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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.

This paper will describe the isolated elastic equations on a bounded planar region

The problems are studied in many applications, e.g., the circular ring problems and the instance problems [

Equations (

The kernels are Kelvin’s fundamental solutions [

The nonlinear integral equations are obtained after the boundary conditions are substituted into (

This kind of nonlinear integral equation has been discussed in many papers. Rodriguez [

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

Richardson extrapolation algorithms (EAs) are pretty effective parallel algorithms which are based on asymptotic expansion about errors. The solutions, which are solved on coarse grid and fine grid, are used to construct high accuracy solutions. The method is known to be of good stability and optimal computational complexity. Cheng et al. [

The reminder of this paper is stated as follows: In Section

We firstly define the notation of the integral operators on boundary

Suppose

The singularity of the kernels

The logarithmic singular operator

The Cauchy singular operator

We can find that

Then the equivalent equation of (

Suppose the boundary

In order to approximate the logarithmic singular operator

For the Cauchy singular operator

The Nyström approximate operator

Thus (

The Cauchy approximate operator combined with the identity operator will be studied for the reversibility. A property about the operator

From Lemma

Equation (

The approximate operator sequences

An asymptotically compact sequence will be proved for operator

We can find that the kernels of

Then we have

Furthermore, we will show that

From Lemma

Since the series

In order to obtain the asymptotically compact convergence of the nonlinear equations, some assumptions should be given [

(1):

(2):

The asymptotically compact convergence will be drawn according to the assumption about the function

In this section, we derive the asymptotic expansion of errors for the solution and construct the extrapolation algorithm to obtain higher accuracy order solutions.

Considering the asymptotic property and

Let (

Consider the components

Note that

Equation (

An auxiliary equation will be introduced with the solution

An asymptotic expansion about the errors in (

Take the mesh widths

Use the values at coarse grids and fine grids to calculate the approximate values at

So we can construct a posteriori error estimate:

We first introduce some notations for

Suppose

The shape of the elliptical body.

We firstly calculate the numerical solutions

The errors and errors ratio of

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The errors and errors ratio of

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From Tables

The normal derivative

The errors and errors ratio of

| 8 | 16 | 32 | 64 | 128 | 256 |
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From Table

The mechanical quadrature methods provide a way to approximate the solutions of the nonlinear boundary value problems with high accuracy. The following conclusions can be drawn.

These methods show that accuracy orders can be three and can be applied with the EAs to obtain higher accuracy. Furthermore, the larger scale of the problems, the more precision of the methods.

To obtain the entry of discrete matrixes is very simple and straightforward, without any singular integrals.

The problems, about Neumann boundary condition or nonlinear boundary condition in polygon elasticity, will be researched in our following work.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This project is supported by Natural Science Foundation of Chongqing (CSTC2013JCYJA00017) and supported by the Educational Council Foundation of Chongqing (KJ1500517 and KJ1600512).