Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

Pan Cheng,Ling Zhang

doi:10.1155/2018/6932164

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.

Highlights

This paper will describe the isolated elastic equations on a bounded planar region Ω in the plane with nonlinear boundary value conditions: σij,j = 0, in Ω, (1)p = −g (x, u) + f (x), on Γ, i, j = 1, 2, where Ω ⊂ R2 is a connected domain with a smooth closed curve Γ, the stress tensors are σij, n = (n1, n2) is the unit outward normal vector on Γ, the tractor vector is assumed given p = (p1, p2)T with pi = σi1n1 + σi2n2, f(x) = (f1(x), f2(x))T and continuous on Γ, and g(x, u) = (g1(x, u), g2(x, u))T, where gi(x, u) is a nonlinear function corresponding to the displaced u
The nonconforming mixed finite element methods are established by Hu and Shi [4]
Talbot and Crampton [5] use a pseudo-spectral method to approach matrix eigenvalue problems which is transformed from the governing partial differential equations

Summary

Introduction

The mechanical quadrature methods (MQMs) are adopted by Cheng et al [10] to solve Steklov eigensolutions in elasticity to obtain high accuracy solutions. Abels et al [16] discussed the convergence in the systems of nonlinear boundary conditions with Holder space Since this is a nonlinear system including the logarithmic singularity and the Cauchy singularity, the difficulty is to obtain the discrete equations appropriately. The displacement vector and stress tensor in Ω can be calculated [17, 18] as follows after the discrete nonlinear equations are solved: ui (y) = ∫ hi∗j (y, x) pj (x) dsx. Huang and Luestablished extrapolation algorithms to obtain high accuracy solutions for solving Laplace equations on arcs [20] and plane problem in elasticity [21].

Mechanical Quadrature Methods

Asymptotically Compact Convergence

Asymptotic Expansions

Numerical Example

Conclusion