A meshless method for solving nonhomogeneous Cauchy problems

Engineering Analysis with Boundary Elements 35 (2011) 499–506

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

A meshless method for solving nonhomogeneous Cauchy problems$ Ming Li a,, C.S. Chen b, Y.C. Hon c a b c

Department of Mathematics, Taiyuan University of Technology, Taiyuan, China Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA Department of Mathematics, City University of Hong Kong, China

a r t i c l e in fo

abstract

Article history: Received 5 March 2010 Accepted 14 August 2010 Available online 8 October 2010

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson’s equation. The main idea is to approximate the solution of Poisson’s equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Inverse problem Cauchy problem Meshless methods Method of fundamental solutions The method of particular solutions Radial basis functions Tikhonov’s regularization

1. Introduction We consider a bounded and connected domain O  Rd , d ¼2,3 and the following classical Cauchy problem:

DuðxÞ ¼ f ðxÞ, x A O,

ð1Þ

with measured Dirichlet and Neumann data on the known fixed boundary G (accessible for data measurement), which is a portion of the boundary @O uðxÞ ¼ gðxÞ, @uðxÞ ¼ hðxÞ, @n

x A G, x A G,

ð2Þ ð3Þ

where n is the unit outward normal vector on G, and f(x), g(x), h(x) are given functions. The above-mentioned Cauchy problem is a typical ill-posed problem in the sense of Hadamard [13] that any small error in the measured data may induce enormous error to the solution. The uniqueness and conditional stability of the solution to the problem (1)–(3) was given by Bukhgeim et al. [3]. The Cauchy problem for the Laplace equation arises from many branches of science and engineering, for example, non-destructive testing [3,16,36]. Recently, many numerical methods were employed to $ The work described in this paper was fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101209).  Corresponding author. E-mail address: [email protected] (M. Li).

0955-7997/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2010.09.003

solve such problem, for instance, Backus-Gilbert algorithm [16], FEM [8], BEM [6] and MFS [33–35]. In general, the solution of the inverse problem does not depend continuously on the initial data, as shown by Wei and Hon [32] for a Cauchy problem which often arises in monitoring the possibility of boundary corrosion in iron melting process. During the past decades, the method of fundamental solution (MFS) is considered as a meshless method and has been proven to be a highly effective boundary meshless method when the fundamental solutions of the governing equations are available [9,12]. Compared with the traditional mesh methods (FDM, FEM and BEM) [10], the meshless methods have the following advantages:

 It is applicable to more complicated domains.  It is readily extendable to solve high-dimensional problems.  It can be extended to solve time-dependent problems with known fundamental solutions. The MFS was first introduced by Kupradze and Aleksidze [22] in 1964. It had been largely applied to solve various types of homogeneous partial differential PDEs [9,12]. For instances, the solutions for potential problems by Mathon and Johnston [26], exterior Dirichlet acoustic scattering problem by Kress and Mohsen [20], and general second order linear elliptic partial differential equations by Clements [7]. Recently, Hon and Wei applied the MFS to solve the Cauchy problem of Laplace equation and heat equation in one-dimension [17], multidimensions [19], and for various kinds of boundary conditions [18]. The MFS also

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M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

has been developed to solve singular direct and inverse problems [24,25,28]. More details on the recent development of the MFS and its applications can be found in [12,9]. However, during the first three decades of development, the MFS was mostly restricted to solving homogeneous equations. In 1994, Golberg and Chen [11] introduced the concept of the method of particular solutions (MPS) using radial basis functions to extend the MFS to solving inhomogeneous and time-dependent problems [12]. More details on the recent development of the MPS can be found in Betcke and Trefethen [2]. Furthermore, due to the increasing attention to the meshless methods in science and engineering community, the MFS, being a meshless method, has become more competitive for solving various types of partial differential equations. More recently, in [4,31], the authors combined the particular solution and homogeneous solution together as a one-stage approach in the solution process for solving partial differential equations. The advantage of such formulation is that it can solve more general elliptic equations, for instance, variable coefficient and inhomogeneous elliptic equation. In [5], the authors introduced a numerical technique based on the radial basis function (RBF) collocation method. Comparing with the iterative schemes, the direct meshless method in [5] shows two advantages: solving ill-posed problems directly without iteration and no initial guess value is required. The above-mentioned one-stage technique using the MPS and MFS is ideal for solving inverse Poisson problems. We believe this is the first approach to apply the MFS to solve inhomogeneous inverse problems. To handle the noisy data, we use the standard Tikhonov regularization technique with L-curve method [14] for choosing the optimal regularized parameter for solving the resultant highly ill-conditioned system of linear equations. This is a novel approach as compared with the method proposed in [17,19]. The main idea of the proposed method is to combine the MPS and MFS to form a one-stage method similar to [1,4,31]. The structure of the paper is as follows. In Section 2, we briefly describe the MPS and the MFS. In Section 3, we add the idea of the formulation by combining the methods mentioned in Section 2 to establish an effective one-stage numerical scheme that will allow us to solve Poisson-type elliptic PDEs with Cauchy boundary conditions. In Section 4, we test three numerical examples to illustrate the stability and accuracy of the proposed method.

2. Standard formulation In this section, we briefly review the methods that will be used later for the formulation of our proposed method.

@uh ðxÞ @up ðxÞ ¼ hðxÞ , @n @n

x A G:

ð7Þ

Eqs. (5)–(7) can be solved by the MFS which will be further elaborated in Section 2.2. The key issue here is how to obtain an approximation to up for a general forcing term f in (4). The classical way of evaluating up is based on the Newtonian potential Z up ðxÞ ¼ GðJxnJÞf ðnÞ dn, ð8Þ O

where GðÞ is the fundamental solution 8 1 > > < 2plnJxnJ, d ¼ 2, GðJxnJÞ ¼ 1 > > , d ¼ 3: : 4pJxnJ

ð9Þ

This domain integration in (8) is difficult to evaluate using the general integration algorithms due to the singular integral kernel GðJxnJÞ. To overcome the difficulty of domain integration in evaluating up, Nardini and Brebbia [27] introduced the dual reciprocity method (DRM). The success of the DRM depends on how the right-hand side is approximated. Typically, this is done by approximating f by a finite sum of radial basis functions f ðxÞ C

n X

aj fðrj Þ,

ð10Þ

j¼1

where rj ¼ Jxxj J and {xj}N 1 are called the centers of trial points and f : R þ -R is a univariate function. The coefficients faj gN 1 are usually obtained by a collocation method; i.e., by solving N X

aj fðrkj Þ ¼ f ðxk Þ, 1r k rN,

ð11Þ

j¼1

where rkj ¼ Jxk xj J, xk A O. As a result, an approximate particular solution, u~ p , to (4) is given by [12] N X

u~ p ðxÞ ¼

aj Fðrj Þ, x A O,

ð12Þ

j¼1

where F is obtained by analytically solving

DF ¼ f:

ð13Þ

An accurate approximation of up depends on how well f is approximated. Consequently, the appropriate choice of basis function f is of considerable importance. Among all types of radial basis functions, multiquadrics (MQ), inverse MQ, and polyharmonic splines are the most popular choices. 2.2. The method of fundamental solutions for Cauchy problems

2.1. The method of particular solution One assumes that, in this paper, the Cauchy problem (1)–(3) has a unique solution u for any given continuous nonhomogeneous term f and the Cauchy data g and h. In the MPS we split the solution u of (1)–(3) into a particular solution up and a homogeneous solution uh. Let u ¼up + uh where up is a particular solution satisfying the nonhomogeneous equation

Dup ðxÞ ¼ f ðxÞ, x A O,

uh ðxÞ ¼ gðxÞup ðxÞ,

Duh ðxÞ ¼ 0, x A O,

ð14Þ

uh ðxÞ ¼ gh ðxÞ,

ð15Þ

@uh ðxÞ ¼ hh ðxÞ, @n

x A G:

ð16Þ

The basic idea of the MFS is that an approximate solution u~ h for homogeneous Cauchy problem (14)–(16) can be expressed as a linear combination of fundamental solutions

ð5Þ x A G,

xA G,

ð4Þ

but does not necessarily satisfy the boundary conditions in (2) and (3). Then the homogeneous solution uh satisfies

Duh ðxÞ ¼ 0, x A O,

We describe the MFS here for solving the homogeneous Cauchy problem:

u~ h ðxÞ ¼ ð6Þ

m X j¼1

bj Gðrj Þ, x A O,

ð17Þ

M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

501

where fnj gm i ¼ 1 are source points and rj ¼ Jxnj J is the Euclidean norm. Note that GðrÞ is the known fundamental solution given in (9). In the MFS, the source points are located on a fictitious ^ outside the physical domain (see Fig. 1) where O  O ^. boundary @O ~ The coefficients fbj gm are determined in such a way that u satisfies h 1 the Dirichlet boundary condition (15) and the Neumann boundary condition (16). Hence, we obtain a system of linear equations: m X

bj Gðrij Þ ¼ gðxi Þ, i ¼ 1, . . . ,M,

ð18Þ

j¼1 m X

bj

@Gðrij Þ @n

j¼1

¼ hðxi Þ,

i ¼ 1, . . . ,M,

ð19Þ

where {xi}M i¼ 1 is the collocation points on the accessible boundary and rij ¼ Jxi nj J. In matrix form, (18)–(19) can be rewritten as follows:

Fig. 1. Boundary collocation points (), interior points (  ), and source points (3) ^. on the fictitious boundary @O m coefficients faj gN j ¼ 1 and fbj gj ¼ 1 simultaneously. We note that DGðrj Þ ¼ 0 for any xA O. Then, from (20) to (23), we have

Ab ¼ b, where

N X

b ¼ ð b1 , . . . , bM Þ T ,

ai fðrli Þ ¼ f~ ðxl Þ, l ¼ 1, . . . ,N,

ð24Þ

i¼1

b ¼ ðgðx1 Þ, . . . ,gðxM Þ,hðx1 Þ, . . . ,hðxM ÞÞT , N X

and 0

Gðrij Þ

1

i¼1

B C A ¼ @ @Gðrij Þ A @n

ai Fðrki Þ þ

:

bj Gðrkj Þ ¼ g~ ðxk Þ, k ¼ N þ 1, . . . ,N þM,

3. Methodology In this section, we propose to apply a recently developed numerical technique [4] for solving Poisson-type Cauchy problems. In practical application, the governing equation (1) and the Dirichlet and Neumann boundary conditions (2)–(3) are given only at some accessible boundary portion which contain measurement noise. Assume that the data are collected at the N interior points {xi}N 1 in O satisfying the governing equation (1) and +M M boundary points {xi}N N + 1 on G satisfying the both Dirichlet and Neumann boundary conditions (1)–(3), respectively, as follows:

Duðxi Þ ¼ f~ ðxi Þ, i ¼ 1,2, . . . ,N,

ð20Þ

uðxi Þ ¼ g~ ðxi Þ,

ð21Þ

i ¼ N þ 1, . . . ,N þ M,

@uðxi Þ ~ Þ, i ¼ N þ 1, . . . ,N þ M, ¼ hðx ð22Þ i @n ~ Þ are randomly perturbed data (Gaussian where f~ ðxi Þ, g~ ðxi Þ, hðx i distributed random vector) of the functions f,g,h with tolerated noise level d 4 0, respectively. In Section 4, we will describe how ~ Þ are generated. the noisy data f~ ðxi Þ, g~ ðxi Þ, hðx i Based on the two methods discussed in the last section, we assume (1)–(3) can be directly approximated by the sum of the particular solution and homogeneous solution [4] N m X X ~ uðxÞ C uðxÞ ¼ aj Fðrj Þ þ bj Gðrj Þ, x A O, ð23Þ j¼1

where rj and rj are defined in the same way as in (10) and (17), respectively. In two-dimensions, the fundamental solution is given by GðrÞ ¼ lnr. Next, we illustrate how to choose the basis function FðrÞ. Fig. 1 shows the distribution of interior collocation points in O, boundary collocation points on G, and source points ^. on the fictitious boundary @O In (23), we apply two different basis functions with two different distance functions to approximate the solution of (1)–(3). Instead of finding the particular solution and homogeneous solution separately, we intend to obtain the undetermined

ð25Þ

j¼1

m @Gðrkj Þ @Fðrki Þ X ~ þ bj ¼ hðx k ¼ N þ1, . . . ,N þ M: k Þ, @n @n i¼1 j¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi If we choose fðrÞ ¼ r 2 þ c2 (MQ), then we obtain N X

ð2MÞm

j¼1

m X

aki

8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > > ð4c2 þ r 2 Þ r 2 þ c2  lnðc þ r 2 þ c2 Þ, > > 9 3 > > >8 3 > c <> > > , FðrÞ ¼ > <3 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi! > > > ð5c2 þ 2r 2 Þ r 2 þ c2 c4 r2 þ r2 þ c2 > > > > þ , ln > > >: > > 24 c 8r2 :

8   r pffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , r 2 þ c2 þ > > > c þ r 2 þ c2 >3 @FðrÞ < 8 0, ¼ > <  > @r  > c4 c4 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 þ r2 > p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,  ln r þ c > > : : 8r c2 þ r 2 8r 2 c

ð26Þ

in 2D, r ¼ 0, in 3D, r 4 0,

in 2D, r ¼ 0, r 40,

in 3D,

where c is the shape parameter of MQ. Other radial basis functions f can also be used and their corresponding F can be derived easily. Since MQ has been widely used for solving PDEs, we will focus on this basis function in this paper. For the implementation, we choose N interior points in O, M boundary collocation points on G, and m source points on the ^ . By the collocation method, we have fictitious boundary @O thus formulated the system of equations (24)–(26) of order (N + 2M)  (N + m). More specifically, we have Al ¼ b~

ð27Þ

with 0

fðrli Þ B Fðr Þ ki B

A¼B @ @Fðrki Þ @n

0

1

Gðrkj Þ C C C, @Gðrkj Þ A

ð28Þ

@n

where i¼1,2,y,N, j ¼N + 1,N + 2,y,N +m, l ¼1,2,y,N and k¼N +1, N +2,y,N + M,

l ¼ ða1 , . . . , aN , b1 , . . . , bm ÞT

ð29Þ

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M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

and T ~ ~ b~ ¼ ðf~ ðx1 Þ, . . . , f~ ðxN Þ, g~ ðxN þ 1 Þ, . . . , g~ ðxN þ M Þ, hðx N þ 1 Þ, . . . , hðxN þ M ÞÞ :

ð30Þ Since the original problem (1)–(3) is ill-posed, the ill-conditioning of the matrix A in (27) still persists. In other words, standard numerical methods cannot achieve reasonable accuracy for solving (27) due to the severe ill-conditioning of the matrix A. In fact, the condition number of matrix A increases dramatically when the total number of nodes becomes large. Several regularization methods have been developed for solving this kind of ill-conditioning problems [14]. In our computation we employed the Tikhonov regularization technique [30] to solve (27). The Tikhonov regularized solution ls for (27) is defined to be the solution to the following penalized least squares problem: ~ 2 þ s2 JlJ2 g, minfJAlbJ

ð31Þ

l

where JJ denotes the usual Euclidean norm and s is the regularization parameter. The determination of a suitable value of the regularization parameter s is crucial and is still under intensive research [29,30]. In our computation we use the L-curve method, which does noise level d, to determine a suitable value for s. The L-curve method was originally applied by Lawson and Hanson [23] to investigate the properties of regularized systems under different values of the regularization parameter s. We briefly describe the L-curve method as follows. Consider the curve L ¼ fðlogðJls J2 Þ,logðJAls bJ2 ÞÞ, s 40g:

ð32Þ

The accuracy of the numerical solution is measured by the following root mean square error (RMSE): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Nt u1 X ~ j Þuðxj ÞÞ2 , RMSE ¼ t ð35Þ ðuðx Nt j ¼ 1 where Nt is the number of testing nodes uniformly distributed in the domain O. 4.1. Two-dimensional examples

Example 1. In this example, we consider the Cauchy problem in (1)–(3) where f,g, and h are imposed based on the exact solution uðx,yÞ ¼ sinðpðxyÞÞ. The perturbed Cauchy data f~ , g~ , and h~ are given based on (33). The domain O is a unit square and the accessible boundary G is defined as

G ¼ fð0,yÞ,ð1,yÞ,ðx,0Þ : 0 r x,yr 1g: The interior points, boundary collocation points and source points for the MFS are shown in Fig. 2. To investigate the application of the proposed method for solving the Poisson-type Cauchy problem, we choose different noise level d. In this example, we choose m ¼40, N ¼64, M ¼30, R¼1.5 and c¼2. The RMSE with different noisy levels d is shown in Fig. 3. Note that even for d ¼ 0:1, which is considered to be quite large, our proposed method still produces decent results. We do not show the figure of approximate solution since the approximate and exact solution are visually indistinguishable. Our method indeed perform quite well in this example.

L is known as the L-curve and a suitable regularization parameter

s is one that corresponds to a regularized solution near the corner of the L-curve [14]. In our computation, we used the Matlab code developed by Hansen [15] for solving the discrete ill-conditioned system s of equations (27). Denote the regularized solution of (27) by l . The approximated solution u~ s for the problem (1)–(3) is then given as uðxÞ C u~ s ðxÞ ¼

N X

ai Fðri Þ þ

i¼1

m X

bj Gðrj Þ, x A O:

j¼1

Fig. 2. Boundary collocation points (), interior points (  ), and source points (3) on the fictitious boundary.

4. Numerical examples To demonstrate the effectiveness and stability of the proposed numerical method, several examples for solving the Poisson equation in 2D and 3D are presented in this section. The noisy data in interior points and accessible boundary points are generated by ð33Þ

where b is the exact data, d is the tolerated noise level and randn is the Gauss random number with variance s ¼ 1. The source points in the MFS are chosen as

n ¼ xþ Rðxx0 Þ,

10–1

RMSE

b~ ¼ bð1 þ randn  dÞ,

100

10–2

ð34Þ

where n is the location of the source points, x represents boundary nodes and x0 is the geometric center of mass of the domain O. The parameter R determines how far away the source points are from the boundary. We refer readers to [12] for further information on how to choose R. How to choose the optimal shape parameter c of MQ could be another issue. We follow the same scheme as suggested in [4] for choosing the optimal shape parameter of MQ.

10–3

10–4 10–6

10–5

10–4

10–3 δ

10–2

Fig. 3. RMSE versus various noise levels d.

10–1

100

M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

503

 Over-specified:

Example 2. In the second example we consider the same problem with exact solution u and domain O as previous example but with different accessible boundaries. We investigate the accuracy of the numerical solutions under three different specified Cauchy data: under-, equally-, and over-specified. The accessible boundary G is given as follows (Fig. 4):

G ¼ fð0,yÞ,ð1,yÞ,ðx,0Þ : 0 rx,yr 1g: The numerical results for under-specified and equally specified boundary are presented in Fig. 5. For over-specified boundary, the results can be seen in Fig. 3 in the previous example where the use of more Cauchy data on the boundary greatly improves the accuracy of numerical solution. Even for the under-specified case, our proposed method did perform reasonably well.

 Under-specified: G ¼ fðx,0Þ : 0 r x r 1g,

 Equally specified:

Example 3. In this example we consider the Cauchy problem in (1)–(3) where f,g, and h are imposed based on the exact solution

G ¼ fðx,0Þ,ðy,0Þ,ðy,1Þ : 0 r x r1,0 r y r 12g,

Fig. 4. Three different accessible boundaries. (a) Under-specified. (b) Equally specified. (c) Over-specified.

100

RMSE

RMSE

100

10–1

10–2 10–6

–4

–2

10

10–2 10–6

0

10

10

10–1

10–4

10–2

100

δ

δ

Fig. 5. The RMSE for various noise levels d for under-specified (left) and equally specified (right) boundaries.

0.06

3

0.04

2

0.02

1

0

0

–0.02

–1

–0.04

–2

–0.06 –0.08

–3 1 y

0

–1

–1

0

1 x

2

–0.1 1 x

0

–1

1

~ Fig. 6. The profiles of exact solution (left) and approximation error uðx,yÞuðx,yÞ (right).

0

–1 y

504

M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx,yÞ ¼ ysinðpxÞþ xsinðpyÞ þ lnð ðx þ1Þ2 þy2 Þ. The perturbed Cauchy data f~ , g~ , and h~ are given based on (33). The solution domain in this example is more irregular and its boundary is defined by the oval of Cassini curve @O ¼ fðx,yÞ : x ¼ rðyÞcos y,y ¼ rðyÞsin y,p r y r pg, where

The distribution of interior point, boundary collocation points, and source points for the MFS are shown in Fig. 1. In this example, we choose m¼50, N ¼627, M ¼50, R¼ 8 and c¼2. In Fig. 6, we show the approximate solution u~ and the profile of the error uu~ in the entire domain O for noisy level d ¼ 0:01. The RMSE is 0.013. 4.2. Three-dimensional examples

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rðyÞ ¼ ðcosð3yÞ þ 2sin2 ð3yÞÞ1=3 :

One of the attractive features of the MFS–MPS is its applicability for solving more challenging problems in 3D. In fact, little effort is required to extend the method from 2D to 3D since the boundary and interior points can be randomly selected. The solution procedure is practically the same for 2D and 3D cases.

The accessible boundary is defined as follows:

G ¼ fðx,yÞ : x ¼ rðyÞcos y,y ¼ rðyÞsin y,12 p r y r 12pg:

Example 4. In this example, we consider the 3D Cauchy–Poisson problem (1)–(3). The boundary conditions f, g, and h in (2)–(3) are given under the pffiffi pffiffi assumption that the exact solution is uðx,y,zÞ ¼ sinðzÞðe 2x þe 2y Þ. The perturbed Cauchy data f~ , g~ , and h~ are given based on (30). We consider the solution domain to be a unit sphere:

O ¼ fðx,y,zÞ : x2 þ y2 þz2 r1g: The accessible boundary G is defined as follows:

G ¼ fðx,y,zÞ : x2 þy2 þ z2 ¼ 1,12p rx r 1,1 ry,z r1,0 r p r 1g:

The collocation points on Γ Fig. 7. Evenly distributed collocation points on the accessible boundary G (left) and shaded surface represents the accessible boundary with p ¼ 0.3 (right).

The source points

Fig. 8. The profile of evenly distributed source points on the fictitious boundary.

Note that the surface area of the accessible boundary G is proportional to p. For instance, Fig. 7 shows the collocation points on G (left) and the shaded accessible boundary with p¼ 0.3. When we increase the value of p, the surface area of G becomes larger. For the numerical implementation, we choose 360 interior points which uniformly distribute in the unit sphere and 200 source points on the fictitious surface of the sphere with radius R¼5 and center at the origin. In order to obtain evenly distributed source points, we employ the method in [21] to generate 200 source points on the fictitious surface (see Fig. 8). The 200 collocation points on the accessible boundary G are generated by using the same algorithm. The effort to generate evenly distributed points on the surface of accessible boundary is to reduce the ill-conditioning of the resultant matrix in the MFS. In Table 1, given the tolerated noise level d ¼ 0:001 and 0.01, respectively, we present the RMSE and maximum error with different values of p. We note that the use of more Cauchy data on the boundary greatly improves the numerical accuracy which is consistent with our intuition and the results in the previous examples. Our proposed method perform well even for small p value. In Table 2, we obtained numerical results using p ¼0.5 and different tolerated noise level d. From Table 2, our method perform very well even for large tolerate noise level.

Table 1 The RMSE and maximum errors using noise level d ¼ 0:001 and 0.01 and different accessible boundary percentage p.

d ¼ 0:001 p 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

d ¼ 0:01 c 0.625 0.788 0.838 0.312 0.726 1 5.750 4.821 7.875

~ 1 JuuJ

RMSE 4

3.528  10 2.789  10  4 8.589  10  4 1.276  10  3 5.324  10  3 1.240  10  2 6.648  10  2 1.580  10  1 5.220  10  1

p 3

1.178  10 1.184  10  3 5.321  10  3 6.816  10  3 2.436  10  3 6.878  10  2 3.386  10  1 7.105  10  1 2.937

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

c 0.891 1.542 1.192 0.625 1.001 1.001 0.969 7.609 0.644

~ 1 JuuJ

RMSE 3

2.331  10 1.945  10  3 4.579  10  3 7.044  10  3 1.635  10  2 2.682  10  2 7.689  10  2 2.441  10  1 4.558  10  1

7.098  10  3 5.841  10  3 2.101  10  2 3.501  10  2 8.442  10  2 1.482  10  1 4.011  10  1 1.0478 1.709

M. Li et al. / Engineering Analysis with Boundary Elements 35 (2011) 499–506

Furthermore, we consider the same problem with a more complicated 3D domain which is enclosed by the following surface: 8 9 x ¼ rðyÞcos y, > > < = ð36Þ @O ¼ ðx,y,zÞ : y ¼ rðyÞsin ysin j, 0 r y o 2p, 0 r j o p , > > : ; z ¼ rðyÞsin ycos j, where 2

rðyÞ ¼ ðcos3y þ ð8sin 3yÞ1=2 Þ1=3 :

505

Table 3 The RSME and maximum errors using different tolerated noise level d.

d

c

RMSE

~ 1 JuuJ

10  5 10  4 10  3 10  2 0.03 0.05 0.1

5.0 3.8 12.9 0.65 3.7 4.8 5.2

1.251  10  4 6.154  10  4 2.095  10  3 6.735  10  2 2.512  10  2 5.493  10  2 9.018  10  2

1.110  10  3 3.813  10  3 1.157  10  2 8.142  10  2 8.147  10  2 1.868  10  1 3.023  10  1

Fig. 9 shows the profile of the 3D graph in (36). In the numerical implementation, we choose 1028 interior points, 181 boundary

Table 2 The RSME and maximum errors using different tolerated noise level d.

d

c

RMSE

~ 1 JuuJ

10  5 10  4 10  3 10  2 0.03 0.05 0.1

0.981 8.687 0.981 1 0.656 8.872 9.652

1.403  10  3 5.014  10  3 3.495  10  3 1.635  10  2 3.544  10  2 3.732  10  2 8.381  10  2

7.946  10  3 3.989  10  2 2.397  10  2 8.442  10  2 1.490  10  1 1.821  10  1 2.961  10  1

points (p ¼0.5) on the accessible boundary G, and 361 source points which are located outside of @O with rðyÞ ¼ 2ðcos 3y þ ð8sin2 3yÞ1=2 Þ1=3 in (36). Fig. 10 shows the distribution of the collocation points (solid dots) on the accessible boundary. In Table 3, we show the numerical results with different noisy levels. We also notice that the numerical results shown in Tables 2 and 3 are consistent. We have demonstrated that the proposed method can handle more complicated 3D domain very well. 5. Conclusions In this paper we applied the newly developed MFS–MPS to solve Poisson-type Cauchy problems. Radial basis functions have been used to obtain the approximate particular solutions. Coupled with the MFS and MPS, the Cauchy problem can be solved directly without iteration. In particular, even for a small accessible boundary, we can obtain reasonable results. Furthermore, such numerical technique is virtually meshless. In particular, no numerical integration is required in the proposed approach. Thus, due to its simplicity, we are able to extend the method to more challenging problems in 3D with little effort which is a very attractive feature in meshless methods. In general, it is not a trivial task to solve a 3D problem using traditional numerical methods due to the indirect approach and constant meshing the domain. Numerical results show that the proposed technique is accurate and reliable for solving inhomogeneous Cauchy problems.

Fig. 9. The profile of 3D graph in (36).

References

1.5 1

Collocation points on Γ

0.5 0 –0.5

–2

–1 –1.5 2

–1 0 1.5

1

0.5

1 0 –0.5

–1 –1.5 2

Fig. 10. The collocation points (solid dots) on the accessible boundary.

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