Mechanical Quadrature Methods and Extrapolation for Solving

Hindawi 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 Pan Cheng

and Ling Zhang

School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China Correspondence should be addressed to Pan Cheng; cheng [email protected] 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. 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 Ω in the plane with nonlinear boundary value conditions: 𝜎𝑖𝑗,𝑗 = 0,

in Ω,

𝑝 = −𝑔 (𝑥, 𝑢) + 𝑓 (𝑥) ,

on Γ, 𝑖, 𝑗 = 1, 2,

(1)

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 = 𝜎𝑖1 𝑛1 + 𝜎𝑖2 𝑛2 , 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. 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. 𝛿𝑖𝑗 2

𝑢𝑗 (𝑦) + ∫ 𝑘𝑖𝑗∗ (𝑦, 𝑥) 𝑢𝑗 (𝑥) 𝑑𝑠𝑥 Γ

=∫

Γ

ℎ𝑖𝑗∗

(𝑦, 𝑥) 𝑝𝑗 (𝑥) 𝑑𝑠𝑥 ,

(2) 𝑦 = (𝑦1 , 𝑦2 ) ∈ Γ,

2

Mathematical Problems in Engineering

where 𝛿𝑖𝑗 is the Kronecker delta for 𝑖, 𝑗 = 1, 2, and the kernels: ℎ𝑖𝑗∗ =

1 [− (3 − 4]) 𝛿𝑖𝑗 ln 𝑟 + 𝑟⋅𝑖 𝑟⋅𝑗 ] , 8𝜋𝜇 (1 − ])

𝑘𝑖𝑗∗ =

1 𝜕𝑟 [ ((1 − 2]) 𝛿𝑖𝑗 + 2𝑟⋅𝑖 𝑟⋅𝑗 ) 4𝜋 (1 − ]) 𝑟 𝜕𝑛

where ∗ = 𝑘𝑖𝑗𝑙

(3)

∗ = ℎ𝑖𝑗𝑙

[(1 − 2]) (𝑟.𝑗 𝛿𝑙𝑖 + 𝑟.𝑖 𝛿𝑙𝑗 − 𝑟.𝑙 𝛿𝑖𝑗 ) + 2𝑟.𝑖 𝑟.𝑗 𝑟.𝑙 ] [4𝜋 (1 − ]) 𝑟]

,

𝜇 𝜕𝑟 (2 [(1 − 2]) 𝑟.𝑙 𝛿𝑖𝑗 2 2𝜋 (1 − ]) 𝑟 𝜕𝑛

+ ] (𝑟.𝑗 𝛿𝑖𝑙 + 𝑟.𝑖 𝛿𝑗𝑙 ) − 4𝑟.𝑖 𝑟.𝑗 𝑟.𝑙 ] + 2] (𝑛𝑖 𝑟.𝑗 𝑟.𝑙

+ (1 − 2]) (𝑛𝑖 𝑟⋅𝑗 − 𝑛𝑗 𝑟⋅𝑖 )] .

(6)

+ 𝑛𝑗 𝑟.𝑖 𝑟.𝑙 ) + (1 − 2]) (2𝑛𝑙 𝑟.𝑗 𝑟.𝑖 + 𝑛𝑗 𝛿𝑖𝑙 + 𝑛𝑖 𝛿𝑗𝑙 ) − (1 The kernels are Kelvin’s fundamental solutions [11]. The Poisson ratio is ] with ] = 𝜆/[2(𝜆 + 𝜇)], 𝑟⋅𝑖 means the derivative to 𝑥𝑖 , and 𝑟 = √(𝑦1 − 𝑥1

)2

)2

+ (𝑦2 − 𝑥2 is ∗ distance of 𝑥 and 𝑦. The parts ∫Γ 𝑘𝑖𝑗 (𝑦, 𝑥)𝑢𝑗 (𝑥)𝑑𝑠𝑥 of (2) the Cauchy singularity and the parts ∫Γ ℎ𝑖𝑗∗ (𝑦, 𝑥)𝑢𝑗 (𝑥)𝑑𝑠𝑥

the are are

the logarithmic singularity. The nonlinear integral equations are obtained after the boundary conditions are substituted into (2): 𝛿𝑖𝑗 2

𝑢𝑗 (𝑦) + ∫ 𝑘𝑖𝑗∗ (𝑦, 𝑥) 𝑢𝑗 (𝑥) 𝑑𝑠𝑥 Γ

= ∫ ℎ𝑖𝑗∗ (𝑦, 𝑥) 𝑔𝑗 (𝑥, 𝑢) 𝑑𝑠𝑥

(4)

Γ

+ ∫ ℎ𝑖𝑗∗ (𝑦, 𝑥) 𝑓𝑗 (𝑥) 𝑑𝑠𝑥 . Γ

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 H¨older 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) = ∫ ℎ𝑖𝑗∗ (𝑦, 𝑥) 𝑝𝑗 (𝑥) 𝑑𝑠𝑥 Γ

− ∫ 𝑘𝑖𝑗∗ (𝑦, 𝑥) 𝑢𝑗 (𝑥) 𝑑𝑠𝑥 , Γ

𝜎𝑖𝑗 (𝑦) = ∫

Γ

−∫

Γ

(𝑦, 𝑥) 𝑝𝑙 (𝑥) 𝑑𝑠𝑥

∗ 𝑘𝑖𝑗𝑙

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 L¨u 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 𝑢ℎ is analyzed and the Richardson extrapolation is applied to achieve the accuracy orders 𝑂(ℎ5 ). 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 Γ as follows to simplify the equations: (𝐾𝑖𝑗 𝑢) (𝑦) = ∫ 𝑘𝑖𝑗∗ (𝑦, 𝑥) 𝑢 (𝑥) 𝑑𝑠𝑥 Γ

(𝐻𝑖𝑗 𝑢) (𝑦) = ∫

Γ

(𝑦, 𝑥) 𝑢𝑙 (𝑥) 𝑑𝑠𝑥 , ∀𝑦 ∈ Ω,

𝑦 ∈ Γ, 𝑖, 𝑗 = 1, 2, (7)

ℎ𝑖𝑗∗

(𝑦, 𝑥) 𝑢 (𝑥) 𝑑𝑠𝑥

𝑦 ∈ Γ, 𝑖, 𝑗 = 1, 2.

So (2) can be simplified as the following operator equations: 1 𝐾12 𝐼0 + 𝐾11 𝑢1 (2 )( ) 1 𝑢2 𝐾21 𝐼 + 𝐾22 2 0

∀𝑦 ∈ Ω, (5)

∗ ℎ𝑖𝑗𝑙

− 4]) 𝑛𝑙 𝛿𝑖𝑗 ) .

=(

𝐻11 𝐻12 𝐻21 𝐻22

)(

𝑔1 + 𝑓1 𝑔2 + 𝑓2

(8)

),

where 𝐼0 is an identity operator. Suppose 𝐶2𝑚 [0, 2𝜋] be the set of 2𝑚 times differentiable periodic functions in which the periods of all functions are

Mathematical Problems in Engineering

3

2𝜋. A regular parameter mapping [0, 2𝜋] → Γ is introduced, so the smooth closed curve 𝑥(𝑠) = (𝑥1 (𝑠), 𝑥2 (𝑠)) satisfying |𝑥󸀠 (𝑠)|2 = |𝑥1󸀠 (𝑠)|2 + |𝑥2󸀠 (𝑠)|2 > 0 with 𝑥𝑖 (𝑠) ∈ 𝐶2𝑚 [0, 2𝜋], 𝑖 = 1, 2. The singularity of the kernels ℎ𝑖𝑗∗ , 𝑘𝑖𝑗∗ will be analyzed as 𝑡 → 𝑠 before the application of discrete methods. Since it is stated in the page 497 of the paper [21] that 𝑛𝑖 𝑟⋅𝑗 − 𝑛𝑗 𝑟⋅𝑖 𝑟

= (−1)𝑖

1 + 𝑂 (𝑡 − 𝑠) , (𝑡 − 𝑠) + 𝑂 (𝑡 − 𝑠)

Then the equivalent equation of (8) will be shown as 1 ( 𝐼 + 𝐶 + 𝑀) 𝑢 = (𝐴 + 𝐵) 𝑔 (𝑢) + 𝑓, 2

where 𝑢(𝑡) = (𝑢1 (𝑥(𝑡)), 𝑢2 (𝑥(𝑡))), 𝑓(𝑡) = (𝑓1 , 𝑓2 )𝑇 with 𝑓𝑖 = 𝐻𝑖,𝑗 𝑓𝑗 (𝑥(𝑡)), 𝑔(𝑢(𝑡)) = 𝑔(𝑥(𝑡), 𝑢(𝑡)), and 𝐼=(

𝑖 ≠ 𝑗,

(9)

so the Cauchy singularity of 𝑘𝑖𝑗∗ comes from the part (𝑛𝑖 𝑟⋅𝑗 − 𝑛𝑗 𝑟⋅𝑖 )/𝑟. Moreover, the logarithmic singularity of ℎ𝑖𝑗∗ comes from the component log |𝑥 − 𝑦| and the other parts of the operators are smooth. So the operators 𝐻𝑖,𝑗 and 𝐾𝑖,𝑗 will be divided into several parts. The logarithmic singular operator 𝐻𝑖𝑗 will be divided into three parts on 𝐶2𝑚 [0, 2𝜋] as follows: 2𝜋 󵄨 󵄨 (𝐴 0 𝑢) (𝑡) = ∫ 𝑎0 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏, 0

(10)

with 𝑎0 (𝑡, 𝜏) = 𝑐0 ln |2𝑒−1/2 sin((𝑡 − 𝜏)/2)|, 𝑐0 = −(3 − 4])/[8𝜋𝜇(1 − ])], and 2𝜋 󵄨 󵄨 (𝐵0 𝑢) (𝑡) = ∫ 𝑏0 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏, 0

(11)

with 𝑏0 (𝑡, 𝜏) = 𝑐0 (ln |𝑥(𝑡) − 𝑥(𝜏)| − ln |2𝑒−1/2 sin((𝑡 − 𝜏)/2)|), and 2𝜋

󵄨 󵄨 (𝐵𝑖𝑗 𝑢) (𝑡) = ∫ 𝑏𝑖𝑗 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏, 0

0

0

(13)

(14)

with 𝑚𝑖𝑖 (𝑡, 𝜏) = 𝑐3 (𝜕𝑟/𝜕𝑛)[(1−2V)+2𝑟⋅𝑖 𝑟⋅𝑖 ]/𝑟, 𝑐3 = −1/[4𝜋(1− ])], and

0

𝐵=( 𝑀=(

0

), 𝐶0

−𝐶0 0

𝐴0 0 0 𝐴0

),

),

(17)

𝐵0 + 𝐵11

𝐵12

𝐵21

𝐵0 + 𝐵22

𝑀11 𝑀12 𝑀21 𝑀22

),

).

Suppose the boundary Γ will be divided into 2𝑛, (𝑛 ∈ 𝑁) equal parts, so the mesh width will be ℎ = 𝜋/𝑛 and the nodes will be 𝑡𝑗 = 𝜏𝑗 = 𝑗ℎ, (𝑗 = 0, 1, . . . , 2𝑛 − 1). The smooth integral operators with the period 2𝜋 can easily obtain high accuracy Nystr¨om’s approximations [22]. For example, the approximation operator 𝐵0ℎ of 𝐵0 can be approximated as follows: 2𝑛−1

(𝐵0ℎ 𝑢) (𝑡) = ℎ ∑ 𝑏0 (𝑡, 𝜏𝑗 ) 𝑢 (𝜏𝑗 ) ,

(18)

𝑗=0

(𝐵0 𝑢) (𝑡) − (𝐵0ℎ 𝑢) (𝑡) = 𝑂 (ℎ2𝑚 ) .

(19)

Nystr¨om’s approximation 𝐵𝑖𝑗ℎ of 𝐵𝑖𝑗 and 𝑀𝑖𝑗ℎ of 𝑀𝑖𝑗 can be approximated similarly. In order to approximate the logarithmic singular operator 𝐴 0 , the continuous approximation kernel 𝑎𝑝 (𝑡, 𝜏) can be defined as follows: for |𝑡 − 𝜏| ≥ ℎ, 𝑎 (𝑡, 𝜏) , { { 0 󵄨 󵄨󵄨 (20) 𝑎𝑝 (𝑡, 𝜏) = { {𝑐 ℎ ln 󵄨󵄨󵄨𝑒−1/2 ℎ 󵄨󵄨󵄨󵄨 , for |𝑡 − 𝜏| < ℎ. 0 󵄨 󵄨 󵄨󵄨 2𝜋 󵄨󵄨 { By Sidi’s quadrature rules [3], the approximation operator can be constructed as follows: 2𝑛−1 󵄨 󵄨 (𝐴ℎ0 𝑢) (𝑡) = ℎ ∑ 𝑎𝑝 (𝑡, 𝜏𝑗 ) 𝑢 (𝜏𝑗 ) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏𝑗 )󵄨󵄨󵄨󵄨 ,

(21)

𝑗=0

2𝜋

󵄨 󵄨 (𝑀𝑖𝑗 𝑢) (𝑡) = ∫ 𝑚𝑖𝑗 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏,

𝐴=(

0 𝐼0

and the error will be

with 𝑐0 (𝑡, 𝜏) = 𝑐2 (𝑛1 𝑟⋅2 − 𝑛2 𝑟⋅1 )/𝑟, 𝑐2 = −(1 − 2])/[4𝜋(1 − ])], and 2𝜋 󵄨 󵄨 (𝑀𝑖𝑖 𝑢) (𝑡) = ∫ 𝑚𝑖𝑖 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏, 𝑖 = 1, 2,

𝐶=(

𝐼0 0

(12)

with 𝑏𝑖𝑗 (𝑡, 𝜏) = 𝑐1 𝑟⋅𝑖 𝑟⋅𝑗 , 𝑐1 = 1/[8𝜋𝜇(1 − ])]. The Cauchy singular operator 𝐾𝑖𝑗 will also be divided into three parts on 𝐶2𝑚 [0, 2𝜋] as follows: 2𝜋 󵄨 󵄨 (𝐶0 𝑢) (𝑡) = ∫ 𝑐0 (𝑡, 𝜏) 𝑢 (𝜏) 󵄨󵄨󵄨󵄨𝑥󸀠 (𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏,

(16)

(15)

𝑖, 𝑗 = 1, 2, 𝑖 ≠ 𝑗, with 𝑚𝑖𝑗 (𝑡, 𝜏) = 𝑐3 (𝜕𝑟/𝜕𝑛)(2𝑟⋅𝑖 𝑟⋅𝑗 )/𝑟. We can find that 𝐵0 , 𝐵𝑖,𝑗 , 𝑀𝑖,𝑗 are smooth operators, 𝐴 0 is a logarithmic singular operator, and 𝐶0 is a Cauchy singular operator.

by which the error estimate will be (𝐴 0 𝑢) (𝑡) − (𝐴ℎ0 𝑢) (𝑡) 𝑚−1 󸀠

𝜍 (−2𝜇) (2𝜇) 𝑢 (𝑡) ℎ2𝜇+1 + 𝑂 (ℎ2𝑚 ) , (2𝜇)! 𝜇=1

= 2∑

where 𝜍󸀠 (𝑡) is the derivative of Riemann zeta function.

(22)

4

Mathematical Problems in Engineering

For the Cauchy singular operator 𝐶0 , L¨u and Huang [21] and Sidi [22] have proposed the operator 𝐶0ℎ to approximate it as shown in Lemma 1.

Theorem 3. The approximate operator sequences {𝐿ℎ1 } and {𝐿ℎ2 } are asymptotically compact sequences and convergent to 𝐿 1 and 𝐿 2 in 𝑉(0) , respectively; i.e.,

Lemma 1. The Nystr¨om approximate operator 𝐶0ℎ can be defined as

𝐿ℎ1 󳨀→ 𝐿 1 ,

2𝑛−1

(𝐶0ℎ 𝑢) (𝑡𝑖 ) = 2𝑐2 𝑎1 (𝑡𝑖 , 𝑡𝑗 ) ℎ ∑ cot (

(𝑡𝑗 − 𝑡𝑖 )

𝑗=0

2

a.c 𝐿ℎ2 󳨀→

) 𝑢 (𝑡𝑗 )

(23)

󵄨 󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑥󸀠 (𝑡𝑗 )󵄨󵄨󵄨󵄨 𝜀𝑖𝑗 , where 𝑎1 (𝑡, 𝑠) = 2[(𝑡−𝑠)+𝑂(𝑡−𝑠)]−1 tan((𝑡−𝑠)/2) when 𝑠 → 𝑡, and 󵄨 󵄨 {1, if 󵄨󵄨󵄨𝑖 − 𝑗󵄨󵄨󵄨 is odd number, 𝜀𝑖𝑗 = { 󵄨 󵄨 0, if 󵄨󵄨󵄨𝑖 − 𝑗󵄨󵄨󵄨 is even number, {

(𝐶0 𝑢) (𝑡𝑖 ) −

(29)

𝐿 2,

a.c

where 󳨀󳨀→ means the asymptotically compact convergence. Proof. An asymptotically compact sequence will be proved for operator {𝐿ℎ2 : 𝑉(0) → 𝑉(0) }, firstly. We can find that the kernels of 𝐵0 and 𝐵𝑖𝑗 (𝑖, 𝑗 = 1, 2, ) are continuous functions, so the collectively compact convergence [10, 20] is true:

(24) c.c

𝐵0ℎ 󳨀→ 𝐵0 ,

so the error estimate will be (𝐶0ℎ 𝑢) (𝑡𝑖 )

a.c

c.c

𝐵𝑖𝑗ℎ 󳨀→ 𝐵𝑖𝑗

2𝑚

= 𝑂 (ℎ ) .

(25)

(30) in 𝐶 [0, 2𝜋] , as 𝑛 󳨀→ ∞.

Thus (16) can be rewritten as follows: 1 ( 𝐼 + 𝐶ℎ + 𝑀ℎ ) 𝑢ℎ − (𝐴ℎ + 𝐵ℎ ) 𝑔 (𝑢ℎ ) = 𝑓ℎ , 2

(26)

where 𝐴ℎ , 𝐵ℎ , 𝐶ℎ , and 𝑀ℎ are the approximate matrixes corresponding to the operators 𝐴, 𝐵, 𝐶, and 𝑀, 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 (1/2)𝐼 + 𝐶ℎ will be presented [10]. −1

Lemma 2. (1/2)𝐼 + 𝐶ℎ is invertible and ((1/2)𝐼 + 𝐶ℎ ) uniformly bounded.

is

From Lemma 2, we can find that the eigenvalues of both 𝐶 and 𝐶ℎ do not include −1/2. So (16) and (26) can be rewritten as follows: find 𝑢 ∈ 𝑉(0) satisfying ̂ (𝐼 + 𝐿 1 ) 𝑢 + 𝐿 2 𝑔 (𝑢) = 𝑓,

(27)

where 𝐿 1 = (𝐼/2 + 𝐶)−1 𝑀, 𝐿 2 = (𝐼/2 + 𝐶)−1 (𝐴 + 𝐵), 𝑓̂ = (𝐼/2 + 𝐶)−1 𝑓, and the space 𝑉(𝑚) = 𝐶(𝑚) [0, 2𝜋] × 𝐶(𝑚) [0, 2𝜋], 𝑚 = 0, 1, 2, . . .. Equation (26) can be rewritten as follows: (𝐼 + 𝐿ℎ1 ) 𝑢ℎ + 𝐿ℎ2 𝑔 (𝑢ℎ ) = 𝑓̂ℎ , −1

−1

(28)

with 𝐿ℎ1 = (𝐼/2 + 𝐶ℎ ) 𝑀ℎ , 𝐿ℎ2 = (𝐼/2 + 𝐶ℎ ) (𝐴ℎ + 𝐵ℎ ), and −1 𝑓̂ = (𝐼/2 + 𝐶ℎ ) 𝑓 . ℎ

ℎ

Since we have constructed the continuous approximate function of 𝑎(𝑡, 𝜏) of 𝑎𝑝 (𝑡, 𝜏), the approximate operator {𝐴ℎ0 } is the a.c

asymptotically compact convergent to 𝐴 0 ; i.e., 𝐴ℎ0 󳨀󳨀→ 𝐴 0 in 𝐶[0, 2𝜋], as 𝑛 → ∞. a.c a.c Then we have 𝐴ℎ 󳨀󳨀→ 𝐴 and 𝐵ℎ 󳨀󳨀→ 𝐵 in 𝑉(0) . It can be concluded that for any bounded sequence {𝑦𝑚 ∈ 𝑉(0) } a convergent subsequence must be found in {(𝐴ℎ + 𝐵ℎ )𝑦𝑚 }. Without loss of generality, we assume (𝐴ℎ + 𝐵ℎ )𝑦𝑚 → 𝑧, as 𝑚 → ∞. According to the asymptotically compact convergence and the errors quadrature formulae [17, 21], we obtain 󵄩󵄩 −1 󵄩 󵄩󵄩 1 󵄩󵄩 ℎ 󵄩󵄩𝐿 2 𝑦𝑚 − ( 𝐼 + 𝐶) 𝑧󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 2 󵄩 󵄩 󵄩󵄩 󵄩 −1 󵄩 󵄩 1 󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩( 𝐼 + 𝐶ℎ ) 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩(𝐴ℎ + 𝐵ℎ ) 𝑦𝑚 − 𝑧󵄩󵄩󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 −1 −1 󵄩 󵄩󵄩 1 󵄩 1 + 󵄩󵄩󵄩󵄩( 𝐼 + 𝐶ℎ ) (𝐶 − 𝐶ℎ ) ( 𝐼 + 𝐶) 𝑧󵄩󵄩󵄩󵄩 2 󵄩󵄩 2 󵄩󵄩 󳨀→ 0,

(31)

as 𝑚 󳨀→ ∞, ℎ 󳨀→ 0,

where ‖ ⋅ ‖ is the norm of m(𝑉(0) , 𝑉(0) ). We proved that {𝐿ℎ2 : 𝑉(0) → 𝑉(0) } is an asymptotically compact sequence. Furthermore, we will show that {𝐿ℎ2 } will be pointwise a.c convergent to 𝐿 2 , as 𝑛 → ∞. We can see that 𝐴ℎ + 𝐵ℎ 󳨀󳨀→ 𝐴 + 𝐵∀𝑦 ∈ 𝑉(0) , so we obtain 󵄩󵄩 ℎ 󵄩 󵄩󵄩(𝐴 + 𝐵ℎ ) 𝑦 − (𝐴 + 𝐵) 𝑦󵄩󵄩󵄩 󳨀→ 0, 󵄩 󵄩

as ℎ 󳨀→ 0.

(32)

Mathematical Problems in Engineering

5

−1

From Lemma 2, ((1/2)𝐼 + 𝐶ℎ ) is uniformly bounded, and the errors of the approximate operators of 𝐴ℎ , 𝐵ℎ , 𝐶ℎ will be substituted into the following equations, then 󵄩 −1 󵄩 󵄩󵄩 󵄩 󵄩󵄩 1 󵄩󵄩 ℎ 󵄩󵄩𝐿 2 𝑦 − 𝐿 2 𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩( 𝐼 + 𝐶ℎ ) 󵄩󵄩󵄩 󵄩 󵄩󵄩 2 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ⋅ 󵄩󵄩󵄩󵄩(𝐴ℎ + 𝐵ℎ ) 𝑦 − (𝐴 + 𝐵) 𝑦󵄩󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 −1 −1 1 󵄩 󵄩 1 + 󵄩󵄩󵄩󵄩( 𝐼 + 𝐶ℎ ) (𝐶ℎ − 𝐶) ( 𝐼 + 𝐶) (𝐴 + 𝐵) 𝑦󵄩󵄩󵄩󵄩 2 󵄩󵄩 󵄩󵄩 2

Proof. Let (16) subtract (26) at 𝑡 = 𝑡𝑖 , then 𝐼 ( + 𝐶 + 𝑀) 𝑢 − (𝐴 + 𝐵) 𝑔 (𝑢) 2 𝐼 − ( + 𝐶ℎ + 𝑀ℎ ) 𝑢ℎ + (𝐴ℎ + 𝐵ℎ ) 𝑔 (𝑢ℎ ) 2

(36)

= 𝑓 − 𝑓ℎ ; (33)

that is, (𝑢 − 𝑢ℎ ) + (𝐶𝑢 − 𝐶ℎ 𝑢ℎ ) + (𝑀𝑢 − 𝑀ℎ 𝑢ℎ ) 2

󳨀→ 0, as ℎ 󳨀→ 0.

(37)

− ((𝐴 + 𝐵) 𝑔 (𝑢) − (𝐴ℎ + 𝐵ℎ ) 𝑔 (𝑢ℎ )) = 𝑓 − 𝑓ℎ .

So we proved that {𝐿ℎ2 } is pointwise convergent to 𝐿 2 , as 𝑛 → ∞. Since the series {𝐿ℎ2 } are asymptotically compact sequences and pointwise convergent to 𝐿 2 , it has been a.c proved that 𝐿ℎ2 󳨀󳨀→ 𝐿 2 . Similar proof can be done for {𝐿ℎ1 }, 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.

Consider the components 𝐶𝑢 − 𝐶ℎ 𝑢ℎ in the equations 𝐶𝑢 − 𝐶ℎ 𝑢ℎ = (𝐶𝑢 − 𝐶ℎ 𝑢) + (𝐶ℎ 𝑢 − 𝐶ℎ 𝑢ℎ ) = (𝐶 − 𝐶ℎ ) 𝑢 + 𝐶ℎ (𝑢 − 𝑢ℎ ) ,

and similar results can be obtained as 𝑀𝑢 − 𝑀ℎ 𝑢ℎ = (𝑀 − 𝑀ℎ )𝑢 + 𝑀ℎ (𝑢 − 𝑢ℎ ). Moreover, using the mean value theorem of differentials, we obtain ((𝐴 + 𝐵) 𝑔 (𝑢) − (𝐴ℎ + 𝐵ℎ ) 𝑔 (𝑢ℎ ))

Assumptions 4. (1): 𝑔(⋅, 𝑢) is measurable and differentiable for 𝑢 ∈ 𝑅, and 𝑔(𝑥, ⋅) is continuous for 𝑥 ∈ Γ. (2): (𝜕/𝜕𝑢)𝑔(𝑥, 𝑢) is Borel measurable and satisfies the inequality 0 < 𝑙 < (𝜕/𝜕𝑢)𝑔(𝑥, 𝑢) < 𝐿 < ∞.

= ((𝐴 + 𝐵 − 𝐴ℎ − 𝐵ℎ ) 𝑔 (𝑢)) + (𝐴ℎ + 𝐵ℎ ) (𝑔 (𝑢) − 𝑔 (𝑢ℎ ))

a.c

−1

(𝐼 + 𝐶ℎ ) (𝑀ℎ + (𝐴ℎ + 𝐵ℎ ) 𝑔󸀠 (𝑢))

(34)

a.c

󳨀→ (𝐼 + 𝐶)−1 (𝑀 + (𝐴 + 𝐵) 𝑔󸀠 (𝑢)) .

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

where 𝑢ℎ = 𝑢ℎ + 𝑡(𝑢 − 𝑢ℎ ) (0 ≤ 𝑡 ≤ 1). Note that 𝑀 and 𝐵 are continuous operators, A is a logarithmic operator, and C is a Cauchy operator, so the approximate errors will be (𝐶 − 𝐶ℎ )𝑢 = 𝑂(ℎ2𝑚 ), (𝑀 − 𝑀ℎ )𝑢 = 𝑂(ℎ2𝑚 ), 𝑓 − 𝑓ℎ = ℎ3 𝜔1 + 𝑂(ℎ5 ), and at 𝑡 = 𝑡𝑖 . (40)

Equation (37) can be simplified as 1 ( 𝐼 + 𝐶ℎ + 𝑀ℎ − (𝐴ℎ + 𝐵ℎ ) 𝑔󸀠 (𝑢ℎ )) (𝑢 − 𝑢ℎ ) 2 3

Theorem 5. Considering the asymptotic property and 𝑥(𝑡), 𝑓(𝑡) ∈ 𝑉2𝑚 [0, 2𝜋], there exists a function 𝜔1 ∈ 𝑉2𝑚−2 [0, 2𝜋] independent of ℎ, such that

= ℎ 𝜔 + 𝑂 (ℎ ) , where 𝜔 = 𝜔1 + 𝜔2 . Since ((1/2)𝐼 + 𝐶ℎ ) uniformly bounded, (41) is rewritten as

(𝐼 + 𝐿ℎ ) (𝑢 − 𝑢ℎ ) = ℎ3 𝜙 + 𝑂 (ℎ5 ) , −1

(35)

(41)

5

−1

4.1. Asymptotic Expansions

󵄨 󵄨 (𝑢 − 𝑢ℎ )󵄨󵄨󵄨𝑡=𝑡𝑗 = ℎ3 𝜔1 󵄨󵄨󵄨󵄨𝑡=𝑡 + 𝑂 (ℎ5 ) 𝑗

+ (𝐴ℎ + 𝐵ℎ ) 𝑔󸀠 (𝑢ℎ ) (𝑢 − 𝑢ℎ )

(𝐴 + 𝐵 − 𝐴ℎ − 𝐵ℎ ) 𝑔 (𝑢) = ℎ3 𝜔2 + 𝑂 (ℎ5 )

4. Asymptotic Expansions and Extrapolation

(39)

= (𝐴 + 𝐵 − 𝐴ℎ − 𝐵ℎ ) 𝑔 (𝑢)

The asymptotically compact convergence will be drawn according to the assumption about the function 𝑔(𝑢): (𝐴ℎ + 𝐵ℎ ) 𝑔󸀠 (𝑢) 󳨀→ (𝐴 + 𝐵) 𝑔󸀠 (𝑢) ,

(38)

exists and is (42)

where 𝐿ℎ = ((1/2)𝐼 + 𝐶ℎ ) (𝑀ℎ − (𝐴ℎ + 𝐵ℎ )𝑔󸀠 (𝑢ℎ )) and 𝜙 = −1 ((1/2)𝐼 + 𝐶ℎ ) 𝜔.

6

Mathematical Problems in Engineering y 0.5

An auxiliary equation will be introduced with the solution 𝜔1 (𝐼 + 𝐿) 𝜔1 = 𝜙,

(43)

and the corresponding approximate equations will be ℎ

(𝐼 + 𝐿 ) 𝜔1ℎ = 𝜙ℎ .

0.3 x

(44)

Substituting (44) into (42) yields the equations: 󵄨 (𝐼 + 𝐿ℎ ) (𝑢ℎ − 𝑢 − ℎ3 𝜔1ℎ )󵄨󵄨󵄨󵄨𝑡=𝑡 = 𝑂 (ℎ5 ) . 𝑗

(45)

Noticing that 𝜔1ℎ ∈ 𝑉2𝑚−2 [0, 2𝜋] and using the results of (42) yield (𝐼 + 𝐿ℎ ) (𝜔1 − 𝜔1ℎ ) (𝑡𝑖 ) = 𝑂 (ℎ3 ) .

(46)

Replace 𝜔ℎl by 𝜔𝑙 and consider 𝐿ℎ is an asymptotic compact operator, 󵄨 (𝑢ℎ − 𝑢 − ℎ3 𝜔1 )󵄨󵄨󵄨󵄨𝑡=𝑡 = 𝑂 (ℎ5 ) ; 𝑗

(47)

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 𝑂(ℎ5 ) can be obtained by computing some coarse grids and fine grids on Γ in parallel. The EAs are described as follows. Step 1. Take the mesh widths ℎ and ℎ/2 to obtain the solution of (26) in parallel, where 𝑢ℎ (𝑡𝑖 ), 𝑢ℎ/2 (𝑡𝑖 ) are the solutions on Γ corresponding to ℎ and ℎ/2, respectively. Step 2. Use the values at coarse grids and fine grids to calculate the approximate values at 𝑡𝑖 𝑢ℎ∗

1 (𝑡𝑖 ) = (8𝑢ℎ/2 (𝑡𝑖 ) − 𝑢ℎ (𝑡𝑖 )) , 7

(48)

and the error is |𝑢ℎ∗ (𝑡𝑖 ) − 𝑢(𝑡𝑖 )| = 𝑂(ℎ5 ). Step 3. So we can construct a posteriori error estimate: 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑢 (𝑡𝑖 ) − 𝑢ℎ/2 (𝑡𝑖 )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨𝑢 (𝑡𝑖 ) − (8𝑢ℎ/2 (𝑡𝑖 ) − 𝑢ℎ (𝑡𝑖 ))󵄨󵄨󵄨󵄨 7 󵄨 󵄨 1 󵄨󵄨 󵄨 󵄨𝑢 (𝑡 ) − 𝑢ℎ (𝑡𝑖 )󵄨󵄨󵄨 7 󵄨 ℎ/2 𝑖 󵄨 󵄨 = 󵄨󵄨󵄨𝑢 (𝑡𝑖 ) − 𝑢ℎ∗ (𝑡𝑖 )󵄨󵄨󵄨 +

+ ≤

(49)

1 󵄨󵄨 󵄨 󵄨𝑢 (𝑡 ) − 𝑢ℎ (𝑡𝑖 )󵄨󵄨󵄨 7 󵄨 ℎ/2 𝑖

1 󵄨󵄨 󵄨 󵄨𝑢 (𝑡 ) − 𝑢ℎ (𝑡𝑖 )󵄨󵄨󵄨 , 7 󵄨 ℎ/2 𝑖

which is useful for constructing self-adaptive algorithms.

Figure 1: The shape of the elliptical body.

5. Numerical Example Example 1. We first introduce some notations for 𝑖 = 1, 2: 𝑒𝑖ℎ (𝑃) = |𝑢𝑖ℎ (𝑃) − 𝑢𝑖 (𝑃)| is the error of the displacement; 𝑟𝑖ℎ (𝑃) = 𝑒𝑖ℎ (𝑃)/𝑒𝑖ℎ/2 (𝑃) is the error ratio; 𝑒ℎ𝑖 (𝑃) = ∗ |𝑢𝑖ℎ (𝑃) − 𝑢𝑖∗ (𝑃)| is the error after one-step EAs; and 𝑝𝑖ℎ (𝑃) = (1/7)|𝑢𝑖ℎ/2 (𝑃) − 𝑢𝑖ℎ (𝑃)| is a posteriori error estimate. Suppose Ω is an isotropic elliptical body with the axis 𝑎 = 0.3, 𝑏 = 0.5 in the plane domain shown in Figure 1. The parameter formulae for the boundary Γ will be described as 𝑥 = 0.3 cos(𝑡), 𝑦 = 0.5 sin(𝑡), 𝑡 ∈ [0, 2𝜋]. And the nonlinear boundary values 𝑝 = (𝑔1 , 𝑔2 )𝑇 are set as 𝑔1 = −𝑢12 + cos(𝑡), 𝑔2 = sin(𝑢2 ) + 2 sin(𝑡), 𝑡 ∈ [0, 2𝜋]. We firstly calculate the numerical solutions 𝑢ℎ = (𝑢1ℎ , 𝑢2ℎ )𝑇 on the boundary Γ following (26). Table 1 lists the approximate values of 𝑢1ℎ (𝑃) at points 𝑃1 = (𝑎 cos(𝜋/8), 𝑏 sin(𝜋/8)) and 𝑃2 = (𝑎 cos(𝜋/4), 𝑏 sin(𝜋/4)). Table 2 lists the approximate values of 𝑢2ℎ (𝑃) at points 𝑃1 = (𝑎 cos(𝜋/8), 𝑏 sin(𝜋/8)) and 𝑃2 = (𝑎 cos(𝜋/4), 𝑏 sin(𝜋/4)). From Tables 1-2, we can numerically see log2 𝑟𝑖ℎ (𝑃) ≈ 3, log2 𝑟ℎ𝑖 (𝑃) ≈ 5,

(50)

which shows that the convergent orders are three for the approximation solutions 𝑢𝑖ℎ and will be improved to five orders after EAs. The normal derivative 𝑝ℎ = (𝑝1ℎ , 𝑝2ℎ )𝑇 on Γ can be obtained when the displacement 𝑢ℎ = (𝑢1ℎ , 𝑢2ℎ )𝑇 on Γ is substituted into the boundary condition of (1). So following (4), the displacement 𝑢ℎ = (𝑢1ℎ , 𝑢2ℎ )𝑇 in Ω can be calculated. Table 3 shows the approximate values of the displacement (𝑢1ℎ (𝑃0 ), 𝑢2ℎ (𝑃0 ))𝑇 at an inner point 𝑃0 = (1/8)(cos(𝜋/3), sin(𝜋/3)) in Ω. From Table 3, we can numerically see that the ℎ3 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.

Mathematical Problems in Engineering

7

Table 1: The errors and errors ratio of 𝑢1ℎ (𝑃) at points 𝑃 = 𝑃1 , 𝑃2 . 𝑛 𝑒1ℎ (𝑃1 ) 𝑟1ℎ (𝑃1 ) 𝑒ℎ1 (𝑃1 ) 𝑟ℎ1 (𝑃1 ) 𝑒1ℎ (𝑃2 ) 𝑟1ℎ (𝑃2 ) 𝑒ℎ1 (𝑃2 ) 𝑟ℎ1 (𝑃2 )

16 8.466𝐸 − 4

32 1.040𝐸 − 4 8.137 4.12𝐸 − 07

6.085𝐸 − 4

7.501𝐸 − 5 8.112 3.91𝐸 − 07

64 1.296𝐸 − 5 8.028 1.29𝐸 − 08 31.92 9.292𝐸 − 6 8.073 1.24𝐸 − 8 31.47

128 1.618𝐸 − 6 8.011 4.03𝐸 − 10 31.99 1.160𝐸 − 6 8.008 4.00𝐸 − 10 31.09

256 2.022𝐸 − 7 8.002 1.24𝐸 − 11 32.41 1.450𝐸 − 7 8.000 1.30𝐸 − 11 30.85

512 2.527𝐸 − 8 8.000 3.93𝐸 − 13 31.68 1.813𝐸 − 8 8.000 4.03𝐸 − 13 32.16

256 9.753𝐸 − 8 8.003 2.45𝐸 − 10 31.77 2.217𝐸 − 7 8.001 2.79𝐸 − 10 30.69

512 1.219𝐸 − 8 8.000 7.75𝐸 − 12 31.65 2.771𝐸 − 8 8.000 8.83𝐸 − 12 31.56

128 1.808𝐸 − 7 8.002 7.04𝐸 − 10 1.809𝐸 − 7 9.599𝐸 − 8 8.005 1.62𝐸 − 9 9.604𝐸 − 8

256 2.260𝐸 − 8 8.000 2.22𝐸 − 11 2.260𝐸 − 8 1.200𝐸 − 8 8.000 5.11𝐸 − 11 1.200𝐸 − 8

Table 2: The errors and errors ratio of 𝑢2ℎ (𝑃) at points 𝑃 = 𝑃1 , 𝑃2 . 𝑛 𝑒2ℎ (𝑃1 ) 𝑟2ℎ (𝑃1 ) 𝑒ℎ2 (𝑃1 ) 𝑟ℎ2 (𝑃1 ) 𝑒2ℎ (𝑃2 ) 𝑟2ℎ (𝑃2 ) 𝑒ℎ2 (𝑃2 ) 𝑟ℎ2 (𝑃2 )

16 4.169𝐸 − 4

32 5.100𝐸 − 5 8.175 7.67𝐸 − 06

9.338𝐸 − 4

1.147𝐸 − 4 8.138 8.30𝐸 − 06

64 6.309𝐸 − 6 8.083 2.44𝐸 − 07 31.44 1.420𝐸 − 5 8.080 2.75𝐸 − 7 30.21

128 7.805𝐸 − 7 8.018 7.80𝐸 − 9 31.29 1.774𝐸 − 6 8.005 8.75𝐸 − 9 31.40

Table 3: The errors and errors ratio of (𝑢1ℎ , 𝑢2ℎ )𝑇 at inner points 𝑃 = 𝑃0 . 𝑛 𝑒1ℎ (𝑃0 ) 𝑟1ℎ (𝑃0 ) 𝑒ℎ1 (𝑃0 ) 𝑝1ℎ (𝑃0 ) 𝑒2ℎ (𝑃0 ) 𝑟2ℎ (𝑃0 ) 𝑒ℎ2 (𝑃0 ) 𝑝2ℎ (𝑃0 )

8 8.156𝐸 − 4

4.372𝐸 − 4

16 9.700𝐸 − 5 8.409 2.13𝐸 − 05 9.727𝐸 − 5 5.145𝐸 − 5 8.498 4.82𝐸 − 05 5.154𝐸 − 5

32 1.172𝐸 − 5 8.276 6.81𝐸 − 07 1.183𝐸 − 5 6.227𝐸 − 6 8.262 1.58𝐸 − 6 6.249𝐸 − 6

64 1.447𝐸 − 6 8.099 2.21𝐸 − 08 1.452𝐸 − 6 7.684𝐸 − 7 8.104 5.09𝐸 − 8 7.688𝐸 − 7

6. Conclusion

Acknowledgments

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.

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

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

References [1] M. Kitahara, Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates, vol. 10, Elsevier, 2014. [2] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, Springer Science and Business Media, Berlin, Germany, 2013. [3] A. S. Saada, Elasticity: Theory and Applications, Elsevier, Amsterdam, The Netherlands, 2013. [4] J. Hu and Z.-C. Shi, “Lower order rectangular nonconforming mixed finite elements for plane elasticity,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 88–102, 2007/08.

8 [5] C. J. Talbot and A. Crampton, “Application of the pseudospectral method to 2D eigenvalue problems in elasticity,” Numerical Algorithms, vol. 38, no. 1-3, pp. 95–110, 2005. [6] Z. Cai and G. Starke, “Least-squares methods for linear elasticity,” SIAM Journal on Numerical Analysis, vol. 42, no. 2, pp. 826–842, 2004. [7] J. T. Chen and H. K. Hong, “Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series,” Applied Mechanics Reviews, vol. 52, no. 1, pp. 17–33, 1999. [8] S. R. Kuo, J. T. Chen, and C. X. Huang, “Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real-part dual BEM,” International Journal for Numerical Methods in Engineering, vol. 48, no. 9, pp. 1401–1422, 2000. [9] X. Li and Q. Nie, “A high-order boundary integral method for surface diffusions on elastically stressed axisymmetric rods,” Journal of Computational Physics, vol. 228, no. 12, pp. 4625–4637, 2009. [10] P. Cheng, J. Huang, and Z. Wang, “Nystr¨om methods and extrapolation for solving Steklov eigensolutions and its application in elasticity,” Numerical Methods for Partial Differential Equations, vol. 28, no. 6, pp. 2021–2040, 2012. [11] C. A. Brebbia and S. Walker, Boundary Element Techniques in Engineering, Elsevier, 2016. [12] J. Rodriguez, “On the solvability of nonlinear discrete boundary value problems,” Journal of Difference Equations and Applications, vol. 9, no. 9, pp. 863–867, 2003. [13] K. Maleknejad and H. Mesgarani, “Numerical method for a nonlinear boundary integral equation,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1006–1009, 2006. [14] K. E. Atkinson and G. Chandler, “Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case,” Mathematics of Computation, vol. 55, no. 192, pp. 451–472, 1990. [15] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Springer Science and Business Media, 2012. [16] H. Abels, N. Arab, and H. Garcke, “On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions,” Journal of Evolution Equations (JEE), vol. 15, no. 4, pp. 913–959, 2015. [17] P. Cheng, X. Luo, Z. Wang, and J. Huang, “Mechanical quadrature methods and extrapolation algorithms for boundary integral equations with linear boundary conditions in elasticity,” Journal of Elasticity: The Physical and Mathematical Science of Solids, vol. 108, no. 2, pp. 193–207, 2012. [18] G. C. Hsiao, E. Schnack, and W. L. Wendland, “A hybrid coupled finite-boundary element method in elasticity,” Computer Methods Applied Mechanics and Engineering, vol. 173, no. 3-4, pp. 287–316, 1999. [19] P. Cheng, J. Huang, and G. Zeng, “Splitting extrapolation algorithms for solving the boundary integral equations of Steklov problems on polygons by mechanical quadrature methods,” Engineering Analysis with Boundary Elements, vol. 35, no. 10, pp. 1136–1141, 2011. [20] J. Huang, T. L¨u, and Z. C. Li, “Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs,” Applied Numerical Mathematics, vol. 59, no. 12, pp. 2908–2922, 2009. [21] T. L¨u and J. Huang, “Mechanical quadrature methods and their extrapolations for solving boundary integral equations of planar

Mathematical Problems in Engineering elasticity problems,” Mathematical numerica sinica, vol. 23, no. 4, pp. 491–502, 2001. [22] A. Sidi, Practical Extrapolation Methods: Theory and Applications, vol. 10 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003. [23] K. Parand and J. A. Rad, “Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5292–5309, 2012. [24] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 1977.

Advances in

Operations Research Hindawi www.hindawi.com

Volume 2018

Advances in

Decision Sciences Hindawi www.hindawi.com

Volume 2018

Journal of

Applied Mathematics Hindawi www.hindawi.com

Volume 2018

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com www.hindawi.com

Volume 2018 2013

Journal of

Probability and Statistics Hindawi www.hindawi.com

Volume 2018

International Journal of Mathematics and Mathematical Sciences

Journal of

Optimization Hindawi www.hindawi.com

Hindawi www.hindawi.com

Volume 2018

Volume 2018

Submit your manuscripts at www.hindawi.com International Journal of

Engineering Mathematics Hindawi www.hindawi.com

International Journal of

Analysis

Journal of

Complex Analysis Hindawi www.hindawi.com

Volume 2018

International Journal of

Stochastic Analysis Hindawi www.hindawi.com

Hindawi www.hindawi.com

Volume 2018

Volume 2018

Advances in

Numerical Analysis Hindawi www.hindawi.com

Volume 2018

Journal of

Hindawi www.hindawi.com

Volume 2018

Journal of

Mathematics Hindawi www.hindawi.com

Mathematical Problems in Engineering

Function Spaces Volume 2018

Hindawi www.hindawi.com

Volume 2018

International Journal of

Differential Equations Hindawi www.hindawi.com

Volume 2018

Abstract and Applied Analysis Hindawi www.hindawi.com

Volume 2018

Discrete Dynamics in Nature and Society Hindawi www.hindawi.com

Volume 2018

Advances in

Mathematical Physics Volume 2018

Hindawi www.hindawi.com

Volume 2018