MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/6932164 6932164 Research Article Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity http://orcid.org/0000-0002-7391-7042 Cheng Pan 1 Zhang Ling 1 Salerno Nunzio School of Mathematics and Statistics Chongqing Jiaotong University Chongqing 400074 China cqjtu.edu.cn 2018 3152018 2018 25 12 2017 24 04 2018 3152018 2018 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.

Natural Science Foundation of Chongqing CSTC2013JCYJA00017 Educational Council Foundation of Chongqing KJ1500517 KJ1600512
1. Introduction

This paper will describe the isolated elastic equations on a bounded planar region Ω in the plane with nonlinear boundary value conditions:(1)σij,j=0,inΩ,p¯=-g¯x,u¯+f¯x,onΓ,i,j=1,2,where ΩR2 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¯=p¯1,p¯2T with p¯i=σi1n1+σi2n2, f¯(x)=f¯1x,f¯2xT and continuous on Γ, and g¯(x,u¯)=g¯1x,u¯,g¯2x,u¯T, where g¯i(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 . Some methods have been proposed for solving the elasticity. The nonconforming mixed finite element methods are established by Hu and Shi  to solve the elasticity with a linear boundary. Talbot and Crampton  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  for obtaining the solution of the elastic problems. Chen and Hong  solved the hyper singular integral equations applying the dual boundary element methods in elasticity. Kuo et al.  solve true and spurious eigensolutions of the circular cavity problems that used dual methods. Li and Nie  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.  to solve Steklov eigensolutions in elasticity to obtain high accuracy solutions.

Equations (1) are converted into the boundary integral equations  (BIEs) with the Cauchy and logarithmic singularities by the variational method.(2)δij2u¯jy+Γkijy,xu¯jxdsx=Γhijy,xp¯jxdsx,y=y1,y2Γ,where δij is the Kronecker delta for i,j=1,2, and the kernels: (3)hij=18πμ1-ν-3-4νδijlnr+r·ir·j,kij=14π1-νrrn1-2νδij+2r·ir·j+1-2νnir·j-njr·i.

The kernels are Kelvin’s fundamental solutions . The Poisson ratio is ν with ν=λ/[2(λ+μ)], r·i means the derivative to xi, and r=y1-x12+y2-x22 is the distance of x and y. The parts Γkij(y,x)u¯j(x)dsx of (2) are the Cauchy singularity and the parts Γhij(y,x)u¯j(x)dsx are the logarithmic singularity.

The nonlinear integral equations are obtained after the boundary conditions are substituted into (2):(4)δij2u¯jy+Γkijy,xu¯jxdsx=Γhijy,xg¯jx,u¯dsx+Γhijy,xf¯jxdsx.

This kind of nonlinear integral equation has been discussed in many papers. Rodriguez  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  to approximate the solution. Atkinson and Chandler  presented product Simpson’s rule methods and two-grid methods to solve the nonlinear equations. Pao  gave a systematic treatment of a class of nonlinear elliptic differential equations and their applications of these problems. Abels et al.  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:(5)u¯iy=Γhijy,xp¯jxdsx-Γkijy,xu¯jxdsx,yΩ,σijy=Γhijly,xp¯lxdsx-Γkijly,xu¯lxdsx,yΩ,where (6)kijl=1-2νr.jδli+r.iδlj-r.lδij+2r.ir.jr.l4π1-νr,hijl=μ2π1-νr22rn1-2νr.lδij+νr.jδil+r.iδjl-4r.ir.jr.l+2νnir.jr.l+njr.ir.l+1-2ν2nlr.jr.i+njδil+niδjl-1-4νnlδij.

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.  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  and plane problem in elasticity .

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.

We firstly define the notation of the integral operators on boundary Γ as follows to simplify the equations:(7)Kijuy=Γkijy,xuxdsxyΓ,i,j=1,2,Hijuy=Γhijy,xuxdsxyΓ,i,j=1,2.So (2) can be simplified as the following operator equations:(8)12I0+K11K12K2112I0+K22u¯1u¯2=H11H12H21H22g¯1+f¯1g¯2+f¯2,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 2π. A regular parameter mapping [0,2π]Γ is introduced, so the smooth closed curve x(s)=(x1(s),x2(s)) satisfying xs2=x1s2+x2s2>0 with xi(s)C2m[0,2π], i=1,2.

The singularity of the kernels hij, kij will be analyzed as ts before the application of discrete methods. Since it is stated in the page 497 of the paper  that(9)nir·j-njr·ir=-1i1+Ot-st-s+Ot-s,ij,so the Cauchy singularity of kij comes from the part (nir·j-njr·i)/r. Moreover, the logarithmic singularity of hij comes from the component logx-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: (10)A0ut=02πa0t,τuτxτdτ,with a0(t,τ)=c¯0ln2e-1/2sint-τ/2, c¯0=-(3-4ν)/[8πμ(1-ν)], and (11)B0ut=02πb0t,τuτxτdτ,with b0(t,τ)=c¯0(lnxt-xτ-ln2e-1/2sint-τ/2), and (12)Bijut=02πbijt,τuτxτdτ,with bij(t,τ)=c1r·ir·j, c1=1/[8πμ(1-ν)].

The Cauchy singular operator Kij will also be divided into three parts on C2m[0,2π] as follows: (13)C0ut=02πc0t,τuτxτdτ,with c0(t,τ)=c2(n1r·2-n2r·1)/r,c2=-(1-2ν)/[4π(1-ν)], and (14)Miiut=02πmiit,τuτxτdτ,i=1,2,with mii(t,τ)=c3r/n[(1-2v)+2r·ir·i]/r,c3=-1/[4π(1-ν)], and (15)Mijut=02πmijt,τuτxτdτ,i,j=1,2,ij,with mij(t,τ)=c3r/n(2r·ir·j)/r.

We can find that B0, Bi,j, Mi,j are smooth operators, A0 is a logarithmic singular operator, and C0 is a Cauchy singular operator.

Then the equivalent equation of (8) will be shown as(16)12I+C+Mu=A+Bgu+f,where u(t)=(u¯1(x(t)),u¯2(x(t))), f(t)=f1,f2T with fi=Hi,jf¯j(x(t)), g(u(t))=g¯(x(t),u(t)), and (17)I=I000I0,C=0C0-C00,A=A000A0,B=B0+B11B12B21B0+B22,M=M11M12M21M22.

Suppose the boundary Γ will be divided into 2n,(nN) 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 Nyström’s approximations . For example, the approximation operator B0h of B0 can be approximated as follows:(18)B0hut=hj=02n-1b0t,τjuτj,and the error will be(19)B0ut-B0hut=Oh2m.Nyström’s approximation Bijh of Bij and Mijh of Mij can be approximated similarly.

In order to approximate the logarithmic singular operator A0, the continuous approximation kernel ap(t,τ) can be defined as follows:(20)apt,τ=a0t,τ,fort-τh,c¯0hlne-1/2h2π,fort-τ<h.By Sidi’s quadrature rules , the approximation operator can be constructed as follows:(21)A0hut=hj=02n-1apt,τjuτjxτj,by which the error estimate will be(22)A0ut-A0hut=2μ=1m-1ς-2μ2μ!u2μth2μ+1+Oh2m,where ς(t) is the derivative of Riemann zeta function.

For the Cauchy singular operator C0, Lü and Huang  and Sidi  have proposed the operator C0h to approximate it as shown in Lemma 1.

Lemma 1.

The Nyström approximate operator C0h can be defined as(23)C0huti=2c2a1ti,tjhj=02n-1cottj-ti2utjxtjεij,where a1(t,s)=2t-s+Ot-s-1tan((t-s)/2) when st, and (24)εij=1,ifi-j  is  odd  number,0,ifi-j  is  even  number,so the error estimate will be(25)C0uti-C0huti=Oh2m.

Thus (16) can be rewritten as follows:(26)12I+Ch+Mhuh-Ah+Bhguh=fh,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 .

Lemma 2.

( 1 / 2 ) I + C h is invertible and ((1/2)I+Ch)-1 is uniformly bounded.

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 uV(0) satisfying(27)I+L1u+L2gu=f^,where L1=(I/2+C)-1M, L2=(I/2+C)-1(A+B), f^=(I/2+C)-1f, and the space V(m)=C(m)[0,2π]×C(m)[0,2π], m=0,1,2,.

Equation (26) can be rewritten as follows:(28)I+L1huh+L2hguh=f^h,with L1h=(I/2+Ch)-1Mh, L2h=(I/2+Ch)-1(Ah+Bh), and f^h=(I/2+Ch)-1fh.

Theorem 3.

The approximate operator sequences {L1h} and {L2h} are asymptotically compact sequences and convergent to L1 and L2 in V(0), respectively; i.e.,(29)L1ha.cL1,L2ha.cL2,where a.c means the asymptotically compact convergence.

Proof.

An asymptotically compact sequence will be proved for operator {L2h: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: (30)B0hc.cB0,Bijhc.cBijinC0,2π,asn.Since we have constructed the continuous approximate function of a(t,τ) of ap(t,τ), the approximate operator {A0h} is the asymptotically compact convergent to A0; i.e., A0ha.cA0 in C[0,2π], as n.

Then we have Aha.cA and Bha.cB in V(0). It can be concluded that for any bounded sequence {ymV(0)} a convergent subsequence must be found in {(Ah+Bh)ym}. Without loss of generality, we assume (Ah+Bh)ymz, as m. According to the asymptotically compact convergence and the errors quadrature formulae [17, 21], we obtain (31)L2hym-12I+C-1z12I+Ch-1Ah+Bhym-z+12I+Ch-1C-Ch12I+C-1z0,asm,h0,where · is the norm of £V0,V0. We proved that L2h:V0V0 is an asymptotically compact sequence.

Furthermore, we will show that {L2h} will be pointwise convergent to L2, as n. We can see that Ah+Bha.cA+ByV(0), so we obtain(32)Ah+Bhy-A+By0,ash0.

From Lemma 2, ((1/2)I+Ch)-1 is uniformly bounded, and the errors of the approximate operators of Ah,Bh,Ch will be substituted into the following equations, then (33)L2hy-L2y12I+Ch-1·Ah+Bhy-A+By+12I+Ch-1Ch-C12I+C-1A+By0,ash0.So we proved that {L2h} is pointwise convergent to L2, as n.

Since the series {L2h} are asymptotically compact sequences and pointwise convergent to L2, it has been proved that L2ha.cL2. Similar proof can be done for {L1h}, 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): g(·,u) is measurable and differentiable for uR, and g(x,·) is continuous for xΓ.

(2): /ug(x,u) is Borel measurable and satisfies the inequality 0<l</ug(x,u)<L<.

The asymptotically compact convergence will be drawn according to the assumption about the function g(u): (34)Ah+Bhgua.cA+Bgu,I+Ch-1Mh+Ah+Bhgua.cI+C-1M+A+Bgu.

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 x(t),f(t)V2m[0,2π], there exists a function ω1¯V2m-2[0,2π] independent of h, such that(35)u-uht=tj=h3ω¯1t=tj+Oh5

Proof.

Let (16) subtract (26) at t=ti, then(36)I2+C+Mu-A+Bgu-I2+Ch+Mhuh+Ah+Bhguh=f-fh;that is,(37)u-uh2+Cu-Chuh+Mu-Mhuh-A+Bgu-Ah+Bhguh=f-fh.

Consider the components Cu-Chuh in the equations (38)Cu-Chuh=Cu-Chu+Chu-Chuh=C-Chu+Chu-uh,and similar results can be obtained as Mu-Mhuh=(M-Mh)u+Mh(u-uh). Moreover, using the mean value theorem of differentials, we obtain (39)A+Bgu-Ah+Bhguh=A+B-Ah-Bhgu+Ah+Bhgu-guh=A+B-Ah-Bhgu+Ah+Bhguh¯u-uhwhere uh¯=uh+t(u-uh)(0t1).

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=h3ω1+O(h5), and (40)A+B-Ah-Bhgu=h3ω2+Oh5att=ti.

Equation (37) can be simplified as(41)12I+Ch+Mh-Ah+Bhguh¯u-uh=h3ω+Oh5,where ω=ω1+ω2. Since ((1/2)I+Ch)-1 exists and is uniformly bounded, (41) is rewritten as(42)I+Lhu-uh=h3ϕ+Oh5,where Lh=((1/2)I+Ch)-1(Mh-Ah+Bhg(uh¯)) and ϕ=((1/2)I+Ch)-1ω.

An auxiliary equation will be introduced with the solution ω1¯(43)I+Lω1¯=ϕ,and the corresponding approximate equations will be(44)I+Lhω¯1h=ϕh.Substituting (44) into (42) yields the equations:(45)I+Lhuh-u-h3ω¯1ht=tj=Oh5.Noticing that ω¯1hV2m-2[0,2π] and using the results of (42) yield(46)I+Lhω1¯-ω¯1hti=Oh3.Replace ω¯lh by ω¯l and consider Lh is an asymptotic compact operator,(47)uh-u-h3ω1¯t=tj=Oh5;we complete the proof.

4.2. Extrapolation Algorithms

An asymptotic expansion about the errors in (35) implies that the extrapolation algorithms  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(48)uhti=178uh/2ti-uhti,and the error is uhti-uti=O(h5).

Step 3.

So we can construct a posteriori error estimate: (49)uti-uh/2tiuti-178uh/2ti-uhti+17uh/2ti-uhti=uti-uhti+17uh/2ti-uhti17uh/2ti-uhti,which is useful for constructing self-adaptive algorithms.

5. Numerical Example Example 1.

We first introduce some notations for i=1,2: eih(P)=uihP-uiP is the error of the displacement; rih(P)=eih(P)/eih/2(P) is the error ratio; e¯ih(P)=uihP-uiP is the error after one-step EAs; and pih(P)=1/7uih/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=0.3cos(t), y=0.5sin(t), t[0,2π]. And the nonlinear boundary values p=g1,g2T are set as g1=-u12+cos(t), g2=sin(u2)+2sin(t), t[0,2π].

The shape of the elliptical body.

We firstly calculate the numerical solutions uh=u1h,u2hT on the boundary Γ following (26). Table 1 lists the approximate values of u1h(P) at points P1=(acosπ/8,bsinπ/8) and P2=(acosπ/4,bsinπ/4). Table 2 lists the approximate values of u2h(P) at points P1=(acosπ/8,bsinπ/8) and P2=(acosπ/4,bsinπ/4).

The errors and errors ratio of u1h(P) at points P=P1, P2.

n 16 32 64 128 256 512
e 1 h ( P 1 ) 8.466 E - 4 1.040 E - 4 1.296 E - 5 1.618 E - 6 2.022 E - 7 2.527 E - 8
r 1 h ( P 1 ) 8.137 8.028 8.011 8.002 8.000
e ¯ 1 h ( P 1 ) 4.12 E - 07 1.29 E - 08 4.03 E - 10 1.24 E - 11 3.93 E - 13
r ¯ 1 h ( P 1 ) 31.92 31.99 32.41 31.68

e 1 h ( P 2 ) 6.085 E - 4 7.501 E - 5 9.292 E - 6 1.160 E - 6 1.450 E - 7 1.813 E - 8
r 1 h ( P 2 ) 8.112 8.073 8.008 8.000 8.000
e ¯ 1 h ( P 2 ) 3.91 E - 07 1.24 E - 8 4.00 E - 10 1.30 E - 11 4.03 E - 13
r ¯ 1 h ( P 2 ) 31.47 31.09 30.85 32.16

The errors and errors ratio of u2h(P) at points P=P1, P2.

n 16 32 64 128 256 512
e 2 h ( P 1 ) 4.169 E - 4 5.100 E - 5 6.309 E - 6 7.805 E - 7 9.753 E - 8 1.219 E - 8
r 2 h ( P 1 ) 8.175 8.083 8.018 8.003 8.000
e ¯ 2 h ( P 1 ) 7.67 E - 06 2.44 E - 07 7.80 E - 9 2.45 E - 10 7.75 E - 12
r ¯ 2 h ( P 1 ) 31.44 31.29 31.77 31.65

e 2 h ( P 2 ) 9.338 E - 4 1.147 E - 4 1.420 E - 5 1.774 E - 6 2.217 E - 7 2.771 E - 8
r 2 h ( P 2 ) 8.138 8.080 8.005 8.001 8.000
e ¯ 2 h ( P 2 ) 8.30 E - 06 2.75 E - 7 8.75 E - 9 2.79 E - 10 8.83 E - 12
r ¯ 2 h ( P 2 ) 30.21 31.40 30.69 31.56

From Tables 1-2, we can numerically see (50)log2rihP3,log2r¯ihP5,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,p2hT 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,u2hT in Ω can be calculated. Table 3 shows the approximate values of the displacement u1hP0,u2hP0T at an inner point P0=(1/8)(cosπ/3,sinπ/3) in Ω.

The errors and errors ratio of u1h,u2hTat inner points P=P0.

n 8 16 32 64 128 256
e 1 h ( P 0 ) 8.156 E - 4 9.700 E - 5 1.172 E - 5 1.447 E - 6 1.808 E - 7 2.260 E - 8
r 1 h ( P 0 ) 8.409 8.276 8.099 8.002 8.000
e ¯ 1 h ( P 0 ) 2.13 E - 05 6.81 E - 07 2.21 E - 08 7.04 E - 10 2.22 E - 11
p 1 h ( P 0 ) 9.727 E - 5 1.183 E - 5 1.452 E - 6 1.809 E - 7 2.260 E - 8

e 2 h ( P 0 ) 4.372 E - 4 5.145 E - 5 6.227 E - 6 7.684 E - 7 9.599 E - 8 1.200 E - 8
r 2 h ( P 0 ) 8.498 8.262 8.104 8.005 8.000
e ¯ 2 h ( P 0 ) 4.82 E - 05 1.58 E - 6 5.09 E - 8 1.62 E - 9 5.11 E - 11
p 2 h ( P 0 ) 5.154 E - 5 6.249 E - 6 7.688 E - 7 9.604 E - 8 1.200 E - 8

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.

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).

Kitahara M. Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates 2014 10 Elsevier Muskhelishvili N. I. Some basic problems of the mathematical theory of elasticity 2013 Berlin, Germany Springer Science and Business Media Saada A. S. Elasticity: Theory and Applications 2013 Amsterdam, The Netherlands Elsevier Hu J. Shi Z.-C. Lower order rectangular nonconforming mixed finite elements for plane elasticity SIAM Journal on Numerical Analysis 2007/08 46 1 88 102 10.1137/060669681 MR2377256 2-s2.0-55549111612 Talbot C. J. Crampton A. Application of the pseudo-spectral method to 2D eigenvalue problems in elasticity Numerical Algorithms 2005 38 1-3 95 110 10.1007/s11075-004-2860-5 MR2128621 Zbl1081.74049 2-s2.0-23944514603 Cai Z. Starke G. Least-squares methods for linear elasticity SIAM Journal on Numerical Analysis 2004 42 2 826 842 10.1137/S0036142902418357 MR2084237 Zbl1159.74419 2-s2.0-10144254912 Chen J. T. Hong H. K. Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series Applied Mechanics Reviews 1999 52 1 17 33 10.1115/1.3098922 2-s2.0-0001799089 Kuo S. R. Chen J. T. Huang C. X. 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 2000 48 9 1401 1422 2-s2.0-0034229431 10.1002/1097-0207(20000730)48:9<1401::AID-NME947>3.0.CO;2-K 10.1002/1097-0207(20000730)48:9<1401::AID-NME947>3.0.CO;2-K Zbl1021.76034 Li X. Nie Q. A high-order boundary integral method for surface diffusions on elastically stressed axisymmetric rods Journal of Computational Physics 2009 228 12 4625 4637 10.1016/j.jcp.2009.03.024 MR2531911 Zbl05572079 Cheng P. Huang J. Wang Z. Nyström methods and extrapolation for solving Steklov eigensolutions and its application in elasticity Numerical Methods for Partial Differential Equations 2012 28 6 2021 2040 10.1002/num.21695 MR2981881 Brebbia C. A. Walker S. Boundary Element Techniques in Engineering 2016 Elsevier Rodriguez J. On the solvability of nonlinear discrete boundary value problems Journal of Difference Equations and Applications 2003 9 9 863 867 10.1080/1023619031000062934 MR1995225 2-s2.0-0042572151 Maleknejad K. Mesgarani H. Numerical method for a nonlinear boundary integral equation Applied Mathematics and Computation 2006 182 2 1006 1009 10.1016/j.amc.2005.12.068 MR2282544 Zbl1107.65109 2-s2.0-33845232721 Atkinson K. E. Chandler G. Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions: the smooth boundary case Mathematics of Computation 1990 55 192 451 472 10.2307/2008428 MR1035924 Pao C. V. Nonlinear Parabolic and Elliptic Equations 2012 Springer Science and Business Media Abels H. Arab N. Garcke H. On convergence of solutions to equilibria for fully nonlinear parabolic systems with nonlinear boundary conditions Journal of Evolution Equations (JEE) 2015 15 4 913 959 10.1007/s00028-015-0287-1 MR3427071 2-s2.0-84948119301 Cheng P. Luo X. Wang Z. Huang J. 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 2012 108 2 193 207 10.1007/s10659-011-9364-z MR2948056 Zbl06062582 2-s2.0-84863722206 Hsiao G. C. Schnack E. Wendland W. L. A hybrid coupled finite-boundary element method in elasticity Computer Methods Applied Mechanics and Engineering 1999 173 3-4 287 316 10.1016/S0045-7825(98)00288-6 MR1687194 2-s2.0-0032654555 Cheng P. Huang J. Zeng G. Splitting extrapolation algorithms for solving the boundary integral equations of Steklov problems on polygons by mechanical quadrature methods Engineering Analysis with Boundary Elements 2011 35 10 1136 1141 10.1016/j.enganabound.2011.05.006 MR2813735 2-s2.0-79958050639 Huang J. T. Li Z. C. Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs Applied Numerical Mathematics 2009 59 12 2908 2922 10.1016/j.apnum.2009.06.006 MR2560824 T. Huang J. Mechanical quadrature methods and their extrapolations for solving boundary integral equations of planar elasticity problems Mathematical numerica sinica 2001 23 4 491 502 MR1881822 Sidi A. Practical Extrapolation Methods: Theory and Applications 2003 10 Cambridge, UK Cambridge University Press Cambridge Monographs on Applied and Computational Mathematics 10.1017/CBO9780511546815 MR1994507 Zbl1041.65001 Parand K. Rad J. A. Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions Applied Mathematics and Computation 2012 218 9 5292 5309 10.1016/j.amc.2011.11.013 MR2870050 Zbl1244.65245 Gilbarg D. Trudinger N. Elliptic Partial Differential Equations of Second Order 1977 Berlin, Germany Springer MR0473443 Zbl0361.35003