A meshless singular boundary method for three‐dimensional elasticity problems

Abstract

SUMMARYThis study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integrationfree attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed.

Electroelastic analysis of two‐dimensional ultrathin layered piezoelectric films by an advanced boundary element method

“…Accurate and efficient electroelastic analysis of piezoelectric materials, however, a formidable task because they are usually made in the forms of ultrathin films (1–10 μm) 4 involving large aspect ratios and regions with relatively high curvature. To maintain reasonable element aspect ratio, the size and computational effort that would be entailed by a naive application of the standard numerical software can be very large or even prohibitive 5‐9 …”

“…To maintain reasonable element aspect ratio, the size and computational effort that would be entailed by a naive application of the standard numerical software can be very large or even prohibitive. [5][6][7][8][9] In the last two decades, there have been increasing efforts in applying the boundary element method (BEM) for modeling thin-structural problems. 6,[10][11][12][13] As has been demonstrated in Liu, 14 Gu,7,15 and Sladek, 6 the BEM can model ultrathin structures very accurately and efficiently, regardless of the thickness of the structures, as long as the nearly singular integrals 6,[16][17][18][19][20] existing in the boundary integral equations (BIEs) are calculated accurately.…”

Error bounds of singular boundary method for potential problems

“…Conversely, by implementing the OIF, the discretization error on the boundary can be corrected, as well as the singular integration at origin of the fundamental solution is avoided. In the literature, the numerical investigations show that the SBM has rapid and stable convergence for stokes flow , elasticity , wave problems , and so forth. However, the mathematical convergence analysis of the SBM is missing, and the available analysis is largely based on numerical experiments.…”

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

“…Similar to Kansa's approach, the unknowns of the approximate solution are determined by the collocation at the inner points of the solution domain and by the collocation of the BCs. More detailed information on the recent developments of the RBF‐based techniques such as the singular boundary method can be found in Chen et al, Racz and Bui, and Gu et al A comparison of 2 techniques applying RBFs—the method based on the direct collocation (Kanza's method) with the method that combines the MFS and the dual reciprocity method—was presented in Li et al The meshless methods for nonlinear PDEs have also been developed by many authors. Zhu et al used a meshless method to solve linear and nonlinear boundary value problems, based on the local boundary integral equation method and the moving least squares approximation.…”