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A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems
International Journal of Heat and Mass Transfer 108 (2017) 945–960
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems Yao Sun a,⇑, Songnian He b a b
Tianjin Key Lab for Advanced Signal Processing, College of Science, Civil Aviation University of China, Tianjin 300300, China College of Science, Civil Aviation University of China, Tianjin 300300, China
a r t i c l e
i n f o
Article history: Received 16 August 2016 Received in revised form 27 November 2016 Accepted 20 December 2016
Keywords: Inverse heat conduction problem Regularization Morozov discrepancy principle
a b s t r a c t This paper documents a meshless method for the three-dimensional inverse heat conduction problems based on the method of fundamental solution (MFS). In order to overcome the ill-posedness of the corresponding problem, the Tikhonov regularization method, as well as Morozov’s discrepancy principle for selecting an appropriate regularization parameter are used. Hence there is to produce a stable and accuracy numerical results. Then some examples are given to check the effectiveness of this method, whilst the sensitive analysis is given. The numerical convergence and stability of this method are also analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Consider a 3D steady-state heat conduction problem in a bounded domain D R3 with a piecewise smooth boundary surface @D; Dc its open complement. C is a portion of @D. In this paper, we refer to isotropic steady heat conduction in the absence of inner heat sources, and the temperature distribution, uðxÞ, in the domain D satisfies the following elliptic partial differential equation, also referred to as the isotropic heat-conduction equation
r2 u ¼ 0;
ð1Þ
in D:
At a point x 2 @D, we denote the unit outward normal vector by nðxÞ, and the normal heat flux qðxÞ defined by
qðxÞ ¼ ru nðxÞ:
ð2Þ
In general, the heat conduction Eq. (1) is solved combining with appropriate Dirichlet, Neumann, or Robin boundary conditions to obtain the temperature distribution in the solution domain D. But in some practice problems, we only get the boundary data on easily accessible boundary. This terms the inverse steady heat conduction problem as follow: the temperature uðxÞ satisfies (1) and the following boundary conditions
e; u¼u
@u ¼e q; @n
on C:
ð3Þ
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Sun). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.079 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
The motivation for our present study concerning the inverse steady heat conduction problem in R3 is a typical problem associated with the iron-making blast furnace which can be described as follows: it is desired to monitor the corroded thickness of the accreted refractory wall in order to avoid a situation that molten metal breaks out. In practice, we can only monitor the temperature and heat flux on easily accessible boundaries. Then, the temperature and the norm heat flux on the remaining part can be recovered by solving an inverse heat conduction problem either for a unsteady or steady state Laplace equation. There are many methods for solving this kind problem, such as boundary element method by Wang et al. [1,2], singular boundary method [3–8], iterative method by Johansson and Lesnic [9,10], boundary particle method by Fu et al. [11], conjugate gradient method by Háo and Lesnic [12], potential function method by Sun et al. [13,14], and method of fundamental solutions by Sun [15], Wei et al. [16] and Young et al. [17], and invariant MFS by Sun [18–20], iterative MFS [21], and cited references here. The MFS, first introduced by Kupradze and Aleksidze [22], is an effective method for solving the direct/inverse problems governed by the partial differential equations. It is to approximate the solution by a linear combination of the fundamental solutions. Then the weighed coefficients will be determined by the boundary collocation method. Thus the MFS is also a meshless boundary collocation method, which belongs to Trefftz-like methods [23,24] and is applicable to boundary value problems in which the fundamental solution of the operator is known. In recent years, the MFS has become very popular primarily because of the ease with which it
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can be implemented, particularly for the problems in complex geometries. The simple implementation of the MFS for the problems with complex boundaries makes it an ideal candidate for the problems in which the boundary is of major importance or requires special attention. Since then, it has been successfully applied to a large variety of physical problems, an account of which can be found in the survey papers [27,25,26]. However, there exist some heat conduction problems for which the simple application of the MFS is not sufficient to obtain an accurate numerical solution, e.g. problems related to domains containing a boundary singularity generated by the presence of a crack or a V-notch, and hence the standard MFS has to be modified/enriched (see [28–34]). In [15], Sun gave a simple and effective numerical method based on the MFS. The main idea is to approximate the solution by the derivative of the Green function, which is a solution of the Laplace Eq. (1). In this paper, the main goal is to extend this method to solve the 3D inverse heat conduction problems, i.e. we approximate the solution u by
uN ðxÞ ¼
N X @ Uðx; yj Þ aj ; @ mðyÞ j¼1
yj 2 @B;
ð4Þ
where B is a bounded domain with piecewise smooth boundary, and mðyÞ is the outward normal at point y 2 @B, Uðx; yÞ denotes 1 . the fundamental solution given by Uðx; yÞ ¼ 4pjxyj In this paper, the main goal is to extend the aforementioned regularized MFS [15] by the same author for the solution of 3D inverse heat conduction problems. Since most practice problems are considered in R3 , it is necessary for checking the effective of this method for solving 3D inverse heat conduction problems.
The matrix arising from the MFS is severely ill-conditioned, thus a regularized solution is obtained by employing a regularization strategy, namely the Tikhonov regularization technique, whilst the optimal regularization parameter is determined by Morozov’s discrepancy principle. The paper is organized as follows: In Section 2, the theoretical results of the solution method employed are then given. In Section 3, we select the Tikhonov regularization technique to give a stable and accuracy results for solving this inverse boundary value problem. Finally, the numerical accuracy and stability of the presented method are given with some numerical examples. 2. The solution method Let B be bounded, connected domain with D B; mðyÞ is the outside unit normal on the boundary @B. Different from the MFS, we approximate the solution u by the following form
Table 1 The values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 400 and various amounts of noise added in the boundary data, for Example 1. Noise level (%)
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
0
5:55 1017
9:51 108
2:35 107
1
3:46 107
8:10 103
9:40 103
5
2:18 106
1:95 102
2:20 102
10
5:76 106
2:93 102
3:25 102
Fig. A.1. The exact solution and the numerical solutions of the temperature on @D n C obtained with various amounts of noise added in the boundary data for Example 1.
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Fig. A.2. The exact solution and the numerical solutions of the normal heat flux on @D n C obtained with various amounts of noise added in the boundary data for Example 1.
uN ðxÞ ¼
N X @ Uðx; yj Þ aj : @ mðyÞ j¼1
ð5Þ
The following theorems are then will give the theoretical results of the present method. From [15], they can be proved with a little modification, thus we omit the proof. Theorem 1. Let ðym Þ 2 @B be a sequence of points, and ðym Þ are dense on @B. Then the normal derivatives of the fundamental solutions
@ Uðx; ym Þ : m ¼ 1; 2; . . . @ mðyÞ
corresponding to the source points located at ðym Þ are complete in L2 ð@DÞ.
Meanwhile, on the boundary @D, we will approximate the normal heat flux of (2) by N X @ 2 Uðx; yj Þ aj ; @nðxÞ@ mðyÞ j¼1
qN ðxÞ ¼
ð7Þ
where nðxÞ denotes the outward unit normal vector. The coefficients aj will be determined by the collocation method from the interpolation conditions at M collocation points. In fact, the coefficients are determined by solving a linear system of M equations, which consist the boundary conditions at the M collocation points. We can rewrite the M collocation points as a system of M linear algebraic equations with N unknowns which can be generically written as
Ad ¼ h: Theorem 2. Let ðym Þ 2 @B be a sequence of points, and ðym Þ are dense on @B. Then
(
@ 2 Uðx; ym Þ : m ¼ 1; 2; . . . @nðxÞ@ mðyÞ
)
corresponding to the source points located at ðym Þ are complete in W, n o R and W is given by w 2 L2 ð@DÞj @D wds ¼ 0 . Then we use this method to solve the inverse Cauchy problem. We will approximate u by the following form N X @ Uðx; yj Þ uN ðxÞ ¼ aj : @ mðyÞ j¼1
ð6Þ
ð8Þ
The matrix A, the unknown vector d and the right-hand side h will be given by the following form:
Table 2 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, d ¼ 5:0, M ¼ N ¼ 300 and various amounts of noise added in the boundary data, for Example 2. Noise level (%) 1 5 10
eR ðuÞ
k 8
eR ð@ n uÞ
7:28 10
1:40 10
3
1:44 102
8:21 107
9:20 103
5:01 102
2:32 106
2:09 102
1:04 101
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@ Uðxi ; yj Þ @ 2 Uðxi ; yj Þ ; AMþi;j ¼ ; dj ¼ aj ; @ mðyÞ @nðxÞ@ mðyÞ ~ðxi Þ; i ¼ 1; . . . ; M; j ¼ 1; . . . ; N: ~ ðxi Þ; hMþi ¼ q hi ¼ u
tool available for the analysis of rank-deficient and discrete illposed problems is the singular value decomposition (SVD). The matrix A in (8) can be SVD-decomposed into
Ai;j ¼
In order to uniquely determine the coefficients, it should be noticed that the number M of the boundary collocation points and the number N of the source points must satisfy 2M P N. 3. Regularization method As following, the Tikhonov regularization is used to solve the system (8) combined Morozov discrepancy principle. In general, the right hand-side vector h of the system (8) is a perturbed vector d
h , namely, we should consider the perturbed equations d
d
Ad ¼ h :
ð9Þ d
More precisely, h is the measured noisy data satisfying d
hi ¼ hi þ drandðiÞhi ; where d is the percentage noise and the number rand(i) is a pseudorandom number drawn from the standard uniform distribution on the interval ½1; 1 generated by Matlab code 1 þ 2 randð1; iÞ. It is well-known that the system (8) is ill-conditioned. Thus this system cannot be solved by the direct methods, such as the leastsquares method, because of which will give a highly unstable solution for a perturbed right-hand side vector. The main numerical
A ¼ WRV > : W and V, which have 2M N and N N dimensions, respectively, are matrices whose columns are the orthonormal vectors W i and V i , the left and right singular vectors, and R ¼ diagðk1 ; . . . ; kN Þ is a diagonal matrix has non-negative diagonal elements in decreasing order, which are the singular values of A. Such a decomposition makes explicit the degree of illconditioning of the matrix A through the ratio between the maximum and the minimum singular value but also allows to write the solution of the system (8) in the following form
d¼
N X W>b i
i¼1
ki
V i:
For ill-conditioned matrix systems, there usually exist extremely small singular values, thus this equation clearly brings out the difficulties to deal with the ill-posed discrete problems. A very popular method for dealing with the ill-posedness of such problem is the Tikhonov regularization. The formly Tikhonov regularized solution of the system (8) is given by the solution as follows
ðkI þ A> AÞdk ¼ A> h: By introducing the regularization operator
Fig. A.3. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 2.
ð10Þ
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Fig. A.4. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 2.
1
Rk :¼ ðkI þ A> AÞ A> ;
we can achieve the regularized solution dk ¼ Rk h of the Eq. (8). From [37, Section 2.5], we can see that dk exists and is the unique minimum of the Tikhonov functional
J k ðdÞ ¼ kAd hk2 þ kkdk2 ;
k > 0:
The discrepancy principle for the determination of k is to compute k ¼ kðdÞ > 0 such that the corresponding Tikhonov solution d
da ; that is, the solution of the equation
ðkI þ A> AÞdk ¼ A> h ; d
d ðkI þ A> AÞ dk dk ¼ dk , as is easily seen by differentiating (11) with respect to k. The computation of k can be carried out with Newton’s method, see Appendix A. In fact, it is very difficult for choosing an appropriate regularization parameter. The regularized solution accuracy critically depends on the regularization parameter, k, and hence giving an appropriate regularization parameter k is crucial to obtain a stable and accurate solution which, at the same time, has a physical meaning. There are several methods to choose an appropriate parameter, such as L-curve [35], Cesàro mean in conjunction with L-curve [36], Morozov discrepancy principle [37], Multi-parameter Tikhonov regularization [38], the Gaussian window together with L-curve [39] and etc. The accuracy of the regularized solution is d
k > 0;
d
ð11Þ
d
that is, the minimum of the Tikhonov functional
J dk ðdÞ ¼ kAd h k2 þ kkdk2 ; d
k > 0:
satisfies the equation d
d
kAdk h k ¼ dkhk:
ð12Þ
The determination of k is thus equivalent to the problem of finding the zero of the monotone function FðkÞ ¼ kAdk h k2 d2 khk2 . The derivative of the mapping d
a # ddk
is
d
given
by
the
solution
of
the
equation
Table 3 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 3. Noise level (%)
k
eR ðuÞ
eR ð@ n uÞ
1
2:36 105
2:80 103
1:15 102
5
3:49 104
6:6 103
2:13 102
10
9:19 104
1:11 102
3:18 102
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depending on the regularization parameter k. Especially when there is a priori information about the amount of noise is available, we can use the Morozov discrepancy principle to give an appropriate regularization parameter. The best-parameter is not the scope of this work. We will choose the regularization parameter k by Morozov discrepancy principle, which was developed in [40,34,33]. The computation of k can be carried out by Newton’s method. When the regularization parameter k is determined, we d
can obtain the regularized solution dk ¼ Rk h .
In this subsection, we give some examples to check the effectiveness of the presented method. The implementation of the algorithm is based on the MATLAB software. As referred by above section, the boundary conditions are the
eK ðuÞ ¼
Nb
l¼1
max juðanÞ ðxl Þj
n P 2 o12 Nb ðanÞ 1 ðxl Þ qðnumÞ ðxl Þ l¼1 q N b
max jqðanÞ ðxl Þj
:
To give a clear expression of the relative RMS error on the sampling points, we define the relative errors as followings
ðanÞ u ðxl Þ uðnumÞ ðxl Þ ; max juðanÞ ðxl Þj
and
EK ð@ n uÞ ¼
ðanÞ q ðxl Þ qðnumÞ ðxl Þ max jqðanÞ ðxl Þj
l ¼ 1; 2 . . . Nb ;
;
l ¼ 1; 2 . . . Nb :
d
measured noisy data h satisfying jh h j < djhj. In order to check the accuracy of the numerical solution, we choose N b boundary points on K # @D n C. We denote the analytical and numerical solution by uðanÞ ðxÞ and uðnumÞ ðxÞ, respectively. Define the normalized relative RMS error on K as following
n P 2 o12 N b ðanÞ 1 u ðxl Þ uðnumÞ ðxl Þ
eK ð@ n uÞ ¼
EK ðuÞ ¼
4. Discussions and examples
d
For the normal heat flux, we denote the analytical and numerical solutions at x 2 K by qðanÞ ðxÞ and qðnumÞ ðxÞ, respectively. Define the normalized relative RMS error on K as following
:
Example 1. As the first example, we consider the solution domain D as a unit sphere D ¼ fðx1 ; x2 ; x3 Þjx21 þ x22 þ x23 < 1g. The exact solution of the Laplace equation is given by uðxÞ ¼ 12 x21 þ 12 x22 x23 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. The source points are fixed on fx : jxj ¼ 6g. Table 1 gives the values of the regularization parameter, k, chosen by Morozov discrepancy
Fig. A.5. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 3.
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Fig. A.6. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 3.
principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The numerical results of the temperature and the normal heat flux are shown in Figs. A.1 and A.2. From the Figs. A.1 and A.2 and Table 1, it can be seen that the numerical solution is a stable approximation to the exact solution, and should be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 2. We consider the solution domain D as a unit cube D ¼ ð0; 1Þ ð0; 1Þ ð0; 1Þ. The exact solution of the Laplace equation is given by uðxÞ ¼ sin x1 ðcosh x2 þ cosh x3 Þ. Here C ¼ fx 2 @D : x1 ¼ 0orx2 ¼ 1orx3 ¼ 0g; R ¼ fx 2 @D : x3 ¼ 1g. We only give the results on R ¼ fx 2 @D : x3 ¼ 1g; R ¼ fx 2 @D : x3 ¼ 1g. The source points are pffiffiffi fixed on fx : jxj ¼ 3 þ 3g. Table 2 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.3 and A.4. From the Figs. A.3 and A.4 and Table 2, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 3. We consider the solution domain D as a ellipsoid defined by
D¼
x 2 R3 :
x21 x22 x23 þ þ < 1 : a2 b2 c2
In particular, we consider a ¼ 1; b ¼ 0:5; c ¼ 0:7. The exact solution of the Laplace equation is given by Here uðxÞ ¼ sin x1 ðcosh x2 þ cosh x3 Þ. C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg; R ¼ @D n C where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. The source points are fixed on x2
x2
x21 þ 0:522 þ 0:732 ¼ 36. Table 3 gives the values of the regularization
Table 4 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 4. Noise level (%)
eR ðuÞ
k
eR ð@ n uÞ 7
1:30 10
4:08 107
3:93 106
1:19 102
1:79 102
10
1:13 105
1:84 102
2:43 102
20
3:18 105
3:15 102
3:25 102
30
5
2
4:04 102
0
2:24 10
5
16
5:47 10
4:44 10
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parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.5 and A.6. From the Figs. A.5 and A.6 and Table 3, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 4. We consider the solution domain D as a torus of radii R1 ; R2 with R1 > R2 , defined by
( D¼
) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 x2R : x1 þ x2 R2 þ x3 < R1 : 3
In particular,we shall consider the axisymmetric toroidal domain generated from this meridian domain with R1 ¼ 1; R2 ¼ 0:5. The exact solution of the Laplace equation is given
Fig. A.7. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 4.
Y. Sun, S. He / International Journal of Heat and Mass Transfer 108 (2017) 945–960
by uðxÞ ¼ x21 x23 þ x2 x3 þ x1 x3 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 2pg; R ¼ @D n C, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðð1 þ 0:5 cos hÞ cos u; ð1 þ 0:5 cos hÞ sin u; 0:5 sin hÞ. The source points are fixed on x21 þ x22 þ x23 ¼ 36. Table 4 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal
953
heat flux are shown in Figs. A.7 and A.8. From the Figs. A.7 and A.8 and Table 4, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well. Example 5. We consider the solution domain D as a bumpy sphere, and has the parametric representation
Fig. A.8. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 4.
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1 D ¼ ðx1 ; x2 ; x3 Þ ¼ q 1 þ sin h ðcos u sin h; sin u sin h; cos hÞ 6 3 2 R ; 0 6 q < 1; 0 6 u < 2p; 0 6 h 6 p: : The exact solution of the Laplace equation is given by uðxÞ ¼ x1 x2 þ x2 x3 þ x1 x3 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg. The source points are fixed on x21 þ x22 þ x23 ¼ 6:52 . Table 4 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding
errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.9 and A.10. From the Figs. A.9 and A.10 and Table 5, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well.
Fig. A.9. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 5.
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Fig. A.10. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 5.
Table 5 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 5.
Table 6 The values of the regularization parameter, k, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 600 and various amounts of noise added in the boundary data, for Example 6.
Noise level (%)
k
eR ðuÞ
eR ð@ n uÞ
Noise level (%)
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
0
3:14 1016
7:83 108
1:40 107
0
2:39 1016
3:33 107
2:24 106
1
4:52 107
5:9 103
4:7 103
1
5:88 107
4:7 103
7:8 103
5
3:02 106
1:71 102
1:45 102
5
5:78 106
1:61 102
2:30 102
10
8:02 106
2:70 102
2:49 102
10
1:49 105
2:71 102
3:72 102
30
4:28 105
6:59 102
6:20 102
30
1:53 104
6:53 102
8:14 102
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Example 6. We consider the solution domain D as a right cylinder defined by
n o D ¼ x 2 R3 : x21 þ x22 < R2 ; h < x3 < h : In particular, we shall consider R ¼ 1; h ¼ 1. The exact solution of the Laplace equation is given by uðxÞ ¼ x1 x2 x3 þ 5x1 þ 5x2 þ 5x3 . Here C ¼ fx 2 @D : x3 ¼ 1g [ fx 2 @D : x2 > 0; 1 < x3 < 1g; R ¼ fx 2 @D : x2 < 0; 1 < x3 < 1g. The source points
are fixed on y ¼ 6x; x 2 @D. Table 6 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.11 and A.12. From the Figs. A.11 and A.12 and Table 6, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as
Fig. A.11. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 6.
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957
Fig. A.12. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 6.
the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well. Table 7 gives the numerical comparations of the present method and the MFS for solving the inverse boundary value problem. From this table, we can see that these two methods are effective for this problem, and they give very accuracy results. There is only a small difference, which is caused by the regularization parameter.
Example 7. In this example, an analytic solution is not available. The solution domain D is to set as a unit sphere D ¼ fðx1 ; x2 ; x3 Þjx21 þ x22 þ x23 < 1g. Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. Consider the following boundary value problem
Du ¼ 0;
in D;
ð13aÞ
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Table 7 The values of the regularization parameter, k, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the IMFS and the MFS, d ¼ 10:0 and various amounts of noise added on u and @ n u, for Example 6. M
d (%)
150
Present method
1
225
300
e@DnC ðuÞ
e@DnC ð@ n uÞ
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
2:39 1010
4:2 103
6:5 103
2:13 1019
3:9 103
6:6 103
3
1:07 109
8:8 103
1:32 102
8:77 109
8:6 103
1:28 102
5
1:99 109
1:25 102
1:86 102
1:60 108
1:24 102
1:96 102
10
4:87 109
2:08 102
3:09 102
3:86 108
2:02 102
3:02 102
1
6:21 1010
3:0 103
4:5 103
5:21 109
2:8 103
4:0 103
3
2:10 109
7:1 103
9:8 103
1:70 108
7:7 103
9:2 103
5
3:71 109
1:08 102
1:46 102
3:00 108
1:04 102
1:39 102
10
8:55 109
1:96 102
2:56 102
6:92 108
1:91 102
2:52 102 3:3 103
1
4:88 1010
2:2 103
3:6 103
4:18 109
2:1 103
3
2:1 109
5:1 103
7:7 103
1:86 108
4:8 103
7:1 103
5
4:20 109
7:4 103
1:09 102
3:45 108
6:9 103
1:01 102
10
9:82 109
1:23 102
1:78 102
7:88 108
1:25 102
1:69 102
Table 8 The values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 400 and various amounts of noise added in the boundary data, for Example 7. Noise level (%)
e@DnC ðuÞ
k
e@DnC ð@ n uÞ
1:29 10
16
1
2:29 10
12
3
2:57 1011
4:25 102
4:01 102
5
11
2
2
0
MFS
k
5:31 10
4
1:44 104
2
1:70 10
1:71 102
1:42 10
5:80 10
5:29 10
u ¼ 10x1 x2 x23 ;
on @D:
ð13bÞ
Different from the previous examples, the input Cauchy data e ¼ 10x1 x2 x23 jC and e q ¼ @ n ujC , in which e q ¼ @ n ujC should be u obtained numerically by solving the direct problem (13). For the solution of this direct problem we use
@B ¼ ð1 þ d1 Þðcos h; sin hÞ;
0 6 h 2p;
whist for the inverse problem we use
@B ¼ ð1 þ dÞðcos h; sin hÞ;
0 6 h 2p:
Fig. A.13. The exact solution and the numerical solutions of the temperature on @D n C obtained with various amounts of noise added in the boundary data for Example 7.
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Fig. A.14. The exact solution and the numerical solutions of the normal heat flux on @D n C obtained with various amounts of noise added in the boundary data for Example 7.
We set d1 ¼ 7; d ¼ 10. To avoid the so-called inverse crime, we use the different artificial pseudo-boundary in the direct problem and the inverse problem. We set the numbers of the source points and boundary collation points M ¼ N ¼ 900. Table 8 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.13 and A.14. From these figures, it can be seen that the relative errors of the numerical solutions are becoming larger than the case which an analytic solution is available. Especially, more boundary collocation points and source points is not useful. We think it is reasonable and meets with expectations.
5. Conclusions In this paper, we study a meshless method [15] to solve the inverse heat conduction problem with the noisy boundary conditions in both convex and non-convex three-dimensional geometries with smooth or piecewise smooth boundaries. To get a stable and accuracy results, the Tikhonov regularization method combined with the Morozov’s discrepancy principle for selecting an appropriate regularization parameter are used. It is shown that
the proposed method is effective and stable even for the data with relatively high noise levels. From the examples, we can see that our proposed method is effective and stable. Acknowledgements We would like to thank the editor and the referees for their careful reading and valuable comments which improved the quality of the original submitted manuscript. The research was supported by the open Research Funds of Tianjin Key Lab for Advanced Signal Processing (No: 2016ASP-TJ02) and the Natural Science Foundation of China (No: 11501566,61401467). Appendix A. The regularization parameter chosen by Morozov discrepancy principle The regularization parameters, k, are computed by Newton’s method as following: 1. Set n ¼ 0, and give an initial regularization parameter k0 > 0; 2. Get dkn from ðA A þ kn IÞdkn ¼ A h ; d
d
d dkn ¼ dkn , from 3. Get ðkn I þ A AÞ dk d
d
0
4. Get Fðkn Þ and F ðkn Þ by
Fðkn Þ ¼ kAdkn h k2 d2 d
d
d
d d d ; dk kn
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and
F 0 ðkn Þ ¼ 2kn kA
d d 2 d d d k þ 2k2n k dkn k2 ; dk kn dk
respectively. nÞ 5. Set knþ1 ¼ kn FFðk . If jknþ1 kn j < eðe 1Þ, end. Else, set 0 ðkn Þ
n ¼ n þ 1 return to 2.
References [1] F. Wang, W. Chen, W. Qu, Y. Gu, A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction, Eng. Anal. Bound. Elem. 63 (2016) 1–14. [2] F. Wang, W. Chen, Y. Gu, Boundary element analysis of inverse heat conduction problems in 2D thin-walled structures, Int. J. Heat Mass Transfer 91 (2015) 1001–1009. [3] Y. Gu, H. Gao, W. Chen, et al., Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, Int. J. Heat Mass Transfer 90 (2015) 291–301. [4] W. Chen, C.J. Liu, Y. Gu, A fast multipole accelerated singular boundary method for potential problems, CMES-Comp. Model. Eng. 105 (4) (2015) 251–270. [5] W. Chen, Z. Fu, X. Wei, Potential problems by singular boundary method satisfying moment condition, CMES-Comp. Model. Eng. 54 (2009) 65–85. [6] W. Chen, Z. Fu, C.S. Chen, Recent Advances on Radial Basis Function Collocation Methods, Springer, Berlin, 2013. [7] W. Chen, Z. Fu, A novel numerical method for infinite domain potential problems, Chin. Sci. Bull. 55 (2010) 1598–1603. [8] Gu Y, Chen W, Zhang C, X. He, A meshless singular boundary method for threedimensional inverse heat conduction problems in general anisotropic media, Int. J. Heat Mass Transfer 84 (2015) 91–102. [9] T. Johansson, D. Lesnic, Reconstruction of a stationary flow from incomplete boundary data using iterative methods, Eur. J. Appl. Math. 17 (2006) 651–663. [10] T. Johansson, D. Lesnic, An iterative method for the reconstruction of a stationary flow, Numer. Methods. Partial. Differ. Equ. 23 (2007) 998–1017. [11] Z. Fu, W. Chen, C. Zhang, Boundary particle method for Cauchy inhomogeneous potential problems, Inverse Prob. Sci. Eng. 20 (2) (2012) 189–207. [12] D.N. Háo, D. Lesnic, The Cauchy problem for Laplace’s equation via the conjugate gradient method, IMA J. Appl. Math. 65 (2000) 199–217. [13] Y. Sun, D. Zhang, F. Ma, A potential function method for the Cauchy problem of elliptic operators, J. Math. Anal. Appl. 395 (2012) 164–174. [14] Y. Sun, Indirect boundary integral equation method for the Cauchy problem of the Laplace equation, J. Sci. Comput. (2016), http://dx.doi.org/10.1007/s10915016-0308-4. [15] Y. Sun, A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems, Int. J. Heat Mass Transfer 97 (2016) 891–907. [16] T. Wei, Y.C. Hon, L. Ling, Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem. 31 (2007) 373–385. [17] D.L. Young, C.C. Tsai, C.W. Chen, C.M. Fan, The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation, Comput. Math. Appl. 55 (2008) 1189–1200. [18] Y. Sun, Modified method of fundamental solutions for the Cauchy problem connected with the Laplace equation, Int. J. Comput. Math. 91 (10) (2014) 2185–2198.
[19] J.T. Chen, J.L. Yang, Y.T. Lee, Y.L. Chang, Formulation of the MFS for the twodimensional Laplace equation with an added constant and constraint, Eng. Anal. Bound. Elem. 46 (2014) 96–107. [20] L. Marin, An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems, Int. J. Heat Mass Transfer 94 (2016) 449–464. [21] L. Marin, An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation, CMES-Comput. Model. Eng. 48 (2009) 121–153. [22] V.D. Kupradze, M.A. Aleksdze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput. Math. Math. Phys. 4 (1964) 82–126. [23] E. Trefftz, Ein Gegenstck zum Ritzschen Verfahren, in: Verhandlungen des 2, Internationalen Kongresses fr Technische Mechanik, Zrich, Switzerland, 1926, pp. 131–137. [24] E. Kita, N. Kamiya, Trefftz method: an overview, Adv. Eng. Softw. 24 (1995) 3– 12. [25] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problem, Adv. Comput. Math. 9 (1998) 69–95. [26] G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem. 27 (2003) 759–769. [27] A. Karageorghis, D. Lesnic, L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng. 19 (2011) 309–336. [28] S.R. Kuo, J.T. Chen, J.W. Lee, Y.W. Chen, Analytical derivation and numerical experiments of degenerate scales for regular N-gon domains in twodimensional Laplace problems, Appl. Math. Comput. 219 (2013) 5668–5683. [29] S.R. Kuo, J.T. Chen, S.K. Kao, Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs, Appl. Math. Letter 26 (2013) 929–938. [30] J.T. Chen, S.R. Kuo, J.H. Lin, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for twodimensional elasticity, Int. J. Numer. Methods Eng. 54 (2002) 1669–1681. [31] J.T. Chen, S.R. Lin, K.H. Chen, Degenerate scale problem when solving Laplace’s equation by BEM and its treatment, Int. J. Numer. Methods Eng. 62 (2005) 233–261. [32] J.T. Chen, W.C. Shen, Degenerate scale for multiply connected Laplace problems, Mech. Res. Commun. 34 (2007) 69–77. [33] Y. Sun, F. Ma, Appropriate implementation of an invariant MFS for inverse boundary determination problem, Inverse Probl. Sci. Eng. 23 (6) (2015) 1040– 1055. [34] Y. Sun, F. Ma, Recovery of the temperature and the heat flux by a novel meshless method from the measured noisy data, Eng. Anal. Bound. Elem. 59 (C) (2015) 112–122. [35] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580. [36] L.Y. Chen, J.T. Chen, H.K. Hong, C.H. Chen, Application of Cesaro mean and the L-curve for the deconvolution problem, Soil Dynam. Earthq. Eng. 14 (1995) 361–373. [37] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd ed., Springer Verlag, New York, 2011. [38] Z. Wang, Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters, J. Comput. Appl. Math. 236 (7) (2012) 1815–1832. [39] J.T. Chen, K.H. Chen, Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions, Appl. Math. Modell. 22 (1998) 703–725. [40] Y. Sun, F. Ma, D. Zhang, An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem, Comput. Methods Appl. Mech. Eng. 271 (2014) 231–252.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems Yao Sun a,⇑, Songnian He b a b
Tianjin Key Lab for Advanced Signal Processing, College of Science, Civil Aviation University of China, Tianjin 300300, China College of Science, Civil Aviation University of China, Tianjin 300300, China
a r t i c l e
i n f o
Article history: Received 16 August 2016 Received in revised form 27 November 2016 Accepted 20 December 2016
Keywords: Inverse heat conduction problem Regularization Morozov discrepancy principle
a b s t r a c t This paper documents a meshless method for the three-dimensional inverse heat conduction problems based on the method of fundamental solution (MFS). In order to overcome the ill-posedness of the corresponding problem, the Tikhonov regularization method, as well as Morozov’s discrepancy principle for selecting an appropriate regularization parameter are used. Hence there is to produce a stable and accuracy numerical results. Then some examples are given to check the effectiveness of this method, whilst the sensitive analysis is given. The numerical convergence and stability of this method are also analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Consider a 3D steady-state heat conduction problem in a bounded domain D R3 with a piecewise smooth boundary surface @D; Dc its open complement. C is a portion of @D. In this paper, we refer to isotropic steady heat conduction in the absence of inner heat sources, and the temperature distribution, uðxÞ, in the domain D satisfies the following elliptic partial differential equation, also referred to as the isotropic heat-conduction equation
r2 u ¼ 0;
ð1Þ
in D:
At a point x 2 @D, we denote the unit outward normal vector by nðxÞ, and the normal heat flux qðxÞ defined by
qðxÞ ¼ ru nðxÞ:
ð2Þ
In general, the heat conduction Eq. (1) is solved combining with appropriate Dirichlet, Neumann, or Robin boundary conditions to obtain the temperature distribution in the solution domain D. But in some practice problems, we only get the boundary data on easily accessible boundary. This terms the inverse steady heat conduction problem as follow: the temperature uðxÞ satisfies (1) and the following boundary conditions
e; u¼u
@u ¼e q; @n
on C:
ð3Þ
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Sun). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.079 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
The motivation for our present study concerning the inverse steady heat conduction problem in R3 is a typical problem associated with the iron-making blast furnace which can be described as follows: it is desired to monitor the corroded thickness of the accreted refractory wall in order to avoid a situation that molten metal breaks out. In practice, we can only monitor the temperature and heat flux on easily accessible boundaries. Then, the temperature and the norm heat flux on the remaining part can be recovered by solving an inverse heat conduction problem either for a unsteady or steady state Laplace equation. There are many methods for solving this kind problem, such as boundary element method by Wang et al. [1,2], singular boundary method [3–8], iterative method by Johansson and Lesnic [9,10], boundary particle method by Fu et al. [11], conjugate gradient method by Háo and Lesnic [12], potential function method by Sun et al. [13,14], and method of fundamental solutions by Sun [15], Wei et al. [16] and Young et al. [17], and invariant MFS by Sun [18–20], iterative MFS [21], and cited references here. The MFS, first introduced by Kupradze and Aleksidze [22], is an effective method for solving the direct/inverse problems governed by the partial differential equations. It is to approximate the solution by a linear combination of the fundamental solutions. Then the weighed coefficients will be determined by the boundary collocation method. Thus the MFS is also a meshless boundary collocation method, which belongs to Trefftz-like methods [23,24] and is applicable to boundary value problems in which the fundamental solution of the operator is known. In recent years, the MFS has become very popular primarily because of the ease with which it
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can be implemented, particularly for the problems in complex geometries. The simple implementation of the MFS for the problems with complex boundaries makes it an ideal candidate for the problems in which the boundary is of major importance or requires special attention. Since then, it has been successfully applied to a large variety of physical problems, an account of which can be found in the survey papers [27,25,26]. However, there exist some heat conduction problems for which the simple application of the MFS is not sufficient to obtain an accurate numerical solution, e.g. problems related to domains containing a boundary singularity generated by the presence of a crack or a V-notch, and hence the standard MFS has to be modified/enriched (see [28–34]). In [15], Sun gave a simple and effective numerical method based on the MFS. The main idea is to approximate the solution by the derivative of the Green function, which is a solution of the Laplace Eq. (1). In this paper, the main goal is to extend this method to solve the 3D inverse heat conduction problems, i.e. we approximate the solution u by
uN ðxÞ ¼
N X @ Uðx; yj Þ aj ; @ mðyÞ j¼1
yj 2 @B;
ð4Þ
where B is a bounded domain with piecewise smooth boundary, and mðyÞ is the outward normal at point y 2 @B, Uðx; yÞ denotes 1 . the fundamental solution given by Uðx; yÞ ¼ 4pjxyj In this paper, the main goal is to extend the aforementioned regularized MFS [15] by the same author for the solution of 3D inverse heat conduction problems. Since most practice problems are considered in R3 , it is necessary for checking the effective of this method for solving 3D inverse heat conduction problems.
The matrix arising from the MFS is severely ill-conditioned, thus a regularized solution is obtained by employing a regularization strategy, namely the Tikhonov regularization technique, whilst the optimal regularization parameter is determined by Morozov’s discrepancy principle. The paper is organized as follows: In Section 2, the theoretical results of the solution method employed are then given. In Section 3, we select the Tikhonov regularization technique to give a stable and accuracy results for solving this inverse boundary value problem. Finally, the numerical accuracy and stability of the presented method are given with some numerical examples. 2. The solution method Let B be bounded, connected domain with D B; mðyÞ is the outside unit normal on the boundary @B. Different from the MFS, we approximate the solution u by the following form
Table 1 The values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 400 and various amounts of noise added in the boundary data, for Example 1. Noise level (%)
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
0
5:55 1017
9:51 108
2:35 107
1
3:46 107
8:10 103
9:40 103
5
2:18 106
1:95 102
2:20 102
10
5:76 106
2:93 102
3:25 102
Fig. A.1. The exact solution and the numerical solutions of the temperature on @D n C obtained with various amounts of noise added in the boundary data for Example 1.
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Fig. A.2. The exact solution and the numerical solutions of the normal heat flux on @D n C obtained with various amounts of noise added in the boundary data for Example 1.
uN ðxÞ ¼
N X @ Uðx; yj Þ aj : @ mðyÞ j¼1
ð5Þ
The following theorems are then will give the theoretical results of the present method. From [15], they can be proved with a little modification, thus we omit the proof. Theorem 1. Let ðym Þ 2 @B be a sequence of points, and ðym Þ are dense on @B. Then the normal derivatives of the fundamental solutions
@ Uðx; ym Þ : m ¼ 1; 2; . . . @ mðyÞ
corresponding to the source points located at ðym Þ are complete in L2 ð@DÞ.
Meanwhile, on the boundary @D, we will approximate the normal heat flux of (2) by N X @ 2 Uðx; yj Þ aj ; @nðxÞ@ mðyÞ j¼1
qN ðxÞ ¼
ð7Þ
where nðxÞ denotes the outward unit normal vector. The coefficients aj will be determined by the collocation method from the interpolation conditions at M collocation points. In fact, the coefficients are determined by solving a linear system of M equations, which consist the boundary conditions at the M collocation points. We can rewrite the M collocation points as a system of M linear algebraic equations with N unknowns which can be generically written as
Ad ¼ h: Theorem 2. Let ðym Þ 2 @B be a sequence of points, and ðym Þ are dense on @B. Then
(
@ 2 Uðx; ym Þ : m ¼ 1; 2; . . . @nðxÞ@ mðyÞ
)
corresponding to the source points located at ðym Þ are complete in W, n o R and W is given by w 2 L2 ð@DÞj @D wds ¼ 0 . Then we use this method to solve the inverse Cauchy problem. We will approximate u by the following form N X @ Uðx; yj Þ uN ðxÞ ¼ aj : @ mðyÞ j¼1
ð6Þ
ð8Þ
The matrix A, the unknown vector d and the right-hand side h will be given by the following form:
Table 2 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, d ¼ 5:0, M ¼ N ¼ 300 and various amounts of noise added in the boundary data, for Example 2. Noise level (%) 1 5 10
eR ðuÞ
k 8
eR ð@ n uÞ
7:28 10
1:40 10
3
1:44 102
8:21 107
9:20 103
5:01 102
2:32 106
2:09 102
1:04 101
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@ Uðxi ; yj Þ @ 2 Uðxi ; yj Þ ; AMþi;j ¼ ; dj ¼ aj ; @ mðyÞ @nðxÞ@ mðyÞ ~ðxi Þ; i ¼ 1; . . . ; M; j ¼ 1; . . . ; N: ~ ðxi Þ; hMþi ¼ q hi ¼ u
tool available for the analysis of rank-deficient and discrete illposed problems is the singular value decomposition (SVD). The matrix A in (8) can be SVD-decomposed into
Ai;j ¼
In order to uniquely determine the coefficients, it should be noticed that the number M of the boundary collocation points and the number N of the source points must satisfy 2M P N. 3. Regularization method As following, the Tikhonov regularization is used to solve the system (8) combined Morozov discrepancy principle. In general, the right hand-side vector h of the system (8) is a perturbed vector d
h , namely, we should consider the perturbed equations d
d
Ad ¼ h :
ð9Þ d
More precisely, h is the measured noisy data satisfying d
hi ¼ hi þ drandðiÞhi ; where d is the percentage noise and the number rand(i) is a pseudorandom number drawn from the standard uniform distribution on the interval ½1; 1 generated by Matlab code 1 þ 2 randð1; iÞ. It is well-known that the system (8) is ill-conditioned. Thus this system cannot be solved by the direct methods, such as the leastsquares method, because of which will give a highly unstable solution for a perturbed right-hand side vector. The main numerical
A ¼ WRV > : W and V, which have 2M N and N N dimensions, respectively, are matrices whose columns are the orthonormal vectors W i and V i , the left and right singular vectors, and R ¼ diagðk1 ; . . . ; kN Þ is a diagonal matrix has non-negative diagonal elements in decreasing order, which are the singular values of A. Such a decomposition makes explicit the degree of illconditioning of the matrix A through the ratio between the maximum and the minimum singular value but also allows to write the solution of the system (8) in the following form
d¼
N X W>b i
i¼1
ki
V i:
For ill-conditioned matrix systems, there usually exist extremely small singular values, thus this equation clearly brings out the difficulties to deal with the ill-posed discrete problems. A very popular method for dealing with the ill-posedness of such problem is the Tikhonov regularization. The formly Tikhonov regularized solution of the system (8) is given by the solution as follows
ðkI þ A> AÞdk ¼ A> h: By introducing the regularization operator
Fig. A.3. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 2.
ð10Þ
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Fig. A.4. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 2.
1
Rk :¼ ðkI þ A> AÞ A> ;
we can achieve the regularized solution dk ¼ Rk h of the Eq. (8). From [37, Section 2.5], we can see that dk exists and is the unique minimum of the Tikhonov functional
J k ðdÞ ¼ kAd hk2 þ kkdk2 ;
k > 0:
The discrepancy principle for the determination of k is to compute k ¼ kðdÞ > 0 such that the corresponding Tikhonov solution d
da ; that is, the solution of the equation
ðkI þ A> AÞdk ¼ A> h ; d
d ðkI þ A> AÞ dk dk ¼ dk , as is easily seen by differentiating (11) with respect to k. The computation of k can be carried out with Newton’s method, see Appendix A. In fact, it is very difficult for choosing an appropriate regularization parameter. The regularized solution accuracy critically depends on the regularization parameter, k, and hence giving an appropriate regularization parameter k is crucial to obtain a stable and accurate solution which, at the same time, has a physical meaning. There are several methods to choose an appropriate parameter, such as L-curve [35], Cesàro mean in conjunction with L-curve [36], Morozov discrepancy principle [37], Multi-parameter Tikhonov regularization [38], the Gaussian window together with L-curve [39] and etc. The accuracy of the regularized solution is d
k > 0;
d
ð11Þ
d
that is, the minimum of the Tikhonov functional
J dk ðdÞ ¼ kAd h k2 þ kkdk2 ; d
k > 0:
satisfies the equation d
d
kAdk h k ¼ dkhk:
ð12Þ
The determination of k is thus equivalent to the problem of finding the zero of the monotone function FðkÞ ¼ kAdk h k2 d2 khk2 . The derivative of the mapping d
a # ddk
is
d
given
by
the
solution
of
the
equation
Table 3 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 3. Noise level (%)
k
eR ðuÞ
eR ð@ n uÞ
1
2:36 105
2:80 103
1:15 102
5
3:49 104
6:6 103
2:13 102
10
9:19 104
1:11 102
3:18 102
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depending on the regularization parameter k. Especially when there is a priori information about the amount of noise is available, we can use the Morozov discrepancy principle to give an appropriate regularization parameter. The best-parameter is not the scope of this work. We will choose the regularization parameter k by Morozov discrepancy principle, which was developed in [40,34,33]. The computation of k can be carried out by Newton’s method. When the regularization parameter k is determined, we d
can obtain the regularized solution dk ¼ Rk h .
In this subsection, we give some examples to check the effectiveness of the presented method. The implementation of the algorithm is based on the MATLAB software. As referred by above section, the boundary conditions are the
eK ðuÞ ¼
Nb
l¼1
max juðanÞ ðxl Þj
n P 2 o12 Nb ðanÞ 1 ðxl Þ qðnumÞ ðxl Þ l¼1 q N b
max jqðanÞ ðxl Þj
:
To give a clear expression of the relative RMS error on the sampling points, we define the relative errors as followings
ðanÞ u ðxl Þ uðnumÞ ðxl Þ ; max juðanÞ ðxl Þj
and
EK ð@ n uÞ ¼
ðanÞ q ðxl Þ qðnumÞ ðxl Þ max jqðanÞ ðxl Þj
l ¼ 1; 2 . . . Nb ;
;
l ¼ 1; 2 . . . Nb :
d
measured noisy data h satisfying jh h j < djhj. In order to check the accuracy of the numerical solution, we choose N b boundary points on K # @D n C. We denote the analytical and numerical solution by uðanÞ ðxÞ and uðnumÞ ðxÞ, respectively. Define the normalized relative RMS error on K as following
n P 2 o12 N b ðanÞ 1 u ðxl Þ uðnumÞ ðxl Þ
eK ð@ n uÞ ¼
EK ðuÞ ¼
4. Discussions and examples
d
For the normal heat flux, we denote the analytical and numerical solutions at x 2 K by qðanÞ ðxÞ and qðnumÞ ðxÞ, respectively. Define the normalized relative RMS error on K as following
:
Example 1. As the first example, we consider the solution domain D as a unit sphere D ¼ fðx1 ; x2 ; x3 Þjx21 þ x22 þ x23 < 1g. The exact solution of the Laplace equation is given by uðxÞ ¼ 12 x21 þ 12 x22 x23 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. The source points are fixed on fx : jxj ¼ 6g. Table 1 gives the values of the regularization parameter, k, chosen by Morozov discrepancy
Fig. A.5. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 3.
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Fig. A.6. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 3.
principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The numerical results of the temperature and the normal heat flux are shown in Figs. A.1 and A.2. From the Figs. A.1 and A.2 and Table 1, it can be seen that the numerical solution is a stable approximation to the exact solution, and should be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 2. We consider the solution domain D as a unit cube D ¼ ð0; 1Þ ð0; 1Þ ð0; 1Þ. The exact solution of the Laplace equation is given by uðxÞ ¼ sin x1 ðcosh x2 þ cosh x3 Þ. Here C ¼ fx 2 @D : x1 ¼ 0orx2 ¼ 1orx3 ¼ 0g; R ¼ fx 2 @D : x3 ¼ 1g. We only give the results on R ¼ fx 2 @D : x3 ¼ 1g; R ¼ fx 2 @D : x3 ¼ 1g. The source points are pffiffiffi fixed on fx : jxj ¼ 3 þ 3g. Table 2 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.3 and A.4. From the Figs. A.3 and A.4 and Table 2, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 3. We consider the solution domain D as a ellipsoid defined by
D¼
x 2 R3 :
x21 x22 x23 þ þ < 1 : a2 b2 c2
In particular, we consider a ¼ 1; b ¼ 0:5; c ¼ 0:7. The exact solution of the Laplace equation is given by Here uðxÞ ¼ sin x1 ðcosh x2 þ cosh x3 Þ. C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg; R ¼ @D n C where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. The source points are fixed on x2
x2
x21 þ 0:522 þ 0:732 ¼ 36. Table 3 gives the values of the regularization
Table 4 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 4. Noise level (%)
eR ðuÞ
k
eR ð@ n uÞ 7
1:30 10
4:08 107
3:93 106
1:19 102
1:79 102
10
1:13 105
1:84 102
2:43 102
20
3:18 105
3:15 102
3:25 102
30
5
2
4:04 102
0
2:24 10
5
16
5:47 10
4:44 10
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parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.5 and A.6. From the Figs. A.5 and A.6 and Table 3, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases.
Example 4. We consider the solution domain D as a torus of radii R1 ; R2 with R1 > R2 , defined by
( D¼
) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 x2R : x1 þ x2 R2 þ x3 < R1 : 3
In particular,we shall consider the axisymmetric toroidal domain generated from this meridian domain with R1 ¼ 1; R2 ¼ 0:5. The exact solution of the Laplace equation is given
Fig. A.7. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 4.
Y. Sun, S. He / International Journal of Heat and Mass Transfer 108 (2017) 945–960
by uðxÞ ¼ x21 x23 þ x2 x3 þ x1 x3 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 2pg; R ¼ @D n C, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðð1 þ 0:5 cos hÞ cos u; ð1 þ 0:5 cos hÞ sin u; 0:5 sin hÞ. The source points are fixed on x21 þ x22 þ x23 ¼ 36. Table 4 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal
953
heat flux are shown in Figs. A.7 and A.8. From the Figs. A.7 and A.8 and Table 4, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well. Example 5. We consider the solution domain D as a bumpy sphere, and has the parametric representation
Fig. A.8. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 4.
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1 D ¼ ðx1 ; x2 ; x3 Þ ¼ q 1 þ sin h ðcos u sin h; sin u sin h; cos hÞ 6 3 2 R ; 0 6 q < 1; 0 6 u < 2p; 0 6 h 6 p: : The exact solution of the Laplace equation is given by uðxÞ ¼ x1 x2 þ x2 x3 þ x1 x3 . Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg. The source points are fixed on x21 þ x22 þ x23 ¼ 6:52 . Table 4 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding
errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.9 and A.10. From the Figs. A.9 and A.10 and Table 5, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well.
Fig. A.9. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 5.
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Fig. A.10. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 5.
Table 5 The values of the regularization parameter, k, and the corresponding errors, eR ðuÞ and eR ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 900 and various amounts of noise added in the boundary data, for Example 5.
Table 6 The values of the regularization parameter, k, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 600 and various amounts of noise added in the boundary data, for Example 6.
Noise level (%)
k
eR ðuÞ
eR ð@ n uÞ
Noise level (%)
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
0
3:14 1016
7:83 108
1:40 107
0
2:39 1016
3:33 107
2:24 106
1
4:52 107
5:9 103
4:7 103
1
5:88 107
4:7 103
7:8 103
5
3:02 106
1:71 102
1:45 102
5
5:78 106
1:61 102
2:30 102
10
8:02 106
2:70 102
2:49 102
10
1:49 105
2:71 102
3:72 102
30
4:28 105
6:59 102
6:20 102
30
1:53 104
6:53 102
8:14 102
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Example 6. We consider the solution domain D as a right cylinder defined by
n o D ¼ x 2 R3 : x21 þ x22 < R2 ; h < x3 < h : In particular, we shall consider R ¼ 1; h ¼ 1. The exact solution of the Laplace equation is given by uðxÞ ¼ x1 x2 x3 þ 5x1 þ 5x2 þ 5x3 . Here C ¼ fx 2 @D : x3 ¼ 1g [ fx 2 @D : x2 > 0; 1 < x3 < 1g; R ¼ fx 2 @D : x2 < 0; 1 < x3 < 1g. The source points
are fixed on y ¼ 6x; x 2 @D. Table 6 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.11 and A.12. From the Figs. A.11 and A.12 and Table 6, it can be seen that the relative errors is a very small, and should also be noted that the numerical solution converges to the exact solution as
Fig. A.11. The exact solution and the relative errors of the temperature on R obtained with various amounts of noise added in the boundary data for Example 6.
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957
Fig. A.12. The exact solution and the relative errors of the normal heat flux on R obtained with various amounts of noise added in the boundary data for Example 6.
the level of noise decreases. We also should conclude that when the noise level is large, the numerical results are also very well. Table 7 gives the numerical comparations of the present method and the MFS for solving the inverse boundary value problem. From this table, we can see that these two methods are effective for this problem, and they give very accuracy results. There is only a small difference, which is caused by the regularization parameter.
Example 7. In this example, an analytic solution is not available. The solution domain D is to set as a unit sphere D ¼ fðx1 ; x2 ; x3 Þjx21 þ x22 þ x23 < 1g. Here C ¼ fx 2 @Dj0 6 hðxÞ 6 p; 0 6 uðxÞ 6 pg, where hðxÞ; uðxÞ is the polar angle of x in the form x ¼ ðcos h sin u; sin h sin u; cos uÞ. Consider the following boundary value problem
Du ¼ 0;
in D;
ð13aÞ
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Table 7 The values of the regularization parameter, k, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the IMFS and the MFS, d ¼ 10:0 and various amounts of noise added on u and @ n u, for Example 6. M
d (%)
150
Present method
1
225
300
e@DnC ðuÞ
e@DnC ð@ n uÞ
k
e@DnC ðuÞ
e@DnC ð@ n uÞ
2:39 1010
4:2 103
6:5 103
2:13 1019
3:9 103
6:6 103
3
1:07 109
8:8 103
1:32 102
8:77 109
8:6 103
1:28 102
5
1:99 109
1:25 102
1:86 102
1:60 108
1:24 102
1:96 102
10
4:87 109
2:08 102
3:09 102
3:86 108
2:02 102
3:02 102
1
6:21 1010
3:0 103
4:5 103
5:21 109
2:8 103
4:0 103
3
2:10 109
7:1 103
9:8 103
1:70 108
7:7 103
9:2 103
5
3:71 109
1:08 102
1:46 102
3:00 108
1:04 102
1:39 102
10
8:55 109
1:96 102
2:56 102
6:92 108
1:91 102
2:52 102 3:3 103
1
4:88 1010
2:2 103
3:6 103
4:18 109
2:1 103
3
2:1 109
5:1 103
7:7 103
1:86 108
4:8 103
7:1 103
5
4:20 109
7:4 103
1:09 102
3:45 108
6:9 103
1:01 102
10
9:82 109
1:23 102
1:78 102
7:88 108
1:25 102
1:69 102
Table 8 The values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method, M ¼ N ¼ 400 and various amounts of noise added in the boundary data, for Example 7. Noise level (%)
e@DnC ðuÞ
k
e@DnC ð@ n uÞ
1:29 10
16
1
2:29 10
12
3
2:57 1011
4:25 102
4:01 102
5
11
2
2
0
MFS
k
5:31 10
4
1:44 104
2
1:70 10
1:71 102
1:42 10
5:80 10
5:29 10
u ¼ 10x1 x2 x23 ;
on @D:
ð13bÞ
Different from the previous examples, the input Cauchy data e ¼ 10x1 x2 x23 jC and e q ¼ @ n ujC , in which e q ¼ @ n ujC should be u obtained numerically by solving the direct problem (13). For the solution of this direct problem we use
@B ¼ ð1 þ d1 Þðcos h; sin hÞ;
0 6 h 2p;
whist for the inverse problem we use
@B ¼ ð1 þ dÞðcos h; sin hÞ;
0 6 h 2p:
Fig. A.13. The exact solution and the numerical solutions of the temperature on @D n C obtained with various amounts of noise added in the boundary data for Example 7.
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Fig. A.14. The exact solution and the numerical solutions of the normal heat flux on @D n C obtained with various amounts of noise added in the boundary data for Example 7.
We set d1 ¼ 7; d ¼ 10. To avoid the so-called inverse crime, we use the different artificial pseudo-boundary in the direct problem and the inverse problem. We set the numbers of the source points and boundary collation points M ¼ N ¼ 900. Table 8 gives the values of the regularization parameter, k, chosen by Morozov discrepancy principle, and the corresponding errors, e@DnC ðuÞ and e@DnC ð@ n uÞ, obtained using the presented method and various amounts of noise added in the boundary data. The exact solutions and the relative errors of the temperature and the normal heat flux are shown in Figs. A.13 and A.14. From these figures, it can be seen that the relative errors of the numerical solutions are becoming larger than the case which an analytic solution is available. Especially, more boundary collocation points and source points is not useful. We think it is reasonable and meets with expectations.
5. Conclusions In this paper, we study a meshless method [15] to solve the inverse heat conduction problem with the noisy boundary conditions in both convex and non-convex three-dimensional geometries with smooth or piecewise smooth boundaries. To get a stable and accuracy results, the Tikhonov regularization method combined with the Morozov’s discrepancy principle for selecting an appropriate regularization parameter are used. It is shown that
the proposed method is effective and stable even for the data with relatively high noise levels. From the examples, we can see that our proposed method is effective and stable. Acknowledgements We would like to thank the editor and the referees for their careful reading and valuable comments which improved the quality of the original submitted manuscript. The research was supported by the open Research Funds of Tianjin Key Lab for Advanced Signal Processing (No: 2016ASP-TJ02) and the Natural Science Foundation of China (No: 11501566,61401467). Appendix A. The regularization parameter chosen by Morozov discrepancy principle The regularization parameters, k, are computed by Newton’s method as following: 1. Set n ¼ 0, and give an initial regularization parameter k0 > 0; 2. Get dkn from ðA A þ kn IÞdkn ¼ A h ; d
d
d dkn ¼ dkn , from 3. Get ðkn I þ A AÞ dk d
d
0
4. Get Fðkn Þ and F ðkn Þ by
Fðkn Þ ¼ kAdkn h k2 d2 d
d
d
d d d ; dk kn
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and
F 0 ðkn Þ ¼ 2kn kA
d d 2 d d d k þ 2k2n k dkn k2 ; dk kn dk
respectively. nÞ 5. Set knþ1 ¼ kn FFðk . If jknþ1 kn j < eðe 1Þ, end. Else, set 0 ðkn Þ
n ¼ n þ 1 return to 2.
References [1] F. Wang, W. Chen, W. Qu, Y. Gu, A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction, Eng. Anal. Bound. Elem. 63 (2016) 1–14. [2] F. Wang, W. Chen, Y. Gu, Boundary element analysis of inverse heat conduction problems in 2D thin-walled structures, Int. J. Heat Mass Transfer 91 (2015) 1001–1009. [3] Y. Gu, H. Gao, W. Chen, et al., Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, Int. J. Heat Mass Transfer 90 (2015) 291–301. [4] W. Chen, C.J. Liu, Y. Gu, A fast multipole accelerated singular boundary method for potential problems, CMES-Comp. Model. Eng. 105 (4) (2015) 251–270. [5] W. Chen, Z. Fu, X. Wei, Potential problems by singular boundary method satisfying moment condition, CMES-Comp. Model. Eng. 54 (2009) 65–85. [6] W. Chen, Z. Fu, C.S. Chen, Recent Advances on Radial Basis Function Collocation Methods, Springer, Berlin, 2013. [7] W. Chen, Z. Fu, A novel numerical method for infinite domain potential problems, Chin. Sci. Bull. 55 (2010) 1598–1603. [8] Gu Y, Chen W, Zhang C, X. He, A meshless singular boundary method for threedimensional inverse heat conduction problems in general anisotropic media, Int. J. Heat Mass Transfer 84 (2015) 91–102. [9] T. Johansson, D. Lesnic, Reconstruction of a stationary flow from incomplete boundary data using iterative methods, Eur. J. Appl. Math. 17 (2006) 651–663. [10] T. Johansson, D. Lesnic, An iterative method for the reconstruction of a stationary flow, Numer. Methods. Partial. Differ. Equ. 23 (2007) 998–1017. [11] Z. Fu, W. Chen, C. Zhang, Boundary particle method for Cauchy inhomogeneous potential problems, Inverse Prob. Sci. Eng. 20 (2) (2012) 189–207. [12] D.N. Háo, D. Lesnic, The Cauchy problem for Laplace’s equation via the conjugate gradient method, IMA J. Appl. Math. 65 (2000) 199–217. [13] Y. Sun, D. Zhang, F. Ma, A potential function method for the Cauchy problem of elliptic operators, J. Math. Anal. Appl. 395 (2012) 164–174. [14] Y. Sun, Indirect boundary integral equation method for the Cauchy problem of the Laplace equation, J. Sci. Comput. (2016), http://dx.doi.org/10.1007/s10915016-0308-4. [15] Y. Sun, A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems, Int. J. Heat Mass Transfer 97 (2016) 891–907. [16] T. Wei, Y.C. Hon, L. Ling, Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem. 31 (2007) 373–385. [17] D.L. Young, C.C. Tsai, C.W. Chen, C.M. Fan, The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation, Comput. Math. Appl. 55 (2008) 1189–1200. [18] Y. Sun, Modified method of fundamental solutions for the Cauchy problem connected with the Laplace equation, Int. J. Comput. Math. 91 (10) (2014) 2185–2198.
[19] J.T. Chen, J.L. Yang, Y.T. Lee, Y.L. Chang, Formulation of the MFS for the twodimensional Laplace equation with an added constant and constraint, Eng. Anal. Bound. Elem. 46 (2014) 96–107. [20] L. Marin, An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems, Int. J. Heat Mass Transfer 94 (2016) 449–464. [21] L. Marin, An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation, CMES-Comput. Model. Eng. 48 (2009) 121–153. [22] V.D. Kupradze, M.A. Aleksdze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput. Math. Math. Phys. 4 (1964) 82–126. [23] E. Trefftz, Ein Gegenstck zum Ritzschen Verfahren, in: Verhandlungen des 2, Internationalen Kongresses fr Technische Mechanik, Zrich, Switzerland, 1926, pp. 131–137. [24] E. Kita, N. Kamiya, Trefftz method: an overview, Adv. Eng. Softw. 24 (1995) 3– 12. [25] G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problem, Adv. Comput. Math. 9 (1998) 69–95. [26] G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem. 27 (2003) 759–769. [27] A. Karageorghis, D. Lesnic, L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng. 19 (2011) 309–336. [28] S.R. Kuo, J.T. Chen, J.W. Lee, Y.W. Chen, Analytical derivation and numerical experiments of degenerate scales for regular N-gon domains in twodimensional Laplace problems, Appl. Math. Comput. 219 (2013) 5668–5683. [29] S.R. Kuo, J.T. Chen, S.K. Kao, Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs, Appl. Math. Letter 26 (2013) 929–938. [30] J.T. Chen, S.R. Kuo, J.H. Lin, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for twodimensional elasticity, Int. J. Numer. Methods Eng. 54 (2002) 1669–1681. [31] J.T. Chen, S.R. Lin, K.H. Chen, Degenerate scale problem when solving Laplace’s equation by BEM and its treatment, Int. J. Numer. Methods Eng. 62 (2005) 233–261. [32] J.T. Chen, W.C. Shen, Degenerate scale for multiply connected Laplace problems, Mech. Res. Commun. 34 (2007) 69–77. [33] Y. Sun, F. Ma, Appropriate implementation of an invariant MFS for inverse boundary determination problem, Inverse Probl. Sci. Eng. 23 (6) (2015) 1040– 1055. [34] Y. Sun, F. Ma, Recovery of the temperature and the heat flux by a novel meshless method from the measured noisy data, Eng. Anal. Bound. Elem. 59 (C) (2015) 112–122. [35] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992) 561–580. [36] L.Y. Chen, J.T. Chen, H.K. Hong, C.H. Chen, Application of Cesaro mean and the L-curve for the deconvolution problem, Soil Dynam. Earthq. Eng. 14 (1995) 361–373. [37] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd ed., Springer Verlag, New York, 2011. [38] Z. Wang, Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters, J. Comput. Appl. Math. 236 (7) (2012) 1815–1832. [39] J.T. Chen, K.H. Chen, Analytical study and numerical experiments for Laplace equation with overspecified boundary conditions, Appl. Math. Modell. 22 (1998) 703–725. [40] Y. Sun, F. Ma, D. Zhang, An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem, Comput. Methods Appl. Mech. Eng. 271 (2014) 231–252.