Generalized Method of Fundamental Solutions (GMFS) for Boundary Value Problems

Engineering Analysis with Boundary Elements 94 (2018) 25–33

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Engineering Analysis with Boundary Elements

journal homepage: www.elsevier.com/locate/enganabound

Generalized method of fundamental solutions (GMFS) for boundary value problems

J.J. Yang a,∗, J.L. Zheng a, P.H. Wen b,∗ a School of Traﬃc and Transportation Engineering, Changsha University of Science and Technology, Changsha 410114, China b School of Engineering and Materials Science, Queen Mary University of London, London E1 4NS, UK

a r t i c l e i n f o a b s t r a c t

Keywords: In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source oﬀset, Meshless methods source location, or a ﬁctitious boundary, a generalized method of fundamental solutions (GMFS) is proposed.

Boundary methods The crucial part of the GMFS is using a generalized fundamental solution approximation (GFSA), which adopts

Method of fundamental solutions (MFS) a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Source oﬀset Then the numerical solution of the GMFS is decided by a group of oﬀsets corresponding to an intervention-point Fictitious boundary

Intervention point diffusion (IPD), instead of the MFS’ offset of a single source. To demonstrate the effectiveness of the proposed approach, five numerical examples are given. The results have shown that the GMFS is more accurate, stable, and has a better convergence rate than the traditional MFS.

1. Introduction fictitious boundary is not needed. However, it is difficult to find the non- singular kernels or general solutions for some practical problems. Even In recent years the method of fundamental solutions (MFS), a bound- though the non-singular kernels or general solutions can be found, the ary meshless method, has attracted great attention for solving homoge- accuracy is normally not very impressive. neous differential equations [1–9] . The MFS is quite simple, efficient, Another proposed method worthy to mention is the non-singular and easy for implementation, and it avoids the singular integrals which method of fundamental solutions (NMFS) [28,29] . For this method, a is necessary in certain boundary meshless methods, such as BNM [10] , desingularization technique is used to regularize the singularity of the LBIE [11] , HBNM [12] , BCM [13] , and BFM [14] . Furthermore, it could fundamental solution. The source points would then be located at the be highly accurate and rapidly convergent when an appropriate offset real boundary, making the fictitious boundary not necessary. Neverthe- is selected [15] . less, drawbacks include the necessity for the boundary nodes be dis- However, despite the effectiveness and simplicity of the MFS, there tributed regularly, desingularization for arbitrary problems may not be are still some outstanding theoretical and numerical issues to be ad- available, and the tedious desingularization procedure compromises the dressed [16–18] . One of the main issues yet to be resolved is the choice simplicity of the method. of the offset. In the MFS, a fictitious boundary outward offset to the real The singular boundary method (SBM) [30–36] uses an origin in- boundary with a distance parameter d is required in order to define the tensity factor (OIF) to substitute the singularity allowing the fictitious source points outside the domain. The offset d is sensitive and vital to boundary not to be necessary. However, choosing the OIFs is not a trivial the accuracy of the MFS. It is possible that we could set a reasonable process, and the given problem must be solved twice. range for the offset based on experience. However, it is not always ef- Moreover, a boundary distributed source (BDS) method [37] should fective, because a good offset for a certain problem could be bad for be mentioned also. For the BDS method, the source points do not neces- another problem. Despite the intensive research, this “offset dilemma ” sarily need to be offset, but they should be distributed. The singular fun- has been an outstanding research topic for the MFS [19,20] . damental solution is integrated firstly over the distributed source cov- In the past, various approaches have been proposed to alleviate this ering the source points. If the distributed source is a simple shape, such difficulty in the MFS such as the BKM [21–23] , BCM [24,25] and BPM as a circle, then the singular integrals could be evaluated analytically. [26,27] . Instead of using the singular fundamental solutions as used in However, the singular integrals are not always analytical, and the solu- the MFS, these methods use non-singular kernels or general solutions. tion is inaccurate near the boundary regions. An improved BDS method As such, the source points can be located on the real boundary, and the [38,39] uses a boundary-integral technique to determine the singular

∗ Corresponding authors. E-mail addresses: [email protected] (J.J. Yang), [email protected] (P.H. Wen). https://doi.org/10.1016/j.enganabound.2018.05.014 Received 2 March 2018; Received in revised form 18 April 2018; Accepted 29 May 2018 0955-7997/© 2018 Elsevier Ltd. All rights reserved. J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Fig. 1. Schematics of the GFSA: (a) regional IPD; (b) linear IPD.

integrals. However, this approach requires the singular integrals to be where hp is the interval for {dp } diffused, e.g., if hp = 0.1, then 𝑅̄ 𝑅̄ calculated directly. {dp } = (0.1,0.2,0.3,0.4,0.5) for Eq. (4). is defined as the normalized In some sense the above-mentioned efforts overcome the old chal- parameter for the boundary dimension ( ) ( ) lenge of the MFS at a price of introducing new obstacles. From our un- ‖ ‖ 1 ‖ max 𝑥 𝑖 − min 𝑥 𝑖 ‖ derstanding, the tenacious barrier of the MFS is still open for improve- 𝑅̄ = √ ‖ ‖ (5) ‖ 2 ‖ ment. Thus, we will try to give another option for the issue of the MFS. 𝐷 ‖ ‖2 in which the subscript “i ”is the component signal of the coordinates, 2. Generalized fundamental solution approximation and D is the number of dimensions.

Obviously, when Np = 1, the GFSA is equivalent to the FSA. In other The MFS uses a fundamental solution approximation (FSA), which words, the GFSA is a generalized FSA. So we use a term of “generalized ” was first proposed by Kupradze and Aleksidze [1,40–42] , as the basis to denote the novel approximation and the corresponding numerical function for solving homogeneous equations. It is notable that, another method. independent work with the same concept was also proposed by Wen [43] which is called the point intensity method (PIM). Let u ( x ) be a 3. Generalized method of fundamental solutions (GMFS) field variable in a given domain Ω bounded by Γ. The basic idea of the FSA is to express u ( x ) as a linear combination of fundamental solutions: Consider the following Laplace equation in a 2D domain Ω bounded by Γ: ∑𝑁 ̄ 𝑢 ( 𝐱 ) = 𝑎 𝐽 𝜓 𝐽 ( 𝐱, 𝐬) 𝐱 ∈ Ω, (1) ∇ 2𝑢 (𝐱 ) = 0 , 𝐱 ∈Ω, (6) 𝐽=1 subject to boundary conditions (BCs): 𝜓 ≡ 𝜓 ≡ 𝜓 where J (x,s) (x, sJ ) (rJ ) is the fundamental solution, 𝑢 𝐱 𝑢̄ 𝐱 , 𝐱 , rJ = ‖x − sJ ‖2 is the Euclidean norm between the measuring point x and ( ) = ( ) ∈Γ𝑢 (7) ̄ ≡ the source point sJ , aJ is the intensity coefficient at sJ , and Ω Ω∪Γ. Being different from the FSA, the generalized fundamental solution 𝜕𝑢 𝑢 ,𝑛 ( 𝐱 ) ≡ ( 𝐱 ) = 𝑞̄ ( 𝐱 ), 𝐱 ∈Γ𝑡 , (8) approximation (GFSA) uses a bilinear combination of fundamental so- 𝜕𝑛 lutions to approximate u(x)as follows: where Γu is the Dirichlet boundary, Γt is the Neumann boundary, Γ=Γu 𝑁 ∑𝑁 ∑𝑝 { } ∪Γ , Γ ∩Γ = ∅, n is the outward normal of the boundary, and 𝑢̄, 𝑞̄ are 𝐽 t u t 𝑢 ( 𝐱 ) = 𝑎 𝐽 𝜓 𝑝 ( 𝐱, 𝐬) , 𝐬 𝑝 ∈Ω𝐽 , (2) the known functions on the boundary. The fundamental solution 𝜓 for 𝐽=1 𝑝 =1 the Laplacian is given by: ̄ where { s } ∉ Ω is the intervention-point diffusion (IPD) of the source ⎧ p −1 node x [44] , N is its point number, and Ω is the diffusion domain ⎪ ln (𝑟 ), 2 𝐷, J p J ⎪ 𝜋 2 centered at xJ which is sheared off by the boundary Γ, as shown in 𝜓 ( 𝑟 ) = ⎨ (9) Fig. 1 (a). Note that we use a superscript “J ”in the function 𝜓 to denote ⎪ 1 , 3 𝐷. ⎪ 𝜋𝑟 a correspondence with the source node xJ . ⎩ 4

Note that the diffusion domain ΩJ could be arbitrarily selected outside the domain. For efficiency, we can also linearly diffuse the IPD, The configuration of IPD (source-point cloud) of the GMFS is shown

in Fig. 2(a). In contrast, the sources of the MFS are shown in Fig. 2(b). such as {sp } ∈ nJ (which is not strict), as shown in Fig. 1(b) where nJ is

The target node xI is indexed for constructing the discrete system equa- the outward normal at xJ , and dp is the oﬀset of an intervention point

tions. We will first try to use the variation method. The functional vari- sp , as ation is given as ‖ ‖ 𝑑 ‖𝐬 𝐱 ‖ , 𝑝 , , , 𝑁 . ∑ ∑ ( ) 𝑝 = ‖ 𝑝 − 𝐽 ‖ = 1 2 … 𝑝 (3) 𝛿 𝛿𝑢 𝑢 𝑢̄ 𝛿𝑢 𝑢 𝑞̄ . 2 Π2 = ( − ) + ,𝑛 ,𝑛 − (10) 𝐱 𝐱 𝐼 ∈Γ𝑢 𝐼 ∈Γ𝑡 In this paper, a diffusion scheme to choose {sp } ∈ nJ is our focus. The ≥ 𝛿 appropriate choice of Np ( 5 is suggested) and offsets {dp } is necessary. Let Π2 = 0, then the BCs given by Eqs. (7) and (8) are satisfied. Then By default, we choose a GMFS1-type system of equations is obtained { } ( ) ̄ ̄ ̄ 𝑑 𝑝 = 0 . 1 ∶ ℎ 𝑝 ∶ 0 . 5 ⋅ 𝑅, (4) 𝐊𝐚 = 𝐅, (11)

26 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Fig. 2. Schematics of the GMFS and the MFS: (a) GMFS; (b) MFS. where 𝐊̄ is the whole “stiffness ”matrix, 𝐅̄ is the whole “load ”ma- ⋅⋅⋅ T trix, and a = [a1 ,a2 , , aN ] are the unknown intensity coefficients in- troduced in Eq. (2) . Furthermore, 𝑇 𝐊̄ = 𝐊 𝑇 𝐊 , 𝐅̄ = 𝐊 𝐅 , (12) where ⎧ 𝑁 𝑝 ( ) ⎪ ∑ 𝐽 ⎪ 𝜓 𝑝 𝐱 𝐼 , 𝐬 , 𝐱 𝐼 ∈Γ𝑢 , ⎪ 𝑝 =1 𝐊 𝐼𝐽 = ⎨ ( ) (13) 𝑁 𝑝 𝐽 ⎪ ∑ 𝜕𝜓 𝑝 𝐱 𝐼 , 𝐬 ⎪ , 𝐱 𝐼 ∈Γ𝑡 , 𝜕𝑛 ⎪ 𝑝 ⎩ =1 { ( ) 𝑢̄ 𝐱 , 𝐱 , ( 𝐼 ) 𝐼 ∈Γ𝑢 𝐅 𝐼 = (14) 𝑞̄ 𝐱 𝐼 , 𝐱 𝐼 ∈Γ𝑡 . On the other hand, we can also directly solve the potential problems using the collocation method, and then a GMFS2-type system equation is obtained 𝐊𝐚 = 𝐅 , (15) Fig. 3. Nodal distribution for the gear-shaped domain problem. where the “stiffness ”matrix K and the “load ”matrix F are defined in Eqs. (13) and (14) . results on the boundary in Fig. 4 . We observe that the GMFS method can From Eq. (11) or (15) , the unknown a can be obtained. Then using obtain a good solution for the problem. the GFSA defined by Eq. (2) , the potential field u for all 𝐱 ∈ Ω̄ can be Next, we test the numerical convergence of the GMFS. We choose attained. the boundary as the analytical path and use 80 test points to evaluate

the numerical errors. An average relative error is defined as 4. Numerical tests √ √ √ ∑𝑚 ( ) ∑𝑚 ( ) 1 √ 𝑛𝑢𝑚 𝑎𝑛𝑎 2 𝑎𝑛𝑎 2 In this section, several numerical examples of 2D potential problems 𝐸𝑟 = 𝜉 − 𝜉 ∕ 𝜉 (19) 𝑚 𝑘 𝑘 𝑘 are presented to demonstrate the effectiveness of the proposed GMFS. 𝑘 =1 𝑘 =1 𝜉𝑛𝑢𝑚 𝜉𝑎𝑛𝑎 Only the traditional MFS will be considered for comparative analysis. In where m is the number of the test points, and 𝑘 , 𝑘 denote the nu- all the examples, the units of geometrical dimensions and field variables merical and analytical solution at the k th test point, respectively. We are assumed to be non-dimensional. choose hp = 0.01 to diffuse the {dp } for the GMFS, and the minimum of

{dp } as the oﬀset of the MFS. The results are shown in Fig. 5. It can be 4.1. A gear-shaped domain with Dirichlet BC seen that the GMFS2 shows the best accuracy and the fast convergence.

We first consider the Laplace equation in a gear-shaped domain 4.2. An epitrochoid domain with mixed BCs whose boundary is defined by Next, the Laplace equation for an epitrochoid domain with mixed Γ= {( 𝑥, 𝑦) |𝑥 = 𝜌 cos 𝜑, 𝑦 = 𝜌 sin 𝜑 } , (16) Dirichlet and Neumann BCs is considered. The parametric equation of where the epitrochoid boundary is defined as 1 1 √ 𝜌 = 2 + sin ( 7 𝜃) , 𝜑 = 𝜃 + sin ( 7 𝜃) , 0 ≤ 𝜃 ≤ 2 𝜋, (17) 2 2 2 𝜌( 𝜃) = ( 𝑎 + 𝑏) + 1 − 2 (𝑎 + 𝑏) cos ( 𝑎𝜃∕ 𝑏) , (20) and the Dirichlet BC is imposed with the following analytical solution: where the shape parameters are taken as a = 3, b = 1. The upper boundary of 0 ≤ 𝜃 ≤ 𝜋 is imposed as the Dirichlet condition while 𝜋< 𝜃< 2 𝜋 𝑢 ( 𝑥, 𝑦) = cos ( 𝑥 ) cosh ( 𝑦 ) + sin ( 𝑥 ) sinh (𝑦 ). (18) is imposed as the Neumann condition. The analytical solution is given

For numerical implementation, we use 50 boundary nodes, as shown by: in Fig. 3 . Then we solve the equation by the GMFS2 and show numerical 𝑢 ( 𝑥, 𝑦) = exp ( 𝑥 ) cos ( 𝑦 ), ( 𝑥, 𝑦 ) ∈ Ω̄ . (21)

27 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Fig. 4. Comparing solutions on the boundary for the gear-shaped domain problem.

Initially, we will use the GMFS2 with 30 boundary nodes to give a tentative solution. The boundary nodes are shown in Fig. 6 (a), and the numerical result is shown in (b). Next, we test the numerical stability. Since the condition number of the GMFS’ system matrix is mainly aﬀected by the max value of its oﬀ-

sets, and then we take the max value dm as the MFS’ offset, and give a comparable solution. Moreover, since the condition number is fluctuat- ing with the number of the boundary nodes N , so N = 100 and N = 200 are both considered in the comparison test. The results are shown in Fig. 7 . We observe that the GMFS always attains a better condition number comparing with the MFS, while the max offset value is used in both methods. Therefore, we can say that the GMFS is more stable than the MFS in terms of condition number. In addition, we find an interesting phenomenon: GMFS1’s condition number is better than that of GMFS2’s for denser nodes ( N = 200), but it is opposite for sparser nodes ( N = 100). This is an indication that the GMFS1 is preferable, but not always. For the convergence test, we choose an analytical path as P = 0.9 𝜌, and use 80 test points to evaluate the numerical errors. Similarly, we

select the mid-value of the GMFS oﬀsets {dp } as the MFS oﬀset. The Fig. 5. Comparing numerical errors for the gear-shaped domain problem. results are shown in Fig. 8 . It can be seen that the GMFS could be highly

Fig. 6. Nodal distribution and numerical result on the domain for the epitrochoid domain problem: (a) nodal distribution; (b) numerical result.

28 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Fig. 7. Comparing condition numbers of the system matrix for the epitrochoid Fig. 9. Numerical errors with the diﬀuse interval for the epitrochoid domain domain problem. problem.

4.3. An amoeba-like domain with multi boundaries and mixed BCs

An amoeba-like domain with multi-boundaries and mixed Dirichlet and Neumann boundary conditions is considered. The inner boundary is a circle with r = 0.2 and center at (0.5, 0.5). The circular boundary is imposed with the Dirichlet condition. The outer amoeba-like boundary is imposed the Neumann condition. The parametric equation of the amoeba-like boundary is deﬁned as follows:

𝜌( 𝜃) = exp ( sin ( 𝜃)) sin 2( 2 𝜃) + exp ( cos ( 𝜃)) cos 2(2 𝜃) , (22) and the analytical solution on the domain is given as √

𝑢 ( 𝑥, 𝑦) = ln ( 𝑥 − 0 . 5) 2 + ( 𝑦 − 0 . 5) 2, ( 𝑥, 𝑦 ) ∈ Ω̄ . (23) In the numerical implementation, we initially use 30 boundary nodes to compute. The distribution of boundary nodes is shown in Fig. 10 (a), and the numerical result on domain is shown in (b). Fig. 8. Comparing numerical errors for the epitrochoid domain problem. Next, we will test the numerical stability. For the MFS, we use an outset d = 0.001, and then embed the outset in the GMFS offsets. We solve the problem with the initial 30 boundary nodes and give the results accurate and the GMFS2 is clearly superior to the GMFS1. Based on this on an analytical path of P = 0.5 𝜌 with 80 measuring points, as shown in and previous tests, GMFS2 is the method of choice. Fig. 11 . The MFS is obviously inaccurate, but the GMFS is still accurate. Finally, we test the effectiveness of the diffuse density. We choose This further illustrates that the GMFS is more stable than the MFS. hp = 0.1, 0.05, 0.01, 0.005, 0.001, which denotes the diffuse density. Finally, we test the numerical convergence. We choice the analytical The numerical errors are shown in Fig. 9 , on the analytical path P = 0.9 𝜌 path of P = 0.5 𝜌 and use 80 measuring points to evaluate the numerical with the 80 test points. Boundary-node number N = 60 and 120 are both errors. Similarly, we select the mid-value of the GMFS offsets as the considered in the comparison solution. It shows that the GMFS has a MFS offset. The results are shown in Fig. 12 . The GMFS2 shows best diffuse-density convergence. convergence, but GMFS1 shows rather lower convergence.

Fig. 10. Nodal distribution and numerical result on the domain for the amoeba-like domain problem: (a) nodal distribution; (b) numerical result.

29 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Fig. 11. Comparing numerical solutions with a bad oﬀset for the amoeba-like Fig. 13. Nodal distribution for the biharmonic problem. domain problem.

4.4. A biharmonic problem with Robin BCs

Consider a biharmonic problem as an extended potential problem, which is deﬁned as

( ) ∇ 4𝑢 (𝐱 ) ≡ ∇ 2 ∇ 2𝑢 = 0 , 𝐱 ∈Ω, (24)

subject to the Robin boundary conditions:

𝑢 (𝐱 ) = 𝑢̄ ( 𝐱 ), ∇ 2𝑢 (𝐱 ) = 𝑄̄ ( 𝐱 ), 𝐱 ∈Γ, (25)

𝑢 (𝐱 ) = 𝑢̄ ( 𝐱 ), 𝑢 ,𝑛 ( 𝐱 ) = 𝑞̄ ( 𝐱 ), 𝐱 ∈Γ, (26)

and the domain is deﬁned in a L-shaped one as shown in Fig. 13 . The boundary condition 𝑢̄ is given as: Fig. 12. Comparing numerical errors for the amoeba-like domain problem.

𝑢̄ ( 𝑥, 𝑦) = 𝑥 2𝑦 3. (27)

Fig. 14. Solutions on the analytical path for the biharmonic problem.

30 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

Table 1 Comparison of numerical errors ( Er ) for GMFS and MFS: Eq. (25) BCs.

Range of {dp } GMFS MFS (0.01–0.1) 𝑅̄ 1.0777e − 04 1.1608e − 04 (0.01–0.2) 𝑅̄ 9.6753e − 05 1.0908e − 04 (0.01–0.3) 𝑅̄ 8.9717e − 05 1.0606e − 04 (0.01–0.4) 𝑅̄ 8.4983e − 05 Collapsed (warning) (0.01–0.5) 𝑅̄ 8.1330e − 05 9.7136e − 04 Fig. 15. Problem sketch of an eccentric annulus.

Table 2 Comparison of numerical errors ( Er ) for GMFS and MFS: Eq. (26) BCs.

Range of {dp } GMFS MFS (0.001–0.1) 𝑅̄ 1.4784e − 04 4.6663e − 04 (0.001–0.2) 𝑅̄ 1.2072e − 04 1.5907e − 04 (0.001–0.3) 𝑅̄ 1.0564e − 04 2.2991e − 04 (0.001–0.4) 𝑅̄ 9.6537e − 05 1.0964 (warning) (0.001–0.5) 𝑅̄ 9.0233e − 05 6.3016e − 04

The fundamental solution 𝜓 for the biharmonic equation is given by:

⎧ − 𝑟 2 ln ( 𝑟 ), 2 𝐷, ⎪ 8 𝜋 𝜓 ( 𝑟 ) = ⎨ (28) ⎪ 𝑟 ⎩ , 3 𝐷. 8 𝜋 Fig. 16. Comparing numerical solutions with a degenerate-scale source for the

Note that the exact solution is not available in this example. For solv- eccentric annulus problem. ing the problem, some special technique should be adopted, detailed in [19,45] . Based on the Maximum Principle [19,45–47] , the maximum error will occur on the boundary. Hence, we can measure the error on accurate. In addition, we noticed that the GMFS showed a convergence the boundary where the boundary condition is known. We solve the with increasing range (diﬀuse-size convergence) in these results. This is problem with 40 boundary nodes and the GFMS2, as shown in Fig. 13. another advantage for the GMFS. We choose the analytical path “stepping perimeter ”, beginning at the left-bottom-corner point and stepping forward in a counterclockwise di- rection on the boundary. Then we give the numerical results with 160 4.5. An eccentric annulus problem with a degenerate scale 𝑅̄ measuring points in the Fig. 14 ({dp } = (0.01–0.1) , Eq. (25) BCs). Next, we compare the performance of the GMFS and the MFS. Us- It is reported that the traditional MFS may fail for the degenerate ing the BCs in Eq. (25), and hp = 0.01 for diﬀused, and the mid-value of scale of the source being contained [48]. Then, we here use the GMFS to

{dp } as the oﬀset for the MFS, the obtained results are shown in Table 1. cope with this kind of MFS’ challenging problem govern by the Laplace

Furthermore, using the BCs in Eq. (26), and hp = 0.001 for diﬀused, and equation. For a doubly-connected domain, an eccentric annulus is con- the mid-value of {dp } for the MFS, the results are shown in Table 2. Evi- sidered as shown in Fig. 15, and O1 = (0.666667, 0), O2 = (0.453333, dently, the GMFS is stable and accurate, but the MFS is unstable and less 0), a1 = 0.533333, a2 = 0.2133333. In this case, the analytical solution

Fig. 17. Sketch of relative locations for the degenerate scale to the IPD.

31 J.J. Yang et al. Engineering Analysis with Boundary Elements 94 (2018) 25–33

and enriches the method being immune to degenerate scale. Moreover, this improvement is simple and less expensive with no other problems apparently surfacing. There are two algorithms presented for the GMFS, GMFS1 and GMFS2. Hence, we would like to recommend the GMFS2 as the standard algorithm here, but the topic of the GMFS1 is still under investigation. Moreover, we have given preliminary standard oﬀsets for the GMFS, as in the Eq. (4) . But, it should be tested and corrected further. In addition, a regional IPD, as shown in Fig. 1 (a), is another option for the GMFS, its eﬀect is also a topic for further research.

Acknowledgments

The authors would like to thank the support from the National Nat- ural Science Foundation of China (No. 51478053 ), and the Open Fund of Key Laboratory of Road Structure and Material of Ministry of Trans- port: CSUST ( KFJ120201 ). Also, we appreciate Prof. C.S. Chen for his valuable advices on the research and Dr. D. Watson for his help on proof Fig. 18. Numerical errors for GMFS’ IPD with or without the degenerate scale. reading of the original manuscript.

Supplementary materials is chosen as: [ ( ) ( )] 𝑢 𝐱 . 𝑈 𝐱, 𝐜 𝑈 𝐱, 𝐜 , Supplementary material associated with this article can be found, in ( ) = 2 88539 2 − 1 + 1 (29) the online version, at doi:10.1016/j.enganabound.2018.05.014 . where, c1 = (−0.4,0) and c2 = (0.4,0) are two focus sources for comput- ing the analytical ﬁeld, and U ( x, s ) is the degenerate kernel [49,50] , References

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