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Singular Boundary Method for Solving Plane Strain Elastostatic Problems

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Singular Boundary Method for Solving Plane Strain Elastostatic Problems International Journal of Solids and Structures 48 (2011) 2549–2556 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr 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 equa- tion 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 À 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 À 2l @x1 @x2 1 À 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 À 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 À l 1 2 1 À 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 meth- od, called the singular boundary method (SBM), was proposed by 1 1 U ðP; QÞ¼ fð3 À 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 À 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 À x1Q Þ þðx2P À x2Q Þ is the distance between the @r xiP ÀxiQ 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.
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