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.
1. Introduction
This paper will describe the isolated elastic equations on a bounded planar region in the plane with nonlinear boundary value conditions:where is a connected domain with a smooth closed curve , the stress tensors are , is the unit outward normal vector on , the tractor vector is assumed given with , and continuous on , and , where is a nonlinear function corresponding to the displaced . Following vector computational rules, the repeated subscripts imply the summation from 1 to 2.
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 [10–12] (BIEs) with the Cauchy and logarithmic singularities by the variational method.where is the Kronecker delta for , and the kernels:
The kernels are Kelvin’s fundamental solutions [11]. The Poisson ratio is with , means the derivative to , and is the distance of and . The parts of (2) are the Cauchy singularity and the parts are the logarithmic singularity.
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 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 is analyzed and the Richardson extrapolation is applied to achieve the accuracy orders . In Section 5 some results are shown regarding approximations for problems by the numerical example.
2. Mechanical Quadrature Methods
We firstly define the notation of the integral operators on boundary as follows to simplify the equations:So (2) can be simplified as the following operator equations:where is an identity operator.
Suppose be the set of times differentiable periodic functions in which the periods of all functions are . A regular parameter mapping is introduced, so the smooth closed curve satisfying with , .
The singularity of the kernels , will be analyzed as before the application of discrete methods. Since it is stated in the page 497 of the paper [21] thatso the Cauchy singularity of comes from the part . Moreover, the logarithmic singularity of comes from the component 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 as follows: with , , and with , and with , .
The Cauchy singular operator will also be divided into three parts on as follows: with ,, and with ,, and with .
We can find that , , are smooth operators, is a logarithmic singular operator, and is a Cauchy singular operator.
Then the equivalent equation of (8) will be shown aswhere , with , , and
Suppose the boundary will be divided into equal parts, so the mesh width will be and the nodes will be , . The smooth integral operators with the period can easily obtain high accuracy Nyström’s approximations [22]. For example, the approximation operator of can be approximated as follows:and the error will beNyström’s approximation of and of can be approximated similarly.
In order to approximate the logarithmic singular operator , 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 bewhere is the derivative of Riemann zeta function.
For the Cauchy singular operator , Lü and Huang [21] and Sidi [22] have proposed the operator to approximate it as shown in Lemma 1.
Lemma 1. The Nyström approximate operator can be defined aswhere when , and so the error estimate will be
Thus (16) can be rewritten as follows:where , , , and are the approximate matrixes corresponding to the operators , , , and , respectively.
3. Asymptotically Compact Convergence
The Cauchy approximate operator combined with the identity operator will be studied for the reversibility. A property about the operator will be presented [10].
Lemma 2. is invertible and is uniformly bounded.
From Lemma 2, we can find that the eigenvalues of both and do not include . So (16) and (26) can be rewritten as follows: find satisfyingwhere , , , and the space , .
Equation (26) can be rewritten as follows:with , , and .
Theorem 3. The approximate operator sequences and are asymptotically compact sequences and convergent to and in , respectively; i.e.,where means the asymptotically compact convergence.
Proof. An asymptotically compact sequence will be proved for operator , firstly.
We can find that the kernels of and are continuous functions, so the collectively compact convergence [10, 20] is true: Since we have constructed the continuous approximate function of of , the approximate operator is the asymptotically compact convergent to ; i.e., in , as .
Then we have and in . It can be concluded that for any bounded sequence a convergent subsequence must be found in . Without loss of generality, we assume , as . According to the asymptotically compact convergence and the errors quadrature formulae [17, 21], we obtain where is the norm of . We proved that is an asymptotically compact sequence.
Furthermore, we will show that will be pointwise convergent to , as . We can see that , so we obtainFrom Lemma 2, is uniformly bounded, and the errors of the approximate operators of will be substituted into the following equations, then So we proved that is pointwise convergent to , as .
Since the series are asymptotically compact sequences and pointwise convergent to , it has been proved that . Similar proof can be done for , so the proof of Theorem 3 is completed.
In order to obtain the asymptotically compact convergence of the nonlinear equations, some assumptions should be given [14, 23] firstly.
Assumptions 4. (1): is measurable and differentiable for , and is continuous for .
(2): is Borel measurable and satisfies the inequality .
The asymptotically compact convergence will be drawn according to the assumption about the function :
4. 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.
4.1. Asymptotic Expansions
Theorem 5. Considering the asymptotic property and , there exists a function independent of , such that
Proof. Let (16) subtract (26) at , thenthat is,Consider the components in the equations and similar results can be obtained as . Moreover, using the mean value theorem of differentials, we obtain where .
Note that and are continuous operators, A is a logarithmic operator, and C is a Cauchy operator, so the approximate errors will be , , , and Equation (37) can be simplified aswhere . Since exists and is uniformly bounded, (41) is rewritten aswhere and .
An auxiliary equation will be introduced with the solution and the corresponding approximate equations will beSubstituting (44) into (42) yields the equations:Noticing that and using the results of (42) yieldReplace by and consider is an asymptotic compact operator,we complete the proof.
4.2. 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 can be obtained by computing some coarse grids and fine grids on in parallel. The EAs are described as follows.
Step 1. Take the mesh widths and to obtain the solution of (26) in parallel, where are the solutions on corresponding to and , respectively.
Step 2. Use the values at coarse grids and fine grids to calculate the approximate values at and the error is .
Step 3. So we can construct a posteriori error estimate: which is useful for constructing self-adaptive algorithms.
5. Numerical Example
Example 1. We first introduce some notations for : is the error of the displacement; is the error ratio; is the error after one-step EAs; and is a posteriori error estimate.
Suppose is an isotropic elliptical body with the axis in the plane domain shown in Figure 1. The parameter formulae for the boundary will be described as , , . And the nonlinear boundary values are set as , , .
We firstly calculate the numerical solutions on the boundary following (26). Table 1 lists the approximate values of at points and . Table 2 lists the approximate values of at points and .
From Tables 1-2, we can numerically see which shows that the convergent orders are three for the approximation solutions and will be improved to five orders after EAs.
The normal derivative on can be obtained when the displacement on is substituted into the boundary condition of (1). So following (4), the displacement in can be calculated. Table 3 shows the approximate values of the displacement at an inner point in .
From Table 3, we can numerically see that the -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.
6. 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This project is supported by Natural Science Foundation of Chongqing (CSTC2013JCYJA00017) and supported by the Educational Council Foundation of Chongqing (KJ1500517 and KJ1600512).