Singular Boundary Method for Solving Plane Strain Elastostatic Problems

International Journal of Solids and Structures 48 (2011) 2549–2556

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International Journal of Solids and Structures

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Singular boundary method for solving plane strain elastostatic problems ⇑ Yan Gu a,b, Wen Chen a,b, , Chuan-Zeng Zhang c a Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing 210098, PR China b State Key Laboratory of Structural Analysis for Industrial Equipment, University of Technology, Dalian 116024, PR China c Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str. 9-11, D-57076 Siegen, Germany article info abstract

Article history: This study documents the ﬁrst attempt to apply the singular boundary method (SBM), a novel boundary Received 15 December 2010 only collocation method, to two-dimensional (2D) elasticity problems. Unlike the method of fundamental Received in revised form 16 April 2011 solutions (MFS), the source points coincide with the collocation points on the physical boundary by using Available online 14 May 2011 an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest. Three benchmark elasticity problems are tested to demonstrate Keywords: the feasibility and accuracy of the proposed method through detailed comparisons with the MFS, bound- Singular boundary method ary element method (BEM), and ﬁnite element method (FEM). Collocation method Ó 2011 Elsevier Ltd. All rights reserved. Singularity Origin intensity factor Inverse interpolation technique Elasticity problem

1. Introduction Thus, over the past decade, some considerable effort was de- voted to eliminating the need for meshing. This led to the develop- The finite element method (FEM) has long been a dominant ment of meshless methods which require neither domain nor numerical technique in the simulation of elasticity problems. How- boundary meshing. They still require discretizations via sets of ever, this method requires tedious domain meshing which is often nodes, but these nodes need not have any connectivity, and the computationally costly and sometimes mathematically trouble- trial functions are built entirely in terms of nodes. Among these some (Brebbia, 1978), especially when the geometry of the body methods, the method of fundamental solutions (MFS) has emerged is not simple. As a domain discretization technique, the FEM is also as a boundary only collocation method with the merit of easy pro- less effective for inverse problems in which measurement is often gramming, high accuracy, and fast convergence (Chen et al., 1998; only accessible on the boundary. As an alternative approach, the Fairweather and Karageorghis, 1998; Karageorghis, 1992). During boundary element method (BEM) has long been touted to avoid the past decade, this method has been successfully applied to the such drawbacks (Brebbia, 1978; Cheng and Cheng, 2005; Cruse, solution of plane isotropic elasticity problems (Liu, 2002; Marin 1988). During the past two decades, this method has rapidly im- and Lesnic, 2004; Redekop, 1982), axisymmetric problems in elas- proved, and is nowadays considered as a competing method to tostatics (Redekop and Thompson, 1983), 3D problems in elasto- FEM. Despite the fact that the BEM requires only meshing on the statics (Redekop and Cheung, 1987), and to problems associated boundary, it involves quite sophisticated mathematics and some with layered elastic materials (Berger and Karageorghis, 2001). difficult numerical integrations of singular functions. It is also In the traditional MFS, a fictitious boundary slightly outside the worth noting that the discretization matrix of the BEM is fully pop- problem domain is required in order to place the source points and ulated due to its global approximation, and thus the total compu- avoid the singularity of the fundamental solution. Despite many tational costs are not as low as expected compared to the costs years of focused research, the determination of the distance be- associated with the local FEM, which results in a sparse matrix. tween the real boundary and the fictitious boundary is based on Moreover, surface meshing in a three-dimensional (3D) domain experience and therefore troublesome (Cheng et al., 2000; Liu, is still a nontrivial task. 2010). In recent years, various efforts have been made aiming to re- move this barrier in the MFS, so that the source points can be placed on the real boundary directly. These methods include, but are not limited to, the boundary knot method (BKM) (Chen and Hon, ⇑ Corresponding author at: Center for Numerical Simulation Software in Engi- neering and Sciences, Department of Engineering Mechanics, Hohai University, 2003; Chen and Tanaka, 2002); the boundary particle method Nanjing 210098, PR China. Tel.: +86 25 8378 6873; fax: +86 25 8373 6860. (BPM) (Chen and Fu, 2009; Fu et al., 2009); the boundary colloca- E-mail address: [email protected] (W. Chen). tion method (BCM) (Chen et al., 2002, 2004); the regularized

0020-7683/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2011.05.007 2550 Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556 meshless method (RMM) (Young et al., 2005, 2007); the modified 1 l @2u ðPÞ @2u ðPÞ 1 @2u ðPÞ 2 1 þ 1 þ 2 ¼ 0; P 2 X; method of fundamental solutions (MMFS) (Sarler, 2009); and the 2 2 1 2l @x1 @x2 1 2l @x1@x2 boundary distributed source (BDS) method (Liu, 2010). ð1Þ Each of the above-mentioned methods has its own merits and demerits. The BKM employs the non-singular general solutions in- 1 @2u ðPÞ @2u ðPÞ 1 l @2u ðPÞ stead of using the singular fundamental solutions to avoid the sin- 1 þ 2 þ 2 2 ¼ 0; P 2 X; 1 2 @x @x @x2 1 2 @x2 gularity of the discretized matrix. This method dealt successfully l 1 2 1 l 2 with many kinds of problems and eliminated the major drawback ð2Þ of fictitious boundary of the MFS. However, like the MFS, the con- subject to the boundary conditions dition number of its discretization matrix worsens quickly with an increasing number of boundary nodes. This method is also mostly uiðPÞ¼ui; P 2 Cu ðdisplacement boundary conditionÞ; ð3Þ applied to interior Helmholtz and diffusion problems because the nonsingular general solution is not available in some cases, such tiðPÞ¼ti; P 2 Ct ðtraction boundary conditionÞ; ð4Þ as the Laplace equation. where l represents the Poisson’s ratio, ti is the component of The RMM employs a subtracting and adding-back technique boundary traction in the ith coordinate direction, and X is a widely used in the BEM-based method to regularize the singulari- 2 bounded domain in R with boundary C = Cu [ Ct which we shall ties of fundamental solutions, based on the fact that the MFS and assume to be piecewise smooth. The over-bar indicates the given the indirect boundary integral formulation are similar in nature. value. The strains eij, i, j = 1, 2, are related to the displacement gra- The condition number of its interpolation matrix remains moder- dients by ate even with a large number of boundary nodes. However, the kernel functions used in the RMM are double-layer potentials in 1 @ui @uj eij ¼ þ ; ð5Þ the potential theory, which lead to troublesome hyper-singular 2 @xj @xi kernels at the origin and may jeopardize its overall accuracy. and the stresses r ; i; j ¼ 1; 2, are related to the strains through Unlike the RMM, the MMFS simply uses the single-layer potential ij Hooke’s law by as its kernel function, where the singular terms are determined by the integration of the fundamental solution on line segments formed rij ¼ kdijekk þ 2Geij; ð6Þ by using neighboring points, and the use of a constant solution to where k and G represent the Lame constants, and d is the Kronecker determine the diagonal coefficients from the derivatives of the ij delta. The boundary tractions t and t are defined in terms of the fundamental solution. This approach is very stable but the integral 1 2 stresses as calculation makes it more complex and less accurate than the MFS. In the BDS method, the singular fundamental solution is inte- ti ¼ rijnj; ð7Þ grated over small regions covering the source points so that the where n is the direction cosine of the unit normal vector at the diagonal elements in the discretized matrix can be evaluated ana- j boundary point P and a summation over repeated subscripts is lytically. However, the analytical expression of the diagonal coeffi- implied. cients for equations with the Neumann boundary condition has to Employing indicial notation for the coordinates of points P and be determined indirectly. Thus, this method is still immature and Q, i.e. x , x and x , x , respectively, the Kelvin fundamental requires further development. 1P 2P 1Q 2Q solution of the systems (1) and (2) can be expressed as In a more recent study, a novel boundary only collocation method, called the singular boundary method (SBM), was proposed by 1 1 U ðP; QÞ¼ fð3 4lÞ log d þ r r gðl; k ¼ 1; 2Þ; Chen (2009), Chen et al. (2009), Chen and Wang (2010). The key lk 8pGð1 lÞ r lk ;l ;k idea of this method is to introduce the concept of the origin inten- ð8Þ sity factor to isolate the singularities of the fundamental solutions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 so that the source points can be placed on the real boundary di- where r ¼ ðx1P x1Q Þ þðx2P x2Q Þ is the distance between the @r xiP xiQ rectly. Then, an inverse interpolation technique is introduced to collocation P and the source point Q, r;i ¼ ¼ expresses the @xiP r determine the above-mentioned origin intensity factors from both derivatives of the distance r with respect to xiP. the fundamental solution and its derivative, without using any ele- The fundamental solutions for the tractions can be obtained by ment or integration concept. This method is mathematically sim- first computing the fundamental solutions for the strains and then ple, easy-to-program, truly integration-free and has been applying Hooke’s law successfully applied to interior and exterior problems with Laplace and Helmholtz equations. 1 TlkðP; QÞ¼ f½ð1 2lÞdlk þ 2r;lr;kr;n In this study, we make the first attempt to extend the SBM to 2D 4pð1 lÞr elasticity problems, which are more difficult than potential prob- þð1 2lÞðr;lnk r;knlÞg ðl; k ¼ 1; 2Þ; ð9Þ lems because of the complexity of the Kelvin fundamental solution. A brief outline of the paper is as follows. Section 2 presents the where r,n = r,1n1 + r,2n2 expresses the derivative of r in the direction SBM formulation for the solution of 2D elasticity problems. In Sec- of the outward normal at the point P. tion 3, the accuracy and stability of the SBM are tested through By using the radial basis functions (RBFs) (Fairweather and three examples in which SBM solutions are compared with the Karageorghis, 1998; Chen and Tanaka, 2002), the displacements MFS, BEM and FEM. Finally, the conclusions and remarks are pro- and tractions can be approximated by a linear combination of fun- vided in Section 4. damental solutions with respect to different source points Qj as follows: 2. The SBM formulation for plane strain elastostatic problems 8 > PN > <> u1ðPiÞ¼ fajU11ðPi; Q jÞþbjU12ðPi; Q jÞg; We consider the Navier equations for the plane strain elasto- j¼1 ð10Þ static problem with neglected body force, in terms of displace- > PN > ments u and u : : u2ðPiÞ¼ fajU21ðPi; Q jÞþbjU22ðPi; Q jÞg; 1 2 j¼1 Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556 2551 8 PN M > In the first step of the IIT, we place a set of sample points fSig > t P a T P ; Q b T P ; Q ; i¼1 < 1ð iÞ¼ f j 11ð i jÞþ j 12ð i jÞg inside the physical domain and use a simple particular solution j¼1 ð11Þ us P , i = 1, 2, as the sample solution of the elasticity equations > PN i ð Þ > :> t2ðPiÞ¼ fajT21ðPi; Q jÞþbjT22ðPi; Q jÞg; (1) and (2). The sample points do not coincide with the source j¼1 points, and the number of sample points M should not be fewer N N than the physical boundary source points number N. By using the where fajg and fbjg are the unknown coefficients, Pi 2 X [ C is j¼1 j¼1 interpolation formulations (10), we can get the ith collocation point, and Qj is the jth source point. 8 > PN In the traditional MFS, a fictitious boundary slightly outside the > s <> u1ðSiÞ¼ fcjU11ðSi; PjÞþdjU12ðSi; PjÞg; problem domain is required for the placement of the source points j¼1 N ð15Þ fQ jgj¼1 to avoid the singularity of fundamental solutions. However, > PN > s despite many years of focused research, the determination of the : u2ðSiÞ¼ fcjU21ðSi; PjÞþdjU22ðSi; PjÞg; distance between the physical boundary and the fictitious bound- j¼1 ary is based on experience and therefore troublesome for some N where fPjgj¼1 are the source points on the physical boundary. The engineering applications, such as problems with complex-shaped N N unknown coefficients fcjgj¼1, fdjgj¼1 and the locations of the sample boundary or multiply-connected domain. points fS gM are determined by minimizing the following function i i¼1 Like the MFS, the SBM also uses the fundamental solution as the XM XN kernel function of its approximation. Unlike the MFS, the colloca- F c N ; d N ; S M c U S ; P ðf jgj¼1 f jgj¼1 f igi¼1Þ¼ f j 11ð i jÞ tion and source points of the SBM are coincident and are placed i¼1 j¼1 on the physical boundary without the need of using a fictitious s 2 þ djU12ðSi; PjÞg u1ðSiÞ boundary. It is clear that the fundamental solutions Ulk(Pi, Pj) and XM XN T (P , P ) are singular as the collocation point P approaches to the lk i j i þ fc U ðS ; P Þ source point P (as r ? 0). When P and P are coincident, it is as- j 21 i j j; i j i¼1 j¼1 sumed in this paper that U (P , P )=T (P , P )=0 if l is not equal lk i i lk i i 2 to k, i.e., s þ djU22ðSi; PjÞg u2ðSiÞ : ð16Þ U12ðPi; PiÞ¼U21ðPi; PiÞ¼0; ð12Þ The minimization of this function is done using the nonlinear T12ðPi; PiÞ¼T21ðPi; PiÞ¼0: least-squares package LMDIF from MINPACK (Garbow et al., In order to verify the above assumption, we here illustrate, for 1980), which minimizes the sum of squares of m non-linear func- example, U12(Pi, Pi) is equal to zero, and the other equations of Eq. tions in n variables using a modified version of the Levenberg–Mar- (12) can be verified in a similar way. From the physical point of N N quard algorithm. Thus, the influence coefficients fcjgj¼1 and fdjgj¼1 view, the fundamental solution U12(Pi, Pi) denotes the displacement can be evaluated. produced at the point Pi by a concentrated unit body force at the Then, replacing the sample points Si with the boundary source same point, in which the first subscript denotes the x1 direction of points Pi and inserting the influence coefficients cj and dj deter- the displacement whereas the second one the x2 direction of the mined from Eq. (16), the SBM interpolation formulations can be unit force. It is reasonable that if we exert a concentrated force at written8 as: a point in a given direction, the displacement at this point along > PN > s the vertical direction will be very close, if not equals, to zero. There- <> u1ðPiÞ¼ fcjU11ðPi; PjÞþdjU12ðPi; PjÞg þ ciu11ðPiÞ; fore, we neglect this tiny displacement and assume U (P , P )=0. j¼1;i–j 12 i i Pi 2 Cu; > PN With the help of the above assumption, the SBM interpolation > s : u2ðPiÞ¼ fcjU21ðPi; PjÞþdjU22ðPi; PjÞg þ diu22ðPiÞ; formulations for 2D elasticity problems can be expressed as: j¼1;i–j 8 > PN ð17Þ > <> u1ðPiÞ¼ fajU11ðPi; PjÞþbjU12ðPi; PjÞg þ aiu11ðPiÞ; j¼1;i–j 8 Pi 2 Cu; > PN > PN > ts ðP Þ¼ fc T ðP ; P Þþd T ðP ; P Þg þ c t ðP Þ; > < 1 i j 11 i j j 12 i j i 11 i : u2ðPiÞ¼ fajU21ðPi; PjÞþbjU22ðPi; PjÞg þ biu22ðPiÞ; j¼1;i–j j¼1;i–j Pi 2 Ct; > PN > s ð13Þ : t2ðPiÞ¼ fcjT21ðPi; PjÞþdjT22ðPi; PjÞg þ dit22ðPiÞ; j¼1;i–j 8 > PN ð18Þ > s <> t1ðPiÞ¼ fajT11ðPi; PjÞþbjT12ðPi; PjÞg þ ait11ðPiÞ; where ti ðPÞ, i = 1, 2, denote the boundary tractions, which corre- j¼1;i–j spond to the differentiation of the sample solution usðPÞ. It is noted > Pi 2 Ct; i > PN that only the origin intensity factors are unknown in the above :> t2ðPiÞ¼ fajT21ðPi; PjÞþbjT22ðPi; PjÞg þ bit22ðPiÞ; j¼1;i – j equations; thus, the origin intensity factors can be calculated with the8 following formulations: ð14Þ > PN > us ðP Þ fc U ðP ;P Þþd U ðP ;P Þg > 1 i j 11 i j j 12 i j where u (P )=U (P , P ), u (P )=U (P , P ), t (P )=T (P , P ), and < j¼1;i–j 11 i 11 i i 22 i 22 i i 11 i 11 i i u11ðPiÞ¼ ; ci t22(Pi)=T22(Pi, Pi) are defined as the origin intensity factors, i.e., ð19Þ > PN > s the diagonal elements of the SBM interpolation matrix. The funda- > u ðP Þ fc U21ðP ;P Þþd U22ðP ;P Þg > 2 i j i j j i j : j¼1;i–j mental assumption of the SBM is the existence of the origin inten- u22ðPiÞ¼ ; di sity factor upon the singularity of the coincident source-collocation 8 nodes for mathematically well-posed problems. Our experimental > PN > ts ðP Þ fc T ðP ;P Þþd T ðP ;P Þg findings are that the origin intensity factor does exist, and it has a > 1 i j 11 i j j 12 i j <> j¼1;i–j t11ðPiÞ¼ c ; finite value depending on the distribution of discrete boundary i ð20Þ nodes and their respective boundary conditions. This study intro- > PN > ts ðP Þ fc T ðP ;P Þþd T ðP ;P Þg > 2 i j 21 i j j 22 i j duces a simple inverse interpolation technique (IIT) to determine : j¼1;i–j t22ðPiÞ¼ ; the above-mentioned origin intensity factors. di 2552 Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556

It is important to note that the origin intensity factors only de- 1 10 SBM ( Sam p le so lut io n 1 ) pend on the distribution of the source points and the fundamental SBM ( Sam p le so lut io n 2 ) solution of the governing equation. Theoretically, the origin inten- Indirect BEM sity factors remain unchanged with different sample solutions MFS (d=1) with the IIT. Therefore, by employing a novel inverse interpolation 0 MFS (d=2) technique, the diagonal elements of the SBM interpolation matrix 10 have been obtained in an indirect way. Once the origin intensity factors are solved, the displacements and stresses at any point inside the domain can be obtained using Eqs. (10) and (11). -1 10 Relative error (%) 3. Numerical results and discussion

To verify the method developed above, three benchmark plane strain problems are studied in which the SBM solutions are com- -2 10 pared with the MFS, BEM, and FEM. The relative errors (%) in this 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 study is deﬁned by x2

jInumerical Iexactj Relative errors ð%Þ¼ 100; ð21Þ Fig. 2. Relative errors (%) of the normal stress r11 at the interior points on the line jIexactj x1 = 1 generated using the proposed SBM, indirect BEM and MFS. where Inumerical represents a numerical result, and Iexact is an analytical solution of the considered problem. values of x2, respectively. The SBM results are compared with those obtained using the BEM and MFS. The BEM solutions are obtained 3.1. Test problem 1: an infinite elastic plate using the indirect boundary integral equations with linear discontinuous boundary elements. It is worth noting that a linear bound- As shown in Fig. 1, an infinite plate under external pressure p is ary element is adequate in this case because it can represent a examined. We assume the length of the plate in the x3 direction is straight line boundary exactly. The number of boundary nodes is large so that this problem can be simplified to a plane strain prob- the same in the SBM, indirect BEM, and MFS. lem. The length L of the plate in the x1 direction and the thickness h As shown in Figs. 2 and 3, the SBM results obtained using the in the x2 direction are constant in this study (L = 16, h = 1). Because two sample solutions are consistent with the exact solutions and the dimension of the length L = 16 is large compared to the dimen- agree pretty well with those of the indirect BEM. We also observe sion of the thickness h = 1, the analytical solution of the whole sys- that the numerical accuracy via the sample solution 1 is slightly tem may be well approximated by considering the whole system as better than that via the sample solution 2. This may be because an infinite elastic plate. Along the boundary x2 = 0, displacement the stress distribution using the first sample solution agrees much components in both the x1 and x2 directions are constrained. better with that of the problem of interest. In addition, the solution In the SBM model, the horizontal boundaries (upper and lower) accuracy of the MFS with the fictitious boundary d = 2 dramatically are divided each into 100 boundary nodes, whereas the vertical deteriorates compared with d = 1, where d is the distance between ones (left and right) into 20 boundary nodes each. Therefore the to- the fictitious boundary and the real boundary. This clearly illus- tal number of boundary nodes is 240. The SBM results are obtained trates the essential role of the fictitious boundary in the determina- using a total of M = 300 sample points uniformly distributed inside tion of the MFS solution accuracy. the domain (7, 7) (0.1, 0.9). To show how the different sample Table 1 presents the computed values of the displacement u2 solutions affect the solution of the SBM, we consider the following at the interior points on the line x1 = 1 generated using the SBM, two cases: indirect BEM, and MFS. As shown in Table 1, both SBM schemes are efficient and perform as well as the indirect BEM. In addi- 3.1.1. Sample solution 1 tion, the computational accuracy of the MFS with d = 1 is higher l The displacement components are given by u1 ¼2G x1, u2 ¼ 1l 2G x2. The corresponding stress components are r11 =0, r22 =1, r12 = r21 = 0, where r11 ¼ rx , r22 ¼ rx , and r12 ¼ sx x . 1 SBM (Sample solution 1) 1 2 1 2 10 SBM (Sample solution 2) 3.1.2. Sample solution 2 Indirect BEM 1l 0 MFS (d=1) The displacement components are given by u1 ¼ 2G x1, 10 l MFS (d=2) u2 ¼2G x2. The corresponding stress components are r11 =1, r22 =0,r12 = r21 =0. -1 Figs. 2 and 3 present the relative error (%) curves of normal 10 stresses r11 and r22 at interior points on the line x1 = 1 for various

-2 10

x2 Relative error (%) p=1 -3 10

h=1 o -4 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x1 L=16m x2

Fig. 1. An inﬁnite plate under constant pressure p (material constants are Fig. 3. Relative errors (%) of the normal stress r22 at the interior points on the line

G = 333333.333 Pa and l = 0.2). x1 = 1 generated using the proposed SBM, indirect BEM and MFS. Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556 2553

Table 1

Relative errors (%) of the displacement u2 at the interior points on the line x1 =1.

x2 Exact Relative errors (%) of SBM Relative errors (%) of BEM and MFS Sample solution 1 Sample solution 2 Indirect BEM MFS (d = 1) MFS (d =2)

0.1 0.1125000E06 0.3369461E01 0.5216847E01 0.1556278E+00 0.1659474E01 0.6739915E+01 0.2 0.2250000E06 0.1167961E01 0.2549632E01 0.1280838E01 0.1394836E01 0.2536028E+01 0.3 0.3375000E06 0.8759023E02 0.7256924E02 0.1061109E01 0.1148725E01 0.1436550E+01 0.4 0.4500000E06 0.6180081E02 0.2354986E01 0.8514924E02 0.9249308E02 0.9773566E+00 0.5 0.5625000E06 0.3753956E02 0.8947545E02 0.6542793E02 0.7090522E02 0.6811443E+00 0.6 0.6750000E06 0.1501412E02 0.5632958E02 0.4712522E02 0.5099739E02 0.4922928E+00 0.7 0.7875000E06 0.5585874E03 0.2368974E02 0.3040524E02 0.3257290E02 0.4017476E+00 0.8 0.9000000E06 0.1542915E02 0.6523152E02 0.2460161E02 0.1613342E02 0.3306049E+00 0.9 0.1012500E05 0.5504158E02 0.8965237E02 0.1920823E01 0.1831339E03 0.3241373E+00

0 p=1 10

-1 10

-2 x2 10

-3 10 o x1 -4

Relative error (%) 10 SBM (Sample solution 1) SBM (Sample solution 2) -5 Indirect BEM 10 MFS (d=0.1) MFS (d=0.2) -6 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ≤≤θπ Fig. 4. An infinite plate with a circular hole under uniform tensile forces at infinity 0/2 (Material constants are G =1 105 Pa and l = 0.3). Fig. 5. Relative errors (%) of the tangential stress rh at the interior points along the circle r = 1 generated using the proposed SBM, indirect BEM and MFS. than that when d = 2. This shows that the appropriate or optimal placement of the fictitious boundary in the MFS can result in a engineering applications, the quadratic element is ideal because very accurate solution. However, unfortunately, the determining it can approximate the geometry of curvilinear boundaries with of optimal fictitious boundary is a difficult task; therefore, the sufficient accuracy. Again, the number of boundary nodes is the fictitious boundary severely limits the applicability of the MFS same in the SBM, indirect BEM, and MFS. to real-world applications. As shown in Fig. 5, the SBM results obtained using the two sample solutions agree pretty well with those of the indirect BEM. 3.2. Test problem 2: an infinite elastic plate with a circular hole Although the solution accuracy of the indirect BEM is slightly better than that of the SBM, the SBM is inherently free of integration As shown in Fig. 4, the body under consideration is an infinite and is more computationally efficient and easier to program than plate with a circular hole under uniform tensile forces p =1 at the BEM. In addition, the MFS solution accuracy with the fictitious infinity. The radius of the circle is r = 1. In the SBM model, there boundary d = 0.1 dramatically deteriorates compared with d = 0.2, are a total of N = 80 boundary nodes that are uniformly distributed where d is the distance between the fictitious boundary and the along the circular boundary. The SBM results are obtained using a real boundary. This shows again that the appropriate or optimal total of M = 160 sample points uniformly distributed on a circle placement of the fictitious boundary in the MFS can result in a very with radius r = 2 and center at the origin. To show how the differ- accurate solution. However, unfortunately, it remains an open is- ent sample solutions affect the solution of the SBM, we consider sue to find the appropriate fictitious boundary for complex the following two cases: problems.

3.2.1. Sample solution 1 3.3. Test problem 3: a cylinder coating on a shaft 1 The displacement components are given by ur ¼ 2Gr, uh = 0. The 2 2 corresponding stress components are rr = 1/r , rh =1/r , srh =0, Finally, the proposed method is used to solve two related sam- where (r, h) denotes the polar coordinate. ple problems of a shaft with a cylinder coating. Plane strain is as- sumed, and the shaft is considered to be rigid when compared to

3.2.2. Sample solution 2 the coating. The shaft and coating have outer radii ra and rb respec- 12l The displacement components are given by ur ¼ 2Gr , uh =0. tively, and the two cases to be considered are: Fig. 5 displays the relative error (%) curves of the tangential stress rh at interior points distributed on the circle r = 1 with (a) the coating is of uniform thickness as shown in Fig. 6(a); 0 6 h 6 p/2. The results obtained using the BEM and MFS are also (b) the coating is of nonuniform thickness; both the shaft and given for the purpose of comparison. The BEM solutions are ob- coating proﬁles remain circle, but their centers are misa- tained using the indirect boundary integral equations with discon- ligned, producing some normalized eccentricity d ¼ rc , rbra tinuous quadratic elements. It is worth noting that in most where rc is the center offset, as shown in Fig. 6(b). 2554 Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556

p p

ra A rb B o rc

a b o x1

Fig. 6. Cross section of a shaft with coatings of uniform and nonuniform thickness (material constants are G = 807692.3 Pa and l = 0.3).

-1 -1 10 10

-3 10 -3 10

-5 10 -5 10

-7 10 SBM Relative error (%)

Indirect BEM Relative error (%) -7 SBM 10 MFS (d=1) BEM -9 10 MFS (d=2) MFS (d=1) MFS (d=2) -9 10 9 8 7 6 5 10 10 9 8 7 6 5 x1 x1

Fig. 7. Relative errors (%) of the radial stress rr at the interior points on the line Fig. 8. Relative errors (%) of the tangential stress rh at the interior points on the line x2 = 0 generated using the proposed SBM, indirect BEM and MFS. x2 = 0 generated using the proposed SBM, indirect BEM and MFS.

In both cases, the coated system is loaded by a pressure p =1 Table 2 presents the computed values of the displacement ur at which is uniformly distributed around the circumference. In addi- the interior points on the line x2 = 0 by using the SBM, MFS and tion, the boundary conditions for both cases, considering the rigid indirect BEM. The proposed method shows the accurate solutions shaft assumption, are ur = uh = 0 for all nodes at the shaft/coating compared with the exact solutions, MFS, and indirect BEM. As interface, where (r, h) denotes the polar coordinate. The SBM dis- the number of boundary nodes increases, Fig. 9 presents the con- cretization uses N = 200 boundary nodes for both sample problems, vergence curves of the stresses rr, rh, and the displacement ur at regardless of the thickness of the structure. point A(7.5, 0) using the proposed method. The numerical solutions converge smoothly as the number of boundary nodes increases, (a) First case: uniform thickness coating indicating that the proposed method works well for this coating system. For the uniform thickness case, the shaft and coating have outer radii ra = 5 and rb = 10, respectively. The SBM results are obtained (b) Second case: nonuniform thickness coating. using a total of M = 300 sample points uniformly distributed on a circle with radius r = 8 and center at the origin. The sample solu- The second important case for analysis is the nonuniform thick- 1 25 tions are given by ur ¼ G ð r 0:4rÞ and uh =0. ness case, which does not demonstrate the symmetry of the ﬁrst Figs. 7 and 8 present the computed values of the radial stress rr sample problem. While no analytical solution exists for d – 0 case, and tangential stress rh at interior points on the line x2 = 0 for var- the asymptotic behavior of the solution as d ? 0 can be checked to ious values x1, respectively. The results obtained using the BEM verify the formulation. In this case, shaft radius is held constant at and MFS are also given for the purpose of comparison. Again the ra = 1 and coating outer radius is also held constant at rb = 1.1. BEM solutions are obtained using the indirect boundary integral However, the eccentricity has been systematically varied over the equations with quadratic discontinuous boundary elements. As range 0.1 6 d 6 0.9. The SBM results are obtained using a total of shown in Figs. 7 and 8, the relative errors (%) of the MFS with M = 300 sample points uniformly distributed on a circle with ra- d = 1 are larger than those with d = 2, which illustrates again that dius r =(rc + 2.1)/2 and center at the origin. The sample solutions 12l the ﬁctitious boundary is vital to the accuracy of the MFS solution. are given by ur ¼ 2Gr and uh = 0, where (r, h) denotes the polar In addition, the results calculated using the proposed SBM are con- coordinate. sistent with the exact solutions and compared well with those of Fig. 10 shows the normalized radial stress rr at boundary node the indirect BEM and MFS. B, and the results obtained using the indirect BEM and FEM are also Y. Gu et al. / International Journal of Solids and Structures 48 (2011) 2549–2556 2555

Table 2

Computed results of the displacement ur at the interior points on the line x2 =0.

Radius r Exact SBM Indirect BEM MFS (d = 1) MFS (d =2)

10 0.4389610E05 0.4389610E05 0.4389582E05 0.4388985E05 0.4389610E05 9 0.3641751E05 0.3641752E05 0.3641764E05 0.3641248E05 0.3641751E05 8 0.2853247E05 0.2853246E05 0.2853112E05 0.2852853E05 0.2853247E05 7 0.2006679E05 0.2006679E05 0.2006847E05 0.2006402E05 0.2006679E05 6 0.1073016E05 0.1073016E05 0.1072959E05 0.1072868E05 0.1073016E05

tence of an origin intensity factor to isolate the singularities of fun- -3 x 10 damental solutions. Then, an inverse interpolation technique is 20 designed to evaluate this origin intensity factor without using Radial stress any element or integration concept. The proposed method circum- Tangential stress vents the major drawback in the MFS while retaining its merits 15 Displacement being mathematically simple, easy-to-program, and truly integration-free. Three benchmark numerical examples are well studied 10 to demonstrate the feasibility and accuracy of the proposed method. The numerical solutions obtained using the proposed method are consistent with the exact solutions and agree pretty well with 5 those of the MFS, indirect BEM, and FEM. The remaining issue with the SBM is as follows. The origin Relative error (%) intensity factor has to be determined indirectly by applying a 0 known sample solution of the problem of interest inside the physical domain, which requires additional computational costs and may cause some problems in the solution of large-scales problems. -5 100 150 200 250 300 350 400 This issue is under intense study, and the results will be reported in Number of elements a subsequent paper.

Fig. 9. Convergence curves of calculated stresses and displacement at point A. Acknowledgements

1.07 The work described in this paper was supported by the National SBM Basic Research Program of China (973 Project No. 2010CB832702), 1.06 Indirect BEM the Opening Fund of the State Key Laboratory of Structural Analysis FEM for Industrial Equipment (Project No. GZ0902), the R&D Special 1.05 Fund for Public Welfare Industry (Hydrodynamics, Project No. 201101014). The corresponding author (Wen Chen) is grateful to 1.04 the research fellowship of the Alexander von Humboldt Founda- tion for experienced researchers. 1.03

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