2 ln ^ ; j Z Z ; Z A n;A 2 vpı vpı vpj 2 vpı vpj 2 vpı
(E4)
with indexes i, j taking values r and z with the appropriate explicit relations given in Appendix A. Case B. Similarly, limiting expression for axisymmetric MQ RBF in Case B is given by A jn;B
Z ln K
r02 : 4ln
(E5)
Partial derivatives of the axisymmetric MQ are given by vA jn;B vl 1 r 2 vl Z n C 20 n ; vpı vpı 4 ln vpı
(E6)
v2 A jn;B 1 r 2 vl vl 1 r 2 v2 l n ZK 30 n n C 1 C 20 : 2 ln vpı vpj 4 ln vpı vpj vpı vpj
(E7)
Limiting expression, and partial derivatives for harmonic axisymmetric MQ RBF are given by 2
ln ^n;B l3n vA j 4 ^ ; Z A jn;B Z 12 vp
vln vp
^n;B ; v2 A j
vpvpj
Z
l2n v2 ln l vl vl C n n n: 4 vpvpj 2 vp vpj (E8)
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[5] Golberg MA. The numerical evaluation of particular solutions in the BEM—a review. BE Commun 1995;6:99–106. [6] Partridge PW. Approximation functions in the dual reciprocity method. BE Commun 1997;8:1–4. [7] Golberg MA, Chen CS. A bibliography on radial basis function approximation. BE Commun 1996;7:155–63. [8] Buhmann MD. Radial basis functions: theory and implementations. Cambridge monographs on applied computational mathematics; 2003. [9] Karur SR, Ramachadran PA. Augmented thin plate spline approximation in DRM. BE Commun 1996;6:55–8. [10] Wrobel LC, Telles CA. A dual reciprocity boundary element method formulation for axisymmetric diffusion problems. In: Tanaka M, Brebbia CA, editors. Boundary elements VIII. Berlin: Springer; 1986. p. 59–69. [11] Masse´ B, Marcouiller L. Axisymmeric flow problems using the dual reciprocity boundary element method. In: Brebbia CA, editor. Boundary elements XIV. CMP Southampton; 1992. p. 639–49. [12] Sˇarler B, Kuhn G. Dual reciprocity boundary element method for convective–diffusive solid–liquid phase change problems, part I: formulation. Eng Anal 1999;21:53–63. [13] Sˇarler B, Kuhn G. Dual reciprocity boundary element method for convective–diffusive solid–liquid phase change problems, part II: numerical examples. Eng Anal 1999;21:65–79. [14] Sˇarler B, Mencinger J. Solution of temperature field in DC cast aluminium alloy billet by the dual reciprocity boundary element method. Int J Numer Methods Heat Fluid Flow 1999;9:269–95. [15] Chen CS, Muleshkov AS, Golberg MA, Mattheij RMM. A mesh-free approach to solving the axisymmetric Poisson’s equation. Numer Methods Part Differ Equat 2005;21:349–67. [16] Golberg MA, Muleshkov AS, Chen CS, Cheng AHD. Polynomial particular solutions for certain partial differential equations. Numer Methods Part Differ Equat 2003;19:112–33. [17] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics, 1. Surface approximations and partial derivative estimates. Comput Math Appl 1990;19: 127–45. [18] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics, 2. Solutions to parabolic, hyperbolic and elliptic partial-differential equations. Comput Math Appl 1990;19:146–61. [19] Golberg MA, Chen CS. The theory of radial basis functions applied to the BEM for inhomogenous partial differential equations. BE Commun 1994; 5:57–61. [20] Powell MJD. The theory of radial basis function approximation in 1990. In: Light W, editor. Advances in numerical analysis, vol. II. Oxford: Oxford Science Publications; 1992. p. 105–210. [21] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. 5th ed. San Diego: Academic Press; 1994. [22] Lo¨sch F. Tables of higher functions. Stuttgart: Teubner; 1966. [23] Korn GA, Korn TA. Mathematical handbook for scientists and engineers. New York: McGraw-Hill; 1968. [24] Abramowitz M, Stegun A. Handbook of mathematical functions. New York: Dover; 1972.
Research note
Axisymmetric multiquadrics Bozˇidar Sˇarler a,*, Nikola Jelic´ b, Igor Kovacˇevic´ a, Mitja Lakner c, Janez Perko a b
a Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Vipavska 13, Nova Gorica 5000, Slovenia Department of Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria and LECAD Group, University of Ljubljana, SI-1000 Ljubljana, Slovenia c Faculty of Civil Engineering and Geodesy, University of Ljubljana, Ljubljana, Slovenia
Received 1 June 2005; received in revised form 7 October 2005; accepted 25 October 2005 Available online 15 December 2005
Abstract This paper reviews the previous axisymmetric global interpolation functions used in the context of the dual reciprocity boundary element method and dual reciprocity method of fundamental solutions connected to axisymmetric Laplace operator. It complements our axisymmetric thin plate splines [1] with the axisymmetric form of the Hardy’s multiquadrics ðr2 C r02 Þm=2 ; mZG1. This new functions can be used in the improved Golberg–Chen–Karur [2] type of approximations. The basic equations are accompanied by a set of related expressions that permit straightforward use of the developed global interpolation functions in a broad spectrum of dual reciprocity boundary element method and method of fundamental solutions, and meshless direct collocation like discrete approximate procedures. q 2005 Elsevier Ltd. All rights reserved. Keywords: Boundary element method; Method of fundamental solutions; Collocation method; Radial basis functions; Dual reciprocity method; Multiquadrics; Axisymmetry
1. Introduction Axisymmetric geometry and field problems occur very frequently in science and engineering. The discrete approximate solutions of the different governing equations in such situations are of pronounced importance. The fusion of the boundary element method and global interpolation emerges in a variety of dual reciprocity (DR) boundary element method (BEM) discrete approximative procedures [3,4] that give reasonable evaluations of the governing equations. Two very comprehensive overviews have been published [5,6] regarding the use of the different global approximation functions in the BEM context. However, the mathematical properties of such methods are nowadays far from being sufficiently understood. Because of the unresolved theoretical answers to related existence, uniqueness, convergence, and stability issues, many numerical experiments and comparisons have been traditionally made in an ad-hoc manner in the DRBEM literature. The problem of global interpolation outside of the BEM context has been much more closely investigated
mathematically [7]. Corresponding analyses show that the use of the radial basis class of functions [8] represents a proper choice for multidimensional global interpolation. Most of the related advances focus on the augmented thin plate splines (ATPS) and multiquadrics (MQ). The ATPS are known to give the minimized curvature of the interpolation and the MQ could, depending on the choice of the free parameter, converge very rapidly. Karur and Ramachadran [9] first gave DRBEM numerical examples with ATPS and claim a superior solution to the heuristic ‘one-plus-r’ global approximation functions in two-dimensional planar problems. Golberg et al. [2] used MQ in the method of fundamental solutions (DRMFS) variant with global interpolation. They demonstrate up to three orders of magnitude of improvement in accuracy over ATPS and ‘oneplus-r’ functions provided that the free parameter is properly chosen. Surprisingly, not many DRBEM solutions structured with the fundamental solution of the Laplace equation deal with axisymmetric problems. In the pioneering work concerning this DRBEM aspect, Wrobel and Telles [10] heuristically use the global approximation functions of the form
* Corresponding author. Fax: C1470 1919. E-mail addresses: [email protected] (B. Sˇarler), [email protected]. at (N. Jelic´), [email protected] (I. Kovacˇevic´), [email protected] (M. Lakner), [email protected] (J. Perko).
1 pr 2 2 1=2 j Z ðp Kp Þ C ðp Kp Þ 1K ; r nr z nz A n 4 pnr
0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2005.10.003
with the notation elaborated in the next chapter. Massee´ and Marcouiller [11] found this function inadequate and after
(1)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
138
several numerical experiments proposed 2 2 p0 ðpr Kpnr Þ2 A jn Z pnr ðpr Kpnr Þ C ðpz Kpnz Þ Cðpz Kpnz Þ
2 1=2
3
C pr ;
(2)
with p0 representing a small positive constant which was set to 0.01. The axisymmetric form of the scaled augmented thin plate splines has been developed in [1]. Its successful implementation and testing in the classical DRBEM is demonstrated in [12,13] where they appear in the context of solving the convective–diffusive problems with non-linear boundary conditions, material properties, and phase-change. These functions have been in addition used in DRBEM solving of the temperature field in DC casting of alluminium alloy billets [14] where they appear in a full-scale industrial context. Chen et al. [15] developed the solution of the Poisson equation based on the DRMFS. To make use of the MFS, it is necessary to calculate a particular solution, which can be subtracted off, so that the MFS can be used to solve the resulting Laplace problem. This presents a novel problem, since the axisymmetric Poisson operator does not have constant coefficients, so previous methods based on radial basis functions cannot be used. To overcome this, the source term is approximated by a two-dimensional polynomial in r and z as in Goldberg et al. [16]. One can then obtain polynomial particular solutions by the method of undetermined coefficients. The principal incitements for this paper are two. The first is that the axisymmetric ATPS cannot be used in the context of transport phenomena that extend with one coordinate to infinity, because they do not decay with growing distance from the collocation point. Such arrangements are of extreme importance for example in environmental transport phenomena. The second fact is the fact that the axisymmetric form of MQ have not been deduced yet and can be applied instead of polynomials (for example in [15]). The present paper thus focuses on a relatively complex derivation of the axisymmetic MQ and related expressions for use in the spectrum of DRBEM, DRMFS, and Kansa [17,18] like discrete approximate procedures. 2. Derivation The interpolation of the scalar function F 2R3 with the three dimensional MQ 3jn could be represented [19,20] in the following form FðpÞ z 3 jn ðpÞzn ;
n Z 1; 2; .; N C 1;
The MQ of interest are qffiffiffiffiffiffiffiffiffiffiffiffiffiffi j Z rn2 C r02 ; 3 n;1
(6)
3 jn;3
Z ðrn2 C r02 Þ3=2 ;
(7)
3 jn;3
1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 rn C r02
(8)
3 jn;K3
Z
1 ; ðrn2 C r02 Þ3=2
with rn Z jrn j; rn Z pKpn ;
3 jNC1
Z 1:
1 ^ ð2r 2 C 5r02 Þ 3 jn;1 Z 24 n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r4 rn2 C r02 C 0 log Rðrn Þ; 8rn
1 ^ ð8r 4 C 26rn2 r02 C 33r02 Þ 3 jn;3 Z 240 n C 1 ^ 3 jn;K1 Z 2
(13)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rn2 C r02
r06 log Rðrn Þ; 16rn
(14)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 rn2 C r02 C 0 log Rðrn Þ; 2rn
(15)
1 log Rðrn Þ; (16) rn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Rðrn Þ h rn C rn2 C r02 . However, instead of working with above functions we prefer to introduce linear combinations 3jn,A and 3jn,B as follows ^
3 jn;K3
3 jn;A
ZK
Z 3 jn;K1 C
r02 j 2 3 n;K3
1 r02 1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C 2 2 2 2 ðrn C r02 Þ3=2 rn C r0
(3)
(4)
3 jn;A
(5)
3 jn;B
r02 j Z 4 3 n;K1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 1 rn2 C r02 K 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 rn2 C r02
(17)
(18)
with corresponding solutions 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rn2 C r02 ;
^
r 1 Z 3 j^ n;K1 C 0 3 j^ n;K3 Z 2 2
^
r2 1 Z 3 j^ n;1 K 0 3 j^ n;K1 Z ðrn2 C r02 Þ3=2 : 12 4
and from the constraint i Z 1; 2; .; N:
(11)
are, respectively
Fðpi Þ Z 3 jn ðpi Þzn ; Z 0;
(10)
The corresponding solutions of Poisson equation in spherical coordinates ^ 1 v 2 v3 j n r (12) Z 3 jn vr r 2 vr
3 jn;B Z 3 jn;1 K
3 jNC1 ðpi Þzi
n Z 1; 2; .; N;
where Pn stands for the position vector of collocation point n. The augmentation function is
where p stands for the position vector and N stands for the number of collocation points. The Einstein summation convention is used. The NC1 coefficients zn are determined from the N collocation equations i Z 1; 2; .; N
(9)
(19)
(20)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
The linear combinations have been introduced because only they provide the closed form solution for axisymmetric problems. It turns out that 1 ^ (21) 3 jn;A Z 3 jn;1 2 and 1 ^ j : (22) 3 jn;B Z 12 3 n;3 The axisymmetry of the function F implies the invariance of the coordinate Pf Fðpðpr ; p4 ; pz ÞÞ Z Fðpðpr ; pz ÞÞ;
(23)
with pr, pf, and pz representing the cylindrical coordinates. For such functions, the interpolation (3) could be written as 0 1 ð 1 Fðpðpr ; pz ÞÞ z @ j ðpðpr ; p4 ; pz ÞÞdp4 Azn ; 2p 3 n (24) 2p n Z 1; 2; .; N C 1: The axisymmetric MQ jn are thus derived by the representation of the three dimensional MQ in cylindrical coordinates and subsequent integration over the symmetry coordinate pr ð 1 2p j ðpðp ; p ÞÞ Z j ðpðpr ; p4 ; pz ÞÞdp4 : (25) r z A n 2p 0 3 n It can be quickly shown that for arbitrary function f(jrj) relation qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 2p f rn2 C r02 dp4 2p 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 p=2 Z f ln 1Kkn2 sin2 a da (26) p 0 is valid, with the axial distances ln and kn defined as pr pnr l2n Z ðpr C pnr Þ2 C ðpz Kpnz Þ2 C r02 ; kn2 Z 4 2 : ln
(27)
Henceforth, the axisymmetric MQ in Cases A and B are Z
2 r2 Kðkn Þ C 03 Dðkn Þ; pln pln
n Z 1; 2; .; N;
(28)
A jn;B Z
2 r2 ln Eðkn ÞK 0 Kðkn Þ; p 2pln
n Z 1; 2; .; N;
(29)
A jn;A
where Kðkn Þ Z and Eðkn Þ Z
ð p=2 0
ð p=2 0
ð1Kkn2 sin2 4ÞK1=2 d4
(30)
ð1Kkn2 sin2 4Þ1=2 d4
(31)
stand for the complete elliptic integral of the first and second kind, respectively, and abbreviation 1 Dðkn Þ Z Eðkn Þ (32) 1Kkn2
139
has been introduced. The adjacent augmentation function is A jNC1 Z 1. Accordingly, the axisymmetric MQ interpolation reads Fðpðpr ; pz ÞÞ z A jn ðpðpr ; pz ÞÞzn ;
n Z 1; 2; .; N C 1:
(33)
The axisymmetric harmonic MQ’s A j^ n are derived by the integration of the three-dimensional harmonic MQ 3 j^ n over the symmetry coordinate pr ð 1 2p ^ ^ j ðpðpr ; p4 ; pz ÞÞdp4 ; (34) A jn ðpðpr ; pz ÞÞ Z 2p 0 3 n which yields in Case A 1 ^ l Eðk Þ; n Z 1; 2; .; N; A jn;A Z p n n and in Case B l3n Cðkn Þ; n Z 1; 2; .; N; 18p with abbreviation Cðkn Þ Z 2 2Kkn2 Eðkn ÞK 1Kkn2 Kðkn Þ: ^
A jn;B
Z
(35)
(36)
(37)
The augmented harmonic function is A j^ NC1 Z 1=6ðp2r C p2z Þ.
3. Related expressions Case A Partial derivatives of the axisymmetric MQ in Case A are given by
vA jn;A 2 k r2 vDðkn Þ vkn Z Dðkn ÞKKðkn Þ C n 20 pln kn vpı 2 ln vkn vpı (38)
2 3r02 vln K 2 Kðkn Þ C 2 Dðkn Þ ; vpı 2ln pln v2 A jn;A vpı vpj
2 2 vDðkn Þ kn r02 v2 Dðkn Þ ZK ðDðkn ÞKKðkn ÞÞK K pln kn kn vkn 2 l2n vkn2
vk vk 2 3k r 2 vDðkn Þ Dðkn ÞKKðkn Þ C n 20 ! n nK 2 vpı vpj pln kn 2 ln vkn
vkn vln vkn vln 4 r2 ! C C 3 Kðkn Þ C3 20 Dðkn Þ vpı vpj vpj vpı pln ln
vl vl 2 k r 2 vDðkn Þ v2 kn ! n nC Dðkn ÞKKðkn Þ C n 20 vpı vpj pln kn 2 ln vkn vpı vpj
2 3 r2 v2 l n K 2 Kðkn Þ C 20 Dðkn Þ ; 2 ln vpı vpj pln (39)
vA j^ n;A l vk 1 vl Z n ½Eðkn ÞKKðkn Þ n C Eðkn Þ n ; vpı pkn vpı p vpı
(40)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
140
v2 A j^ n;A l vk vk Z n 2 ½Kðkn ÞKDðkn Þ n n vpı vpj vpı vpj pkn 1 vkn vln vkn vln C ½Eðkn ÞKKðkn Þ C pkn vpı vpj vpj vpı 2
C
2
ln v kn 1 v ln ½Eðkn ÞKKðkn Þ C Eðkn Þ pkn vpı vpj p vpı vpj
v2 ^ 1 A jNC1 Z ; 3 vp2ı
(47)
v2 ^ jZ0: vpr pz
(48)
(41)
with indexes , j taking values r and z with the appropriate explicit relations given in Appendix A. Case B. Partial derivatives of the axisymmetric MQ in Case B are given by
vk vA jn;B 2l 1 r2 n Z n Eðkn ÞKKðkn ÞK 20 Dðkn ÞKKðkn Þ 4 ln vpı pkn vpı
2 1 r2 vln Eðkn Þ C 20 Kðkn Þ C ; ð42Þ p 4 ln vpı
4. Conclusions The present paper develops all necessary expressions for using the axisymmetric MQ in connection with the axisymmetric Laplace operator in DRBEM, DRMFS, and Kansa-like procedures. The corresponding double precision Fortran-77 subroutines can be obtained from the first author upon the request. The developed functions are currently being used in mesh-reduced solutions of various diffusion problems. A related publication follows. Acknowledgements
2
v A jn;B vpı vpj Z
2ln 1 r02 kn r02 vDðkn Þ Kðk ÞKDðk ÞC ðDðk ÞKKðk ÞÞK n n n n 2 l2n 4 l2n vkn pkn2
vkn vkn 2 1 r02 C Eðkn ÞKKðkn ÞC 2 ðDðkn ÞKKðkn ÞÞ ! 4 ln vpı vpj pkn (43) vkn vln vkn vln 1 r02 vln vln ! C K 3 Kðkn Þ p ln vpı vpj vpj vpı vpı vpj
2 2ln 1 r02 v kn C Eðkn ÞKKðkn ÞK 2 ðDðkn ÞKKðkn ÞÞ 4 ln pkn vpı vpj
2 2 1 r02 v ln C Eðkn ÞC 2 Kðkn Þ ; p 4 ln vpı vpj
vA j^ n;B 1 3 vCðkn Þ vkn 1 2 vl l l Cðkn Þ n ; Z C 18p n vkn vpı 6p n vpı vpı
v2 A j^ n;B l3 v2 Cðkn Þ vkn vkn l2 vCðkn Þ Z n C n 2 vpı vpj 18p vkn vpı vpj 6p vkn vkn vln vkn vln l vl vl ! C C n Cðkn Þ n n vpı vpj vpj vpı 3p vpı vpj l3n
2
l2n
(44)
(45)
2
vCðkn Þ v kn v ln C C Cðkn Þ : 18p vkn vpı vpj 6p vpı vpj The related partial derivatives of the augmented harmonic function are v ^ 1 (46) A jNC1 Z pı ; vpı 3
A part of the present research has been performed in the framework of the project J2-6403-1540-04: Modelling and Simulation of Solid-Liquid Systems, sponsored by the Slovenian Ministry of High Education, Science and Technology. Appendix A The partial derivatives of kn used are pr C pnr vkn 1 Z kn K ; 2pr vpr l2n
(A1)
vkn p Kp ZKkn z 2 nz ; vpz ln
(A2)
ðpr C pnr Þ2 1 pr C pnr v2 k n 1 Z kn 3 K 2K 2K ; ln 4pr vp2r l4n pr l2n
(A3)
v2 k n ðpz Kpnz Þ2 1 Z k 3 K ; n l2n vp2z l4n
(A4)
pr C pnr v2 k n pz Kpnz 1 Z kn 3 K : 2pr vpz pr l2n l2n
(A5)
The partial derivatives of ln used are pr C pnr vln Z ; vpr ln
(A6)
vln p Kpnz Z z ; vpz ln
(A7)
v2 ln 1 ðpr C pnr Þ2 ðpz Kpnz Þ2 C r02 Z K Z ; ln vp2r l3n l3n
(A8)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
ðpr C pnr Þ2 C r02 v2 l n 1 ðpz Kpnz Þ2 Z K Z ; ln vp2z l3n l3n ðpr C pnr Þðpz Kpnz Þ v2 l n ZK : vpr pz l3n
(A9)
(A10)
Appendix B The partial derivatives of C(kn) used are vCðkn Þ vCðkn Þ vkn Z ; vpı vkn vpı
(B1)
v2 Cðkn Þ v2 Cðkn Þ vkn 2 vCðkn Þ v2 kn Z C ; vkn vp2ı vpı vp2ı vkn2
(B2)
v2 Cðkn Þ v2 Cðkn Þ vkn vkn vCðkn Þ v2 kn Z C ; vpr pz vkn vpr vpz vkn2 vpr vpz
(B3)
vCðkn Þ 3 Z ð1K2kn2 ÞEðkn ÞKð1Kkn2 ÞKðkn Þ ; vkn kn
(B4)
v Cðkn Þ 3 ZK 2 ð1 C 4kn2 ÞEðkn ÞKð1 C 2k2 ÞKðkn Þ ; 2 vkn kn
(B5)
EðkÞ Z
141
2
p 1 1 3 2 k4 1 3 5 2 k6 1K K K/ : k2 K 2 2 2 4 2 4 6 3 5 (C2)
Expansions used when lim k/1 LK1 9 7 ð1Kk2 Þ C LK ð1Kk2 Þ2 KðkÞ Z L C 4 64 6 25 37 LK C ð1Kk2 Þ3 C/; 256 30
1 1 3 13 2 LK ð1Kk Þ C LK EðkÞ Z1 C ð1Kk2 Þ2 2 2 16 12 15 6 LK ð1Kk2 Þ3 C/ C ðC4Þ 128 5 with LZ log 4K1=2 logð1Kk2 Þ.
Appendix D Suitable approximations [24] for K(k) and E(k) are in the form
2
GðkÞ Z a0 C a1 k 0 C a2 k 0 2 C a3 k 0 3 C a4 k 0 4 C ½b0
based on [21–24] vKðkn Þ 1 1 Z Eðkn ÞK Kðkn Þ; 2 vkn kn kn ð1Kkn Þ
(B6)
v2 Kðkn Þ 1K3kn2 1K2kn2 ZK 2 Eðkn Þ C 2 Kðkn Þ; 2 2 2 kn ð1Kkn Þ kn ð1Kkn2 Þ vkn
(B7)
vEðkn Þ 1 1 Z Eðkn ÞK Kðkn Þ; vkn kn kn
(B8)
v2 Eðkn Þ 1 1 ZK 2 Eðkn Þ C 2 Kðkn Þ: vkn2 kn ð1Kkn2 Þ kn
(B9)
(B10)
v2 Dðkn Þ 1 Z 2 ½ðK1 C 7kn2 C 2kn4 ÞEðkn Þ 2 vkn kn ð1Kkn2 Þ3 C ð1K6kn2 C 5kn4 ÞKðkn Þ:
C b1 k 0 C b2 k 0 2 C b3 k 0 3 C b4 k 0 4log
1 C eðkÞ; k0
(D1)
with k 0 Z1Kk2, je(k)j!2!10K8, and coefficients in Table D1.
Similarly vDðkn Þ 1 C kn2 1 Z Eðkn ÞK Kðkn Þ; 2 2 vkn kn ð1Kkn Þ kn ð1Kkn2 Þ
(C3)
Table D1 Coefficients used in the approximation (74) of the complete elliptic integrals K(k) and E(k) G
K
E
a0 a1 a2 a3 a4 b0 b1 b2 b3 b4
1.38629 436112 0.09666 344259 0.03590 092383 0.03742 563713 0.01451 196212 0.50000 000000 0.12498 593597 0.06880 248576 0.03328 355346 0.00441 787012
1.00000 000000 0.44325 141463 0.06260 601220 0.04757 383546 0.01736 506451 0.00000 000000 0.24998 368310 0.09200 180037 0.04069 697526 0.00526 449639
Appendix E ðB11Þ
Appendix C The most important limiting expressions [23] are Expansions used when lim k/0
2 p 1 1 3 2 4 1 3 5 2 6 2 1C KðkÞ Z k C k C k C/ ; 2 2 2 4 2 4 6 (C1)
Limiting expressions in case prZ0 Case A. Limiting expression of the axisymmetric MQ RBF is given by A jn;A Z
1 r2 C 02 : ln 2ln
(E1)
Partial derivatives of the axisymmetric MQ are given by vA jn;A 1 vl 3 r 2 vl ZK 2 n K 40 n ; vpı ln vpı 2 ln vpı
(E2)
B. Sˇarler et al. / Engineering Analysis with Boundary Elements 30 (2006) 137–142
142
v2 A jn;A 2 3r 2 vln vln 1 3 r 2 v2 ln Z 3 1 C 20 K 2 1 C 20 : 2 ln vpı vpj vpı vpj ln ln vpı vpj ln (E3) Limiting expression, and partial derivatives for harmonic axisymmetric MQ RBF are given by ln vA j^ n;A 1 vln v2 A j^ n;A 1 v2 ln ^ ; j Z Z ; Z A n;A 2 vpı vpı vpj 2 vpı vpj 2 vpı
(E4)
with indexes i, j taking values r and z with the appropriate explicit relations given in Appendix A. Case B. Similarly, limiting expression for axisymmetric MQ RBF in Case B is given by A jn;B
Z ln K
r02 : 4ln
(E5)
Partial derivatives of the axisymmetric MQ are given by vA jn;B vl 1 r 2 vl Z n C 20 n ; vpı vpı 4 ln vpı
(E6)
v2 A jn;B 1 r 2 vl vl 1 r 2 v2 l n ZK 30 n n C 1 C 20 : 2 ln vpı vpj 4 ln vpı vpj vpı vpj
(E7)
Limiting expression, and partial derivatives for harmonic axisymmetric MQ RBF are given by 2
ln ^n;B l3n vA j 4 ^ ; Z A jn;B Z 12 vp
vln vp
^n;B ; v2 A j
vpvpj
Z
l2n v2 ln l vl vl C n n n: 4 vpvpj 2 vp vpj (E8)
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