a relaxation method of an alternating iterative algorithm for the cauchy problem in linear isotropic...

Comput. Methods Appl. Mech. Engrg. 199 (2010) 3179–3196

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Comput. Methods Appl. Mech. Engrg.

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A relaxation method of an alternating iterative algorithm for the Cauchy problemin linear isotropic elasticity

Liviu Marin a,⇑, B. Tomas Johansson b

a Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille, P.O. Box 1-863, 010141 Bucharest, Romaniab School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

a r t i c l e i n f o

Article history:Received 11 October 2009Received in revised form 24 February 2010Accepted 23 June 2010Available online 30 July 2010

Keywords:Linear elasticityInverse problemCauchy problemAlternating iterative algorithmRelaxation proceduresBoundary element method (BEM)

0045-7825/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cma.2010.06.024

⇑ Corresponding author. Tel./fax: +40 (0) 21 312673E-mail addresses: [email protected], liviu@

[email protected] (B.T. Johansson).

a b s t r a c t

We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary dis-placements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary inthe case of the alternating iterative algorithm of Kozlov et al. [16] applied to Cauchy problems in linearelasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. Thenumerical results obtained using these procedures, in conjunction with the boundary element method(BEM), show the numerical stability, convergence, consistency and computational efficiency of the pro-posed method.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

In direct problems in solid mechanics, one has to determine theresponse of a system when the governing system of partial differ-ential equations (equilibrium equations), the constitutive andkinematics equations, the initial and boundary conditions for thedisplacement and/or traction vectors and the geometry of the do-main occupied by the solid are all known. The existence anduniqueness of the solution to such problems have been well estab-lished, see e.g. Knops and Payne [13]. However, if at least one of theconditions enumerated above is partially or entirely lacking thenone has a so-called inverse problem. Moreover, it is well known thatinverse problems are in general unstable, in the sense that smallmeasurement errors in the input data may amplify significantlythe errors in the solution, see e.g. Hadamard [8]. Over the last dec-ades, inverse problems in solid mechanics have been extensivelytreated; an overview of these developments is available in Bonnetand Constantinescu [3].

A classical example of an inverse boundary value problem insolid mechanics is represented by the Cauchy problem. In this case,boundary conditions are incomplete in the sense that a part of theboundary of the domain occupied by the solid is over-specified byprescribing on it both the displacement and traction vectors,

ll rights reserved.

6.imsar.bu.edu.ro (L. Marin),

whilst the remaining boundary is under-specified and boundaryconditions on the latter boundary have to be determined. Mani-atty et al. [19] used simple diagonal regularization, in conjunctionwith the finite element method (FEM), to determine the tractionboundary condition. Spatial regularization was introduced to-gether with the boundary element method (BEM) by Zabaraset al. [38] and with the FEM by Schnur and Zabaras [31]. Later,Maniatty and Zabaras [20] applied the Bayesian statistical theoryfor general inverse problems to inverse elasticity problems andalso compared it to the method proposed in Schnur and Zabaras[31]. The Cauchy problem in elasticity was studied theoreticallyby Yeih et al. [37], who analysed its existence, uniqueness andcontinuous dependence on the data and proposed an alternativeregularization procedure, namely the fictitious boundary indirectmethod, based on simple and double layer potential theory. Thenumerical implementation of the aforementioned method wasundertaken by Koya et al. [14], who employed the BEM and theNyström method for discretizing the integrals. However, this for-mulation has not yet removed the problem of evaluating multipleintegrals. Turco [34] used the BEM to discretize the problem alongwith a strategy based on the Tikhonov regularization methodcompleted by the generalized cross-validation (GCV) criterion inorder to make the solution process entirely automatic. The BEMwas successfully combined with the CGM and a stopping criterionbased on a Monte-Carlo simulation of the GCV by Turco [35].Recently, Bilotta and Turco [2] solved the Cauchy problem intwo-dimensional isotropic linear elasticity by using a standard

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3180 L. Marin, B.T. Johansson / Comput. Methods Appl. Mech. Engrg. 199 (2010) 3179–3196

FEM approach, in conjunction with the Tikhonov regularizationmethod and the GCV criterion for the optimal choice of the regu-larization parameter.

The alternating iterative algorithm of Kozlov et al. [16], whichreduces the Cauchy problem to solving a sequence of well-posedboundary value problems, was implemented numerically usingthe BEM for isotropic and anisotropic linear elastic materials byMarin et al. [21], and Comino et al.[5], respectively. Huang and Shih[9], Marin et al. [23], Marin and Lesnic [24] used both the conjugategradient method (CGM) and the Tikhonov regularization methodcombined with the BEM, in order to solve the same problem. Thesingular value decomposition (SVD), in conjunction with theBEM, was employed by Marin and Lesnic [25] to determine thenumerical solutions to Cauchy problems in linear elasticity. Fourregularization methods for solving stably the Cauchy problem inlinear elasticity, namely the Tikhonov regularization, the SVD, theCGM and the alternating iterative algorithm of Kozlov et al. [16],were compared in Marin et al. [22]. It was found in Marin et al.[22] that the truncated SVD outperforms the Tikhonov regulariza-tion method, whilst the latter outperforms the CGM. Marin andLesnic [26] and Marin[27] proposed a meshless method, namelythe method of fundamental solutions (MFS), in conjunction withthe Tikhonov regularization method, for solving the Cauchy prob-lem in two-and three-dimensional isotropic linear elasticity,respectively. The Cauchy problem in elasticity with L2-boundarydata was approached by combining the BEM with the Landwe-ber–Fridman method and the minimal error method by Marinand Lesnic [28] and Marin [29], respectively. Delvare and Hanus[6] introduced an evanescent regularization method combinedwith the FEM to solve the Cauchy problem for two-dimensionallinear elastic materials. Recently, Andrieux and Baranger [1] refor-mulated the Cauchy problem for three-dimensional elastic mediaas an energy error minimization problem, with the unknowns gi-ven by the surface displacement and traction vectors on the un-der-specified boundary of the solid. It should be mentioned thattheoretical investigations, as well as formulae for the solution ofthe Cauchy problem in linear elasticity, were given by Makhmudov[18], Shlapunov [32], and Tarkhanov [33], however, these formulaehave yet to be tested numerically.

Jourhmane and Nachaoui [11] and Jourhmane et al. [12] pro-posed the relaxation of the given Dirichlet data in the case ofthe alternating iterative algorithm of Kozlov et al. [16] appliedto the Cauchy problem for steady-state heat conduction in isotro-pic and anisotropic media, respectively. This procedure drasticallyreduced the number of iterations required to achieve convergencefor the inverse problems considered. Recently, see Ellabib andNachaoui [7], a relaxation of the alternating method in elasticitywas numerically investigated. Encouraged by their results, wedo further investigations and propose alternative ways of relaxa-tion of both the prescribed displacements and tractions on theover-specified boundary. Moreover, importantly, we prove theconvergence of these schemes and introduce appropriate optimalstopping rules.

The paper is organized as follows: Section 2 is devoted to themathematical formulation of the inverse problem investigated, aswell as the introduction of the function spaces used herein. Thealternating iterative algorithms with relaxation for the Cauchyproblem in linear elasticity are then presented in Section 3, whilethe proof of the convergence theorem for these procedures is givenin Section 4. The implementation of the proposed numerical meth-ods is realized using the BEM and this is briefly discussed in Sec-tion 5. In Section 6, the algorithms introduced in Section 3 areapplied to solving three Cauchy problems with noisy Cauchy datafor a two-dimensional isotropic linear elastic material. Finally,some concluding remarks and possible future work are providedin Section 7.

2. Mathematical formulation

2.1. Notation and function spaces

Consider a homogeneous linear elastic material which occupiesa bounded Lipschitz domain X � Rd, where d is the dimension ofthe space where the problem is posed, usually d 2 {1,2,3}. LetC0 – £ be an arc of @X such that meas(C0) – 0 and setC1 ¼ @X n C0. Let H1(X) be the Sobolev space of real valued func-tions in X endowed with the standard norm. We denote byH1

0ðXÞ and H1CiðXÞ; i ¼ 0;1, the subspaces of functions from H1(X)

that vanish on @X and Ci, i = 0,1, respectively.The space of traces of functions from H1(X) to @X is denoted by

H1/2(@X), while the restrictions of the functions belonging to thespace H1/2(@X) to the subset Ci � @X, i = 0,1, define the space H1/

2(Ci), i = 0, 1. The set of real valued functions in @X with compactsupport in Ci, i = 0, 1, and bounded first-order derivatives are densein H1/2(Ci), i = 0, 1. Furthermore, we also define the spaceH1=2

00 ðCiÞ; i ¼ 0;1, that consists of functions from H1/2(@X) and van-ishing on C1�i, i = 0, 1. The space H1=2

00 ðCiÞ; i ¼ 0;1, is a subspace ofH1/2(@X) with the norm given by:

kfkH1=200 ðCiÞ

¼Z

Ci

f 2ðxÞdistðx;CiÞ

dCðxÞþZ

Ci

ZCi

jf ðxÞ� f ðyÞj2

jx�yjddCðxÞdCðyÞ

!1=2

:

ð1Þ

It should be mentioned that the space of restrictions fromH1=2

00 ðCiÞ to Ci, i = 0, 1, is dense in H1/2(Ci), i = 0, 1. Nonetheless,H1=2

00 ðCiÞ – H1=2ðCiÞ. In this paper, we use the following notation:

H1ðXÞd ¼ H1ðXÞ � � � � � H1ðXÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}d times

;

as well as similar notations for the other spaces used. Finally, we de-note by (H1(Ci)d)* the dual space of H1=2

00 ðCiÞd; i ¼ 0;1.

2.2. The Cauchy problem

In the absence of body forces, the equilibrium equations are gi-ven by, see e.g. Landau and Lifshitz [17],

LuðxÞ � �r � r uðxÞð Þ ¼ 0; x 2 X: ð2Þ

Here L is the Lamé (Navier) differential operator, r(u(x)) =[rij(u(x))]16i,j,6d is the stress tensor associated with the displacementvector u(x) = (u1(x), . . . ,ud(x))T, whilst on assuming small deforma-tions, the corresponding strain tensor e(u(x)) = [eij(u(x))]16i,j,6d isgiven by the kinematic relations:

e uðxÞð Þ ¼ 12ruðxÞ þ ruðxÞT� �

; x 2 X ¼ X [ @X: ð3Þ

These tensors are related by the constitutive law, namely

r uðxÞð Þ ¼ Ce uðxÞð Þ; x 2 X; ð4Þ

where C = [Cijkl]16i,j,k,l6d is the fourth-order elasticity tensor which issymmetric and positive definite.

We let n(x) = (n1(x), . . . ,nd(x))T be the outward unit normal vec-tor at x 2 @X, and NuðxÞ � tðxÞ ¼ ðt1ðxÞ; . . . ; tdðxÞÞT be the tractionvector at a point x 2 @X, defined by

NuðxÞ � tðxÞ ¼ r uðxÞð Þ � nðxÞ; x 2 @X; ð5Þ

where N is the boundary-differential operator associated with theLamé (Navier) differential operator, L, and Neumann boundary con-ditions on @X. In the direct problem formulation, the knowledge ofthe displacement and/or traction vectors on the whole boundary @Xgives the corresponding Dirichlet, Neumann, or mixed boundaryconditions which enable us to determine the displacement vectorin the domain X. Then, the strain tensor, e, can be calculated from

L. Marin, B.T. Johansson / Comput. Methods Appl. Mech. Engrg. 199 (2010) 3179–3196 3181

Eq. (3) and the stress tensor, r, is determined using the constitutivelaw (4).

If it is possible to measure both the displacement and tractionvectors on a part of the boundary @X, say C0, then this leads tothe mathematical formulation of the Cauchy problem consistingof the partial differential equation (2) and the boundary conditions

uðxÞ ¼ uðxÞ; NuðxÞ � tðxÞ ¼ wðxÞ; x 2 C0; ð6Þ

where u 2 H1/2(C0)d and w 2 H1=200 ðC0Þd

� ��are prescribed vector-

valued functions. In the above formulation of the boundary condi-tions (6), it can be seen that the boundary C0 is over-specified byprescribing both the displacement ujC0

¼ u and the tractiontjC0¼ w vectors, whilst the boundary C1 is under-specified since

both the displacement ujC1and the traction tjC1

vectors areunknown and have to be determined. Hence on using relations(2)–(6) the Cauchy problem under investigation may be recast as

LuðxÞ ¼ 0; x 2 X;

uðxÞ ¼ uðxÞ; x 2 C0;

NuðxÞ ¼ wðxÞ; x 2 C0:

8><>: ð7Þ

We assume that data are chosen such that there exists a solution tothis Cauchy problem. This solution is unique according to the so-called unique continuation properties for elliptic equations.

It should be mentioned that in the case of a homogeneous iso-tropic linear elastic material, the components of the fourth-orderelasticity tensor, C, are given by

Cijkl ¼ G2m

1� mdijdkl þ dikdjl þ dildjk

� �; ð8Þ

where G is the shear modulus, m is Poisson’s ratio and dij is theKronecker delta tensor. Consequently, the constitutive law forhomogeneous isotropic linear elastic materials can be expressed as

r uðxÞð Þ ¼ G ruðxÞ þ ruðxÞT� �

þ 2mG1� m

r � uðxÞð ÞI; x 2 X; ð9Þ

with I = [dij]16i,j6d the identity matrix in Rd�d, whilst the Lamé (Na-vier) differential operator, L, and its corresponding boundary-differ-ential operator associated with Neumann conditions, N , given byrelations (2) and (5), respectively, may be recast as:

LuðxÞ¼� Gr� ruðxÞþruðxÞT� �

þ 2mG1�2m

r r�uðxÞð Þ� �

x2X

ð10Þ

and

NuðxÞ¼ G ruðxÞþruðxÞT� �

þ 2mG1�2m

r�uðxÞð ÞI� �

�nðxÞ; x2 @X;

ð11Þ

respectively.

3. Alternating iterative algorithms with relaxation

In this section we propose two alternating iterative algorithmswith relaxation which aim to reduce the computational time of thealternating iterative algorithm introduced by Kozlov et al. [16], andat the same time maintaining the accuracy of the numerical resultsobtained with the latter.

Alternating iterative algorithm with relaxation I:

Step 1.1. If k = 1 then choose an arbitrary function

nð1Þ 2 H1=200 ðC1Þd

� ��.

Step 1.2. If k P 2 then solve the direct problem

Luð2k�2ÞðxÞ ¼ 0; x 2 X;

tð2k�2ÞðxÞ � r uð2k�2ÞðxÞ

� nðxÞ ¼ wðxÞ; x 2 C0;

uð2k�2ÞðxÞ ¼ gðk�1ÞðxÞ; x 2 C1;

8><>: ð12Þ

where g(k�1)(x) = u(2k�3)(x), x 2 C1 to obtain u(2k�2)(x), x 2X andt(2k�2)(x) � r(u(2k�2)(x)), x 2 C1.Step 2. Provided that k P 2 update the unknown Neumann data onC1 as:

nðkÞðxÞ ¼ htð2k�2ÞðxÞ þ ð1� hÞnðk�1ÞðxÞ; x 2 C1; ð13Þ

where the relaxation factor, 0 6 h 6 2, is fixed. For k P 1 solve thedirect problem8


uð2k�1ÞðxÞ ¼ uðxÞ; x 2 C0;


� nðxÞ ¼ nðkÞðxÞ; x 2 C1

><>: ð14Þ

to determine u(2k�1)(x), x 2X and u(2k�1)(x), x 2 C1.Step 3. Set k = k + 1 and repeat Steps 1 and 2 until a prescribed stop-ping criterion is satisfied.

Remark 3.1. The value h = 1 in Eq. (13) corresponds to the alter-nating iterative algorithm introduced by Kozlov et al. [16] withan initial guess for the Neumann data, whilst values h 2 (0,1) andh 2 (1,2) in Eq. (13) correspond to the alternating iterative algo-rithm introduced by Kozlov et al. [16] with an initial guess forthe Neumann data and a constant under-and over-relaxation fac-tor, respectively.

Alternating iterative algorithm with relaxation II:

Step 1.1. If k = 1 then choose an arbitrary function g(1) 2 H1/2(C0)d.Step 1.2. If k P 2 then solve the direct problem


uð2k�2ÞðxÞ ¼ uðxÞ; x 2 C0;


� nðxÞ ¼ nðk�1ÞðxÞ; x 2 C1;

8><>:ð15Þ

where n(k�1)(x) = t(2k�3)(x), x 2 C1, to obtain u(2k�2)(x), x 2X, andu(2k�2)(x), x 2 C1.Step 2. Provided that k P 2 update the unknown Dirichlet data onC1 as:

gðkÞðxÞ ¼ huð2k�2ÞðxÞ þ ð1� hÞgðk�1ÞðxÞ; x 2 C1; ð16Þ

where the relaxation factor, 0 6 h 6 2, is fixed. For k P 1 solve thedirect problem8



� nðxÞ ¼ wðxÞ; x 2 C0;

uð2k�1ÞðxÞ ¼ gðkÞðxÞ; x 2 C1

><>: ð17Þ

to determine u(2k�1)(x), x 2X, and t(2k�1)(x), x 2 C1.Step 3. Set k = k + 1 and repeat Steps 1 and 2 until a prescribed stop-ping criterion is satisfied.

Remark 3.2. The value h = 1 in Eq. (16) corresponds to the alternat-ing iterative algorithm introduced by Kozlov et al. [16] with an ini-tial guess for the Dirichlet data, whilst values h 2 (0,1) and h 2 (1,2)in Eq. (16) correspond to the alternating iterative algorithm intro-duced by Kozlov et al. [16] with an initial guess for the Dirichlet dataand a constant under-and over-relaxation factor, respectively.

4. Convergence of the alternating iterative algorithms withrelaxation

Following the ideas of Jourhmane and Nachaoui [11] we shallprove:


Theorem 4.1. Let u 2 H1/2(C0)d and w 2 H1=200 ðC0Þd

� ��. Assume that

the Cauchy problem (7) has a solution u 2 H1(X)d. Let u(k) be the kthapproximate solution in the alternating procedure I described above.Then there exists a number 1 < b 6 2 such that when the relaxationparameter h is chosen with 1 6 h 6 b, then

limk!1ku� uðkÞkH1ðXÞd ¼ 0 ð18Þ

for any initial data element nð1Þ 2 H1=200 ðC1Þd

� ��.

Proof. First, let u(1) be the solution to Eq. (14) with k = 1 for givenfunctions n(1) = n and u = 0. Then, let u(2) be the solution to Eq. (12)for k = 2 with w = 0 and u(2) = gð1Þ on C1, where gð1Þ ¼ uð1ÞjC1

. Definethe linear operator Bh : H1=2

00 ðC1Þd� ��

! H1=200 ðC1Þd

� ��for h P 0 by

Bhn ¼ h r uð2ÞðnÞ

� n� �

jC1þ ð1� hÞn; ð19Þ

which is well-defined. In a similar way, let v(2) be the element ob-tained from the second approximation in the proposed relaxationof the alternating procedure, with the initial guess n(1) = 0, and de-fine the element Gh(u,w) by

Ghðu;wÞ ¼ h r vð2Þ

� n� �

jC1: ð20Þ

The Cauchy problem (7) can then be written (according to Koz-lov and Maz’ya [15] and Kozlov et al. [16]) as a fixed point equation

Bhnþ Ghðu;wÞ ¼ n: ð21Þ

Note that n is a control variable; choosing n as a fixed point ofEq. (21), by construction, the Cauchy conditions on C0 are satisfied(since for such a n then u(1) is equal to u(2) in X due to the fact thatthey satisfy the same Cauchy conditions on C1 and similar for v(1)

and v(2)). Thus, to show the convergence of the procedure, we haveto investigate the operator Bh.

To show properties of the operator Bh, we introduce thefollowing inner product in H1=2

00 ðC1Þd� ��

ðn; fÞ ¼Z

Xru � rvdx; ð22Þ

where u solves (14) with n(k) = n and u = 0, and similarly v solves(14) with n(k) = f and u = 0. The corresponding norm is denoted byk � k.

In Kozlov and Maz’ya [15] and Kozlov et al. [16] it was shownthat the operator B = B1 is self-adjoint, positive definite, non-expansive and one is not an eigenvalue. Note that

Bhn ¼ hBnþ ð1� hÞn: ð23Þ

Clearly then Bh is self-adjoint for h P 0.Let us then show that the operator Bh is positive definite for

0 6 h 6 1/(1 � c0) for some 0 < c0 < 1. In Kozlov and Maz’ya [15]and Kozlov et al. [16] it was shown that

ðBn; nÞP c0knk2 ð24Þ

for some positive number c0 and we can, without any loss of gener-ality, assume that c0 < 1. Now, by definition,

ðBhn; nÞ ¼ hðBn; nÞ þ ð1� hÞðn; nÞ ð25Þ

and clearly, for 0 6 h 6 1, Bh is positive definite. For h > 1, combiningEqs. (24) and (25) implies

ðBhn; nÞP c0hðn; nÞ þ ð1� hÞðn; nÞ:

Thus, if 1 < h < 1/(1 � c0), the operator Bh is still positive definite.Next, we prove that Bh is non-expansive for 0 6 h 6 2. We

obtain

kBhnk2 ¼ h2kBnk2 þ 2hð1� hÞðBn; nÞ þ ð1� hÞ2ðn; nÞ: ð26Þ

By definition

kBnk2 ¼Z

Xjruð3Þj2dx; ð27Þ

where u(3) is the solution to Eq. (14) for k = 2 with u = 0 andNuð3ÞðxÞ ¼ Nuð2ÞðxÞ on C1. Using integration by parts, and notingthat u(3) = 0 on C0 and Nuð3ÞðxÞ ¼ Nuð2ÞðxÞ on C1, we obtainZ

Xruð2Þ � ruð3Þdx ¼

ZC1

Nuð2Þ

uð3ÞdCðxÞ

¼Z

C1

Nuð3Þ

uð3ÞdCðxÞ ¼Z

Xjruð3Þj2dx: ð28Þ

This impliesZXjrðuð3Þ � uð2ÞÞj2dx ¼

ZXruð2Þ 2dx�

ZXruð3Þ 2dx: ð29Þ

Thus combining relations (27) and (29),

kBnk2 ¼Z

Xruð3Þ 2dx 6


Moreover, from the definition of the inner product (22)

ðBn; nÞ ¼Z

Xruð3Þ � ruð1Þdx: ð31Þ

Using integration by parts in the right-hand side in Eq. (31), andnoting that u(2) = u(1) on C1;Nuð3ÞðxÞ ¼ Nuð2ÞðxÞ on C1, and u(1) = 0on C0 and Nuð2ÞðxÞ ¼ 0 on C0,

ðBn; nÞ ¼Z

C1

Nuð3Þ

uð1ÞdCðxÞ ¼Z

C1

Nuð2Þ

uð2ÞdCðxÞ

¼Z

Xruð2Þ 2dx: ð32Þ

Using Eqs. (30) and (32) in Eq. (26), we obtain

kBhnk26 h2 ruð2Þ

�� 2

L2ðXÞ þ 2hð1� hÞ ruð2Þ�� 2

L2ðXÞ þ ð1� hÞ2ðn; nÞ

¼ hð2� hÞ ruð2Þ�� 2

L2ðXÞ þ ð1� hÞ2knk2: ð33Þ

Similar to (29), one can check thatZXrðuð2Þ � uð1ÞÞ 2dx ¼

ZXruð1Þ 2dx�


Combining this with the definition of the norm generated from Eq.(22), the following estimate is obtained

ruð2Þ�� 2

L2ðXÞ 6 ruð1Þ�� 2

L2ðXÞ ¼ knk2: ð35Þ

For 0 6 h 6 2, employing the estimate (35) in Eq. (33) gives

kBhnk26 hð2� hÞknk2 þ ð1� hÞ2knk2 ¼ knk2

: ð36Þ

Thus Bh is non-expansive for 0 6 h 6 2.Finally, the number one is not an eigenvalue for the operator Bh

for h P 0. Indeed, if one is an eigenvalue it follows from thedefinition of Bh that one is an eigenvalue of the operator B, butaccording to Kozlov and Maz’ya [15] and Kozlov et al. [16], one isnot an eigenvalue for B.

To prove convergence, according to Eq. (21), it is sufficient toconsider the case when u = 0 and w = 0, i.e. to show that u(k) tendsto zero. From definition (19) we find that nðkÞ ¼ Bk

hnð1Þ. Thus from

the above properties of the operator Bh, we conclude thatlimk?1n(k) = 0 in the norm k � k, which in turn implies thatlimk?1u(2k+1) = 0 in H1(X)d. One can then check, using integrationby parts together with the relation u(2k+2) = u(2k+1) on C1 similar toEq. (34), thatZ

Xrðuð2kþ2Þ � uð2kþ1ÞÞ 2dx ¼

ZXruð2kþ1Þ 2dx�

ZXruð2kþ2Þ 2dx:


Since the left-hand side of this relation is positive andlimk?1u(2k+1) = 0 in H1(X),d we finally conclude that alsolimk?1u(2k+2) = 0 in H1(X)d, which completes the proof. h

Remark 4.1. Let u0 be the initial guess of the displacement andchoose the traction

n ¼ r D�1u0

� �jC1;

where D�1 gives the solution to the Dirichlet problem with u = u0

on C1 and u = u on C0. Starting the alternating iterative algorithmwith relaxation I with this traction as guess, the second approxima-tion will be precisely our first initial approximation. Thus, from theabove theorem, convergence is settled also for the alternating iter-ative algorithm II.

Remark 4.2. To present a stopping rule for the case of relaxation Iwith noisy data, let w(2) be the element obtained from the secondapproximation in the proposed alternating procedure, with the ini-tial guess u = 0, and define the element E(u,w) by

Eðu;wÞ ¼ t wð2Þ

jC1: ð37Þ

Then, with noisy data ud and wd, where d > 0, and

Eðud;wdÞ � Eðu;wÞ�� 6 d; ð38Þ

the discrepancy principle can be employed as a stopping rule, seeVainikko and Veretennikov [36, Chapter 3, Section 3]. This impliesin particular that if k = k(d) is the smallest integer with

tðuð2kþ2Þd Þ � tðuð2kÞ

d Þ�� 6 bd ð39Þ

for given b > 1, then uðkðdÞÞd converges to the exact solution of (7)when d ? 0.

5. Boundary element method

In this section we describe the BEM for homogeneous isotropiclinear elastic materials in two-dimensions, i.e. d = 2, although sim-ilar arguments apply for homogeneous anisotropic linear elasticmaterials, as well as in higher dimensions, i.e. d > 2. The Lamé sys-tem (2) or (10) in the two-dimensional case can be formulated inintegral form with the aid of the Second Theorem of Betti, seee.g. Landau and Lifshitz [17], namely

cijðxÞujðxÞ þZ--@X

Tijðx; yÞujðyÞdCðyÞ ¼Z@X

Uijðx; yÞtjðyÞdCðyÞ; ð40Þ

for i; j ¼ 1;2;x 2 X, and y 2 @X, where the first integral is taken inthe sense of the Cauchy principal value, cij(x) = 1 for x 2X andcij(x) = 1/2 for x 2 @X (smooth), and Uij and Tij are the fundamentaldisplacements and tractions for the two-dimensional isotropic lin-ear elasticity given by

Uijðx; yÞ ¼ C1 C2 ln rðx; yÞdij �@rðx; yÞ@yi

@rðx; yÞ@yj

!ð41Þ

and

Tijðx; yÞ ¼C3

rðx; yÞ C4dij þ 2@rðx; yÞ@yi

@rðx; yÞ@yj

!@rðx; yÞ@nðyÞ

"

�C4@rðx; yÞ@yi

njðyÞ �@rðx; yÞ@yj

niðyÞ !#

; ð42Þ

respectively. Here r(x,y) represents the distance between the node/collocation point x and the field point y, whilst the constants C1, C2,C3 and C4 are given by

C1 ¼ �1=½8pGð1� mÞ�; C2 ¼ 3� 4m;C3 ¼ �1=½4pð1� mÞ�; C4 ¼ 1� 2m; ð43Þ

where m ¼ m for the plane strain state and m ¼ m=ð1þ mÞ for the planestress state.

A BEM with continuous and discontinuous linear boundary ele-ments for domains with smooth and piecewise smooth boundaries,respectively, see e.g. Brebbia et al. [4], is employed in order to dis-cretise the integral Eq. (40). If the boundaries C0 and C1 are discre-tised into N0 and N1 (dis) continuous linear boundary elements,respectively, such that N = N0 + N1, then on applying the boundaryintegral Eq. (40) at each node/collocation point, we arrive at thefollowing system of linear algebraic equations

AU ¼ BT: ð44Þ

Here A and B are matrices which depend solely on the geometry ofthe boundary @X and material properties, i.e. the Poisson ratio, m,and the shear modulus, G, and can be calculated analytically, andthe vectors U and T consist of the discretised values of the boundarydisplacements and tractions, respectively. The BEM system of linearalgebraic equations (44) can be re-written as

Að00Þ Að01Þ

Að10Þ Að11Þ

" #Uð0Þ

Uð1Þ

!¼ Bð00Þ Bð01Þ

Bð10Þ Bð11Þ

" #Tð0Þ

Tð1Þ

!; ð45Þ

where the vectors Uð0Þ ¼ ðuð1Þ; . . . ;uðN0ÞÞT 2 R2N0 andTð0Þ ¼ ðtð1Þ; . . . ; tðN0ÞÞT 2 R2N0 contain the values of the displacementand traction vectors, respectively, at the nodes/collocation pointson the under-specified boundary C0, while the vectors Uð1Þ ¼ðuðN0þ1Þ; . . . ;uðN0þN1ÞÞT 2 R2N1 and Tð1Þ ¼ ðtðN0þ1Þ; . . . ; tðN0þN1ÞÞT 2 R2N1

consist of the values of the displacement and traction vectors,respectively, at the nodes/collocation points on the over-specifiedboundary C1. Here, the matrices AðijÞ 2 R2Ni�2Nj andBðijÞ 2 R2Ni�2Nj ; i; j ¼ 0;1, contain the elements of the BEM matricesA and B, respectively, corresponding to the decomposition of theglobal vectors U and T, with the convention that the indices i andj indicate the fact that the nodes/collocation points belong to theboundary Ci, i = 0, 1, and the field points are located on the bound-ary Cj, j = 0, 1, respectively.

It should be mentioned that at each step of the two alternatingiterative algorithms with relaxation presented in Section 3, two di-rect mixed well-posed problems are solved using the BEM. Conse-quently, the general form of the BEM system of linear algebraicequations associated with these direct problems may be recast as

CX ¼ F; ð46Þ

where

C¼ Að00Þ �Bð01Þ

Að10Þ �Bð11Þ

" #; X¼ Uð2k�2Þ

Tð2k�2Þ

!; F¼ Bð00Þ �Að01Þ

Bð10Þ �Að11Þ

" #W

Eðk�1Þ

� �;

ð47:1Þ

Uð2k�2Þ ¼ uð2k�2;1Þ; . . . ;uð2k�2;N0Þ T

;

Tð2k�2Þ ¼ tð2k�2;N0þ1Þ; . . . ; tð2k�2;N0þN1Þ T

; ð47:2Þ

W ¼ wð1Þ; . . . ;wðN0Þ T

;

Eðk�1Þ ¼ gðk�1;N0þ1Þ; . . . ;gðk�1;N0þN1Þ T ð47:3Þ

for the direct mixed well-posed problems (12) and

C¼ �Bð00Þ Að01Þ

�Bð10Þ Að11Þ

" #; X¼ Tð2k�1Þ

Uð2k�1Þ

!; F¼ �Að00Þ Bð01Þ

�Að10Þ Bð11Þ

" #U

NðkÞ

� �;

ð48:1Þ


Tð2k�1Þ ¼ tð2k�1;1Þ; . . . ; tð2k�1;N0Þ T

;

Uð2k�1Þ ¼ uð2k�1;N0þ1Þ; . . . ;uð2k�1;N0þN1Þ T

; ð48:2Þ

U ¼ uð1Þ; . . . ;uðN0Þ T

;NðkÞ ¼ nðk;N0þ1Þ; . . . ; nðk;N0þN1Þ� �T

; ð48:3Þ

for the direct mixed well-posed problems (14), in the case of thealternating iterative algorithm with relaxation I. Formulae (47)and (48) are also valid for the direct mixed well-posed problems(17) and (15), respectively, corresponding to the alternating itera-tive algorithm with relaxation II, with the mention that the pairsof indices (2k � 2,k � 1) and (2k � 1,k) are interchanged.

6. Numerical results and discussion

It is the purpose of this section to present the numerical imple-mentation of the alternating iterative methods described in Sec-tion 3, using the BEM presented in Section 5, for two-dimensional isotropic linear elastic materials and analyse thenumerical convergence and stability of this procedure, as well asthe influence of the constant relaxation parameter, h.

6.1. Examples and their parameters

We consider an isotropic linear elastic medium characterised bythe material constants G = 3.35 � 1010 N/m2 and m = 0.34 corre-sponding to a copper alloy, and we solve the Cauchy problem (7)for three typical examples in both smooth and piecewise smooth,as well as simply and doubly connected geometries:

Example 1. [Doubly connected, smooth geometry] We considerthe following analytical solution for the displacements:

uðanÞi ðx1; x2Þ ¼

12Gð1þ mÞ Vð1� mÞxi �Wð1þ mÞ xi

x21 þ x2

2

� �xi;

i ¼ 1;2; ð49Þ

with

V ¼ �ror2o � rir2

i

r2o � r2

i

; W ¼ ðro � riÞr2or2

i

r2o � r2

i

;

ri ¼ 1:0� 1010 N=m2; ro ¼ 2:0� 1010 N=m2; ð50Þ

in the annular domain X ¼ fx 2 R2jri < qðxÞ < rog, where qðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

1 þ x22

qis the radial polar coordinate of x, ri = 1 and ro = 4, which

corresponds to constant internal and external pressures ri and ro,respectively, for which the stress tensor is given by

rðanÞ11 ðx1; x2Þ ¼ V þW

x21 � x2

2

ðx21 þ x2

2Þ2 ;

rðanÞ22 ðx1; x2Þ ¼ V �W

x21 � x2

2

ðx21 þ x2

2Þ2 ;

rðanÞ12 ðx1; x2Þ ¼ 2W

x1x2

ðx21 þ x2

2Þ2 : ð51Þ

Here C0 = Co = {x 2 @Xjq(x) = ro} and C1 = Ci = {x 2 @Xjq(x) = ri}.

Example 2. [Simply connected, smooth geometry] We considerthe following analytical solution for the displacements:

uðanÞi ðx1; x2Þ ¼

1� m2Gð1þ mÞr0xi; i ¼ 1;2;

r0 ¼ 1:5� 1010 N=m2; ð52Þ

in the disk X ¼ fx 2 R2jqðxÞ < rg, where r = 1, which corresponds tothe uniform hydrostatic stress

rðanÞij ðx1; x2Þ ¼ r0dij; i; j ¼ 1;2: ð53Þ

Here C0 = {x 2 @Xj0 6 h(x) < p/8} [ {x 2 @Xj3p/8 < h(x) < 2p} andC1 = {x 2 @Xjp/8 6 h(x) 6 3p/8}, where h(x) is the angular polarcoordinate of x.

Example 3. [Simply connected, piecewise smooth geometry] Weconsider the following analytical solution for the displacements:

uðanÞ1 ðx1; x2Þ ¼

12Gð1þ mÞr0x1;

uðanÞ2 ðx1; x2Þ ¼ �

m2Gð1þ mÞr0x2; r0 ¼ 1:5� 1010 N=m2 ð54Þ

in the square X = (�1,1) � (�1,1), which corresponds to a uniformtraction stress given by

rðanÞ11 ðx1; x2Þ ¼ r0; rðanÞ

12 ðx1; x2Þ ¼ rðanÞ22 ðx1; x2Þ ¼ 0: ð55Þ

Here C0 = [ � 1,1] � { ± 1} [ { � 1} � (�1,1) and C1 = {1} � (�1,1).For the inverse problems considered in this study, the BEM sys-

tem of linear algebraic Eq. (44) has been solved for each of thewell-posed, direct, mixed boundary value problems that occur ateach iteration, k, of the algorithms presented in Section 3 to pro-vide simultaneously the unspecified boundary displacements andtractions on C1. In this study, the type and number of boundaryelements used for discretising the boundary @X were taken asfollows:

(i) N0 = 40 and N1 = 120 continuous linear boundary elementsfor Example 1;

(ii) N0 = 80 and N1 = 160 continuous linear boundary elementsin the case of Example 2;

(iii) N0 = 40 and N1 = 120 discontinuous linear boundary ele-ments for Example 3.

It is also important to mention that for the inverse problemsinvestigated in this paper, as well as the alternating iterative algo-rithms I and II, the initial guesses n(1) and g(1) for the traction tjC1

and displacement ujC1vectors, respectively, were taken to be

nð1ÞðxÞ ¼ 0; x 2 C1 ð56:1Þ

and

gð1ÞðxÞ ¼ 0; x 2 C1; ð56:2Þ

respectively. Moreover, all numerical computations have been per-formed in FORTRAN 90 in double precision on a 3.00 GHz Intel Pen-tium 4 machine.

6.2. Results obtained with exact data: Convergence of the algorithms

In order to analyse the accuracy, convergence and stability ofthe proposed alternating iterative algorithms with relaxation, fork P 1 we introduce the following errors

euðkÞ ¼

uð2k�1Þ � uðanÞ��

L2ðC1Þdfor the alternating iterative

algorithm with relaxation I;uð2kÞ � uðanÞ��


algorithm with relaxation II

8>>>><>>>>:ð57:1Þ

1 10 100 1000Number of iterations, k

10-8

10-6

10-4

10-2

100

Accu

racy

erro

r, e u

= 0.10 = 0.50 = 1.00 = 1.50 = 1.80

1 10 100 1000Number of iterations, k

10-8

10-6

10-4

10-2

100

Accu

racy

erro

r, e t

= 0.10 = 0.50 = 1.00 = 1.50 = 1.80

Fig. 1. The accuracy errors (a) eu and (b) et, as functions of the number of iterations, k, obtained using the alternating iterative algorithm I, exact Cauchy data and severalvalues of the parameter h, namely h 2 {0.10,0.50, 1.00,1.50,1.80}, for the Cauchy problem given by Example 1.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u1

AnalyticalNumerical

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u2

AnalyticalNumerical

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 1/10

10

AnalyticalNumerical

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 2/10

10

AnalyticalNumerical

Fig. 2. The analytical and numerical displacements (a) u1jC1and (b) u2jC1

, and tractions (c) t1jC1and (d) t2jC1

, obtained using the alternating iterative algorithm I, thediscrepancy principle, h = 1.80 and exact Cauchy data, i.e. pu = pt = 0%, for the Cauchy problem given by Example 1.



and

etðkÞ ¼

tð2kÞ � tðanÞ�� L2ðC1Þd

for the alternating iterative

algorithm with relaxation I;tð2k�1Þ � tðanÞ��


algorithm with relaxation II:

8>>>><>>>>:ð57:2Þ

Here u(2k�1) (u(2k)) and t(2k) (t(2k�1)) are the displacement and trac-tion vectors retrieved on the under-specified boundary C1 after kiterations using the alternating iterative algorithm with relaxationI (II), respectively, with the mention that each iteration consists ofsolving two direct mixed well-posed problems, namely equations(12) and (14) for the alternating iterative algorithm with relaxationI (Eqs. (15) and (17) for the alternating iterative algorithm withrelaxation II).

Figs. 1(a) and (b) show, on a logarithmic scale, the accuracy er-rors eu and et, as functions of the number of iterations, k, obtainedusing the alternating iterative algorithm I, exact Cauchy data andvarious values of the relaxation parameter h, in the case of Exam-ple 1. It can be seen from these figures that, for all values of therelaxation parameter used in this paper, both errors eu and et keepdecreasing until a specific number of iterations, after which theconvergence rate of the aforementioned accuracy errors becomesvery slow so that they reach a plateau. As expected, for each value

1 5 10 50 100 500Number of iterations, k

0.006

0.01

0.03

0.06

0.1

0.3

Accu

racy

erro

r, e u

pu = 1%pu = 2%pu = 3%

1 5 10Number o

0.01

0.05

0.1

0.5

1.0

5.0

10.0

Con

verg

ence

erro

r, E

Fig. 3. The accuracy errors (a) eu and (b) et, and the convergence error (c) E, as functionh = 1.50 and various amounts of noise added into ujC0

, i.e. pu 2 {1%,2%,3%} and pt = 0%, f

of the relaxation parameter employed, eu(k) < et(k) for all k P 1,i.e. displacements are more accurate than tractions; also, the lar-ger the parameter h, the lower the number of iterations and, con-sequently, computational time are required for obtaining accuratenumerical results for both the displacement and the traction vec-tors on C1. Therefore, choosing h 2 (1,2) in the alternating itera-tive algorithms I and II results in a significant reduction of thenumber of iterations as compared with the corresponding originalalternating iterative algorithms proposed by Kozlov et al. [16], i.e.for h = 1.

The same conclusions can be drawn from Figs. 2(a) and (b),which illustrate the analytical and numerical displacements u1jC1

and u2jC1, respectively, obtained with h = 1.80 after k = 1000, and

Figs. 2(c) and (d), which present graphically the correspondinganalytical and numerical values for the tractions t1jC1

and t2jC1,

respectively. From Figs. 1 and 2, it can be concluded that the alter-nating iterative algorithm with relaxation I described in Section 3provides excellent approximations for the unknown Dirichlet andNeumann data on C1 and is convergent with respect to increasingthe number of iterations, k, if exact Cauchy data are prescribed onthe over-specified boundary C0. Although not presented, it shouldbe mentioned that similar results have been obtained for Examples2 and 3, and all admissible values of the relaxation parameter, aswell as the alternating iterative algorithm with relaxation II ap-plied to all examples investigated in this study.


0.06

0.1

0.3

0.6

1.0

Accu

racy

erro

r, e t

pu = 1%pu = 2%pu = 3%

50 100 500f iterations, k

pu = 1%pu = 2%pu = 3%

s of the number of iterations, k, obtained using the alternating iterative algorithm I,or the Cauchy problem given by Example 1.


6.3. Regularizing stopping criterion

Once the convergence of the numerical solution to the exactsolution, with respect to number of iterations performed, k, hasbeen established, we investigate the stability of the numericalsolution for the examples considered. To do so and in order to sim-ulate the inherent inaccuracies in the measured data on C0, we as-sume that various levels of Gaussian random noise, pu and pt, havebeen added into the exact displacement ujC0

¼ u and tractiontjC0¼ w data, respectively, so that the following perturbed dis-

placements and tractions are available:

ud 2 L2ðC0Þd : kuðanÞjC0�udkL2ðC0Þd

¼ d ð58:1Þ

and

wd 2 L2ðC0Þd : ktðanÞjC0� wdkL2ðC0Þd

¼ d: ð58:2Þ

Figs. 3(a) and (b) present, on a logarithmic scale, the accuracyerrors eu and et, respectively, as functions of the number of itera-tions, k, obtained using the alternating iterative algorithm I,h = 1.50 and pu 2 {1%,2%,3%}, for the Cauchy problem given byExample 1. From these figures it can be seen that, for each fixed va-lue of pu, the errors in predicting the displacement and tractionvectors on the under-specified boundary C1 decrease up to a cer-tain iteration number and after that they start increasing. If the


0.001

0.005

0.01

0.05

0.1

0.5

1.0

Accu

racy

erro

r, e u

pt = 1%pt = 2%pt = 3%

1 5 10Number o

10-3

10-2

10-1

100

101

Con

verg

ence

erro

r, E

Fig. 4. The accuracy errors (a) eu and (b) et, and the convergence error (c) E, as functionsh = 1.50 and various amounts of noise added into tjC0

, i.e. pu = 0% and pt 2 {1%,2%,3%}, fo

iterative process is continued beyond this point then the numericalsolutions lose their smoothness and become highly oscillatory andunbounded, i.e. unstable. Therefore, a regularizing stopping crite-rion must be used in order to cease the iterative process at thepoint where the errors in the numerical solutions start increasing.

To define the stopping criterion required for regularizing/stabi-lizing the iterative methods analysed in this paper, for k P 1, thefollowing convergence error is introduced:

EðkÞ ¼ kAUðkÞ � BTðkÞk; ð59Þ

where A and B are the BEM matrices. Here the vectors U(k) and T(k)

are given as follows:

(i) For the alternating iterative algorithm with relaxation I

0.0

0.0

0.

0.

1.

5.

10.

Accu

racy

erro

r, e t

5f iterat

pt = 1%pt = 2%pt = 3%

of the nr the Ca

UðkÞ ¼U

Uð2k�1Þ

� �;

Ud ¼ uðd;1Þ; . . . ;uðd;N0Þ T

;

Uð2k�1Þ ¼ uð2k�1;1Þ; . . . ;uð2k�1;N0Þ T

; ð60:1Þ

TðkÞ ¼W

Tð2kÞ

� �; Wd ¼ wðd;1Þ; . . . ;wðd;N0Þ

T;

Tð2kÞ ¼ tð2k;N0þ1Þ; . . . ; tð2k;N0þN1Þ T

: ð60:2Þ


1

5

1

5

0

0

0

pt = 1%pt = 2%pt = 3%

0 100 500ions, k

umber of iterations, k, obtained using the alternating iterative algorithm II,uchy problem given by Example 1.


(ii) For the alternating iterative algorithm with relaxation II

Fig. 5.discrep

Table 1The valalgorithproblem

h

0.10

0.50

1.00

1.50

1.80

UðkÞ ¼U

Uð2kÞ

� �; Ud ¼ uðd;1Þ; . . . ;uðd;N0Þ

T;

Uð2kÞ ¼ uð2k;1Þ; . . . ;uð2k;N0Þ T

; ð61:1Þ

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u1

Analyticalpu = 1%pu = 2%pu = 3%

-1-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u2

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 1/10

10


-1-1.0

-0.5

0.0

0.5

1.0

t 2/10

10

The analytical and numerical displacements (a) u1jC1and (b) u2jC1

, and tractions (c) tancy principle, h = 1.50 and various amounts of noise added into ujC0

, i.e. pu 2 {1%,2%,3%

ues of the optimal iteration number, kopt, the corresponding accuracy errors, eu(kopt) andm I, the discrepancy principle, various amounts of noise added into ujC0

, i.e. pu 2 {1%,2%, 3%}given by Example 1.

pu (%) pt (%) kopt eu(kopt

1 0 203 0.49472 0 175 0.93363 0 158 0.13541 0 40 0.47992 0 35 0.89873 0 32 0.12951 0 20 0.45302 0 17 0.87293 0 16 0.12371 0 13 0.43342 0 12 0.80233 0 11 0.11611 0 11 0.41552 0 10 0.77973 0 9 0.1134

TðkÞ ¼W

Tð2k�1Þ

� �;

Wd ¼ wðd;1Þ; . . . ;wðd;N0Þ T

;

Tð2k�1Þ ¼ tð2k�1;N0þ1Þ; . . . ; tð2k�1;N0þN1Þ T

: ð61:2Þ

.0 -0.8 -0.6 -0.4 -0.2 0.0/2


.0 -0.8 -0.6 -0.4 -0.2 0.0/2


1jC1and (d) t2jC1

, obtained using the alternating iterative algorithm I, the} and pt = 0%, for the Cauchy problem given by Example 1.

et(kopt), and the computational time, obtained using the alternating iterativeand pt = 0%, and various values for the relaxation parameter, h, for the Cauchy

) et(kopt) CPU time [sec]

6 � 10�2 0.47071 � 10�1 397.822 � 10�2 0.86381 � 10�1 350.183 � 10�1 0.12277 � 100 303.750 � 10�2 0.45935 � 10�1 76.288 � 10�2 0.84196 � 10�1 67.595 � 10�1 0.11952 � 100 63.534 � 10�2 0.44108 � 10�1 40.396 � 10�2 0.82284 � 10�1 35.510 � 10�1 0.11567 � 100 36.454 � 10�2 0.42491 � 10�1 27.677 � 10�2 0.77212 � 10�1 26.518 � 10�1 0.11010 � 100 24.796 � 10�2 0.40854 � 10�1 24.627 � 10�2 0.74962 � 10�1 23.106 � 10�1 0.10779 � 100 21.60


The alternating iterative algorithms I and II described in Sec-
tion 3 are ceased according to the discrepancy principle of Morozov[30], see also Marin et al. [21] and Comino et al. [5], namely at theoptimal iteration number, kopt, which is the smallest integer with
EðkÞ OðdÞ: ð62Þ

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u1

Analyticalpt = 1%pt = 2%pt = 3%

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 1/10

10



, and tractiodiscrepancy principle, h = 1.50 and various levels of noise added into tjC0

, i.e. pu = 0% an

Table 2The values of the optimal iteration number, kopt, the corresponding accuracy errors, eu(ko

algorithm I, the discrepancy principle, various amounts of noise added into tjC0, i.e. pu = 0%

problem given by Example 1.

h pu (%) pt (%) kopt

0.10 0 1 1520 2 1230 3 106

0.50 0 1 300 2 250 3 22

1.00 0 1 150 2 130 3 11

1.50 0 1 100 2 90 3 8

1.80 0 1 90 2 70 3 7

Fig. 3(c) presents the evolution of the convergence error E withrespect to the number of iterations performed, k, using the alter-nating iterative algorithm I, h = 1.50 and various levels of Gauss-ian noise added into the displacements on the over-specifiedboundary C0, namely pu 2 {1%,2%,3%}, for the Cauchy problem

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u2


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 2/10

10


ns (c) t1jC1and (d) t2jC1

, obtained using the alternating iterative algorithm I, thed pt 2 {1%,2%,3%}, for the Cauchy problem given by Example 1.

pt) and et(kopt), and the computational time, obtained using the alternating iterativeand pt 2 {1%,2%,3%}, and various values for the relaxation parameter, h, for the Cauchy

eu(kopt) et(kopt) CPU time [sec]

0.92226 � 10�2 0.73617 � 10�1 283.120.18373 � 10�1 0.14605 � 100 224.780.27449 � 10�1 0.21775 � 100 194.010.89794 � 10�2 0.76813 � 10�1 58.140.17090 � 10�1 0.14596 � 100 49.820.25018 � 10�1 0.21342 � 100 44.500.81957 � 10�2 0.77429 � 10�1 31.680.14745 � 10�1 0.13865 � 100 28.010.24051 � 10�1 0.22624 � 100 24.840.73335 � 10�2 0.77341 � 10�1 22.670.12527 � 10�1 0.12976 � 100 21.010.19075 � 10�1 0.19747 � 100 19.620.54442 � 10�2 0.60551 � 10�1 20.070.13645 � 10�1 0.15775 � 100 17.310.16103 � 10�1 0.18018 � 100 17.32

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u1


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u2


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 1/10

10


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 2/10

10




, obtained using the alternating iterative algorithm II, thediscrepancy principle, h = 1.50 and various amounts of noise added into ujC0

, i.e. pu 2 {1%,2%,3%} and pt = 0%, for the Cauchy problem given by Example 1.


given by Example 1. By comparing Figs. 3(a)–(c), it can be seenthat selecting the optimal iteration number, kopt, according tothe stopping rule (62) captures very well the minimum valuesfor the accuracy errors eu and et. Therefore, Eq. (62) representsa stabilizing stopping criterion for the alternating iterative algo-rithm with relaxation I, at the same time being consistent withthe findings of Marin et al. [21] and Comino et al. [5], who ana-lysed a particular case of the aforementioned algorithm, namelyh = 1, for two-dimensional isotropic and anisotropic linear elasticmaterials, respectively. Although not illustrated, it is importantto mention that similar results and conclusions have been ob-tained for the other examples considered and h 2 (0,2). The sameconclusions can be drawn if the alternating iterative algorithmwith relaxation II is applied to solving the Cauchy problem givenby Example 1, using h = 1.50 and various amounts of noise addedinto the traction vector on C0, namely pt 2 {1%,2%,3%}, seeFigs. 4(a)–(c).

As mentioned in Section 6.2, for exact data the iterative processis convergent with respect to increasing the number of iterations,k, since the accuracy errors eu and et keep decreasing even after alarge number of iterations, see Fig. 1. It should be noted that, in thiscase, a stopping criterion is not necessary since the numerical solu-tion is convergent with respect to increasing the number of itera-tions. However, even exact Cauchy data on C0 the errors E, eu

and et have a similar behaviour and the error E may be used to stop

the iterative process at the point where the rate of convergence isvery small and no substantial improvement in the numerical solu-tion is obtained if the iterative process is continued. Hence it canbe concluded that the regularizing stopping criterion proposedfor the alternating iterative algorithms with relaxation I and II isvery efficient in locating the point where the errors start increasingand the iterative process should be ceased.

6.4. Results obtained with noisy data: Stability of the algorithms

Based on the stopping criterion (62) described in Section 6.3,the numerical results obtained for the x1– and x2 – componentsof the displacement vector, obtained on the under-specifiedboundary C1 using the alternating iterative algorithm I andh = 1.50, and their corresponding analytical values are presentedin Figs. 5(a) and (b), respectively, for various amounts of noiseadded into the displacement vector ujC0

, i.e. pu 2 {1%,2%,3%}, inthe case of Example 1. The associated analytical and numerical val-ues for t1jC1

and t2jC1, retrieved using the alternating iterative algo-

rithm I and the regularizing stopping criterion (62), are illustratedin Figs. 5(c) and (d), respectively. It can be seen from Figs. 5(a)–(c)that the numerical solution for both the displacement and tractionvectors is a stable approximation to the corresponding exact solu-tion, free of unbounded and rapid oscillations, and it converges tothe exact solution as pu decreases.

Table 3The values of the optimal iteration number, kopt, the corresponding accuracy errors, eu(kopt) and et(kopt), and the computational time, obtained using the alternating iterativealgorithm II, the discrepancy principle, various amounts of noise added into ujC0

, i.e. pu 2 {1%,2%,3%} and pt = 0%, and various values for the relaxation parameter, h, for the Cauchyproblem given by Example 1.

h pu (%) pt (%) kopt eu(kopt) et(kopt) CPU time [sec]

0.10 1 0 423 0.40043 � 10�2 0.43267 � 10�1 777.072 0 372 0.73929 � 10�2 0.78421 � 10�1 678.933 0 343 0.10528 � 10�1 0.11040 � 100 611.15

0.50 1 0 68 0.39913 � 10�2 0.43260 � 10�1 125.152 0 60 0.73697 � 10�2 0.78449 � 10�1 115.823 0 55 0.10486 � 10�1 0.11010 � 100 106.93

1.00 1 0 23 0.38741 � 10�2 0.42215 � 10�1 48.702 0 20 0.71255 � 10�2 0.75619 � 10�1 42.183 0 19 0.10197 � 10�1 0.10806 � 100 40.92

1.50 1 0 9 0.38849 � 10�2 0.44053 � 10�1 21.952 0 8 0.89069 � 10�2 0.78816 � 10�1 19.513 0 8 0.12481 � 10�1 0.11835 � 100 19.18

1.80 1 0 25 0.14737 � 10�1 0.18721 � 100 51.032 0 22 0.27429 � 10�1 0.35237 � 100 44.103 0 19 0.38212 � 10�1 0.48591 � 100 41.67


The values of the optimal iteration number, kopt, the corre-sponding accuracy errors, eu(kopt) and et(kopt), and the CPU time,obtained using the alternating iterative algorithm I, the stoppingcriterion (62), various levels of noise added into the Dirichlet dataon C0 and various values of the relaxation parameter, h 2 (0,2), forthe Cauchy problem given by Example 1, are presented in Table 1.The following major conclusions can be drawn from Table 1:

(i) For all fixed values of the relaxation parameter h 2 (0,2),both accuracy errors eu(kopt) and et(kopt) decrease as pu

decreases (i.e. the algorithm I is stable with respect todecreasing the level of noise added into the Dirichlet dataon C0), while the optimal number of iterations kopt and, con-sequently, the CPU time required for the alternating iterativealgorithm I to reach the numerical solutions for theunknown displacement and traction vectors on C1 increaseas pu decreases;

(ii) For all fixed amounts of noise added into the displacementson the over-specified boundary C0, pu 2 {1%,2%,3%}, theaccuracy errors eu(kopt) and et(kopt), the optimal number ofiterations, kopt and the CPU time required for the alternatingiterative algorithm I to reach the numerical solutions for theunknown displacement and traction vectors on C1 decreaseas h ? 2, i.e. as more over-relaxation is introduced in thealgorithm I. However, it should be stressed that the differ-ences, in terms of accuracy, between the numerical resultsfor both ujC1

and tjC1, obtained for various values of the

relaxation parameter, h, are not very significant.

In order to assess the performance of the alternating iterativealgorithm I with under-, no and over-relaxation, we exemplify byconsidering Example 1 with pu = 1%: In this case, the CPU timesneeded for the alternating iterative algorithm I with h = 0.50 (un-der-relaxation), h = 1.00 (no relaxation) and h = 1.50 (over-relaxa-tion) to reach the numerical solutions for the displacement andtraction vectors on C1 were found to be 76.28, 40.39 and 27.67 s,respectively, while the corresponding values for the optimal itera-tion number required, kopt, were found to be 40, 20 and 13, respec-tively. This means that, to attain the numerical solutions for theunknown Dirichlet and Neumann data on C1, the alternating iter-ative algorithm I with over-relaxation (h = 1.50) requires a reduc-tion in the number of iterations performed and CPU time byapproximately 35% and 67% with respect to those correspondingto the standard iterative algorithm I as proposed by Kozlov et al.[16], i.e. without relaxation (h = 1.00), and the alternating iterativealgorithm I with under-relaxation (h = 0.50), respectively.

Similar conclusions to those obtained from Figs. 5(a)–(d) can bedrawn from Figs. 6(a)–(c), which present the numerical values forthe displacement and traction vectors obtained on the under-spec-ified boundary C1, in comparison with their analytical counter-parts, using the alternating iterative algorithm I, the regularizingstopping criterion (62), h = 1.50 and various amounts of noiseadded into the traction vector tjC0

, i.e. pt 2 {1%,2%,3%}, for Example1. By comparing Figs. 5 and 6, it can be observed that, as expected,the alternating iterative algorithm I applied to Example 1 is moresensitive to noise added into the traction vector tjC0

than to pertur-bations of the displacement vector ujC0

since the former containfirst-order derivatives of the latter.

Table 2 tabulates the values of the optimal iteration number,kopt, according to the discrepancy principle (62), the correspondingaccuracy errors given by equations (57.1) and (57.2), and the CPUtime, obtained using the alternating iterative algorithm I, variouslevels of noise added into the Neumann data on C0 and various val-ues of the relaxation parameter, h 2 (0,2), for the Cauchy problemgiven by Example 1. From Tables 1 and 2 it can be noticed that thesensitivity of the alternating iterative algorithm I with respect tonoisy Dirichlet and Neumann data on C0, for Example 1, resultsin the following:

(i) More inaccurate numerical results for both ujC1and tjC1

areobtained for perturbed tractions on C0 than for noisy dis-placements on C0;

(ii) The optimal number of iterations kopt and, consequently, theCPU time required for the alternating iterative algorithm I toreach the numerical solutions for the unknown displace-ment and traction vectors on C1 for perturbed displace-ments on C0 are larger that those corresponding to noisytractions on C0.

Accurate, convergent and stable numerical results for the un-known displacements and tractions on C1 have also been obtained,in the case of the Cauchy problem associated with Example 1,when using the alternating iterative algorithm II, h = 1.50 and var-ious amounts of noise added into the displacement vector ujC0

, andthey are illustrated in Figs. 7(a)–(d). On comparing Figs. 5 and 7, itcan be seen that the numerical results for ujC1

and tjC1obtained

using the alternating iterative algorithms I and II, h = 1.50 and var-ious values for pu, in the case of Example 1, have almost the samedegree of accuracy. The same conclusions can be drawn fromTables 1 and 3, with the mention that the latter presents the valuesfor kopt, eu(kopt), et(kopt) and the CPU time, retrieved by employingthe alternating iterative algorithm II, various values for h and pu, in

-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u1


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

u2


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 1/10

10


-1.0 -0.8 -0.6 -0.4 -0.2 0.0/2

-1.0

-0.5

0.0

0.5

1.0

t 2/10

10




, obtained using the alternating iterative algorithm II, thediscrepancy principle, h = 1.50 and various levels of noise added into tjC0

, i.e. pu = 0% and pt 2 {1%,2%,3%}, for the Cauchy problem given by Example 1.

Table 4The values of the optimal iteration number, kopt, the corresponding accuracy errors, eu(kopt) and et(kopt), and the computational time, obtained using the alternating iterativealgorithm II, the discrepancy principle, various amounts of noise added into tjC0

, i.e. pu = 0% and pt 2 {1%,2%,3%}, and various values for the relaxation parameter, h, for the Cauchyproblem given by Example 1.


0.10 0 1 333 0.52271 � 10�2 0.42072 � 10�1 594.540 2 281 0.10454 � 10�1 0.82974 � 10�1 535.070 3 251 0.15605 � 10�1 0.12291 � 100 458.62

0.50 0 1 54 0.50156 � 10�2 0.38912 � 10�1 108.640 2 46 0.98775 � 10�2 0.75368 � 10�1 91.530 3 41 0.15164 � 10�1 0.11457 � 100 79.03

1.00 0 1 18 0.53090 � 10�2 0.36990 � 10�1 40.120 2 16 0.89634 � 10�2 0.62103 � 10�1 33.700 3 14 0.15688 � 10�1 0.10621 � 100 29.28

1.50 0 1 8 0.56415 � 10�2 0.29823 � 10�1 19.920 2 8 0.86669 � 10�2 0.60374 � 10�1 19.810 3 7 0.22843 � 10�1 0.14299 � 100 16.62

1.80 0 1 19 0.52737 � 10�2 0.70820 � 10�1 38.370 2 16 0.11055 � 10�1 0.19004 � 100 32.560 3 16 0.15828 � 10�1 0.24963 � 100 32.07


the case of Example 1. From Table 3 the following comments canbe made regarding the alternating iterative algorithm II:

(i) For all fixed values of the relaxation parameter h 2 (0,2), theevolution of the accuracy errors eu(kopt) and et(kopt), the opti-mal number of iterations kopt and the CPU time required forthe alternating iterative algorithm II to reach the numerical

solutions for the unknown displacement and traction vec-tors on C1 with respect to pu is similar to that associatedwith the alternating iterative algorithm I, see also Table 1;

(ii) For all fixed levels of noise added into the displacements onthe over-specified boundary C0, pu 2 {1%,2%,3%}, the accu-racy errors eu(kopt) and et(kopt), the optimal number of itera-tions kopt and the CPU time required for the alternating

0.1 0.15 0.2 0.25 0.3 0.35 0.4/2

-0.05

0.0

0.05

u1

Analyticalpu = pt = 1%pu = pt = 2%pu = pt = 3%

0.1 0.15 0.2 0.25 0.3 0.35 0.4/2

0.075

0.08

0.085

0.09

0.095

0.1

u 2


0.1 0.15 0.2 0.25 0.3 0.35 0.4/2

-2

-1

0

1

2

t 1/10

10


0.1 0.15 0.2 0.25 0.3 0.35 0.4/2

0.8

1.0

1.2

1.4

1.6

t 2/10

10




, obtained using the alternating iterative algorithm I, thediscrepancy principle, h = 1.50 and various amounts of noise added into the Cauchy data, i.e. pu = pt 2 {1%,2%,3%}, for the Cauchy problem given by Example 2.


iterative algorithm II to reach the numerical solutions for theunknown displacement and traction vectors on C1 decreaseuntil the relaxation parameter, h, reaches a threshold over-relaxation value, say eh 2 ð1;2Þ, after which all of these quan-tities increase as h ? 2. It should be emphasised, however,that the differences, in terms of accuracy, between thenumerical results for both ujC1

and tjC1, obtained for

h 2 0; eh� �, are not very significant.

The performance of the alternating iterative algorithm II withunder-, no and over-relaxation is exemplified by consideringExample 1 with pt = 1%: in this case, the CPU times needed forthe alternating iterative algorithm II with h = 0.50 (under-relaxa-tion), h = 1.00 (no relaxation) and h = 1.50 (over-relaxation) toreach the numerical solutions for ujC1

and tjC1were found to be

125.15, 48.70 and 21.95 s, respectively, while the correspondingvalues for the optimal iteration number required, kopt, were foundto be 68, 23 and 9, respectively. This means that, in order to attainthe numerical solutions for the unknown Dirichlet and Neumanndata on C1, the alternating iterative algorithm II with over-relaxa-tion (h = 1.50) requires a reduction in the number of iterations per-formed, as well as CPU time, by approximately 60% and 86% withrespect to those corresponding to the standard iterative algorithmII as proposed by Kozlov et al. [16], i.e. without relaxation(h = 1.00), and the alternating iterative algorithm II with under-relaxation (h = 0.50), respectively.

Figs. 8(a)–(d) show the numerical values for the displacementand traction vectors obtained on the under-specified boundaryC1, in comparison with their corresponding analytical solutions,using the alternating iterative algorithm II, the discrepancy prin-ciple (62), h = 1.50 and various amounts of noise added into thetraction vector tjC0

, in the case of Example 1. By comparingFigs. 7 and 8, it can be noticed that, as expected, the alternatingiterative algorithm II provides more inaccurate numerical resultsfor ujC1

and tjC1for noisy tractions on C0 than for perturbed dis-

placements on C0 since the former contain first-order derivativesof the latter. From Figs. 6 and 8, it can be seen that the numer-ical results for the displacement and traction vectors on the un-der-specified boundary C1, obtained using the alternatingiterative algorithms I and II, h = 1.50 and various values for pt,in the case of Example 1, have almost the same degree ofaccuracy.

Similar conclusions can be drawn from Tables 2 and 4, wherethe latter presents the values for kopt, eu(kopt), et(kopt) and theCPU time retrieved by employing the alternating iterative algo-rithm II, various values for h and pt, in the case of Example 1. Also,from Tables 3 and 4 the remarks made concerning the sensitivity ofthe alternating iterative algorithm II with respect to noisy Dirichletand Neumann data on C0, for Example 1, are analogous to thosemade regarding the sensitivity of the alternating iterative algo-rithm I with respect to noisy displacements and tractions on C0,for Example 1, when comparing Tables 1 and 2.

Table 5The values of the optimal iteration number, kopt, the corresponding accuracy errors, eu(kopt) and et(kopt), and the computational time, obtained using the alternating iterativealgorithm I, the discrepancy principle, various amounts of noise added into the Cauchy data, i.e. pu = pt 2 {1%,2%,3%, 4%,5%}, and various values for the relaxation parameter, h, forthe Cauchy problem given by Example 2.


0.20 1 1 617 0.64808 � 10�1 0.19698 � 100 329.642 2 385 0.68957 � 10�1 0.24174 � 100 211.733 3 274 0.73787 � 10�1 0.32295 � 100 150.924 4 203 0.79149 � 10�1 0.41627 � 100 111.455 5 155 0.84899 � 10�1 0.50528 � 100 88.09

0.50 1 1 247 0.64808 � 10�1 0.19693 � 100 133.392 2 155 0.68955 � 10�1 0.24156 � 100 86.063 3 110 0.73786 � 10�1 0.32335 � 100 60.984 4 82 0.79145 � 10�1 0.41650 � 100 48.155 5 63 0.84892 � 10�1 0.50545 � 100 35.31

1.00 1 1 124 0.64807 � 10�1 0.19695 � 100 67.962 2 78 0.68953 � 10�1 0.24145 � 100 45.233 3 56 0.73777 � 10�1 0.32248 � 100 31.844 4 42 0.79132 � 10�1 0.41561 � 100 24.605 5 32 0.84889 � 10�1 0.50737 � 100 18.87

1.50 1 1 83 0.64807 � 10�1 0.19696 � 100 46.262 2 52 0.68953 � 10�1 0.24168 � 100 31.823 3 38 0.73768 � 10�1 0.32161 � 100 23.344 4 28 0.79137 � 10�1 0.41866 � 100 17.465 5 22 0.84874 � 10�1 0.50684 � 100 14.64

1.80 1 1 69 0.64807 � 10�1 0.19682 � 100 38.622 2 44 0.68948 � 10�1 0.24104 � 100 25.713 3 32 0.73773 � 10�1 0.32109 � 100 18.844 4 24 0.79178 � 10�1 0.41569 � 100 14.735 5 20 0.84958 � 10�1 0.49521 � 100 12.67

-1.0 -0.5 0.0 0.5 1.0x2

0.138

0.14

0.142

0.144

0.146

0.148

0.15

u1


-1.0 -0.5 0.0 0.5 1.0x2

-0.04

-0.02

0.0

0.02

0.04

u2


-1.0 -0.5 0.0 0.5 1.0x2

1.4

1.6

1.8

2.0

t 1/10

10


-1.0 -0.5 0.0 0.5 1.0x2

-0.6

-0.4

-0.2

0.0

0.2

0.4

t 2/10

10




, obtained using the alternating iterative algorithm I, thediscrepancy principle, h = 1.50 and various amounts of noise added into the Cauchy boundary data, i.e. pu = pt 2 {1%,2%,3%}, for the Cauchy problem given by Example 3.



6.5. Limitations of the algorithms

The numerical values for the displacements u1jC1and u2jC1

, ob-tained using the alternating iterative algorithm I, the discrepancyprinciple and h = 1.50, and their corresponding analytical valuesare presented in Figs. 9(a) and (b), respectively, for variousamounts of noise added into the Cauchy data, i.e.pu = pt 2 {1%,2%,3%}, in the case of the Cauchy problem given byExample 2. Figs. 9(c) and (d) illustrate the associated analyticaland numerical values for t1jC1

and t2jC1, respectively, retrieved

using the alternating iterative algorithm I and the regularizingstopping criterion (62). Although the errors in the numerical re-sults obtained for both the displacement and traction vectors onthe under-specified boundary C1 decrease with respect to decreas-ing the level of noise added into the Cauchy data on C0, it can beseen from Figs. 9(a)–(d) that the numerically retrieved displace-ment and traction vectors on C1 still remain inaccurate approxi-mations for their corresponding analytical counterparts. Inaddition, the components of the numerical traction vector on C1

are very inaccurate approximations for their analytical counter-parts, as well as highly oscillatory, at the endpoints of the under-specified boundary C1.

Table 5 presents the values of the optimal iteration number, thecorresponding accuracy errors given by Eqs. (57.1) and (57.2), andthe CPU time, obtained using the alternating iterative algorithm I,pu = pt 2 {1%,2%,3%} and various values of the relaxation parame-ter, h 2 (0,2), for the Cauchy problem given by Example 2. Similarconclusions regarding the accuracy and stability of the alternatingiterative algorithm I applied to the Cauchy problem with noisyCauchy data on C0 in the simply connected domain with smoothboundary, as given by Example 2, can be drawn from this table.Figs. 9(a)–(d) and Table 5 clearly show the difficulty of the alter-nating iterative algorithm I in reconstructing the unknown dis-placements and tractions on the under-specified boundary C1

from noisy Cauchy measurements on the remaining boundary C0

in the case of a simply connected domain and hence the limitationof the proposed numerical procedure for such geometries. For thistype of problems, special treatment is required for the displace-ment vector at the common endpoints of the over-and under-specified boundaries, i.e. points belonging to C0 \ C1. One mayuse weighted functions at each iteration of the algorithm in orderto cancel the singularity, see e.g. Johansson and Marin [10], but thiswill be investigated in a future work. Similar results have beenobtained when solving the Cauchy problem given by Example 3,i.e. simply connected domain with piecewise smooth boundary,using the alternating iterative algorithm I, h = 1.50 andpu = pt 2 {1%,2%,3%}, as can be seen from Figs. 10(a)–(d).

7. Conclusions

In this paper, we proposed two algorithms involving the relax-ation of either the given Dirichlet data (displacement vector) or theprescribed Neumann data (traction vector) on the over-specifiedboundary in the case of the alternating iterative algorithm of Koz-lov et al. [16] applied to Cauchy problems in linear elasticity. Aconvergence proof of these relaxation methods was given, as wellas a regularizing stopping criterion. The aforementioned alternat-ing iterative algorithms with relaxation were implemented, fortwo-dimensional isotropic linear elastic materials, by employingcontinuous and discontinuous linear boundary elements. Thenumerical results obtained using these procedures, in the case ofdoubly connected domains, i.e. domains whose over-and under-specified boundaries have no common points, showed thenumerical stability, convergence, accuracy, consistency and com-putational efficiency of the proposed method. More specifically,

both alternating iterative algorithms with constant over-relaxationof either the given displacement vector or the prescribed tractionvector on the over-specified boundary significantly reduced thenumber of iterations performed in order to achieve the numericalsolutions for the displacement and traction vectors on the under-specified boundary, as well as the CPU time allocated for thispurpose.

The limitation of the proposed algorithms is related tonumerically solving Cauchy problems in two-dimensional simplyconnected domains, i.e. geometries for which the over-andunder-specified boundaries have common endpoints. In suchsituations, the present alternating iterative methods fail to produceaccurate approximations for the boundary displacements andtractions on the under-specified boundary from measured Cauchydata available on the remaining boundary. Future work is relatedto adapting the present procedures to Cauchy problems for linearelastic materials in domains with corners by employing weightedfunctions.

Acknowledgement

The financial support received by L. Marin from the RomanianMinistry of Education, Research and Innovation through IDEI Pro-gramme, Exploratory Research Projects, Grant PN II-ID-PCE-1248/2008, is gratefully acknowledged.

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a relaxation method of an alternating iterative algorithm for the cauchy problem in linear isotropic...

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