Engineering Analysis with Boundary Elements 43 (2014) 95–104
Contents lists available at ScienceDirect
Engineering Analysis with Boundary Elements
journal homepage: www.elsevier.com/locate/enganabound
Application of the Trefftz method, on the basis of Stroh formalism,
to solve the inverse Cauchy problems of anisotropic elasticity
in multiply connected domains
Tao Zhang a,b, Leiting Dong d,n, Abdullah Alotaibi c, Satya N. Atluri b
a
School of Aeronautic Science and Engineering, Beihang University, China
Center for Aerospace Research & Education, University of California, Irvine, USA
c
Department of Mathematics, King Abdulaziz University, Jeddah
d
Department of Engineering Mechanics, Hohai Uniersity, China
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 29 August 2013
Received in revised form
11 March 2014
Accepted 28 March 2014
In this paper, the Trefftz collocation method is applied to solve the inverse Cauchy problem of
anisotropic elasticity, wherein both tractions as well as displacements are prescribed at a small part of
the boundary of an arbitrary simply/multiply connected anisotropic elastic domain. The Stroh formalism
is used to construct the Trefftz basis functions. Negative and positive power series are used together with
conformal mapping to approximate the complex potentials of the Stroh formalism. For inverse problems
where noise is present in the measured data, Tikhonov regularization is used together with the L-curve
parameter selection method, in order to mitigate the inherent ill-posed nature of inverse problems. By
several numerical examples, we show that this simple and elegant method can successfully solve inverse
problems of anisotropic elasticity, with noisy measurements, in both simply and multiply connected
domains.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
Anisotropic elasticity
Stroh formalism
Characteristic length
Ill-posed
Inverse problem
1. Introduction
Computational modeling of solid/fluid mechanics, heat transfer,
electromagnetics, and other physical, chemical & biological
sciences have experienced an intense development in the past
several decades. Tremendous efforts have been devoted to solving
the so-called direct problems, where the boundary conditions are
generally of Dirichlet, Neumann, or Robin type. Existence, uniqueness, and stability of the solutions have been established for many
of these direct problems. Numerical methods, such as finite
elements, boundary elements, finite volume, meshless methods,
have been successfully developed and are available in many offthe shelf commercial softwares, see [1]. On the other hand, inverse
problems, although being more difficult to tackle and being less
studied, have equal, if not greater importance in the applications
of engineering and sciences.
One of the many types of inverse problems is to identify sources
or inaccessible boundary fields with over-specified measurements at
only part of the boundary, i.e. the Cauchy problem. Take elasto-static
solid mechanics as an example. Consider a domain of interest Ω,
n
Corresponding author at: Department of Engineering Mechanics, Hohai
University, China.
E-mail address: dong.leiting@gmail.com (L. Dong).
http://dx.doi.org/10.1016/j.enganabound.2014.03.012
0955-7997/& 2014 Elsevier Ltd. All rights reserved.
displacements ui are prescribed at Su , and tractions t i are prescribed
at St . If Su and St form a complete division of ∂Ω, i.e.
Su [ St ¼ ∂Ω; Su \ St ¼ ∅, then a direct problem is to be solved.
Otherwise, if both the displacements ui as well as tractions t i are
only prescribed at part of the boundary Sc , then an inverse Cauchy
problem is to be solved.
In spite of its wide popularity, the finite element method is
unsuitable for solving inverse problems. This is mainly because the
symmetric Galerkin weak form prohibits one from prescribing
both displacements as well as tractions at the same part of
boundary. Therefore, in order to solve the inverse problem by
FEM, one needs to iteratively solve a direct problem, and minimize
the difference between the solution and measurement by adjusting the guessed boundary fields, see [2–4] for example. Recently,
simple non-iterative methods have been under development for
solving inverse problems without using the symmetric Galerkin
weak-form: with global RBF as the trial function, collocation of the
differential equation and boundary conditions leads to the global
primal RBF collocation method [5,6]; with Kelvin’s solutions as the
trial function, collocation of the boundary conditions leads to the
method of fundamental solutions, see [7,8]; with non-singular
general solutions as the trial function, collocation of the boundary
conditions leads to the boundary particle method [9–11]; with
Trefftz trial functions, collocation of the boundary conditions leads
to the Trefftz collocation method [12–15]. The common idea they
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T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
share is that the collocation method is used to satisfy either the
differential equations and/or the boundary conditions at discrete
points. Collocation method is one of the most simple, efficient, and
flexible methods, which allow both the tractions and displacements to be prescribed at the same location. Moreover, collocation
method is also more suitable for inverse problems because
measurements are most often made at discrete locations.
Among the various methods, the Trefftz method has shown
extremely high efficiency and accuracy, provided that a relative
complete trial function is used [14–18]. For a two-dimensional
problem such as 2D Laplace equations and linear elasticity, the
general solutions can mostly be expressed as analytic functions of
complex variables. The completeness of the Trefftz trial functions
therefore solely depends on how an analytic function should be
approximated in a complex plane. Based on a detailed discussion
in [14], a generalized Trefftz method is proposed to solve twodimensional isotropic linear elasticity with arbitrarily shaped
multiply connected domains. The later successful application of
it in the direct numerical solution (DNS) of heterogeneous materials considering a large number of voids or inclusions [18–21], also
demonstrated the ability of generalized Trefftz method in solving
problems of 2D and 3D multiply connected domains.
In this paper, we combine and follow the work of [14] and [22–24]
to apply the Trefftz method on the basis of Stroh Formalism to solve
inverse problems of anisotropic elasticity, which was firstly dealt with
in [25], in multiply connected domains. In Section 2, we introduce the
Stroh formalism for two-dimensional anisotropic elasticity, with
special attention being paid to how complex potentials f α ðzα Þ should
be selected to construct the Trefftz trial functions. In Section 3, we give
the detailed algorithm of Trefftz collocation method for inverse
problems of anisotropic elasticity. Specifically, a simple regularization
algorithm is given to increase the robustness of the algorithm when
noise is considered. After that, several numerical examples are given in
Section 4, to study the accuracy, convergence, and robustness of the
proposed method. At last, some concluding remarks are made in
Section 5.
This leads to the Navier’s equations with displacements as
primary variables, for a homogenous elastic body:
C ijkl uk;li þ f j ¼ 0
ð2Þ
Here C ijkl are the components of the fourth-order elasticity tensor
for a homogenous solid.
For plane problems where body forces are negligible, the
general solution of the Navier’s equation (2) can be expressed
through the Stroh Formalism. According to Ting’s monograph [24],
we have
4
u ¼ ∑ aα f α ðzα Þ
ð3Þ
zα ¼ x1 þ pα x2
ð4Þ
q¼1
f α ðzα Þ is an arbitrary analytic functions of zα , and pα and aα are the
eigenvalues and eigenvectors of the following eigen-equation:
fQ þpðR þ RT Þ þ p2 T ga ¼ 0
which is equivalent to
"
T1
T 1 RT
RT
1
T
R Q
ðT
1
ð5Þ
# a
T T
R Þ
b
Considering a linear elastic solid undergoing infinitesimal
elasto-static deformations, the equations of linear and angular
momentum balance, constitutive equations, and compatibility
equations can be written as
sij;i þ f j ¼ 0;
sij ¼ sji
1
2
εij ¼ ðui;j þuj;i Þ uði;jÞ
b
Q ik ¼ C i1k1 ; Rik ¼ C i1k2 ; T ik ¼ C i2k2
ð6Þ
ð7Þ
Eq. (5) will give two pairs of conjugate solutions:
pα þ 2 ¼ pα ; aα þ 2 ¼ aα ; bα þ 2 ¼ bα ;
α ¼ 1; 2
ð8Þ
Letting f α þ 2 ¼ f α , then Eq. (3) can be re-written as
2
u ¼ 2Re ∑ aα f α ðzα Þ
α¼1
ð9Þ
And corresponding stresses can be expressed as
si1 ¼ Φi;2 ; si2 ¼ Φi;1
2
Φ ¼ 2Re ∑ bα f α ðzα Þ
ð10Þ
Now that general expressions for displacements and tractions have
been worked out, the main issue is how the function f α ðzα Þ should
be approximated for numerical implementation.
According to [13], when a simply connected domain is considered, it is reasonable to express the complex potentials with
positive power series, representing modes of tension, shear,
bending, etc.:
N
f α ðzα Þ ¼ ∑ ðiAnαo þ Bnαo Þðzα zoα Þn :
sij ¼ C ijkl εkl
a
For plane elasticity, Q, R, T are 2 by 2 real matrices given by
α¼1
2. Stroh formalism for anisotropic elasticity
¼p
ð11Þ
n¼0
ð1Þ
where the Einstein summation convention on repeated indices
is used.
where zoα ¼ x1 þ pα x2 with ðx1 ; x2 Þ being the source point placed
inside the domain.
For a doubly connected domain, we can locate the source point
inside the cavity, and apply conformal mapping from zα plane to
Fig. 1. Conformal mapping from an ellipse to a unit circle.
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
97
ξα plane, where the cavity is mapped to a unit circle. For example,
This regularization technique is applied to many FEM and BEMbased methods, which leads to the regularized solution:
1
α ¼ AT A þ γ I
AT b
ð21Þ
zα zcα ¼ ωcα ðζ α Þ ¼ a 2ipα bζ α þ a þ2ipα b ζ1c
α
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
c
ðzα zα Þ 7 ðzα zcα Þ2 a2 p2α b
:
ζ cα ¼ ωcα 1 ðzα zcα Þ ¼
a ipα b
c
c
c
z α ¼ x 1 þ pα x 2
In this paper, Tikhonov Regularization is combined with the
L-curve method to enhance the robustness of the solution by
Trefftz method. The regularization parameter γ is determined by
the corner of the L-curve, which is the plot of the norm of residual
with respect to the norm of the solution vector. Detailed demonstration of the L-curve method is given in Case 4 of the numerical
examples presented in the next section.
if an elliptical cavity c0 is considered with semi-axes a and b, and
center ðxc1 ; xc2 Þ, see Fig. 1, the conformal mapping can be constructed as
c
c
ð12Þ
And the complex potentials are expressed as
f α ðzα Þ ¼
N
∑ ðiAnαc þ Bnαc Þξα :
c
ð13Þ
n ¼ M
4. Numerical examples
Furthermore, if multiple cavities are considered with centers
ðxc1 ; xc2 Þ; c ¼ 1; 2; …; K, the complex potentials can be represented as
N
K
f α ðzα Þ ¼ ∑ ðiAnαo þ Bnαo Þðzα zoα Þn þ ∑
N
∑ ðiAnαc þ Anαc Þξα
c
ð14Þ
c ¼ 1 n ¼ M
n¼0
However, it should be noted that, the exponential growth of Trefftz
basis functions as given in (11)–(14) will commonly give an illconditioned system of equations. This can be easily resolved by
introducing a characteristic length to scale the Trefftz basis
functions, see [14] for a detailed discussion.
3. Trefftz collocation method for inverse problems with
regularization
uðx1 ; x2 Þ ¼ Nðx1 ; x2 Þα
rðx1 ; x2 Þ ¼ Sðx1 ; x2 Þα
tðx1 ; x2 Þ ¼ Rðx1 ; x2 Þα ¼ nSðx1 ; x2 Þα
ð15Þ
at SC
ð16Þ
at SC
We collocate the displacements and tractions at points P k :
ðxk1 ; xk2 Þ A SC ; k ¼ 1; 2; …,
Ni ðxk1 ; xk2 Þα ¼ ui ðxk1 ; xk2 Þ;
k ¼ 1; 2::::
wR i ðxk1 ; xk2 Þα ¼ wt i ðxk1 ; xk2 Þ;
k ¼ 1; 2…;
ð17Þ
ð18Þ
where w is a parameter to make the coefficient matrices of
Eqs. (17) and (18) to have the same norm. The resulting systems
of equations can be rewritten in matrix/vector form:
Aα ¼ b
In the first case, a beam-shaped simply connected domain is
considered, which can be described by Ω ¼ ðx1 ; x2 Þj0 r x1 r
L; c r x2 r cg. As shown in Fig. 2, the geometrical parameters
are L¼ 24, c ¼ 2. Arbitrarily we consider the elastic stiffness
2
3 2
3
C 11 C 12 C 16
3:0638 0:2553
0
6C
7
6
7
0 5. And the
matrix C ¼ 4 21 C 22 C 26 5 ¼ 4 0:2553 1:0213
C 62
C 66
0
0
0:43
compliance matrix is determined by A ¼ C 1 .
The analytical solution of a cantilever beam subjected to shear load
at the free end is considered, which is given by Lekhnitskii in [28]:
2
2
2
u1 ¼ Px
6I ð3x1 a11 ð2L x1 Þ þ ða12 a66 Þðx2 c ÞÞ
where n represents the unit normal vector at the boundary.
For the inverse Cauchy problem we are considering in this
study, the displacements as well as tractions are prescribed at only
a small portion of the boundary:
nj sij ¼ t i
Case 1. Cantilever beam
C 61
With the detailed algorithm of constructing Trefftz basis functions, we can represent the trial functions as
ui ¼ ui
In this section, we use some numerical examples to demonstrate the effectiveness of the proposed method. All the numerical
examples are conducted with MATLAB on a PC with i7 processor.
P
ð 3a12 x22 ðL x1 Þ ð2a66 a12 Þc2 x1 þa11 x21 ð3L x1 ÞÞ
6I
Px
P
s11 ¼ 2 ðL x1 Þ; s22 ¼ 0; s12 ¼ ðc2 x22 Þ
2I
I
u2 ¼
ð22Þ
So we raise a problem like this: can we identify the displacements,
strains, and stresses in the whole domain, if both the displacements and the tractions at only a small portion of the boundary are
measured (over-specified)?
We solve this problem by using the Trefftz method of Stroh
formalism with Positive power series truncated at the order of 3.
The displacement and tractions are measured at the 10 red circles
as shown in Fig. 2, which are uniformly distributed at the lower
edge SC ¼ ðx1 ; x2 Þj16 r x1 r 24; x2 ¼ 2 . Collocations of displacements and tractions are made at the locations of measurements.
The identified stress component s11 and vertical displacement
v on the lower edge Sd of the cantilever beam by Tefftz method
are plotted in Fig. 3 (normalized to their maximum values).
The computed results agree perfectly with the analytical solution.
ð19Þ
It is well known that the inverse problems are ill-posed. A very
small perturbation of the measurement data can lead to a
significant change of the solution. In order to mitigate the illposedness of the inverse problem, regularization techniques can
be used. For example, following the work of Tikhonov [26], many
regularization techniques were developed. Hansen et al. [27] have
given an explanation that the Tikhonov regularization of ill-posed
linear algebra equations is a trade-off between the size of the
regularized solution, and the quality to fit the given data. With a
positive regularization parameter γ , the solution is found by
min jjAαbjj22 þ γ jjαjj22
ð20Þ
Fig. 2. The problem of a cantilever beam under a shear load at the free end and the
adopted collocation points. (For interpretation of the references to color in this
figure caption, the reader is referred to the web version of this paper.)
98
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
cantilever beam
1
0.9
0.8
Analytical
STROH FORMALISM
Normalized σ11
0.7
0.6
0.5
0.4
0.3
0.2
Fig. 4. A plate with a circular hole under uniform tension. (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of
this paper.)
0.1
0
0
0.2
0.4
0.6
0.8
1
X1
Case 3. a plate with an elliptical hole
cantilever beam
In this example, a doubly connected domain is considered.
n
o
2 2
As shown in Fig. 7,Ω ¼ ðx1 ; x2 Þ c r x1 ; x2 r c; xa1 þ xb2 r 1
1
0.9
is defined by one elliptical hole in a 20 by 20 square plate
under uniform tension P, where c ¼ 10 and P ¼ 1. Arbitrarily
2
3
C 11 C 12 C 16
6
7
we consider the stiffness matrix C ¼ 4 C 21 C 22 C 26 5 ¼
C 61 C 62 C 66
2
3
1:2030 0:0427
0
6 0:0427 0:6015
0 7
4
5. And three different ratios of semi0
0
0:07
axes of the ellipse are used here for demonstration:
Normalized Displacement v
0.8
0.7
0.6
0.5
0.4
0.3
a ¼ 2b ¼ c=3 ðor c=10; c=20Þ:
0.2
Analytical
STROH FORMALISM
0.1
0
0.2
0.4
0.6
0.8
1
X1
Fig. 3. Comparison of the normalized identified stress component s11 and vertical
displacement v at the lower edge Sd of the cantilever beam, Sd ¼ fðx1 ; x1 Þj0 r
x1 r 24; x2 ¼ 2g.
Case 2. a plate with a circular hole
As shown in Fig. 4, we also consider a doubly connected domain,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
o
x21 þ x22 r R . R ¼1 denotes the
Ω ¼ ðx1 ; x2 Þj 10 rx1 ; x2 r 10;
radius of the hole. Arbitrarily we consider the elastic stiffness
2
3 2
3
C 11 C 12 C 16
1:2030 0:0427
0
6C
7
6
0 7
C ¼ 4 21 C 22 C 26 5 ¼ 4 0:0427 0:06015
5. The analytical
C 61 C 62 C 66
0
0
0:07
solution of an infinite plate with a circular hole under uniform
tension, given by [28], is used here. We collocate both the
displacements and tractions at 10 points uniformly distributed at
part of the lower edge, i.e. SC ¼ fðx1 ; x2 Þj 10 r x1 r 6;
x2 ¼ 10g. The collocation points are shown as red circles in
Fig. 4.
After solving this inverse Cauchy problem with Trefftz method,
we plot the stress components s11 , s22 and horizontal displacement u at the positive x1 -axis, and we plot the stress components
s11 , s22 and vertical displacement v at the positive x2 -axis. As we
can see from Figs. 5 and 6, the computed results agree well with
the analytical solutions.
The analytical solution of an infinite plate with an elliptical hole
under uniform tension by Lekhnitskii’s work in [28] is used here as
the solution to be sought for. We measure the displacements and
tractions at 10 points, shown as red circles in the Fig. 7, at the
same locations as those of case 2, i.e. SC ¼ fðx1 ; x2 Þj 10 r
x1 r 6; x2 ¼ 10g. We solve this problem by using the Trefftz
method with Stroh formalism as trial functions. Positive power
series complete to the order of 2 is used with negative power
series complete to the order of 1. We plot the stress components
s11 , s22 and horizontal displacement u along the positive x2 -axis.
And we plot stress components s11 , s22 and vertical displacement
v along the positive x2 -axis. As can be seen in Figs. 8, 9 and 10,
good agreements can be found with analytical solutions.
Case 4. a plate with an elliptical hole with different level of
white noise
In this case, we consider a 20 by 20 anisotropic square plate
n
2
with an elliptical hole Ω ¼ ðx1 ; x2 Þ 10 r x1 ; x2 r10; xa1 þ
o
x 2
2
r 1 , subject to uniform tension P. Geometrical parameters
b
a ¼ 6; b ¼ 5 are considered in this case. Arbitrarily we take the
same stiffness matrix C as that for case 3. The analytical solution of
a plate with an elliptical hole under uniform tension can be found
by Lekhnitskii’s work in [28], and we use it here as the solution to
be sought for.
When dealing with inverse problems, usually the measured
data is perturbed by noise and the solutions are unstable due to
the ill-posedness of inverse problems. Thus, through case 4, we
investigate properly the following important features of the
proposed method.
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
99
R=1
R=1
0.9
5.5
0.8
5
0.7
4.5
4
Analytical
Stroh Formalism
0.5
σ11
σ11
0.6
0.4
3.5
3
0.3
2.5
0.2
2
0.1
1.5
0
Analytical
Stroh Formalism
1
2
3
4
5
6
7
8
9
1
10
1
2
3
4
5
X1
R=1
7
8
9
10
R=1
0.1
0.2
0
0.18
−0.1
0.16
Analytical
Stroh Formalism
Analytical
Stroh Formalism
0.14
−0.2
0.12
−0.3
σ22
σ22
6
X2
−0.4
0.1
0.08
−0.5
0.06
−0.6
0.04
−0.7
−0.8
0.02
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
X1
6
7
8
9
10
X2
R=1
R=1
9.5
−0.7
9
−0.8
8.5
Displacement v
Displacement u
8
7.5
7
6.5
Analytical
Stroh Formalism
−1
−1.1
6
5.5
−1.2
Analytical
Stroh Formalism
5
4.5
−0.9
1
2
3
4
5
6
7
8
9
10
X1
−1.3
1
2
3
4
5
6
7
8
9
10
X2
Fig. 5. Comparison of the identified stress components s11 , s22 and the horizontal
displacement u at the positive x1 -axis for the plate with circular hole.
Fig. 6. Comparison of the identified stress component s11 , s22 and the vertical
displacement v at the positive x2 -axis for the plate with circular hole.
4.1. Influence of noisy measurements
Both displacements and tractions are measured at 10 uniformly
distributed points at two edges: fðx1 ; x2 Þj 10 r x1 r2; x2 ¼ 10g
and fðx1 ; x2 Þjx1 ¼ 10; 2 r x2 r 10g: For this problem, the truncation number for the Trefftz basis functions is 10. We analyze the
Various levels (pu and pt ) of white noise are added into
the measured displacements and tractions, respectively.
100
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
a=c/20
1
0.9
0.8
Analytical
Stroh Formalism
0.7
σ11
0.6
0.5
0.4
0.3
0.2
Fig. 7. a plate with an elliptical hole under uniform tension. (For interpretation of
the references to color in this figure caption, the reader is referred to the web
version of this paper.)
0.1
0
0
2
4
6
8
10
X1
1
a=c/20
0.9
a=c/20
0.1
0.8
0
0.7
a=c/10
−0.1
Analytical
Stroh Formalism
−0.2
0.5
a=c/3
0.4
σ22
σ11
0.6
−0.3
−0.4
0.3
Analytical
Stroh Formalism
0.2
−0.5
0.1
−0.6
0
0
2
4
6
8
−0.7
10
X1
−0.8
Fig. 8. Comparison of the identified stress component s11 at the positive x1 -axis
with different sizes of the elliptical hole.
0
2
4
6
8
10
X1
a=c/20
4.2. Influence of number of collocation points
In order to find out the effect of the number of collocation
points, we consider three different collocation schemes. In the
following three cases, measurements and collocations are uniformly distributed at nc ¼ 3, 10, and 80 points on each of the two
sides ðx1 ; x2 Þj0 r x1 r 2;x2 ¼ 10 and ðx1 ; x2 Þjx1 ¼ 10; 2 r
9
8
7
Displacement u
numerical solutions retrieved from three levels of noises (1%, 3%
and 5%) added to: (i) the Dirichlet data (displacements); (ii) the
Neumann data (tractions); and (iii) the Cauchy data (displacement
and tractions), respectively.
Figs. 11–13 present the stress components s11 , s22 and the
horizontal displacement u along the positive x2 -axis obtained by
using the Trefftz method under various levels of noise added into
the measured displacements, i.e. pu A f1%; 3%; 5%g; the measured
tractions, i.e. pt A f1%; 3%; 5%g; both displacements and tractions i.
e. pt ¼ pu A f1%; 3%; 5%g, respectively. It can be seen that, for each
fixed level of noise, the numerical solutions are stable approximations to the corresponding exact solution, free of unbounded and
rapid oscillations, and it converges to the exact solution as the
noise level decreases. By comparing Figs. 11–13, it can be seen that
the numerical solutions with noisy Neumann dada agree less with
the analytical solution than those numerical solutions with noisy
Dirichlet data.
6
5
4
Analytical
Stroh Formalism
3
2
1
0
2
4
6
8
10
X1
Fig. 9. Comparison of the identified stress components s11 , s22 and the horizontal
displacement u at the positive x1 -axis with a ¼ ð1=20Þc.
x2 r 0:g. And measured displacements and tractions are contaminated by 5% noise, i.e. pu ¼ pt ¼ 5%. For this problem, the truncation number for the Trefftz basis functions is 10.
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
101
a=6, b=5
a=c/20
5
3.5
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
4.5
3
Analytical
Stroh Formalism
4
3.5
σ11
σ11
2.5
2
3
2.5
2
1.5
1.5
1
1
0
2
4
6
8
5
10
6
7
X2
9
10
a=6, b=5
a=c/20
0.16
0.09
0.14
0.08
0.07
0.12
Analytical
Stroh Formalism
0. 1
σ22
0.06
σ22
8
X2
0.05
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
0.08
0.06
0.04
0.04
0.03
0.02
0.02
0
0.01
−0.02
0
0
2
4
6
8
5
6
7
10
8
9
10
X2
X2
a=6, b=5
Fig. 10. Comparison of the identified stress components s11 , s22 and the vertical
displacement v at the positive x2 -axis with a ¼ ð1=20Þc.
−5
−5.1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 NE
Es ¼
∑ ½ðsi si11 Þ2 þ ðsi22 si22 Þ2 þ ðsi12 si12 Þ2 =
N E i ¼ 1 11
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 NE
∑ ½ðsi Þ2 þ ðsi22 Þ2 þ ðsi12 Þ2 N E i ¼ 1 11
where ui1 ; ui2 ; si11 ; si22 ; si12 represent the numerically identified
solutions at xi , a set of N E uniformly distributed points in the
domain Ω.
Table 1 presents the numerical accuracy of the solutions with
different numbers of the collocation points. It can be seen that the
numerical accuracy of the identified displacements and stresses
are improved with the increased number of collocation points.
It should be pointed out that, the Tikhonov regularization as
shown in Section 4 is employed at here, with its regularization
parameter determined by the L-curve criterion [27]. Take the
numerical simulation using three collocation points (nc ¼ 3)
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
−5.2
−5.3
Displacement v
In order to analyze the overall accuracy of solution, we
introduce the following root mean-square (RMS) errors:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 NE
1 NE
Eu ¼
∑ ½ðui ui1 Þ2 þ ðui2 ui2 Þ2 =
∑ ½ðui Þ2 þðui2 Þ2 NE i ¼ 1 1
NE i ¼ 1 1
−5.4
−5.5
−5.6
−5.7
−5.8
−5.9
−6
5
6
7
8
9
10
X2
Fig. 11. The analytical and numerically identified stress components s11 , s22 , and
the vertical displacement v along the positive x2 -axis with various levels of noise
added to displacement measurements for case 4.
at each edge for example, the regularization parameter
pffiffiffi
γ ¼ 0:0093252 is determined by the corner of the L-curve shown
in Fig. 14, i.e. γ ¼ 8:6959e 5.
102
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
a=6, b=5
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
4.5
4
4.5
3.5
σ11
σ11
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
4
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
a=6, b=5
5
5
1
5
6
7
8
9
10
5
6
7
8
9
10
X2
X2
a=6, b=5
0.18
0.16
0.16
0.14
0.14
a=6, b=5
0.12
0.12
0.1
σ22
σ22
0. 1
0.08
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
0.06
0.06
0.04
0.04
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
0.02
0
−0.02
0.08
5
6
7
8
9
0.02
0
−0.02
10
5
6
7
8
X2
a=6, b=5
−4. 6
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
−5
−5.1
Analytical
Stroh Formalism With 5% Noise Level
Stroh Formalism With 3% Noise Level
Stroh Formalism With 1% Noise Level
−5.2
Displacement v
−5.3
Displacement v
10
a=6, b=5
−5
−4. 8
9
X2
−5. 2
−5. 4
−5. 6
−5.4
−5.5
−5.6
−5.7
−5. 8
−5.8
−6
−6. 2
−5.9
5
6
7
8
9
10
−6
5
6
7
Fig. 12. The analytical and numerically identified stress components s11 ,s22 , and
the vertical displacement v along the positive x2 -axis with various amounts of
noise added to traction measurements for case 4.
4.3. Influence of the size of the accessible boundary SC .
In this subsection, we investigate how the size of the accessible
boundary SC affects the accuracy of the numerical solution. Three
8
9
10
X2
X2
Fig. 13. The analytical and numerically identified stress components s11 , s22 , and
vertical displacement v along the positive x2 -axis with various levels of noise added
to displacement and tractions measurements for case 4.
cases are considered here, with different sizes of SC :
SC 1 ¼ fðx1 ; x2 Þj0 r x1 r 2; x2 ¼ 10g
[ fðx1 ; x2 Þjx1 ¼ 10; 2 r x2 r0g;
T. Zhang et al. / Engineering Analysis with Boundary Elements 43 (2014) 95–104
Table 1
The accuracy of numerical solution based on Stroh formalism using different
number of collocation points (nc ¼ 3; 10; 80) for case 4 with 5% noise level.
Stroh Formalism With 1% Noise Level
1
10
0
nc
Eu
Es
γ
3
10
80
1.341e 1
5.500e 2
5.800e 3
2.904e 1
1.242e 1
1.930e 2
8.6959e 5
2.2357e 5
2.4103e 5
10
Eu
Eσ
−1
10
Errors
L−curve, Tikh. corner at 0.0093252
3
103
−2
10
10
−3
10
−4
solution norm || α ||2
10
0.00065503
2
10
−5
10
0.0016053
2.0885
0
−1
0
1
10
residual norm || A α − b ||
10
2
10
2
Fig. 14. Using L-curve to determine the regularization parameter of the Tikhonov
Regularization when measured data are contaminated by noises.
Table 2
The accuracy of numerical solution based on Stroh formalism
using different sizes of accessible boundary (S1c ; S2c ; S3c ) for case
4 with 1% noise level.
Sc
Eu
S1c
1.330e 2
2.150e 2
2.300e 3
8.100e 3
2.268e 4
1.400e 3
S2c
S3c
10
15
20
25
30
Fig. 15. The accuracy of numerical solution based on Stroh formalism with different
truncation number of Trefftz basis functions for case 4 with 1% noise level.
0.85221
1
10
10
5
trunction number of Trefftz basis functions (NT)
0.003934
0.1419
10
−2
10
0
Es
SC 2 ¼ ðx1 ; x2 Þj 2 r x1 r2; x2 ¼ 10
[ ðx1 ; x2 Þjx1 ¼ 10; 2 r x2 r 2 ;
SC 3 ¼ ðx1 ; x2 Þj 2 r x1 r2; x2 ¼ 10
[ ðx1 ; x2 Þjx1 ¼ 10; 2 r x2 r 2
[ fðx1 ; x2 Þj 2 r x1 r 2; x2 ¼ 10;g
[ fðx1 ; x1 Þjx1 ¼ 10; 2 r x2 r 2g
Eighty collocation points are uniformly distributed at each of the
two sides, SC ¼ fðx1 ; x1 Þj0 r x1 r 2; x2 ¼ 10g [ fðx1 ; x1 Þjx1 ¼ 10;
2 r x2 r0g. And measured displacements and tractions are contaminated with 1% noises.
Fig. 15 presents, on a logarithmic scale, the numerical error as a
function of the truncation number, N T , of the power series. It can
be observed that Eu and Es decrease with respect to the truncation
number of the Trefftz basis functions. After the number of basis
functions reaches a certain level, the numerical error stays
stable, which may be mainly due to the existence of the noise
contamination.
5. Conclusion
The Tefftz collocation method, based on the Stroh formalism, is
developed to solve the inverse problems of anisotropic plane
elasticity, for arbitrary simply/multiply connected domains. Negative power series and positive power series are used together with
conformal mapping to approximate the complex potentials of the
Stroh formalism. When noise is present in the measured Cauchy
data, Tikhonov regularization is employed to enhance the robustness of the solution, with L-curve method to select the regularization parameter. Several numerical examples are given to
demonstrate the accuracy, convergence, and robustness of the
proposed numerical method.
Acknowledgment
In each case, displacements and tractions are collocated at 10
points of each side. Measured displacements and tractions are
contaminated by 1% noise, i.e. pu ¼ pt ¼ 1%. For this problem, the
truncation number for the Trefftz basis functions is 10. Numerical
errors in Eu and Es for each case are given in Table 2. It can be seen
that the numerical accuracy of the identified displacements and
stresses are improved with increased size of the accessible
boundary SC .
The first author acknowledges the financial support of the
China Scholarship Council (Grant no.201206020059); the National
High-tech R&D Program of China (Grant no.2012AA112201); the
National Natural Science Foundation of China (Grant no.10772013);
the Aeronautical Science Foundation of China (Grant no.20100251007).
This work was partially funded by the Deanship of Scientific
Research (DSR), King Abdulaziz University, under grant no.
(3-130-25-HiCi). The authors, therefore, acknowledge technical
and financial support of KAU.
4.4. Influence of the truncation number NT of Trefftz basis functions
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