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Numerical integration of singularities in meshless implementation of local boundary integral equations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Sládek, Vladimír Sladek, Jan Atluri, Satya N. Keer, Roger Van |
| Abstract | The necessity of a special treatment of the numerical integration of the boundary integrals with singular kernels is revealed for meshless implementation of the local boundary integral equations in linear elasticity. Combining the direct limit approach for Cauchy principal value integrals with an optimal transformation of the integration variable, the singular integrands are recasted into smooth functions, which can be integrated by standard quadratures of the numerical integration with suf®cient accuracy. The proposed technique exhibits numerical stability in contrast to the direct integration by standard Gauss quadrature. 1 Introduction A lot of attention has been paid during the past decade to meshless implementations of both the formulations based originally on variational principles (weak form) (Belytschko et al., 1996) and/or boundary integral equations (Zhu et al., 1998; Mukherjee and Mukherjee, 1997). Recall that, by using an approach based on the BIE, the dimension of the integration region is reduced by one as compared with the dimension of the domain in which a boundary value problem is solved. Beside this most evident attractive property of the BIE formulations one could bring other advantages, such as good conditioning and high accuracy, resulting from the use of singular kernels. Sometimes the appearance of singular integrals has been considered as a handicap of the BIE formulations because of the relative complexity of accurate numerical integration. The problem of singularities has been resolved successfully in boundary element implementations of the BIE formulations (see e.g., Sladek and Sladek, 1998) when the boundary densities are approximated within ®nite size elements polynomially. Having known the boundary densities in a closed form, one can regularize the integrands involving singular kernels before utilizing quadratures for numerical integration (Tanaka et al., 1994). Nevertheless, the question of singularities is to be reconsidered in meshless implementations of the BIE. Now, instead of the de®nition of ®nite size elements by grouping nodal points on the boundary, the nodal points are spread throughout the whole domain including its boundary. When the coupling among the nodal points is satis®ed via the moving least-squares (MLS) approximation of physical ®elds (such as potential, displacements), the boundary densities are not known in a closed form any more, because the shape functions are evaluated only digitally at any required point. Thus, the peak-like factors in singular kernels cannot be smoothed by cancellation of divergent terms with vanishing ones in boundary densities before the numerical integration. The proposed method consists in the use of direct limit approach and utilization of an optimal transformation of the integration variable. The smoothed integrands can be integrated with suf®cient accuracy even by using standard quadratures of numerical integration. Section 2 summarizes the important equations of the local BIE formulation for solution of boundary value problems of linear elasticity. Section 3 deals with the derivation of nonsingular integrands in the meshless implementation of the LBIE endowed with the MLS approximation of displacements. Finally, in Sect. 4, the proposed technique of the numerical integration is tested in a numerical example with comparison of numerical results with those obtained by using standard Gauss quadrature without any elaboration of the integrand. 2 The local boundary integral equations for linear elasticity Let us consider a linear elastostatical problem on the domain X bounded by the boundary C. Then, the displacements are governed by the Navier equations (Balas et al., 1989) cijkluk;jl bi 0 1 in which bi is the body force and cijkl is the tensor of material parameters, which are reduced to two constants in the case of isotropic and homogeneous elastic continua with cijkl l 2v 1ÿ 2v dijdkl dikdjl dildjk ; 2 Computational Mechanics 25 (2000) 394±403 Ó Springer-Verlag 2000 |
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| Alternate Webpage(s) | https://www.depts.ttu.edu/coe/CARES/pdf/39.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Approximation algorithm Boundary element method Computation Computational mechanics Conditioning (Psychology) DNA Integration Elasticity (data store) Exhibits as Topic Gauss Gaussian quadrature Gauss–Hermite quadrature Gauss–Jacobi quadrature Hyperkeratosis, Epidermolytic KRT10 wt Allele Meshfree methods Moving least squares Navier–Stokes equations Nodal analysis Node - plant part Numerical analysis Numerical integration Numerical method Numerical stability Singular value decomposition Smoothing (statistical technique) Springer (tank) Variational principle Weak formulation density suf protein, zebrafish |
| Content Type | Text |
| Resource Type | Article |
