the method of fundamental solutions for the cauchy problem in two-dimensional linear elasticity

International Journal of Solids and Structures 41 (2004) 3425–3438

www.elsevier.com/locate/ijsolstr

The method of fundamental solutions for the Cauchyproblem in two-dimensional linear elasticity

Liviu Marin a,*, Daniel Lesnic b

a School of the Environment, University of Leeds, Leeds LS2 9JT, UKb Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received 20 January 2004; received in revised form 20 January 2004

Available online 11 March 2004

Abstract

In this paper, the application of the method of fundamental solutions to the Cauchy problem in two-dimensional

isotropic linear elasticity is investigated. The resulting system of linear algebraic equations is ill-conditioned and

therefore its solution is regularised by employing the first-order Tikhonov functional, while the choice of the regu-

larisation parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise

smooth geometries, as well as for constant and linear stress states. The convergence and the stability of the method with

respect to increasing the number of source points and the distance between the source points and the boundary of the

solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Meshless method; Method of fundamental solutions; Cauchy problem; Linear elasticity; Regularisation; Inverse problem

1. Introduction

The method of fundamental solutions (MFS) was originally introduced by Kupradze and Aleksidze

(1964), whilst its numerical formulation was first given by Mathon and Johnston (1977). The main idea of

the MFS consists of approximating the solution of the problem by a linear combination of fundamental

solutions with respect to some singularities/source points which are located outside the domain. Then, theoriginal problem is reduced to determining the unknown coefficients of the fundamental solutions and the

coordinates of the source points by requiring the approximation to satisfy the boundary conditions and

hence solving a non-linear problem. If the source points are a priori fixed then the coefficients of the MFS

approximation are determined by solving a linear problem. An excellent survey of the MFS and related

methods over the past three decades has been presented by Fairweather and Karageorghis (1998).

The advantages of the MFS over domain discretisation methods, such as the finite difference (FDM) and

the finite element methods (FEM), are very well documented (see e.g. Fairweather and Karageorghis,

* Corresponding author. Tel.: +44-0113-3436744; fax: +44-0113-3436716.

E-mail address: [email protected] (L. Marin).

0020-7683/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2004.02.009

mail to: [email protected]

3426 L. Marin, D. Lesnic / International Journal of Solids and Structures 41 (2004) 3425–3438

1998). In addition, the MFS has all the advantages of boundary methods, such as the boundary element

method (BEM), as well as several advantages over other boundary methods. For example, the MFS does

not require an elaborate discretisation of the boundary, integrations over the boundary are avoided, the

solution in the interior of the domain is evaluated without extra quadratures, its implementation is veryeasy and only little data preparation is required. The most arguable issue regarding the MFS is still the

location of the source points. However, this problem can be overcome by employing a non-linear least-

squares minimisation procedure. Alternatively, the source points can be prescribed a priori (see Bala-

krishnan and Ramachandran, 1999; Golberg and Chen, 1996, 1999) and the post-processing analysis of the

errors can indicate their optimal location.

The MFS has been successfully applied for solving a wide variety of boundary value problems. Kara-

georghis and Fairweather (1987) have solved numerically the biharmonic equation using the MFS and later

their method has been modified in order to take into account the presence of boundary singularities in boththe Laplace and the biharmonic equations by Poullikkas et al. (1998a). Furthermore, Poullikkas et al.

(1998b) have investigated the numerical solutions of the inhomogeneous harmonic and biharmonic

equations by reducing these problems to the homogeneous corresponding cases and subtracting a particular

solution of the governing equation. The MFS has been formulated for three-dimensional Signorini

boundary value problems and it has been tested on a three-dimensional electropainting problem related to

the coating of vehicle roofs in Poullikkas et al. (2001). Whilst Karageorghis and Fairweather (2000) have

studied the use of the MFS for the approximate solution of three-dimensional isotropic materials with

axisymmetrical geometry and both axisymmetrical and arbitrary boundary conditions. The application ofthe MFS to two-dimensional problems of steady-state heat conduction and elastostatics in isotropic and

anisotropic bimaterials has been addressed by Berger and Karageorghis (1999, 2001). Karageorghis (2001)

has investigated the calculation of the eigenvalues of the Helmholtz equation subject to homogeneous

Dirichlet boundary conditions for circular and rectangular geometries by employing the MFS, whilst

Poullikkas et al. (2002) have successfully applied the MFS for solving three-dimensional elastostatics

problems. The MFS, in conjunction with singular value decomposition, has been employed by Rama-

chandran (2002) in order to obtain numerical solutions of the Laplace and the Helmholtz equations. Re-

cently, Balakrishnan et al. (2002) have proposed an operator splitting-radial basis function method as ageneric solution procedure for transient nonlinear Poisson problems by combining the concepts of operator

splitting, radial basis function interpolation, particular solutions and the MFS.

In most boundary value problems in solid mechanics, the governing system of partial differential

equations, i.e. the equilibrium, constitutive and kinematics equations, have to be solved with the appro-

priate initial and boundary conditions for the displacement and/or traction vectors, i.e. Dirichlet, Neumann

or mixed boundary conditions. These problems are called direct problems and their existence and

uniqueness have been well established (see for example Knops and Payne, 1986). However, there are other

engineering problems which do not belong to this category. For example, the material properties and/or theexternal sources are unknown, the geometry of a portion of the boundary is not determined or the

boundary conditions are incomplete, either in the form of underspecified and overspecified boundary

conditions on different parts of the boundary or the solution is prescribed at some internal points in the

domain. These are inverse problems, and it is well known that they are generally ill-posed, i.e. the existence,

uniqueness and stability of their solutions are not always guaranteed (see e.g. Hadamard, 1923).

A classical example of an inverse problem in elasticity is the Cauchy problem in which both displacement

and traction boundary conditions are prescribed only on a part of the boundary of the solution domain,

whilst no information is available on the remaining part of the boundary. This problem has been studied byYeih et al. (1993), who have analysed its existence, uniqueness and continuous dependence on the data and

have proposed an alternative regularisation procedure, namely the fictitious boundary indirect method,

based on the simple or double layer potential theory. The numerical implementation of the aforementioned

method has been undertaken by Koya et al. (1993), who have used the BEM and the Nystr€om method for

L. Marin, D. Lesnic / International Journal of Solids and Structures 41 (2004) 3425–3438 3427

discretising the integrals. However, this formulation has not yet removed the problem of multiple inte-

grations. Marin et al. (2001) have determined the approximate solutions to the Cauchy problem in linear

elasticity using an alternating iterative BEM which reduced the problem to solving a sequence of well-posed

boundary value problems and they have later extended this numerical method to singular Cauchy problems(see Marin et al., 2002a). Whilst Huang and Shih (1997) and Marin et al. (2002b) have both used the

conjugate gradient method combined with the BEM, in order to solve the same problem. Recently, the

Tikhonov regularisation method and the singular value decomposition, in conjunction with the BEM, have

been employed by Marin and Lesnic (2002a,b) to solve the two-dimensional Cauchy problem in linear

elasticity.

To our knowledge, the application of the MFS to the Cauchy problem in linear elasticity has not been

investigated yet. The MFS discretised system of equations is ill-conditioned and hence it is solved by

employing the first-order Tikhonov regularisation method (see e.g. Tikhonov and Arsenin, 1986), whilst thechoice of the regularisation parameter is based on the L-curve criterion (see Hansen, 2001). Three

benchmark examples for the two-dimensional isotropic linear elasticity involving smooth and piecewise

smooth geometries and dealing with constant and linear stress states, i.e. linear and quadratic displace-

ments, respectively, are investigated and the convergence and stability of the method with respect to the

location and the number of source points and the amount of noise added into the Cauchy input data,

respectively, are analysed.

2. Mathematical formulation

Consider an isotropic linear elastic material which occupies an open bounded domain X � R2 and as-

sume that X is bounded by a piecewise smooth curve C ¼ oX, such that C ¼ C1 [ C2, where C1;C2 6¼ ; and

C1 \ C2 ¼ ;. In the absence of body forces, the equilibrium equations with respect to the displacement

vector uðxÞ, also known as the Lam�e or Navier equations, are given by (see e.g. Landau and Lifshits, 1986)

Go2uiðxÞoxjoxj

þ G1 2m

o2ujðxÞoxioxj

¼ 0; x 2 X; i ¼ 1; 2; ð1Þ

where G and m are the shear modulus and Poisson ratio, respectively, and the customary standard Cartesiannotation for summation over repeated indices is used. The strains eijðxÞ, i; j ¼ 1, 2, are related to the dis-

placement gradients by the kinematic relations

eijðxÞ ¼1

2

ouiðxÞoxj

�þ oujðxÞ

oxi

�; x 2 X; i; j ¼ 1; 2; ð2Þ

while the stresses rijðxÞ, i; j ¼ 1; 2, are related to the strains through the constitutive law (Hooke’s law),

namely

rijðxÞ ¼ 2G eijðxÞ�

þ m1 2m

ekkðxÞ�; x 2 X; i; j ¼ 1; 2; ð3Þ

with dij the Kronecker delta tensor. We now let nðxÞ be the outward normal vector at C and tðxÞ be the

traction vector at a point x 2 C whose components are defined by

tiðxÞ ¼ rijðxÞnjðxÞ; x 2 C; i ¼ 1; 2: ð4Þ

In the direct problem formulation, the knowledge of the displacement and/or traction vectors on the whole

boundary C gives the corresponding Dirichlet, Neumann, or mixed boundary conditions which enables usto determine the displacement vector in the domain X. Then, the strain tensor eij can be calculated from the

kinematic relations (2) and the stress tensor is determined using the constitutive law (3).


If it is possible to measure both the displacement and traction vectors on a part of the boundary C, say

C1, then this leads to the mathematical formulation of an inverse problem consisting of Eq. (1) and the

boundary conditions

uiðxÞ ¼ ~uiðxÞ; tiðxÞ ¼ ~tiðxÞ; x 2 C1; i ¼ 1; 2; ð5Þ

where eu and et are prescribed vector valued functions. In the above formulation of the boundary conditions(5), it can be seen that the boundary C1 is overspecified by prescribing both the displacement ujC1

¼ eu and

the traction tjC1¼ et vectors, whilst the boundary C2 is underspecified since both the displacement ujC2

and

the traction tjC2vectors are unknown and have to be determined. This problem, termed the Cauchy

problem, is much more difficult to solve both analytically and numerically than the direct problem, since the

solution does not satisfy the general conditions of well-posedness. Although the problem may have a

unique solution, it is well known (see e.g. Hadamard, 1923) that this solution is unstable with respect to

small perturbations in the data on C1. Thus the problem is ill-posed and we cannot use a direct approach,

such as the Gauss elimination method, in order to solve the system of linear equations which arises from thediscretisation of the partial differential equation (1) and the boundary conditions (5). Therefore, regular-

isation methods are required in order to solve accurately the Cauchy problem in linear elasticity.

3. Method of fundamental solutions

The fundamental solutions U ðjÞ, j ¼ 1; 2, of the Lam�e system (1) in the two-dimensional case are given by

(see e.g. Berger and Karageorghis, 2001)

U ðjÞðx; yÞ ¼ Uijðx; yÞei; x 2 X; y 2 R2 n X; j ¼ 1; 2; ð6Þ

where y is a source point, ei, i ¼ 1; 2, is the unit vector along the xi-axis and

Uijðx; yÞ ¼ 1

8pGð1 mÞ ð3�

4mÞ ln rðx; yÞdij ðxi yiÞðxj yjÞ

r2ðx; yÞ

�; i; j ¼ 1; 2: ð7Þ

Here m ¼ m for the plane strain state and m ¼ m=ð1þ mÞ in the plane stress state, rðx; yÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 y1Þ2 þ ðx2 y2Þ2

qrepresents the distance between the domain point x and the source point y and dij is

the Kronecker delta tensor.

The main idea of the MFS consists of the approximation of the displacement vector in the solu-

tion domain by a linear combination of fundamental solutions with respect to M source points yj in the

form

uðxÞ � uMða; b;Y ; xÞ ¼ ajUð1Þðx; yjÞ þ bjU

ð2Þðx; yjÞ; x 2 X; ð8Þ

where a ¼ ða1; . . . ; aMÞ, b ¼ ðb1; . . . ; bMÞ and y is a 2M-vector containing the coordinates of the source

points yj, j ¼ 1; . . . ;M . On taking into account the kinematic relations (2), the constitutive law (3), the

definitions of the components of the traction vector (4) and the fundamental solutions (6) then the traction

vector can be approximated on the boundary C by

tðxÞ � tMða; b;Y ; xÞ ¼ ajTð1Þðx; yjÞ þ bjT

ð2Þðx; yjÞ; x 2 C; ð9Þ

where

TðjÞðx; yÞ ¼ Tijðx; yÞei; x 2 C; y 2 R2 n X; j ¼ 1; 2 ð10Þ


and

T1jðx; yÞ ¼ 2G1 m1 2m

oU1jðx; yÞox1

�þ m

1 2moU2jðx; yÞ

ox2

�n1ðxÞ

þ GoU1jðx; yÞ

ox2

�þ oU2jðx; yÞ

ox1

�n2ðxÞ; j ¼ 1; 2;

T2jðx; yÞ ¼ GoU1jðx; yÞ

ox2

�þ oU2jðx; yÞ

ox1

�n1ðxÞ

þ 2Gm

1 2moU1jðx; yÞ

ox1

�þ 1 m

1 2moU2jðx; yÞ

ox2

�n2ðxÞ; j ¼ 1; 2:

ð11Þ

If N collocation points xi, i ¼ 1; . . . ;N , are chosen on the overspecified boundary C1 of the solution domain

X and the location of the source points yj, j ¼ 1; . . . ;M , is set then Eqs. (8) and (9) recast as a system of 4Nlinear algebraic equations with 2M unknowns which can be generically written as

AX ¼ F; ð12Þ
where the MFS matrix A, the unknown vector X and the right-hand side vector F are given by
Aðk1ÞNþi;ðl1ÞMþj ¼ Uklðxi; yjÞ; Aðkþ1ÞNþi;ðl1ÞMþj ¼ Tklðxi; yjÞ;Xj ¼ aj; XMþj ¼ bj; Fðk1ÞNþi ¼ ukðxiÞ; Fðkþ1ÞNþi ¼ tkðxiÞ;i ¼ 1; . . . ;N ; j ¼ 1; . . . ;M ; k; l ¼ 1; 2:

ð13Þ

It should be noted that in order to uniquely determine the solution X of the system of linear algebraic

equations (12), i.e. the coefficients aj and bj, j ¼ 1; . . . ;M , in the approximations (8) and (9), the number Nof boundary collocation points and the number M of source points must satisfy the inequality M 6 2N .However, the system of linear algebraic equation (12) cannot be solved by direct methods, such as the least-

squares method, since such an approach would produce a highly unstable solution. Most of the standard

numerical methods cannot achieve a good accuracy in the solution of the system of linear algebraic

equation (12) due to the large value of the condition number of the matrix A which increases dramatically

as the number of boundary collocation points and source points increases. It should be mentioned that for

inverse problems, the resulting systems of linear algebraic equations are ill-conditioned, even if other well-

known numerical methods (FDM, FEM or BEM) are employed. Although the MFS system of linear

algebraic equation (12) is ill-conditioned even when dealing with direct problems, the MFS has no longerthis disadvantage in comparison with other numerical methods and, in addition, it preserves its advantages,

such as the lack of any mesh, the high accuracy of the numerical results, etc.

3.1. Tikhonov regularisation method

Several regularisation procedures have been developed to solve such ill-conditioned problems (see for

example Hansen, 1998). However, we only consider the Tikhonov regularisation method (see Tikhonov and

Arsenin, 1986) in our study since it is simple, non-iterative and it provides an explicit solution, see Eq. (16)below. Furthermore, the Tikhonov regularisation method is feasible to apply for large systems of equations

unlike the singular value decomposition which may become prohibitive for such large problems (see

Hansen, 2001).

The Tikhonov regularised solution to the system of linear algebraic equation (12) is sought as

Xk : TkðXkÞ ¼ minX2R2M

TkðXÞ; ð14Þ


where Tk represents the kth order Tikhonov functional given by

Tkð�Þ : R2M ! ½0;1Þ; TkðXÞ ¼ jjAX Fjj22 þ k2jjRðkÞX jj22; ð15Þ
the matrix RðkÞ 2 Rð2MkÞ�2M induces a Ck-constraint on the solution X and k > 0 is the regularisation
parameter to be chosen. Formally, the Tikhonov regularised solution Xk of the problem (14) is given as the

solution of the regularised equation

ðATAþ k2RðkÞTRðkÞÞX ¼ ATF: ð16Þ
Regularisation is necessary when solving ill-conditioned systems of linear equations because the simple
least-squares solution, i.e. k ¼ 0, is completely dominated by contributions from data errors and rounding

errors. By adding regularisation we are able to damp out these contributions and maintain the norm

jjRðkÞXjj2 to be of reasonable size.

3.2. Choice of the regularisation parameter

The choice of the regularisation parameter in Eq. (16) is crucial for obtaining a stable solution and this isdiscussed next. If too much regularisation, or damping, i.e. k2 is large, is imposed on the solution of Eq. (16)

then it will not fit the given data F properly and the residual norm jjAX Fjj2 will be too large. If too little

regularisation is imposed on the solution of equation (16), i.e. k2 is small, then the fit will be good, but the

solution will be dominated by the contributions from the data errors, and hence jjRðkÞX jj2 will be too large.

It is quite natural to plot the norm of the solution as a function of the norm of the residual parametrised by

the regularisation parameter k, i.e. fjjAXk Fjj2; jjRðkÞXkjj2; k > 0g. Hence, the L-curve is really a trade-off

curve between two quantities that both should be controlled and, according to the L-curve criterion, the

optimal value kopt of the regularisation parameter k is chosen at the ‘‘corner’’ of the L-curve (see Hansen,1998, 2001).

As with every practical method, the L-curve has its advantages and disadvantages. There are two main

disadvantages or limitations of the L-curve criterion. The first disadvantage is concerned with the recon-

struction of very smooth exact solutions (see Tikhonov et al., 1998). For such solutions, Hanke (1996)

showed that the L-curve criterion will fail, and the smoother the solution, the worse the regularisation

parameter k computed by the L-curve criterion. However, it is not clear how often very smooth solutions

arise in applications. The second limitation of the L-curve criterion is related to its asymptotic behaviour as

the problem size 2M increases. As pointed out by Vogel (1996), the regularisation parameter k computed bythe L-curve criterion may not behave consistently with the optimal parameter kopt as 2M increases.

However, this ideal situation in which the same problem is discretised for increasing 2M may not arise so

often in practice. Often the problem size 2M is fixed by the particular measurement setup given by N , and if

a larger 2M is required then a new experiment must be undertaken since the inequality M 6 2N must be

satisfied. Apart from these two limitations, the advantages of the L-curve criterion are its robustness and

ability to treat perturbations consisting of correlated noise (see for more details Hansen, 2001).

4. Numerical results and discussion

Although neither convergence nor estimate proofs are as yet available for the MFS, the numerical results

presented in this section for the Cauchy problem in two-dimensional isotropic linear elasticity indicate that

the proposed method is feasible and efficient. In order to present the performance of the MFS in con-

junction with the first-order Tikhonov regularisation method, we consider an isotropic linear elastic

medium characterised by the material constants G ¼ 3:35� 1010 N/m2 and m ¼ 0:34 corresponding to acopper alloy and we solve the Cauchy problem (1) and (5) for three typical examples in both smooth and

piecewise smooth geometries:


Example 1. We consider the following analytical solution for the displacements:

uðanÞi ðx1; x2Þ ¼1 m

2Gð1þ mÞ r0xi; i ¼ 1; 2; r0 ¼ 1:5� 1010N=m2; ð17Þ

in the unit disk X ¼ fx ¼ ðx1; x2Þjx21 þ x22 < 1g, which corresponds to a uniform hydrostatic stress which is

given by

rðanÞij ðx1; x2Þ ¼ r0dij; i; j ¼ 1; 2: ð18Þ

Here C1 ¼ fx 2 Cj06 hðxÞ6 p=4g and C2 ¼ fx 2 Cjp=4 < hðxÞ < 2pg, where hðxÞ is the angular polarcoordinate of x.


uðanÞ1 ðx1; x2Þ ¼1

2Gð1þ mÞ r0x1; uðanÞ2 ðx1; x2Þ ¼ m2Gð1þ mÞ r0x2; ð19Þ

in the square X ¼ fx ¼ ðx1; x2Þj 1 < x1; x2 < 1g which corresponds to a uniform traction stress which is

given by

rðanÞ11 ðx1; x2Þ ¼ r0; rðanÞ

12 ðx1; x2Þ ¼ rðanÞ22 ðx1; x2Þ ¼ 0: ð20Þ

Here C1 ¼ fx 2 Cjx1 ¼ 1;16 x2 6 1g [ fx 2 Cj 16 x1 6 1; x2 ¼ 1g and C2 ¼ fx 2 Cjx1 ¼ 1;16 x2 <1g [ fx 2 Cj 16 x1 < 1; x2 ¼ 1g:


uðanÞ1 ðx1; x2Þ ¼1

2Gð1þ mÞ r0x1x2; uðanÞ2 ðx1; x2Þ ¼ 1

4Gð1þ mÞ r0½ðx21 1Þ þ mx22�; ð21Þ

in the square X ¼ fx ¼ ðx1; x2Þj 1 < x1; x2 < 1g which corresponds to a linear traction stress state which is

given by

rðanÞ11 ðx1; x2Þ ¼ r0x2; rðanÞ

12 ðx1; x2Þ ¼ rðanÞ22 ðx1; x2Þ ¼ 0: ð22Þ

Here C1 ¼ fx 2 Cjx1 ¼ 1;16 x2 6 1g [ fx 2 Cj 16 x1 6 1; x2 ¼ 1g and C2 ¼ fx 2 Cjx1 ¼ 1;16 x2 <1g [ fx 2 Cj 16 x1 < 1; x2 ¼ 1g:

It should be noted that for the examples considered, the Cauchy data is available on a portion C1 of the

boundary C such that measðC1Þ ¼ measðCÞ=4 in the case of Example 1 and measðC1Þ ¼ measðCÞ=2 in the

case of Examples 2 and 3. The Cauchy problems investigated in this study have been solved using a uniform

distribution of both the boundary collocation points xi, i ¼ 1; . . . ;N , and the source points yj, j ¼ 1; . . . ;M ,

with the mention that the later were located on the boundary of the disk Bð0;RÞ, where the radius R > 0 was

chosen such that X � Bð0;RÞ. Furthermore, the number of boundary collocation points was set to N ¼ 40

for the Example 1 and N ¼ 80 for the Examples 2 and 3.

4.1. Stability of the method

In order to investigate the stability of the MFS, the displacement vector ujC1¼ uðanÞjC1

has been per-

turbed as

~ui ¼ ui þ dui; i ¼ 1; 2; ð23Þ

(a) (b)

Fig. 1. (a) The L-curves obtained for various levels of noise added into the displacement data ujC1, namely pu ¼ 1% (––), pu ¼ 3% (– –)

and pu ¼ 5% (� � �), and (b) the accuracy errors eu (––) and et (– –) as functions of the regularisation parameter k, obtained for various

levels of noise added into the displacement ujC1, namely pu ¼ 1% ð�Þ, pu ¼ 3% ð�Þ and pu ¼ 5% ðMÞ, with M ¼ 10 source points,

N ¼ 40 boundary collocation points and R ¼ 5:0 for the Example 1.


where dui is a Gaussian random variable with mean zero and standard deviation ri ¼ maxC1juijðpu=100Þ,

generated by the NAG subroutine G05DDF, and pu% is the percentage of additive noise included in the

input data ujC1in order to simulate the inherent measurement errors.

Fig. 1(a) presents the L-curves obtained for the Cauchy problem given by Example 1 using the first-order

Tikhonov regularisation method, i.e. k ¼ 1 in (15), to solve the MFS system (12), M ¼ 10 source points,

R ¼ 5:0 and with various levels of noise added into the input displacement data. From this figure it can be

seen that for each amount of noise considered the ‘‘corner’’ of the corresponding L-curve can be clearlydetermined and k ¼ kopt ¼ 1:0� 103 and k ¼ kopt ¼ 3:16� 103 for pu ¼ 1 and pu 2 f3; 5g, respectively.

In order to analyse the accuracy of the numerical results obtained, we introduce the errors eu and et givenby

euðkÞ ¼1

L

XL

l¼1

kuðanÞðxlÞ(

uðkÞðxlÞk22

)1=2

; etðkÞ ¼1

L

XL

l¼1

ktðanÞðxlÞ(

tðkÞðxlÞk22

)1=2

; ð24Þ

where xl, l ¼ 1; . . . ; L, are L uniformly distributed points on the underspecified boundary C2, uðanÞ and tðanÞ

are the analytical displacement and traction vectors, respectively, and uðkÞ and tðkÞ are the numerical dis-

placement and traction vectors, respectively, obtained for the value k of the regularisation parameter. Fig.

1(b) illustrates the accuracy errors eu and et given by relation (24), as functions of the regularisation

parameter k, obtained with various levels of noise added into the input displacement data ujC1for the

Cauchy problem given by Example 1. From this figure it can be seen that both errors eu and et decrease as

the level of noise added into the input displacement data decreases for all the regularisation parameters kand eu < et for all the regularisation parameters k and a fixed amount pu of noise added into the inputdisplacement data, i.e. the numerical results obtained for the displacements are more accurate than those

retrieved for the tractions on the underspecified boundary C2. Furthermore, by comparing Fig. 1(a) and (b),

(a) (b)

Fig. 2. (a) The analytical uðanÞ1 (––) and the numerical uðkÞ1 displacements, and (b) the analytical tðanÞ2 (––) and the numerical tðkÞ2 tractions,

retrieved on the underspecified boundary C2 with M ¼ 10 source points, N ¼ 40 boundary collocation points, R ¼ 5:0, k ¼ kopt and

various levels of noise added into the displacement ujC1, namely pu ¼ 1% (–�–), pu ¼ 3% (–�–) and pu ¼ 5% (–M–), for the Example 1.


it can be seen, for various levels of noise, that the ‘‘corner’’ of the L-curve occurs at about the same value of

the regularisation parameter k where the minimum in the accuracy errors eu and et is attained. Hence thechoice of the optimal regularisation parameter kopt according to the L-curve criterion is fully justified.

Similar results have been obtained for the Cauchy problems given by Examples 2 and 3 and therefore they

are not presented here.

Fig. 2(a) and (b) illustrates the analytical and the numerical results for the displacement u1 and the

traction t2, respectively, obtained on the underspecified boundary C2 using the optimal regularisation

parameter k ¼ kopt chosen according to the L-curve criterion, M ¼ 10 source points, R ¼ 5:0 and various

(a) (b)

Fig. 3. (a) The analytical uðanÞ2 (––) and the numerical uðkÞ2 displacements on the underspecified boundary x2 ¼ 1, and (b) the analytical

tðanÞ1 (––) and the numerical tðkÞ1 tractions on the underspecified boundary x1 ¼ 1, retrieved with M ¼ 20 source points, N ¼ 80

boundary collocation points, R ¼ 5:0, k ¼ kopt and various levels of noise added into the displacement ujC1, namely pu ¼ 1% (–�–),

pu ¼ 3% (–�–) and pu ¼ 5% (–M–), for the Example 2.

(a) (b)


tðanÞ1 (––) and the numerical tðkÞ1 tractions on the underspecified boundary x1 ¼ 1, retrieved with M ¼ 20 source points, N ¼ 80

boundary collocation points, R ¼ 5:0, k ¼ kopt and various levels of noise added into the displacement ujC1, namely pu ¼ 1% (–�–),

pu ¼ 3% (–�–) and pu ¼ 5% (–M–), for the Example 3.


levels of noise added into the input displacement data ujC1, namely pu 2 f1; 3; 5g, for the Cauchy problem

given by Example 1. From these figures we can conclude that the numerical solutions retrieved for Example

1 are stable with respect to the amount of noise pu added into the input displacement data ujC1. Moreover, a

similar conclusion can be drawn from Figs. 3 and 4 which present the numerical results for the displacement

u2 on the underspecified boundary x2 ¼ 1 and the traction t2 on the underspecified boundary x1 ¼ 1 in

comparison with their analytical values, obtained using k ¼ kopt chosen according to the L-curve criterion,

M ¼ 20 source points, R ¼ 5:0 and various levels of noise added into the input displacement data ujC1,

namely pu 2 f1; 3; 5g, for the Cauchy problems given by Examples 2 and 3, respectively.

4.2. Convergence and accuracy of the method

In order to investigate the influence of the number M of source points on the accuracy and stability of

the numerical solutions for the displacement and the traction vectors on the underspecified boundary C2,

we set R ¼ 5:0 for the first two examples considered in this study and pu ¼ 5 and 3 for the Cauchy problems

given by Examples 1 and 2, respectively. Although not presented here, it should be noted that similar results

have been obtained for the Cauchy problem given by Example 3. In Fig. 5(a) and (b) we present the

accuracy errors et for the Example 1 and eu for the Example 2, respectively, as functions of the number M ofsource points, obtained using k ¼ kopt given by the L-curve criterion. It can be seen from these figures that

both accuracy errors tend to zero as the number M of source points increases and, in addition, these errors

do not decrease substantially for M P 5 in the case of Example 1 and M P 10 in the case of Example 2.

These results indicate the fact that accurate numerical solutions for the displacement and the traction

vectors on the underspecified boundary C2 can be obtained using a relatively small number M of source

points. Fig. 6(a) and (b) presents the analytical and the numerical solutions for the displacement u2 and the

traction t1, respectively, obtained on the boundary C2 with various numbers of source points, namely

M 2 f3; 5; 10; 15g, for the inverse problem given by Example 1. From this figure we may conclude that thenumerical solutions for the displacement u2 and the traction t1 approach their corresponding analytical

(a) (b)

Fig. 5. The accuracy errors (a) et obtained with N ¼ 40 boundary collocation points, R ¼ 5:0, k ¼ kopt and pu ¼ 5% for the Example 1,

and (b) eu obtained with N ¼ 80 boundary collocation points, R ¼ 5:0, k ¼ kopt and pu ¼ 3% for the Example 2, as functions of the

number M of source points.

(a) (b)

Fig. 6. (a) The analytical uðanÞ2 (––) and the numerical uðkÞ2 displacements, and (b) the analytical tðanÞ1 (––) and the numerical tðkÞ1 tractions,

retrieved on the underspecified boundary C2 with N ¼ 40 boundary collocation points, R ¼ 5:0, k ¼ kopt, pu ¼ 5% and various numbers

of source points, namely M ¼ 3 (–�–), M ¼ 5 (–�–), M ¼ 10 (–M–) and M ¼ 15 (–*–), for the Example 1.


values as the number M of source points increases. A similar conclusion can be drawn from Fig. 7(a) and(b) which present the numerical solutions for the displacement u1 on the underspecified boundary x1 ¼ 1

and the traction t2 on the underspecified boundary x2 ¼ 1, respectively, obtained for Example 2 with

various numbers of source points, namely M 2 f5; 10; 15; 20g, in comparison with their analytical values.

From Fig. 5–7 it can be seen that the MFS in conjunction with the first-order Tikhonov regularisation

method provides accurate and convergent numerical solutions with respect to increasing the number of

source points, with the mention that a small number of source points is required in order for the accuracy of

the numerical displacements and tractions to be achieved.

(a) (b)


tðanÞ2 (––) and the numerical tðkÞ2 tractions on the underspecified boundary x1 ¼ 1, retrieved with N ¼ 80 boundary collocation points,

R ¼ 5:0, k ¼ kopt, pu ¼ 3% and various numbers of source points, namely M ¼ 5 (–�–), M ¼ 10 (–�–), M ¼ 15 (–M–) and M ¼ 20

(–*–), for the Example 2.

(a) (b)

Fig. 8. The accuracy errors (a) eu obtained with M ¼ 10 source points, N ¼ 40 boundary collocation points, k ¼ kopt and pu ¼ 5% for

the Example 1, and (b) et obtained with M ¼ 20 source points, N ¼ 80 boundary collocation points, k ¼ kopt and pu ¼ 34% for the

Example 2, as functions of the distance R between the source points and the boundary C of the solution domain X.


Next, we analyse the convergence of the numerical method proposed with respect to the position of thesource points. To do so, we set M ¼ 10 and pu ¼ 5, and M ¼ 20 and pu ¼ 3 for the Cauchy problems given

by Examples 1 and 2, respectively, while at the same time varying the radius R. Fig. 8(a) and (b) illustrates

the accuracy errors eu for the Example 1 and et for the Example 2, respectively, as functions of R, obtainedusing k ¼ kopt given by the L-curve criterion. From these figures it can be seen that the larger is the distance

from the source points to the boundary of the solution domain X, i.e. the larger is R, the better the accuracy

in the numerical displacements and tractions. It should be noted that the value R ¼ 10 was found to be

sufficiently large such that any further increase of the distance between the source points and the boundary


C did not significantly improve the accuracy of the numerical solutions for the examples tested in this paper.

However, the optimal choice of R still remains an open problem, as pointed out by Katsurada and

Okamoto (1996), and it requires further research.

5. Conclusions

In this paper, the Cauchy problem in two-dimensional isotropic linear elasticity has been investigated by

employing the MFS. The resulting ill-conditioned system of linear algebraic equations has been regularised

by using the first-order Tikhonov regularisation method, while the choice of the optimal regularisation

parameter was based on the L-curve criterion. Three benchmark examples involving smooth and piecewise

smooth geometries and dealing with constant and linear stress states, i.e. linear and quadratic displace-

ments, respectively, have been analysed. The numerical results obtained show that the proposed method isconvergent with respect to increasing the number of source points and the distance from the source points

to the boundary of the solution domain and stable with respect to decreasing the amount of noise added

into the input data. Moreover, the method is efficient, easy to adapt to three-dimensional Cauchy problems

in linear elasticity, as well as to irregular domains and stress concentration problems, but these investi-

gations are deferred to future work.

Acknowledgements

L. Marin would like to acknowledge the financial support received from the EPSRC. The authors wouldlike to thank Dr. L. Elliott and Prof. D.B. Ingham from the Department of Applied Mathematics at the

University of Leeds for all their encouragement in performing this research work. The authors would also

like to thank the referees for their constructive suggestions.

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the method of fundamental solutions for the cauchy problem in two-dimensional linear elasticity

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