Commun Nonlinear Sci Numer Simulat

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1 Commun Nonlinear Sci Numer Simulat 6 (2) Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: The scaled boundary FEM for nonlinear problems Zhiliang Lin, Shijun Liao * State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai JiaoTong University, Shanghai 224, China article info abstract Article history: Received 5 February 2 Received in revised form 4 March 2 Accepted 7 March 2 Available online 5 March 2 Keywords: Scaled boundary finite-element Homotopy analysis method Nonlinear problem The traditional scaled boundary finite-element method (SBFEM) is a rather efficient semianalytical technique widely applied in engineering, which is however valid mostly for linear differential equations. In this paper, the traditional SBFEM is combined with the homotopy analysis method (HAM), an analytic technique for strongly nonlinear problems: a nonlinear equation is first transformed into a series of linear equations by means of the HAM, and then solved by the traditional SBFEM. In this way, the traditional SBFEM is extended to nonlinear differential equations. A nonlinear heat transfer problem is used as an example to show the validity and computational efficiency of this new SBFEM. Ó 2 Elsevier B.V. All rights reserved.. Introduction The scaled boundary finite-element method (SBFEM), developed recently by Wolf and Song [ 3], is a novel semi-analytical method to solve partial differential equations. In the frame of the SBFEM, a scaled boundary coordinate system is introduced. The weighted residual approximation of finite elements is applied in the circumferential direction, and then the governing partial differential equations are transformed to ordinary differential equations in the radial direction. Those ordinary differential equations can be solved analytically in the radial direction. Like the boundary element method, only the boundary of the problem domain is discretized, but no fundamental solution is required. Thus, this semi-analytical method combines the advantages of the finite element and boundary element methods, and besides, it presents appealing features of its own. The SBFEM has been used for the problems of elasto-statics and elasto-dynamics, especially for soil-structure interaction problems in unbounded domains [3,4]. Recently, the SBFEM has been extended to the fluid flow problems [5 8]. Meanwhile, researchers reported successful algorithms for the coupling of SBFEM with other numerical techniques [9 ] to analyze geomechanics problems. By coupling the finite element method and the SBFEM, Doherty and Deeks [9] captured the nonlinearity of problems in the near field. However, the SBFEM only accurately modeled the linear elastic far field response. It must be emphasized that, up to now, all problems solved by SBFEM are governed by linear partial differential equations. The homotopy analysis method (HAM), which is an analytic approach to get series solutions of strongly nonlinear equations, was first proposed by Liao [2] in 992. Thereafter, the HAM has been improved step by step [3 9]. Briefly speaking, by means of the HAM, one constructs a continuous mapping of an initial guess approximation to the exact solution of considered equations. An auxiliary linear operator is chosen to construct such kind of continuous mapping, and a convergencecontrol parameter is used to ensure the convergence of solution series. More importantly, this method enjoys great freedom in choosing initial guesses and auxiliary linear operators. By means of this kind of freedom, a complicated nonlinear problem can be transferred into an infinite number of simple, linear sub-problems, as shown by Liao and Tan [7]. Thus, if the SBFEM is successfully combined with the HAM, it could be extended to many engineering problems governed by nonlinear partial differential equations. This is the motivation of our present work. * Corresponding author. addresses: linzhiliang@sjtu.edu.cn (Z. Lin), sjliao@sjtu.edu.cn (S. Liao) /$ - see front matter Ó 2 Elsevier B.V. All rights reserved. doi:.6/j.cnsns.2.3.5

2 64 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) The aim of this paper is to combine the SBFEM with the HAM and then apply it to solve nonlinear problems. Shortly speaking, we first transfer an original nonlinear problem into an infinite number of linear sub-problems using the HAM, and then solve all those sub-problems by the SBFEM. To show its validity, 2D nonlinear Poisson-type problems are solved, and numerical results demonstrate the efficiency and accuracy of this approach. 2. Theoretical description 2.. Governing equations Consider a 2D Poisson equation with a varying conductivity kðx; yþ, which can be expressed as r ½kðx; yþruðx; yþš ¼ f ðx; yþ; ðx; yþ 2X; ðþ under boundary conditions u ¼ g ðx; yþ; ðx; yþ 2C ¼ g 2ðx; yþ; ðx; yþ 2C 2 ; ð3þ where r is the Hamilton operator, uðx; yþ is the dependent variable, the real functions f ðx; yþ; g ðx; yþ and g 2 ðx; yþ are prescribed functions or given values, the x and y are spatial coordinates, n denotes the normal to the boundary, X the domain and C ¼ C [ C 2 the boundary. Obviously, Eq. () can be rewritten as kðx; yþr 2 uðx; yþþrkðx; yþruðx; yþ ¼f ðx; yþ; ðx; yþ 2X: ð4þ Assuming that kðx; yþ ¼ þ uðx; yþ, i.e., the conductivity is dependent on the unknown uðx; yþ. Then Eq. (4) becomes r 2 uðx; yþþuðx; yþr 2 uðx; yþþruðx; yþruðx; yþ ¼f ðx; yþ; ðx; yþ 2X: ð5þ Thus, the 2D nonlinear Poisson-type problems which we will solve are governed by Eq. (5) with the boundary conditions (2) and (3) Homotopy transformation According to Eq. (5), a nonlinear operator N is defined as follows: N½UŠ ¼r 2 U þ Ur 2 U þ ru ru f ðx; yþ: ð6þ Then Eq. (5) can be written as N½uðx; yþš ¼ ; ðx; yþ 2 X: ð7þ In the frame of the HAM, we first construct the zeroth-order deformation equation ð qþl½uðx; y; qþ u ðx; yþš ¼ c qn½uðx; y; qþš; q 2½; Š; c ; ðx; yþ 2X ð8þ with the boundary conditions Uðx; y; qþ ¼g ðx; yþ; ðx; yþ 2C ; y; ¼ g 2 ðx; yþ; ðx; yþ 2C 2 ; ðþ where c is the convergence-control parameter, u ðx; yþ is an initial guess of uðx; yþ; L is the auxiliary linear operator, and q 2 ½; Š is called homotopy-parameter, respectively. As q increases from to, the solution Uðx; y; qþ deforms continuously from the initial guess u ðx; yþ to the solution uðx; yþ of Eqs. (5), (2) and (3). The homotopy-series solution is given by uðx; yþ ¼u ðx; yþþ Xþ u m ðx; yþ; ðx; yþ 2X: ðþ m¼ u m ðx; yþ ðm P Þ are governed by the so-called mth-order deformation equations L½u m ðx; yþ v m u m ðx; yþš ¼ c d m ðx; yþ; ðx; yþ 2X; ð2þ with boundary conditions u m ðx; yþ ¼; ðx; yþ 2C ; m ðx; ¼ ; ðx; yþ 2C 2 ; ð4þ

3 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) where d m ðx; yþ m N½Uðx; y; qþš ðm m q¼ ð5þ and ; m 6 ; v m ¼ ; otherwise: ð6þ According to Eqs. (6) and (5), we can obtain d m ðx; yþ ¼r 2 u m ðx; yþþ Xm u j ðx; yþr 2 u m j ðx; yþþ Xm ru j ðx; yþru m j ðx; yþ ð v m Þf ðx; yþ: j¼ j¼ ð7þ Therefore, the original nonlinear problem governed by Eqs. (5), (2) and (3) has been transferred into an infinite number of linear sub-problems governed by Eqs. (2) (4) Implementation of SBFEM It should be emphasized that, using the HAM, one has great freedom to choose the auxiliary linear operators and an initial guess under some basic rules [5,7]. Here, we select r 2 as the auxiliary linear operator. Then Eq. (2) can be rewritten as where r 2 u m ðx; yþ ¼R m ðx; yþ; ðx; yþ 2X; ð8þ R m ðx; yþ ¼v m r 2 u m ðx; yþþc d m ðx; yþ: ð9þ That is, the mth-order deformation Eq. (2) becomes a 2D linear Poisson equation and all the linear sub-problems become 2D linear Poisson-type problems which can be solved by the scaled boundary finite-element method. To describe the implementation of the SBFEM, the 2D linear Poisson equation is written as r 2 uðx; yþ ¼Rðx; yþ in X ð2þ with boundary conditions u ¼ u on C u ¼ u n on C m ; ð22þ where Rðx; yþ is the so-called non-homogeneous term and the overbar denotes a prescribed value. The weak form of the weighted-residual statement is formulated as Z Z Z r T wrudx þ wrdx wu n dc ¼ ; ð23þ X X C m where w is the weighting function, and it is the starting point of the derivation of the SBFEM. In the frame of the SBFEM, a scaled boundary coordinate system is introduced at first. As shown in Fig., the circumferential coordinate s is anticlockwise along the boundary and the dimensionless radial coordinate n is a scaling factor, defined as at the scaling centre O and at the boundary S. The two straight sections s ¼ s and s ¼ s are termed side-faces. If the boundary is closed, the side-faces will coincide. Thus, the whole solution domain X is in the range of n 6 n 6 n and s 6 s 6 s. Bounded domains containing the scaling centre are modeled by taking n ¼ and n ¼ ; whereas, unbounded domains are modeled by taking n ¼ and n ¼. The mapping between the Cartesian coordinate system and this scaled boundary coordinate system is expressed by the scaling equations x ¼ x þ n x s ðsþ; y ¼ y þ n y s ðsþ; ð24þ where ðx ; y Þ is the scaling center and x s ðsþ and y s ðsþ are C continuous piecewise smooth functions describing the boundary S in Cartesian coordinates relative to the scaling center. Using conventional techniques the operator r can be mapped to the scaled boundary coordinate system as [3] r ¼ þ n ; ð25þ where b ðsþ and b 2 ðsþ are dependent only on the definition of the boundary S:

4 66 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) b ðsþ ¼ jjj ( ) y s ðsþ ;s x s ðsþ ;s ; b 2 ðsþ ¼ jjj Fig.. The scaled boundary coordinate definition. y s ðsþ x s ðsþ ; ð26þ and jjj is the Jacobian at the boundary jjj ¼x s ðsþy s ðsþ ;s y s ðsþx s ðsþ ;s : In the SBFEM, the unknown variables u at any point ðn; sþ in the domain are sought in the form uðn; sþ ¼NðsÞaðnÞ; ð27þ ð28þ where NðsÞ is the shape function, the vector aðnþ is analogous to the nodal values in the standard finite-element method. The weighting function w can be chosen as the same shape function as (28) by Galerkin approach wðn; sþ ¼NðsÞwðnÞ ¼wðnÞ T NðsÞ T : ð29þ Substituting Eqs. (25), (28) and (29) into Eq. (23) and integrating the domain integrals containing wðnþ ;n by parts with respect to n using Green s theorem, and noting that dx ¼jJjndnds, yields Z qðn Þ¼ NðsÞ T u n ðn ; sþn ds; ð3þ SZ qðn Þ¼ NðsÞ T u n ðn ; sþn ds; ð3þ S E n 2 aðnþ ;nn þðe þ E T E ÞnaðnÞ ;n E 2 aðnþþfðnþ ¼; ð32þ where and qðnþ ¼E naðnþ ;n þ E T aðnþ; FðnÞ ¼nF s ðnþ n 2 F b ðnþ; B ðsþ ¼b ðsþnðsþ; B 2 ðsþ ¼b 2 ðsþnðsþ ;s ; Z E ¼ B ðsþ T B ðsþjjjds; ZS E ¼ B 2 ðsþ T B ðsþjjjds; ZS E 2 ¼ B 2 ðsþ T B 2 ðsþjjjds; S Z F b ðnþ ¼ NðsÞ T RjJjds; S F s ðnþ ¼Nðs Þ T u n ðn; s ÞjJðs Þj þ Nðs Þ T u n ðn; s ÞjJðs Þj: ð33þ ð34þ ð35þ ð36þ ð37þ ð38þ ð39þ ð4þ ð4þ Eq. (32) is the so-called scaled boundary finite-element equation. By introducing the shape function, the Poisson equation has been weakened in the circumferential direction in a finite element manner, but remains strong in the radial direction. Thus, the governing partial differential equation is transformed to an ordinary matrix differential equation in the radial direction. The rank of matrices E ; E ; E 2 and vectors aðnþ; FðnÞ; F b ðnþ; F s ðnþ is N (where N is the number of nodes in the boundary S). The solution technique of the ordinary matrix differential equation is described in Appendix A.

5 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) Thus, when a proper initial guess u is chosen, we can gradually solve the mth-order deformation equations by means of SBFEM as m increases. It must be noted that the right-hand side term d m of the mth-order deformation Eq. (2) becomes more and more complicated as m increases. It means that the non-homogeneous term FðnÞ will become more and more complicated. This will result in a significant reduction in computational efficiency. To overcome this drawback, Chebyshev interpolation is employed to obtain a polynomial approximation P n ðnþ of degree n to FðnÞ, where n is the highest order of Chebyshev polynomials. Replacing FðnÞ with P n ðnþ, we can solve the scaled boundary finite element equations with high efficiency. For details of Chebyshev interpolation, please refer to [2] Optimal c The Mth-order approximation of uðx; yþ is given by ~u M ðx; yþ ¼u ðx; yþþ XM m¼ u m ðx; yþ: Note that, ~u M ðx; yþ is not only dependent upon x and y, but also upon the parameter c. Recently, some researches [9,2] have been done to find out the optimal convergence-control parameter c to get a faster convergent series solution. At the Mth-order approximation, the exact square residual error is defined as follow: Z Z n Z s Z n Z s D M ¼ ðn½~u M ðx; yþšþ 2 dx ¼ ðn½~u M ðn; sþšþ 2 njjjdsdn ¼ Ef ðn; sþdsdn; ð43þ s where X n s n ð42þ Ef ðn; sþ ¼ðN½~u M ðn; sþšþ 2 njjj: ð44þ However, a large amount of CPU time is needed to calculate the exact residual error D M, especially for high order of approximation. Thus, to decrease the CPU time, the so-called averaged residual error is defined by E M ¼ X I I J i¼ X J j¼ Ef ðn þ idn; s þ jdsþ; ð45þ where Dn ¼ðn n Þ=I and Ds ¼ðs s Þ=J. E M is the function of the unknown convergence-control parameter c. Obviously, the more quickly E M decreases to zero, the faster the corresponding homotopy-series solution converges. For a given M, the corresponding optimal values of the convergence-control parameter c are given by the minimum of E M, corresponding to an algebraic equation de M dc ¼ : ð46þ 3. Numerical examples To demonstrate the validity of the homotopy-based scaled boundary FEM, assume that the solution of Eq. (5) is exactly equal to a prescribed function uðx; yþ. And then the non-homogeneous term f ðx; yþ can be determined by substituting the exact solution into Eq. (5). Meanwhile, the functions g ðx; yþ and g 2 ðx; yþ can be set according to uðx; yþ. In this way, we can compare our numerical results with the known exact solution. All of our calculations were done on a laptop PC with a 2.49 GHz CPU and 2 GB RAM. 3.. For a bounded domain First of all, we consider the nonlinear 2D Poisson equation (5) defined in a unit bounded domain X ¼fðx; yþ : x 2 þ y 2 6 g with the Dirichlet-type boundary condition uðx; yþ ¼gðx; yþ on its boundary C. Let uðx; yþ ¼x 2 þ y 2 be the solution of Eq. (5), then we have f ðx; yþ ¼4 þ 8ðx 2 þ y 2 Þ and gðx; yþ ¼. Due to the symmetry, only one quarter of this problem is modeled, as illustrated in Fig. 2. The boundary condition remains the same as uðx; yþ ¼g ðx; yþ ¼onC, but the boundary condition u ;n ¼ g 2 ðx; yþ ¼ is enforced on C 2 due to the symmetry, where the comma in the subscript p designates partial derivative with respect to the variable following the comma. The initial guess is chosen as u ðx; yþ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2, which satisfies the boundary condition automatically. The scaling centre is chosen at the original of the Cartesian coordinates, and then the boundary C 2 is the so-called side-faces which need not to be discretized. The boundary C is discretized with three-noded quadratic elements. For a quarter unite circle, we have x s ðsþ ¼cosðsÞ; y s ðsþ ¼sinðsÞ; ð47þ where s is in the range of ½; p=2š. One kind of mesh is constructed by binary subdivision of the boundary C, i.e. the mesh consists of two scaled boundary elements, as shown in Fig. 2. To calculate the averaged residual error E M (45), we use I ¼ 2 and J ¼ 2.

6 68 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) Fig. 2. Bounded domain. pffiffiffiffiffiffi Firstly, let n, the highest order of Chebyshev polynomials, equal to. The curves of logð E MÞ versus c at different order of approximation M ¼ 4; 8; 2; 6 and 2 are shown in Fig. 3, which indicate that the optimal value of c is about -.6. In case of n ¼ 2, the optimal value of c is close to.63, as shown in Table. For different values of n, the optimal value of c changes very little. As long as the choice of c is near the optimal value, the homotopy-series solution could converge faster. For the sake of comparison, c is set as :6. The variations of the square root of E m and the CPU time with the order of approximation M at different n ¼ ; 5; 2; 25 are plotted in Figs. 4 and 5, respectively. Fig. 4 illustrates that, for a given n, the value of E M decreases as M increases, but when M increases to a certain value M p ; E M is almost the same. The lowest value of E M is denoted as E M, which depends on n. The larger value of n, the smaller the value of E M.InFig. 5, for a given order of approximation M, more CPU time is needed as n increases. It means that the highest order of Chebyshev polynomials affect not only the accuracy but also the computational efficiency. p As shown in Figs. 4 and 5, in the case of n ¼ 25 and M ¼, the CPU time is about 9236 seconds with ffiffiffiffiffiffi E M ¼ 3:6 7 ; pffiffiffiffiffiffi in the case of n ¼ and M ¼ 2; E M has decreased to : 5, and the CPU time is about 37 s, less than 3 min. In the frame of the SBFEM, properties of the standard finite-element method are maintained in the circumferential direction, but the analytical solution is provided in the radial direction. As long as the high precision is ensured in the radial direction and the number of surface elements is large enough in the circumferential direction, we could guarantee the accuracy of the numerical solution in the whole domain. In this example, since the boundary S is a quarter of circle and the exact solution on any path from the scaling centre to the boundary is the same, the impact from the discretization of the boundary can be ignored. So we only need to compare the numerical approximations with the exact solution in a given radial path. As shown in Figs. 6 and 7, the higher the order of approximation, the less the absolute error between the numerical approximation and the exact solution n= ) log ( EM M=4,8,2,6, C pffiffiffiffiffiffi Fig. 3. The logarithm of E M versus c in the case of n ¼ for the bounded domain.

7 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) Table Optimal value of c in the case of n ¼ 2 for the bounded domain. Order of approximation M Optimal value of c pffiffiffiffiffiffi Minimum value of E M E E E E E E E E E 6-2 C = n= EM -8 - n= n=2-6 n= M Fig. 4. The square root of E M versus M in the case of c ¼ :6 for the bounded domain. 4 CPU time (sec.) 3 2 n=,5,2,25 C = M Fig. 5. The CPU time versus M in the case of c ¼ :6 for the bounded domain.

8 7 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) n=2 c = -.6 u M ~.8.6 M= (initialguess) M= M=2 M=6 exactsolution Fig. 6. The approximation ~u M versus n in the case of n ¼ 2 and c ¼ :6 for the bounded domain. ξ.7 7E-6.6 n = 2 c = -.6 6E-6 n=2 c = E-6 Err.4 Err 4E-6.3 M=4 3E-6 M=6.2. M=8 M= ξ 2E-6 E-6 M=2 M= ξ Fig. 7. The error (j~u M u exact j) versus n in the case of n ¼ 2 and c ¼ :6 for the bounded domain For an unbounded domain Secondly, the nonlinear 2D Poisson equation (5) defined in an unbounded domain X ¼fðx; yþ : x 2 þ y 2 P g with the Dirichlet-type boundary is considered. Let the exact solution uðx; yþ equal to =ðx 2 þ y 2 Þ, then we have f ðx; yþ ¼4ðx 2 þ y 2 þ 2Þ=ðx 2 þ y 2 Þ 3 and gðx; yþ ¼gðx; yþ ¼. Similarly, due to the symmetry, only one quarter of this problem is modeled, as illustrated in Fig. 8. The boundary condition remains the same as uðx; yþ ¼g ðx; yþ ¼onC, and the condition u ;n ¼ g 2 ðx; yþ ¼ is enforced on C 2 due to the sym- Fig. 8. Unbounded domain.

9 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) p metry. The initial guess is chosen as u ðx; yþ ¼= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2. Since the boundary S is a quarter of unit circle, the scaled boundary transformation is the same as the bounded domain, but the scaling factor n varies from to þ. In order to calculate E M, we use Dn ¼ :, I ¼ and J ¼ 2 in Eq. (45). So the optimal convergence-control parameter c can be calculated by Eq. (46). As shown in Table 2, in the case of n ¼, the optimal value of c is about.59. Fig. 9 shows that if c is chosen near the optimal value, E M decreases faster as the order of approximation M increases. The value of the highest order of Chebyshev polynomials has a great influence on the computational efficiency and accuracy, as shown in Figs. and. In the case of n ¼ 2 and c ¼ :6, as M increases to 2, the square root of the averaged residual error E M decreases to 8:6 7, but the total CPU time is only about 42 s. Similarly, let n equal to 2 and the convergence-control parameter c equal to.6, then we compare the numerical approximations with the exact solution in a given radial path. The exact solution uðn; sþ is equal to =n 2, which tends to zero as n!. Figs. 2 and 3 illustrate that the absolute error between the numerical approximation and the exact solution decreases quickly as M increases. 4. Conclusions A new approach has been proposed for 2D nonlinear Poisson-type problems by combining the scaled boundary finite-element method with the homotopy analysis method. In this approach, the original nonlinear problem is transferred into an infinite number of linear Poisson-type sub-problems by the HAM, and then all those linear Poisson-type sub-problems are solved by means of the SBFEM. Numerical examples governed by a 2D nonlinear Poisson equation, including a bounded Table 2 Optimal value of c in the case of n ¼ for the unbounded domain. Order of approximation M Optimal value of c pffiffiffiffiffiffi Minimum value of E M E E E E E E E E E n= log ( EM ) M=4,8,2,6, C pffiffiffiffiffiffi Fig. 9. The logarithm of E M versus c in the case of n ¼ for the unbounded domain.

10 72 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) C = n= E M - n= n=2-7 n= M Fig.. The square root of E M versus M in the case of c ¼ :6 for the unbounded domain. 4 3 n=,5,2,25 CPU time (sec.) 2 C = M Fig.. The CPU time versus M in the case of c ¼ :6 for the unbounded domain. domain and an unbounded domain, are solved by this new approach, respectively. All the numerical results demonstrate the high efficiency and accuracy of our approach. Note that the HAM enjoys great freedom in choosing initial guess u. If the convergence-control parameter c is selected properly, the Mth-order approximation ~u M is better than the free selected initial approximation. So, it can provide us with a family of iterative formulae. Clearly, this new approach has great potential to the nonlinear problems. In this paper, we have successfully extended the SBFEM to solve nonlinear problems. Meanwhile, the idea of the HAM has been further applied to numerical calculation. Though more research effort is needed to develop and improve this new approach, it may not be far away to apply our approach to solve more nonlinear engineering problems with increased complexity.

11 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) n =2 c = M= (initialguess) M= M=2 M=6 exactsolution ~ u M ξ Fig. 2. The approximation ~u M versus n in the case of n ¼ 2 and c ¼ :6 for the unbounded domain..7 7E-6.6 n= 2 c =-.6 6E-6 n=2 c = E-6 Err.4 M=4 Err 4E-6 M=6.3 3E-6.2 2E-6 M=8. M=8 M=6 2 3 ξ E-6 M=2 2 3 ξ Fig. 3. The error (j~u M u exact j) versus n in the case of n ¼ 2 and c ¼ :6 for the unbounded domain. Acknowledgements The work is partly supported by National Natural Science Foundation of China (Approval No ) and State Key Lab of Ocean Engineering (Approval No. GKZD2). Thanks to Prof. Tao (Newcastle University) for the suggestive discussions and valuable comments. Appendix A. Solution technique of SBFEM By introducing the variable XðnÞ ¼ aðnþ ; ða:þ qðnþ Eq. (32) is transformed into the first-order ordinary differential equations nxðnþ ;n ¼ ZXðnÞ FðnÞ ða:2þ with a Hamiltonian matrix " # Z ¼ E ET E : ða:3þ E 2 þ E E ET E E

12 74 Z. Lin, S. Liao / Commun Nonlinear Sci Numer Simulat 6 (2) The eigenvalue problem is formulated as ZU ¼ UK ða:4þ with the eigenvalues of the Hamiltonian matrix K ¼ K ; K ¼dk i c ða:5þ K and the eigenvector matrix partitioned conformably U ¼ U U 2 : ða:6þ U 2 U 22 The real parts of all elements k i are non-positive. The solution of Eq. (A.2) equals XðnÞ ¼ U " # U 2 n K C ðnþ ; ða:7þ U 2 U 22 n K C 2 ðnþ where Z n C ðnþ ¼c C 2 ðnþ ¼c 2 Z n s s K A 2 FðsÞds; s s K A 22 FðsÞds; c ; c are integration constants and A ¼ A A 2 ¼ U : A 2 A 22 ða:8þ ða:9þ ða:þ Substituting (A.7) into (A.) yields aðnþ ¼U n K C ðnþþu 2 n K C 2 ðnþ qðnþ ¼U 2 n K C ðnþþu 22 n K C 2 ðnþ ða:þ ða:2þ For a bounded domain, aðnþ at n ¼ must remain finite, leading to C 2 ðnþj n¼ ¼, then c 2 ¼ Z s s K A 22 FðsÞds: ða:3þ Substituting (A.3) into (A.9) yields Z n C 2 ðnþ ¼ s s K A 22 FðsÞds: And then substituting Eqs. (A.8) and (A.4) into Eqs. (A.) and (A.2) results in Z Z n aðnþ ¼U n K c þ U 2 n K n s s K A 2 FðsÞds s s K A 22 FðsÞds ða:4þ ða:5þ and qðnþ ¼U 2 n K c þ Z s s K A 2 FðsÞds U 22 n K Z n n s s K A 22 FðsÞds: ða:6þ Note that c is determined by the boundary condition aðnþj n¼ ¼ U c U 2 Z s s K A 22 FðsÞds ða:7þ or qðnþj n¼ ¼ U 2 c U 22 Z s s K A 22 FðsÞds: ða:8þ For an unbounded domain, aðnþ as n!must remain finite, leading to C ðnþj n! ¼. The solution procedure is similar, please refer to [3].

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