Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equationswill be converted into nonlinear boundary integral equations by the potential theory, inwhich 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.

The problems are studied in many applications, e.g., the circular ring problems and the instance problems [1,2], and the bending of prismatic bars [3].Some methods have been proposed for solving the elasticity.The nonconforming mixed finite element methods are established by Hu and Shi [4] to solve the elasticity with a linear boundary.Talbot and Crampton [5] use a pseudo-spectral method to approach matrix eigenvalue problems which is transformed from the governing partial differential equations.The least-square methods are introduced by Cai and Starke [6] for obtaining the solution of the elastic problems.Chen and Hong [7] solved the hyper singular integral equations applying the dual boundary element methods in elasticity.Kuo et al. [8] solve true and spurious eigensolutions of the circular cavity problems that used dual methods.Li and Nie [9] researched the stressed axis-symmetric rods problems by a high-order integration factor method.And the mechanical quadrature methods (MQMs) are adopted by Cheng et al. [10] to solve Steklov eigensolutions in elasticity to obtain high accuracy solutions.
Equations (1) are converted into the boundary integral equations [10][11][12] (BIEs) with the Cauchy and logarithmic singularities by the variational method.where   is the Kronecker delta for ,  = 1, 2, and the kernels: The kernels are Kelvin's fundamental solutions [11].The Poisson ratio is ] with ] = /[2( + )],  ⋅ means the derivative to   , and The nonlinear integral equations are obtained after the boundary conditions are substituted into (2): This kind of nonlinear integral equation has been discussed in many papers.Rodriguez [12] established sufficient conditions for the existence of solutions to nonlinear, discrete boundary value problems.Wavelet-Galerkin's methods constructed from Legendre wavelet functions are used by Maleknejad and Mesgarani [13] to approximate the solution.Atkinson and Chandler [14] presented product Simpson's rule methods and two-grid methods to solve the nonlinear equations.Pao [15] gave a systematic treatment of a class of nonlinear elliptic differential equations and their applications of these problems.Abels et al. [16] discussed the convergence in the systems of nonlinear boundary conditions with Hölder 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: where 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. [19] harnessed extrapolation algorithms to obtain high accuracy order for Steklov eigenvalue of Laplace equations.Huang and Lü established extrapolation algorithms to obtain high accuracy solutions for solving Laplace equations on arcs [20] and plane problem in elasticity [21].
The reminder of this paper is stated as follows: In Section 2 the MQMs are constructed for the nonlinear boundary integral equations and obtain a discrete nonlinear system.In Section 3 the asymptotically compact convergence is proved.In Section 4 an asymptotic expansion of the error about  ℎ is analyzed and the Richardson extrapolation is applied to achieve the accuracy orders (ℎ 5 ).In Section 5 some results are shown regarding approximations for problems by the numerical example.

Mechanical Quadrature Methods
We firstly define the notation of the integral operators on boundary Γ as follows to simplify the equations: So ( 2) can be simplified as the following operator equations: ( where  0 is an identity operator.Suppose  2 [0, 2] be the set of 2 times differentiable periodic functions in which the periods of all functions are The singularity of the kernels ℎ *  ,  *  will be analyzed as  →  before the application of discrete methods.Since it is stated in the page 497 of the paper [21] that so the Cauchy singularity of  *  comes from the part (   ⋅ −    ⋅ )/.Moreover, the logarithmic singularity of ℎ *  comes from the component log | − | and the other parts of the operators are smooth.So the operators  , and  , will be divided into several parts.
The logarithmic singular operator   will be divided into three parts on  2 [0, 2] as follows: with and with with The Cauchy singular operator   will also be divided into three parts on  2 [0, 2] as follows: with with and with   (, ) =  3 (/)(2 ⋅  ⋅ )/.We can find that  0 ,  , ,  , are smooth operators,  0 is a logarithmic singular operator, and  0 is a Cauchy singular operator.
In order to approximate the logarithmic singular operator  0 , the continuous approximation kernel   (, ) can be defined as follows: By Sidi's quadrature rules [3], the approximation operator can be constructed as follows: by which the error estimate will be where   () is the derivative of Riemann zeta function.
For the Cauchy singular operator  0 , Lü and Huang [21] and Sidi [22] have proposed the operator  ℎ 0 to approximate it as shown in Lemma 1.
Lemma 1.The Nyström approximate operator  ℎ 0 can be defined as where so the error estimate will be Thus ( 16) can be rewritten as follows: where  ℎ ,  ℎ ,  ℎ , and  ℎ are the approximate matrixes corresponding to the operators , , , and , respectively.

Asymptotically Compact Convergence
The Cauchy approximate operator combined with the identity operator will be studied for the reversibility.A property about the operator (1/2) +  ℎ will be presented [10].
In order to obtain the asymptotically compact convergence of the nonlinear equations, some assumptions should be given [14,23] firstly.

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

Extrapolation
Algorithms.An asymptotic expansion about the errors in (35) implies that the extrapolation algorithms [24] can be applied to the solution of (2) to improve the approximate order.The high accuracy order (ℎ 5 ) can be obtained by computing some coarse grids and fine grids on Γ in parallel.The EAs are described as follows.
From Table 3, we can numerically see that the ℎ 3 -Richardson extrapolation algorithms are effective in obtaining high accuracy approximate solutions in which just the values at coarse grids and fine grids are applied.Furthermore, the a posteriori error estimate can be used to reckon the errors of the numerical solutions.

Conclusion
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.

Figure 1 :
Figure 1: The shape of the elliptical body.