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.
IntroductionA 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 ...
Abstract.A new approximation scheme is presented for the mathematical model of convectiondiffusion and adsorption. The method is based on the relaxation method and the method of characteristics. We prove the convergence of the method and present some numerical experiments in 1D. The results can be applied to the model of contaminant transport in porous media with multi-site, equilibrium and non-equilibrium type of adsorption.Mathematics Subject Classification. 65M25, 65M12.
SUMMARYA new meshless method for solving transient elastodynamic boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation (MLS), is proposed in this paper. The LBIE with the MLS is applied to both transient and steady-state (Laplace transformed) elastodynamics. Applying the MLS approximation for spatially dependent terms in the ÿrst approach, the LBIEs are transformed into a system of ordinary di erential equations for nodal unknowns. This system of ordinary di erential equations is solved by the Houbolt ÿnite di erence scheme. In the second formulation, the time variable is eliminated by using the Laplace transformation. Unknown Laplace transforms of displacements and traction vectors are computed from the LBIEs with the MLS approximation. The time-dependent values are obtained by the Durbin inversion technique.
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