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Mathematical Problems in Engineering
Volume 2018, Article ID 6932164, 8 pages
https://doi.org/10.1155/2018/6932164
Research Article

Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Correspondence should be addressed to Pan Cheng; moc.anis@ssap_gnehc

Received 25 December 2017; Accepted 24 April 2018; Published 31 May 2018

Academic Editor: Nunzio Salerno

Copyright © 2018 Pan Cheng and Ling Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.