A novel boundary element approach for solving the 2D elasticity problems

Keywords: BEM Elastic plane problem Exact geometrical element IBIE Nonsingular IBIE The presentation is mainly devoted to the research on the regularized BEM formulations efﬁcient and gen- employed whole as tw egories: the local and the global strategies. The local strategies are employed to calculate the singular integrals d They usually include, but are not limited to, analytical and semi-analytical one [5–8], new Gaussian quadrature [9] formation method [10–12], the local regularization method [8,13,14], the interval subdivision [15,16] and ﬁnite-part i [17,18], etc. The analytical algorithm needs enormous works of deduction and is generally considered more difﬁcult for http://dx.doi.org/10.1016/j.amc.2014.01.071 0096-3003/� 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author at: Institute of Applied Mathematics, Shandong University of Technology, Zibo 255049, Shandong Province, PR China. E-mail address: [email protected] (Y.M. Zhang). Applied Mathematics and Computation 232 (2014) 568–580 Contents lists available at ScienceDirect Applied Mathematics and Computation are evaluated. Regularization, or singularity removal before numerical computation, can make the BEM an erally well-conditioned numerical solution procedure [1–4]. For a better understanding of the proposed algorithm in this paper, a brief review of the other schemes dle the singularity of the integrals is necessary and these proposed methods can be summarized on the to han- o cat- irectly. , trans- ntegral 1. Introduction The boundary element method (BEM) has become an efﬁcient numerical method in solving practical engineering prob- lems. It is now not only available for many relevant linear problems of applied mechanics (e.g. elastostatics, elastodynamics, plate bending, etc.) but also for a number of nonlinear problems, such as elastoplasticity, etc. The main advantage of this method is the reduction of the dimension of boundary value problems. On the other hand, it is known as well that the stan- dard BEM formulations include singular integrals. Hence, the integrations should be performed very carefully and the overall accuracy of the BEM is largely dependent on the precision with which the various integrals, especially the singular integrals, with indirect unknowns for two-dimensional (2D) elasticity problems. A novel regulariza- tion technique, in which regularized forms of the gradient equations without involving the direct calculation of CPV and HFP integrals are derived and shown to be independent of dis- placement equations, is pursued in this paper. After that, a numerically systematic scheme with generality is established by adopting quadratic Lagrange’s elements. Moreover, con- sidering the special geometric domain, such as circular or elliptic arcs, a new boundary geometric approximate technique, named as exact elements, is presented, and thus by the utilization of these elements the error of the results will arise mainly from the approx- imation of boundary quantities. Numerical examples show that a better precision and high computational efﬁciency can be achieved. � 2014 Elsevier Inc. All rights reserved. A novel boundary element approach for solving the 2D elasticity problems Y.M. Zhang a,b,⇑, Z.Y. Liu c, X.W. Gao b, V. Sladek d, J. Sladek d a Institute of Applied Mathematics, Shandong University of Technology, Zibo 255049, Shandong Province, PR China b State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China c School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, PR China d Slovak Academy of Sciences, Institute of Construction and Architecture, 84503 Bratislava, Slovakia a r t i c l e i n f o a b s t r a c t journal homepage: www.elsevier .com/ locate /amc merel ani et the ex mainl soluti ginal r derive ities in ulariz emphasis on hypersingular integrals and divergent series. Ref. [2] has summarized the regularization procedure for singular to this conclusion. A n Comp of the the nu dratic used i eleme ples w Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 569 be achieved by the presentation, and especially the numerical results of boundary quantities match the exact solutions very well. The presentation mainly deals with the calculation of boundary stress and does not consider crack problem (see Fig. 9 and Table 7). 2. Basic theorems In this paper, we always assume that X is a bounded domain in R2; Xc its open complement, and C ¼ @X their common boundary. tðxÞ and nðxÞ (or t and n) are the unit tangent and outward normal vectors of C to the domain X at the point x respectively. The fundamental solutions of displacement and traction (plane strain) can be expressed as u�ikðx; yÞ ¼ k0 2l ½�k1 ln rdik þ r;ir;k� ð1Þ p�ikðx; yÞ ¼ � k0 r @r @n ðk2dik þ 2r;ir;kÞ � k2ðr;ink � r;kniÞ � � ð2Þ where k0 ¼ 1=ð4pð1� vÞÞ; k1 ¼ 3� 4v; k2 ¼ 1� 2v . Lemma 1. Let C be a piecewise smooth curve. Then for the fundamental solutions (1) and (2), there holdsZ C @ ln r @nx dCx ¼ 2p; y 2 X 0; y 2 Xc � ; Z C p�ikðx; yÞdCx ¼ �dik; y 2 X 0; y 2 Xc � Theorem 1. Let C be a piecewise smooth curve, then there are Z C @ðr2;kÞ @nx dCx ¼ �2pþ 2 RC @ ln r@xk nkðxÞdCx; y 2 X 2 R C @ ln r @xk nkðxÞdCx; y 2 Xc ( ðk ¼ 1;2Þ ð3Þ Z C @ ln r @tx dCx ¼ 0; Z C @ðr2;kÞ @tx dCx ¼ 0; Z C @ðr;1r;2Þ @tx dCx ¼ 0; 8y 2 X [Xc ð4Þ Z C @ðr;1r;2Þ @nx dCx ¼ 2 Z C n1ðxÞ @ ln r @x2 dCx ¼ 2 Z C n2ðxÞ @ ln r @x1 dCx; 8y 2 X [Xc ð5Þ with excluding the summation rule in Eq. (3). merical implementation, without loss of generality, a numerically systematic scheme is established by adopting qua- Lagrange’s elements. Besides, in engineering applications, the structures with circle or ellipse boundaries are widely n order to avoid stress concentration. Based on these geometrical characteristics, we proposed an exactly geometrical nt so that the error only arises from the approximation of boundary quantities by interpolation. Some numerical exam- ill be applied to validate the current scheme. It is shown that a better precision and high computational efﬁciency can placement equations and, as such, can be collocated at the same locations as the displacement equations. This provides addi- tional and concurrently useable equations for various purposes. Secondly, the proposed IBEM need not deal with or regularize the HFP integrals, so it is simpler and more efﬁcient. Thirdly, in the calculation of displacements or stresses near boundary, the boundary integrals may show only weakly near or strong near singularity without hyper near singularity. In ovel regularization technique is pursued in the present paper, in which the regularized indirect BIE is established. ared with previous direct BEMs, the proposed indirect BEM (IBEM) has many advantages. Firstly, a unique feature gradient BIEs expressed by density functions is that, unlike the traction equations, they are independent of the dis- integral equation. Ref. [26] has discussed the non-singular nature of the BIEs in BEM and reviewed the research effort leading y utilized to treat weakly singularities. Among these methods, the local regularization technique proposed by Guiggi- al. is extensively used to handle various orders of singularities. The main drawback of this technique is that it requires pansion of every quantity involved in the integrand as Taylor’s series about the local distance. The global strategies are y adopted to calculate the singular integrals indirectly, such as the null ﬁeld method [19–21], method of fundamental on, the virtue BEM [22,23] and the simple solution method [24–28], etc. Sladek et al. have obtained amount of the ori- esults in this ﬁeld. In Refs. [29,30], a class of stress (or displacement derivative) nonsingular integral representation is d. Another fully regularized representation by singularity separation was proposed by Sladek et al. for hyper-singular- Refs. [2,4,31]. The above two regularized formulations are theoretically fully regularized and are nonsingular. In reg- ed BEM ﬁeld, readers can refer to [1,2,26]. Ref. [1] has made a detail review for dual boundary element methods with curved elements. The coordinate transformation method, which is invalid to the strongly and hyper-singular integrals, is X^ ¼ X For where ify th C C d ¼ infx2Cjy � xj. If hd 6 K1 (with constant K1), then there holds any e > 0; the difference between the left and right hand side of Eq. (9) is Th Using jD j 6 K K R 1 dC , and further t ¼ jx� x^j denote the bowstring corresponding to the arc xx^_ . Let x^ be a smooth point By No Z � � 570 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 jD2j � Cc x1 � y1 ðx1 � y1Þ2 þ ðx2 � y2Þ2 � x1 � x^1 ðx1 � x^1Þ2 þ ðx2 � x^2Þ2 �� jwðxÞ � wðx^ÞjdCx noting that d has been ﬁxed and x^ does not belong to Cc , therefore by taking jy � x^j sufﬁciently small it is obvious that jD2j � e=2. h 0 t choosing sufﬁciently small d one has jD1j � e=2: w we estimate D2� � 1 2 3 Cd jx�x^j1�a x on C. Hence it is obvious that the ratio of the length of arc xx^ _ to that of bowstring t is bounded. i.e., jdCxj 6 K4dt (with con- stant K4), therefore jD1j 6 K2K3K4 Z d 1 1�a dt � K5da K5 ¼ 1 a K2K3K4 is a constant � � Besides, noting wðxÞ 2 C0;aðCÞ; i.e., jwðxÞ � wðx^Þj 6 K3jx� x^ja (for constants a;K3 with 0 < a � 1), it yields D ¼ Z C x1 � y1 jx� yj2 � x1 � x^1 jx� x^j2 " # wðxÞ � wðx^Þ½ �dCx e integral D is divided into two parts: D1 over Cd ¼ fx 2 Cjjx� x^j � dg and D2 over Cc ¼ C� Cd. Firstly we estimate D1. jyk � x^kj 6 K1jy � xj; we have x1 � y1 jx� yj2 � x1 � x^1 jx� x^j2 �� K2jx� x^j ðwith constant K2Þ Proof. For the case that k ¼ 1;we will give the strict deduction here .The other can be performed in an almost same way. For lim y!x^ Z C xk � yk jx� yj2 ½wðxÞ � wðx^Þ�dCx ¼ Z C xk � x^k jx� x^j2 ½wðxÞ � wðx^Þ�dCx ðk ¼ 1;2Þ ð9Þ Theorem 2. Let wðxÞ 2 C0;aðCÞ; and x^ be a smooth point on C: For x^ 2 C; taking the limit y ! x^, suppose h ¼ jy � x^j and Z @u�ikðx; yÞ @nx dCx ¼ � dikl þ k0 l Z niðxÞ @ ln r @xk dCx; y 2 X ð8Þ Z C @u�ikðx; yÞ @nx dCx ¼ k0l Z C nkðxÞ @ ln r @xi dCx ¼ k0l Z C niðxÞ @ ln r @xk dCx; y 2 Xc ð7Þ Z C @u�ikðx; yÞ @tx dCx ¼ 0; 8y 2 X [Xc ð6Þ Ce 1 2R C @r2 ;2 @nx dCx ¼ � R C @r2 ;1 @nx dCx, since r2;1 þ r2;2 ¼ 1. The proof of Eqs. (4) and (5) can be performed in an almost same way. h Corollary 1. Let C be a piecewise smooth curve, then we have ði; k ¼ 1;2Þ P ¼ 2ðx1�y1Þ2ðx2�y2Þr4 ; Q ¼ � 2ðx1�y1Þ 3 r4 ; applying Green formula in domain X^;we have R C[Ce Pdx1 þ Qdx2 ¼ 0: It is easy to ver- at R Pdx þ Qdx ¼ 2p; then we can derive Eq. (3). As for k ¼ 2, we can deduce the conclusion from Z C @r2;1 @nx dCx ¼ 2 Z C @ ln r @x1 n1ðxÞdCx þ Z C Pdx1 þ Qdx2 �X \ Be;Ce ¼ @Be the proof of Eq. (3), as k ¼ 1, there is Proof. For conciseness, we merely give the proof of Eq. (3) for the case y 2 X: As for y 2 Xc; it can be performed in a more simpler way. Let BeðyÞ be brieﬂy written as Be; denote a circle with radius e and center at point y including X: Setting Corollary 2. Under the conditions as in Theorem 2, there holdsZ 3 Z 3 k l Corol Rema 3. Reg In t where /kðxÞðk ¼ 1;2Þ are the indirect boundary unknowns. ^ C Z � Z � Z �� By Sub Z Z � Z �� k Z @ ln r Z @ ln r � Z @ ln r �� If t Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 571 uiðyÞ ¼ Z C /kðxÞu�ikðx; yÞdCx; y 2 C ði; k ¼ 1;2Þ ð13Þ diately obtain the regularized BIEs with replacing y by x. Thus, the non-singular BIEs on Xc can be expressed as the following simultaneous equations þ 0 l nðx^Þ C ½nkðxÞ � nkðx^Þ� @xi dCx þ nkðx^Þ C ½tiðxÞ � tiðx^Þ� @tx dCxþnkðx^Þ C ½niðxÞ � niðx^Þ� @nx dCx lðx^Þ 2 C0;aðCÞ;nlðx^Þ 2 C0;aðCÞ ðl ¼ 1;2Þ; and /ðxÞ 2 C0;aðCÞ; therefore, as y ! x^; according to Corollary 4, we can imme- ryuiðyÞ ¼ C ½/kðxÞ � /kðx^Þ�ryu�ikðx; yÞdCx � /kðx^Þ C ½tðxÞ � tðx^Þ� @uikðx; yÞ @tx dCx þ C ½nðxÞ � nðx^Þ� @uikðx; yÞ @nx dCx � /kðx^Þ C ½tðxÞ � tðx^Þ� @uikðx; yÞ @tx dCx þ C ½nðxÞ � nðx^Þ� @uikðx; yÞ @nx dCx þ nðx^Þ C @uikðx; yÞ @nx dCx ð11Þ using Corollary 1, it is easy to deduceZ C @u�ikðx; yÞ @nx dCx ¼ k0l Z C ½nkðxÞ � nkðx^Þ� @ ln r @xi dCx þ nkðx^Þ Z C ½tiðxÞ � tiðx^Þ� @ ln r @tx dCx þ nkðx^Þ Z C ½niðxÞ � niðx^Þ� @ ln r @nx dCx � � ð12Þ stituting (12) into (11), we obtain For any given point x 2 C; we have ryuiðyÞ ¼ Z ½/kðxÞ � /kðx^Þ�ryu�ikðx; yÞdCx rigid body displacement and the body forces, the displacement formulations on Xc can be expressed as [22,33] uiðyÞ ¼ Z C /kðxÞu�ikðx; yÞdCx; y 2 Xc ði ¼ 1;2Þ ð10Þ ularized indirect boundary integral equations his section, we will establish the regularized BIEs with unknowns on Xc . The one on X is similar. Without regard to the limit y ! x�, with y ¼ x� hnðxÞ, x� ¼ limh!þ0ðx� hnðxÞÞ. lim y!x^ C ½wðxÞ � wðx^Þ� @uikðx; yÞ @nx dCx ¼ C ½wðxÞ � wðx^Þ� @uikðx; xÞ @nx dCx rk 1. The conclusions of Theorem 2, and Corollaries 2–4 include the especial limit process: for any x^ 2 C, taking the ^ ^ ^ ^ ^ ^ Z � Z � ^ y!x^ C @xl C @xl lim y!x^ Z C ½wðxÞ � wðx^Þ� @u � ikðx; yÞ @tx dCx ¼ Z C ½wðxÞ � wðx^Þ� @u � ikðx; x^Þ @tx dCx lim Z ½wðxÞ � wðx^Þ� @u � ikðx; yÞdCx ¼ Z ½wðxÞ � wðx^Þ� @u � ikðx; x^Þ dCx lim y!x^� C ½wðxÞ � wðx^Þ� jx� yj4 dCx ¼ C ½wðxÞ � wðx^Þ� jx� x^j4 dCx lary 4. Under the conditions as in Theorem 2, then we have (k; l ¼ 1;2) Corollary 3. Under the conditions as in Theorem 2, then there are (k; l ¼ 1;2; and; k–l) Z ðxk � y Þ2ðxl � y Þ Z ðxk � x^kÞðxl � x^lÞ2 lim y!x^ C ½wðxÞ � wðx^Þ� ðxk � ykÞ jx� yj4 dCx ¼ C ½wðxÞ � wðx^Þ� ðxk � x^kÞ jx� x^j4 dCx ðk ¼ 1;2Þ Z Z� Z �� Similarly, the non-singular BIEs on X can be expressed as 1 1 � Esp Rema where (18) s 4. Nu consid 4.1. Q We quadr where of sou 572 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 atic element. Let wpðnÞ ðp ¼ 1;2;3Þ denote the shape function of quadratic Lagrange interpolation. i.e. wpðnÞ ¼ ðn� n qÞðn� nrÞ ðnp � nqÞðnp � nrÞ ðp; q; r ¼ 1;2;3; q–r; p–q; rÞ n is dimensionless coordinate and nkðk ¼ p; q; rÞ represent the coordinate of node k. Let n and g denote the coordinate rce point x and ﬁeld point y; respectively, then there are xk ¼ P3 p¼1w pðnÞxpk ; yk ¼ P3 p¼1w pðgÞxpk ; and wise parabolic curve, while the distribution of the boundary quantity on each element is approximated by a discontinuous er here because of involving only normal integrals. uadratic element will give a general numerical integration strategy here. The boundary geometric is modeled by a continuous piece- In this section, we shall perform the numerical integration strategy for Eqs. (13)–(18). The key problem is how to evaluate the integral when the ﬁeld point is located at one of the nodes of the element under integration, while the other need not merical implementation piðyÞ ¼ /iðyÞ þ C ½/kðxÞ � /kðyÞ�p^ikðx; yÞdCx � k0/kðyÞ C r � ½nðxÞ � nðyÞ� � ðk2dik þ 2r;ir;kÞ�k2 r ½nkðxÞ � nkðyÞ� þ k2 r^;kr ½niðxÞ � niðyÞ� � dCx; y 2 C ð18Þ rk 2. If /ðxÞ 2 C0;a1 ðCÞ;nðxÞ 2 C0;a2 ðCÞ; tðxÞ 2 C0;a3 ðCÞ, i.e. j/ðxÞ � /ðyÞj 6 M1jx� yja1 ; jnðxÞ � nðyÞj 6 M2jx� yja2 ; jtðxÞ � tðyÞj 6 M3jx� yja3 a1;a2;a3;M1;M2;M3 are constants, and 0 < a1;a2;a3 6 1, then the singularities of integrals in Eqs. (14), (15), (17) and hould have been removed. Z � Z rxr� r^;i þ k0 l nðyÞ Z C ½nkðxÞ � nkðyÞ� @ ln r @xi dCx þ nkðyÞ Z C ½tiðxÞ � tiðyÞ� @ ln r @tx dCx � þnkðyÞ Z C ½niðxÞ � niðyÞ� @ ln r @nx dCx �� ; y 2 C ði; k ¼ 1;2Þ ð17Þ ecially ryuiðyÞ ¼ /kðyÞnðyÞl dik � 2ð1� vÞnkðyÞniðyÞ þ C½/kðxÞ � /kðyÞ�ryuikðx; yÞdCx � /kðyÞ Z C ½tðxÞ � tðyÞ� @u � ikðx; yÞ @tx dCx þ Z C ½nðxÞ � nðyÞ� @u � ikðx; yÞ @nx dCx � � � Z uiðyÞ ¼ Z C /kðxÞu�ikðy; xÞdCx; y 2 C ði; k ¼ 1;2Þ ð16Þ ð14Þ Especially, the traction BIEs can be expressed as piðyÞ ¼ Z C ½/kðxÞ � /kðyÞ�p^�ikðy; xÞdCx � k0/kðyÞ Z C rxr r � ½nðxÞ � nðyÞ� � � ðk2dik þ 2r;ir;kÞ�k2 r^;ir ½nkðxÞ � nkðyÞ� þ k2 r^;kr ½niðxÞ � niðyÞ� � dCx; y 2 C ði; k ¼ 1;2Þ ð15Þ where p^�ikðy; xÞ ¼ � k0 r @r @ny ðk2dik þ 2r^;ir^;kÞ þ k2ðr^;in^k � r^;kn^iÞ � � ; with r^;l ¼ yl � xlr ; n^l ¼ nlðyÞ ðl ¼ 1;2Þ: þk0 l nðyÞ C ½nkðxÞ�nkðyÞ�@ lnr @xi dCxþnkðyÞ C ½tiðxÞ� tiðyÞ�@ lnr @tx dCxþnkðyÞ C ½niðxÞ�niðyÞ�@ lnr @nx dCx ; y2C ryuiðyÞ¼ Z C ½/kðxÞ�/kðyÞ�ryu�ikðx;yÞdCx�/kðyÞ Z C ½tðxÞ� tðyÞ�@u � ikðx;yÞ @tx dCxþ Z C ½nðxÞ�nðyÞ�@u � ikðx;yÞ @nx dCx � and / beside with J Th where Acc since The it (17), ( In ﬁrst tw eleme Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 573 this section, four examples of 2D elastostatics with curved boundaries are given to test the proposed method. In the o examples, the boundary geometry is depicted by the exact elements, while in the last two ones by the quadratic nts. Meanwhile, the unknown boundary quantities is approximated by quadratic discontinuous elements. 5. Numerical examples ems ½� � �� cannot lead to singular integral. So the singularities of the integrals on the right side of Eqs. (14), (15) and Eqs. 18) are eliminated. r ¼ 2 sin 2 � � a2 sin 2 þ b cos2 2 all these terms are proportional to r�1 and ag � an�� ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 ag þ an 2 ag þ anr @yj ¼ j sin½ðag � anÞ=2�j ½� � ��; ðj ¼ 1;2Þ ð19Þ ording to the fundamental solutions, it is easy to deduce that @u�ðx; yÞ 1 t2ðyÞ� t2ðxÞ¼n1ðyÞ�n1ðxÞ¼bh2�h1JðnÞ sin ag�an 2 sin anþag 2 þh2�h1 2JðgÞ cosag a2 cos n g2 ðsinanþsinagÞ�b sin n g2 ðcosanþcosagÞ JðnÞþ JðgÞ x1 ¼ a cosan; x2 ¼ b sinan; with an ¼ 2 h1 þ 2 h2; �1 � n � 1 a; b are the major and minor semi-axis respectively, therefore t1ðyÞ� t1ðxÞ¼�ðn2ðyÞ�n2ðxÞÞ¼ ah2�h1JðnÞ sin ag�an 2 �cosanþag 2 þh2�h1 2JðgÞ sinag a2 cosanþag2 ðsinanþsinagÞ�b 2 sinanþag2 ðcosanþcosagÞ JðnÞþ JðgÞ ( ) a þa 2 a þa( ) 4.2. Exact element In engineering applications, the structures with circle or ellipse boundaries are widely used in order to avoid stress con- centrations around sharp corners. Based on these geometrical characteristics, we proposed a Natural Boundary Approximate Method, i.e. the elliptic arc elements, which can exactly depict these boundaries. For elliptic domain, its boundary geometric can be expressed as ð1� nÞ ð1þ nÞ k k JðnÞ kþ1 JðgÞ½JðgÞ þ JðnÞ� kþ1 kþ1 en, the singularities of the integrals in Eqs. (14), (15), (17), and (18) should have been eliminated. n ðxÞ � n ðyÞ ¼ ð�1Þ kþ1q a � ðgþ nÞaiai þ 2aibi ðga þ b Þ � � nkðxÞ ¼ JðnÞ dn ¼ ð�1Þ k kJðnÞ ; nkðyÞ ¼ JðgÞ dn �n¼g ¼ ð�1Þ k kJðgÞ ðnÞ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dxi dn dxi dn q , and ð�Þ3 ¼ ð�Þ1. Thus, tkðxÞ � tkðyÞ ¼ qJðnÞ ak � ðgþ nÞaiai þ 2aibi JðgÞ½JðgÞ þ JðnÞ� ðgak þ bkÞ � � k JðnÞ dn JðnÞ k JðgÞ dn n¼g JðgÞ ð�1Þkþ1 dxkþ1 kþ1 na þb ð�1Þkþ1 dxkþ1 �� kþ1 ga þb s t ðxÞ ¼ 1 dxk ¼ nakþbk ; t ðyÞ ¼ 1 dxk �� ¼ gakþbk ; q ¼ n� g;U ðn;gÞ ¼ ðnp � nqÞðnp � nrÞ ðp–q; r; q–rÞ ðxÞ � /ðyÞ ¼ qP3p¼1Upðn;gÞ/pk , where q and Up are deﬁned as p nþ g� nq � nr dxk dn ¼ X3 p¼1 dwpðnÞ dn xpk ¼ X3 p¼1 2n� nq � nr ðnp � nqÞðnp � nrÞ x p k ¼ nak þ bk where xpkðk ¼ 1;2; p ¼ 1;2;3Þ is Cartesian coordinate, and ak ¼ P3 p¼1 2xp k npqnpr , bk ¼ � P3 p¼1 nqþnr npqnpr x p k , with n pq ¼ np � nq and (p; q; r ¼ 1;2;3; q–r; p–q; r). Then xk � yk ¼ X3 p¼1 wpðnÞxpk � X3 p¼1 wpðgÞxpk ¼ q X3 p¼1 Upðn;gÞxpk Example 1. A thick-walled cylinder subjected to uniform radial internal pressures p ¼ 10; whose inner and outer radii are 5 and 10, respectively, is considered in this example, as shown in Fig. 1. The material constants are the elastic shear modulus l ¼ 807692:3 N=cm2 and the Poisson’s ratio v ¼ 0:2307692. There are 25 exact elements divided along the whole boundaries, 15 equally spaced elements over inner boundary, and 10 equally spaced elements over outer boundary. Quadratic discontinuous interpolation, i.e. discontinuous element, is adopted to approximate the boundary functions. The numerical results of the radial and tangent stresses rr ;rh of interior points along x1 axis are listed in Table 1, from which it can be seen that compared with the exact solution, the present results are satisfactory despite using small compu- tational element number. In addition, with the increase of the discretized boundary elements, the tangent stresses rh and its relative errors (RE (%)) related to the exact solutions are shown in Table 2, from which we can observe that the convergence speeds of the computed tangent stresses rh are quite fast. Example 2. The second example considers an inﬁnite plate with an elliptical hole subjected to the uniform tensile forces p ¼ 1 at inﬁnity, as shown in Fig. 2. The semi axes of the elliptical hole are a ¼ 10 and b ¼ 8, and the material constants are the same as in Example 1. The elliptical boundary is discretized into 20 exact elements and the boundary quantities are approximated by the dis- continuous quadratic elements. The results of tangent stresses rh and its percentage errors with respect to the exact solutions along the quarter boundary ð0 p=2Þ are listed in Table 3. It can be seen from Table 3 that the accuracy of the results are high and stable with the largest relative error less than 0.08%. Moreover, Fig. 3 depicts that with the increase of the discretized boundary elements, the maximum relative errors (%) of tangent stresses change, from which we can see that the convergence rates of the present method are fast. Example 3. As shown in Fig. 4, this example is concern with an inﬁnite plate with two equivalent circular holes subjected to the uniform tensile forces p ¼ 1 at inﬁnity. The radius of each circle is R ¼ 10 and the distance between the centers of two circles is d ¼ 40. In this example, the elastic shear modulus is l ¼ 807692:3 N=cm2 and the Poisson’s ratio is v ¼ 0:3: There are 40 uniform quadratic boundary elements divided along each circular boundary. y 574 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 x Fig. 1. Thick cylinder subjected to the uniform radial pressure p at inside and outside boundary. Table 1 Radial and tangent stresses rr , rh of some points (boundary and internal) along x1 axis. Points(r) Present method Exact solution Numerical results RE (%) rr rh rr rh rr rh 5.0 0.1665561E+02 0.6633357E�01 0.1666667E+02 6.0 �0.5925318E+01 0.1258897E+02 0.1025165E�01 0.2875743E�01 �0.5925926E+01 0.1259259E+02 6.5 �0.4555143E+01 0.1121990E+02 0.2349029E�01 0.2655909E�01 �0.4556213E+01 0.1122288E+02 7 �0.3468341E+01 0.1013341E+02 0.3017112E�01 0.2604759E�01 �0.3469388E+01 0.1013605E+02 7.5 �0.2591667E+01 0.9256861E+01 0.3569934E�01 0.2590609E�01 �0.2592593E+01 0.9259259E+01 8 �0.1874196E+01 0.8539450E+01 0.4285566E�01 0.2595263E�01 �0.1875000E+01 0.8541667E+01 8.5 �0.1279567E+01 0.7944865E+01 0.5542756E�01 0.2615715E�01 �0.1280277E+01 0.7946943E+01 9 �0.7812418E+00 0.7446580E+01 0.8328165E�01 0.2657546E�01 �0.7818930E+00 0.7448560E+01 10 0.6664560E+01 0.3159914E�01 0.6666667E+01 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 575 Table 2 Tangent stresses rh and RE (%) under different element numbers. Points (r) Inside B:12E; outside B:16E Inside B:24E; outside B:32E Inside B:36E; outside B:48E r RE (%) r RE (%) r RE (%) When the forces act, respectively, in x1 direction, x2 direction, both x1 and x2 directions, as shown in Fig. 4, the results of the tangent stresses rh along the quarter boundary ðp p=2Þ from point C to point B are shown in Tables 4–6. From the description of these three tables, it is easy to see that the results of Tables 4 and 5 are somewhat inferior to Table 6, and among all of these boundary points, the accuracy of point C is the worst in both Tables 4 and 5. In order to show the h h h 5.0 0.1665980E+02 0.4119320E�01 0.1666561E+02 0.6362836E�02 0.1666633E+02 0.2002221E�02 6.0 0.1259038E+02 0.1757747E�01 0.1259223E+02 0.2896136E�02 0.1259248E+02 0.9223006E�03 6.5 0.1122101E+02 0.1666921E�01 0.1122256E+02 0.2838637E�02 0.1122278E+02 0.9132414E�03 7 0.1013440E+02 0.1632598E�01 0.1013577E+02 0.2809264E�02 0.1013596E+02 0.9061646E�03 7.5 0.9257762E+01 0.1616621E�01 0.9259001E+01 0.2784878E�02 0.9259176E+01 0.8993649E�03 8 0.8540288E+01 0.1613678E�01 0.8541431E+01 0.2762942E�02 0.8541590E+01 0.8927843E�03 8.5 0.7945652E+01 0.1625021E�01 0.7946725E+01 0.2747222E�02 0.7946873E+01 0.8866230E�03 9.0 0.7447326E+01 0.1656684E�01 0.7448355E+01 0.2752268E�02 0.7448494E+01 0.8827074E�03 10.0 0.6665312E+01 0.2032274E�01 0.6666448E+01 0.3275946E�02 0.6666597E+01 0.1041696E�02 p p Fig. 2. An inﬁnite plate with an elliptical hole subjected to the uniform tensile forces. Table 3 Tangent stresses rh along the quarter boundary ð0 p=2Þ . Boundary points Exact solution Numerical results RE (%) x1 x2 rh rh rh 0.9876883E+01 0.1251476E+01 0.3330269E+01 0.3330304E+01 �0.1033251E�02 0.9723699E+01 0.1867563E+01 0.3128217E+01 0.3129215E+01 �0.3190384E�01 0.8910065E+01 0.3631924E+01 0.2201365E+01 0.2201182E+01 0.8334001E�02 0.8526402E+01 0.4179989E+01 0.1835972E+01 0.1836008E+01 �0.1949664E�02 0.7071068E+01 0.5656854E+01 0.7560976E+00 0.7559353E+00 0.2146036E�01 0.6494480E+01 0.6083248E+01 0.4322026E+00 0.4318894E+00 0.7247015E�01 0.4539905E+01 0.7128052E+01 �0.3588374E+00 �0.3588906E+00 �0.1484376E�01 0.3826834E+01 0.7391036E+01 �0.5547604E+00 �0.5549375E+00 �0.3193255E�01 0.1564345E+01 0.7901507E+01 �0.9288950E+00 �0.9288917E+00 0.3486595E�03 0.7845910E+00 0.7975339E+01 �0.9822318E+00 �0.9822520E+00 �0.2053748E�02 576 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 0.006 0.008 0.01 0.012 m R E( % ) efﬁciency of the proposed algorithm, we will present the convergence curves of point C in Figs. 5 and 6. Exact solutions, ex- pressed as a series in Ref. [32], are also listed in these three tables on the purpose of a comparison. Nevertheless, the exact solutions are also approximate results, since we merely calculate the ﬁnite partial sums of the series. Figs. 5 and 6 both depict that the relative error of tangent stress at point C changes with the increase of element numbers. In Fig. 5 the plate is subjected to the tensile forces in x1 direction, while in Fig. 6 the one in x2 direction. Fig. 7 describes that under the act of the uniform tensile forces in x2 direction, the tangent stresses of point C and B change with the increase of the distance d between the centers of two circles, while Fig. 8 describes the case under the act of two uniform tensile forces in x1 and x2 directions. 20 40 60 80 100 0 0.002 0.004 Element numbers M ax im u Fig. 3. Maximum RE (%) of tangent stresses changes with the increase of element numbers. 2x 1x p p R A B C θ d Fig. 4. An inﬁnite plate with two equivalent circular holes subjected to the uniform tensile forces. Table 4 The inﬁnite plate subjected to the uniform tensile forces only in x1 direction. Boundary points (h) Exact solutions Present method RE (%) rh rh rh Tangent stresses rh along the quarter boundary ðp p=2Þ from point C to point B p �0.617435 �0.6094029 0.130088E+01 19p/20 �0.577658 �0.5714791 0.106965E+01 9p/10 �0.447469 �0.4449452 0.564017E+00 17p/20 �0.203849 �0.2037269 0.598973E�01 4p/5 0.166596 0.1659558 0.384283E+00 3p/4 0.648801 0.6472998 0.231381E+00 7p/10 1.195303 1.192243 0.256002E+00 3p/5 2.205903 2.202824 0.139580E+00 11p/20 2.540776 2.539467 0.515197E�01 p/2 2.703115 2.702791 0.119862E�01 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 577 Table 5 The inﬁnite plate subjected to the uniform tensile forces only in x2 direction. Boundary points (h) Exact solutions Present method RE (%) rh rh rh Tangent stresses rh along the quarter boundary ðp p=2Þ from point C to point B p 3.029126 3.020343 0.289952E+00 19p/20 2.954308 2.947316 0.236671E+00 9p/10 2.728855 2.725568 0.120453E+00 17p/20 2.355034 2.354619 0.176218E�01 4p/5 1.849875 1.850901 �0.554632E�01 Example 4. As an extension of Example 3, this example considers an inﬁnite plate with inﬁnite equivalent circular holes subjected to the uniform tensile forces p ¼ 1 at inﬁnity. This problem has broad application backgrounds, such as the analysis of the stress concentration of dams with close holes where water drains away, etc. Therefore, accurate evaluation of this example plays an important role in some engineering applications. With regard to the numerical calculation, it is impossible to consider all of the circular holes exactly, here; we will utilize 21 circles as an approximation. In this example, the radius of each circle is R ¼ 10 and the distance between the centers of two adjacent circles is d ¼ 40: The elastic shear modulus G and the Poisson’s ratio v are the same as in Example 3. Every circular boundary is discretized into 20 uniform quadratic boundary elements. The boundary quantities are approximated by the discontinuous quadratic elements. 3p/4 1.253478 1.255717 �0.178623E+00 7p/10 0.627160 0.6308217 �0.583854E+00 3p/5 �0.435529 �0.4321102 0.784976E+00 11p/20 �0.754149 �0.7523450 0.239210E+00 p/2 �0.882164 �0.8813614 0.909808E�01 Table 6 The inﬁnite plate subjected to the uniform tensile forces in both x1 and x2 directions. Boundary points (h) Exact solutions Present method RE (%) rh rh rh Tangent stresses rh along the quarter boundary ðp p=2Þ from point C to point B p 2.411691 2.410940 0.311399E�01 19p/20 2.376650 2.375837 0.342078E�01 9p/10 2.281386 2.280623 0.334446E�01 17p/20 2.151184 2.150892 0.135739E�01 4p/5 2.016471 2.016857 �0.191424E�01 3p/4 1.902278 1.903017 �0.388482E�01 7p/10 1.822463 1.823064 �0.329773E�01 3p/5 1.770374 1.770714 �0.192050E�01 11p/20 1.786626 1.787122 �0.277618E�01 p/2 1.820952 1.821430 �0.262500E�01 40 60 80 100 120 0.4 0.6 0.8 1 1.2 1.4 Element numbers R E of p oi nt C (% ) Fig. 5. RE (%) of point C with forces in x1 direction changes along with element numbers. 40 60 80 100 120 0.22 0.24 0.26 0.28 0.3 Element numbers R E of p oi nt C (% ) Fig. 6. RE (%) of point C with forces in x2 direction changes along with element numbers. 30 40 50 60 70 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 The distance d Ta ng en t s tre ss es point C point B Fig. 7. Changes of tangent stresses with the increase of distance d under the act of the uniform tensile forces in x2 direction. 30 40 50 60 70 1.6 1.8 2 2.2 2.4 2.6 2.8 3 The distance d Ta ng en t s tre ss es point C point B Fig. 8. Changes of tangent stresses with the increase of distance d under the act of two uniform tensile forces in x1 and x2 directions. 578 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 Y.M. Zhang et al. / Applied Mathematics and Computation 232 (2014) 568–580 579 The stress state is symmetric concerning both x-axis and the origin. We merely list the tangent stresses rh at the bound- ary points with angles h ¼ 00 and h ¼ 900 in Table 7. The results of Ref. [32] are also listed in Table 7 for the purpose of com- parison. What needs to be stressed here is that the results of Ref. [32], obtained in terms of a series, are also approximations, since a series cannot be computed inﬁnitely in a numerical sense. 6. Conclusions A novel regularized BEMmethod, in which the regularized BIEs with indirect unknowns have been derived, is proposed in this paper. Compared with existing approaches, the presented one is simple, ﬂexible and applicable. Firstly, it is not neces- sary to deal with the HFP so that it is simpler, and with better precision and high computational efﬁciency. Secondly, it is adaptable to different combination of the resulting BIEs according to the boundary conditions of considered problems. Thirdly, from Theorem 1 we can see that this paper may provide a technique applicable for the calculation of CPV integrals. Furthermore, for boundary value problems with circle or ellipse boundaries, we have also proposed an exactly geometrical element method with which the boundary can be modeled exactly, and thus the error of the results will arise mainly from the approximation of boundary quantities. Four examples of 2D elastostatics with curved boundaries have been presented to show the effectiveness of the proposed method with excellent results. The presented method is also general and can be ap- plied to other problems in BEM, which will be discussed later. Acknowledgement y xA B Fig. 9. An inﬁnite plate with inﬁnite equivalent circular holes subjected to the uniform tensile forces. Table 7 Tangent stresses rh at two boundary points. Boundary points Ref. [32] Present method Relative error (%) h = 0� (A) 3.27 0.3246794E+01 0.709664 h = 90� (B) �0.500 �0.4993839E+00 0.12322 The support from the Opening Fund of the State Key Laboratory of Structural Analysis for Industrial Equipment (GZ1017) and the National Natural Science Foundation of Shangdong Province of China (ZR2010AZ003) are gratefully acknowledged. 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Banerjee, R. Butterﬂeld, Boundary Element Methods in Engineering Science, McGraw-Hill Book Company(UK) Limited, 1981. A novel boundary element approach for solving the 2D elasticity problems 1 Introduction 2 Basic theorems 3 Regularized indirect boundary integral equations 4 Numerical implementation 4.1 Quadratic element 4.2 Exact element 5 Numerical examples 6 Conclusions Acknowledgement Appendix A Supplementary data References

A novel boundary element approach for solving the 2D elasticity problems

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