-
Research Article
Received 29 November 2010 Published online 10 November 2011 in
Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.1533MOS subject
classification: 49Q12; 49K40; 65K10; 49N45; 35Q93
Inverse thermal imaging in materials withnonlinear conductivity
by material and shapederivative method
I. Cimrk*
Communicated by H. Ammari
The material and shape derivative method is used for an inverse
problem in thermal imaging. The goal is to identify theboundary of
unknown inclusions inside an object by applying a heat source and
measuring the induced temperature nearthe boundary of the sample.
The problem is studied in the framework of quasilinear elliptic
equations. The explicit formis derived of the equations that are
satisfied by material and shape derivatives. The existence of weak
material derivativeis proved. These general findings are
demonstrated on the steepest descent optimization procedure.
Simulations involv-ing the level set method for tracing the
interface are performed for several materials with nonlinear heat
conductivity.Copyright 2011 John Wiley & Sons, Ltd.
Keywords: sensitivity analysis; shape optimization; speed
method, level set method
1. Introduction
The internal thermal properties of an object, the presence of
cracks or voids, or the shape of some unknown portion of the
boundarycan be determined by a technique called thermal imaging.
This technique is widely utilized in non-destructive testing and
evaluation.A heat source is used on an object, and the resulting
temperature response is observed near the objects surface. Thermal
imaging hasbeen significantly investigated as a method for
detecting damage or corrosion in industrial machines, vehicles, or
aircrafts. Industrialnon-destructive testing uses this technique
for broad range of materials ranging from composite materials to
electronics [6, 14]. Wemention the reconstruction of small
inclusions from boundary measurements of temperature [1] and study
of conductivity interfaceproblems by layer potential techniques
[3]. These authors have studied also other types of thermal
imaging.
We elaborate a specific problem of crack, voids, and impurities
identification inside an object with nonlinear thermal
conductivity.We attempt to identify the inhomogeneities from the
measurements of the heat equilibrium. The model equation is thus
steady-stateheat equation with unknown u and with coefficients
nonlinearly dependent on u.
2. Mathematical model
First, we introduce some notation. Let D be a bounded domain in
R2 with C2 boundary and D its proper subdomain with C2boundary. We
use classical Sobolev spaces W1,2.D/, W1,20 .D/, the space with
square integrable functions L
2.D/ and the space L1.D/of bounded functions. The scalar product
in L2.D/ is denoted by ., /. The norm in L2.D/, L1.D/ is denoted by
k k2, k k1 and thenorm in general space X by k kX . The vectors
inRd will be denoted by bold symbols, for example, x or by couples
(in 2D) or triples (in3D), for example, x D .x1, y1, z1/T . The
scalar product of two vectors u, v inRd will be denoted by u v.
Partial derivative of f .x, y/ withrespect to x is denoted either
by @f
@x or by fx . We frequently use the restriction of a function.
Therefore to simplify the notation, we use
the expression f 2 L2./ even for f : D !R instead of a longer
notation f j 2 L2./.D represents the object under consideration
inside that there are some inhomogeneities. The domain represents
these inhomo-
geneities. Note that can consist of several disjoint parts.
Their number, position, and shape are to be determined. Consider a
functionu : D !R representing a temperature distribution and assume
that the function b : D R!R is defined piecewise by
NaM2 Research Group, Department of Mathematical Analysis, Ghent
University, Galglaan 2, B-9000, Belgium*Correspondence to: I.
Cimrk, NaM2 Research Group, Department of Mathematical Analysis,
Ghent University, Galglaan 2, B-9000, Belgium.E-mail:
[email protected] work was supported by the Fund for
Scientific Research - Flanders FWO, Belgium
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I. CIMRK
b.x, s/D
b1.s/ for x 2,b2.s/ for x 2 D n (1)
where b1, b2 are smooth nonlinear functions. The material
occupying D n has nonlinear thermal conductivity represented by a
non-linear function b2, satisfying some properties listed later. In
the case of crack or void identification, the domain representing
the voidsis filled with air, water, or some other liquid or
gas.
Forwardmodel for steady-state temperature distribution inside D
with a heat source represented by a function f andwith
boundarieskept at constant temperature uc reads as
r .b.x, u/ru/D f .x/ in D, u D uc on @D, (2)with the following
interface conditions
uj@ D 0, b.x, u/ru nj@ D 0, (3)where v is the jump of a quantity
v across the interface @ and n the unit outward normal to the
boundary @. The interfaceconditions at the boundary of the voids
reflect the continuity of the temperature on the interface and of
the heat flux through theinterface.
In some literature concerning parameter determination in heat
conduction problems, the authors consider the parabolic
heatequations of the form
@u
@t r .a.x/ru/D f .x, u/ in D, u D 0 on @D.
We emphasize that in this model, the coefficient in front of the
highest derivative is a function independent of u, whereas in
ourmodel it is a u-dependent function. In these equations, the
nonlinearity appears outside the divergence operator, and they are
easierto treat as in our case when the nonlinearity appears under
the divergence operator.
The ultimate goal of this work is the reconstruction of if we
possess the measurements of the temperature distribution u on
aspecific part ! D. Measurements are typically available near the
boundary of D. In this paper, we consider ! not as a part of
theboundary @D but as a part of domain D. The measurements are ! is
also allowed to intersect the interface @.
Weak formulation of the direct problem reads as: For given, find
u 2 W1,2.D/ such that u uc 2 W1,20 .D/ and.b.u/ru,r'/D .f ,'/ ,
(4)
is satisfied for all ' 2 W1,20 .D/. This weak formulation is
derived bymultiplication of (2) with the test function ' and
integrating by parts.All the boundary integrals, explicitly
appearing when we carry out the integration by parts, disappear
because of Dirichlet boundaryconditions on @D and because of
(3).
We denote the given data by NK.x/. On !, the values of NK.x/
correspond to the measurements and outside ! they are extended by
0.We construct the cost functional measuring the fidelity of the
computed solution to the measurements
J./D 12
Z!
u./ NK 2 , (5)
where u./ is the solution of (4) for given.The inverse problem
of determination frommeasurements NK will be solved by minimization
of the above functional.We employ gradient-type minimization method
to minimize the cost function. For this, we need to compute the
gradient DJ of J.
In earlier works concerning different applications, we have used
formal differentiation techniques to obtain DJ [7, 10, 11]. We
employthe shape sensitivity analysis using the material and shape
derivative as tools for computation of DJ. The shape and material
deriva-tive has been widely used in the shape optimization, among
others, we refer [2, 23, 26] and the references therein. This
concept hasbeen applied in the shape sensitivity for unilateral
problems describing such physical phenomena as contact problems in
elasticity,elasto-plastic torsion problems, obstacle problems, and
others.
We use the same notations as in aforementioned references. In
Section 4, we derive basic developments including the form of
theequation that is satisfied by the material derivative, the proof
of the weak as well as of the strong convergence of material
derivative.
Further, in Section 5, we determine the form of the equation
that must be satisfied by the shape derivative. Finally, in Section
6, weuse the adjoint method to explicitly express the derivative of
the cost functional DJ.
In Section 7, we employ the obtained results. We elaborate the
level set method to represent the interface @, and we show how DJis
used in practice. We describe the optimization algorithm that
eventually finds by minimization of the cost function J./.
Finally in Section 8, we show the implementation of the
minimization algorithm, and we present the numerical results.
3. Analysis of the direct problem
When the interface @ is smooth enough, the solution of the
interface problem is also smooth in individual regions separated by
thediscontinuities. The global regularity is, however, very low, we
have only u 2 W1,2.D/. For regularity studies in the case of linear
equa-tions, see, for example, [17]. These results have been used in
the finite element (FE) approximations to show the convergence and
errorestimates for FE methods [4].
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The literature concerning the case of semi-linear or more
generally quasilinear equations is very rich. We refer to [13, 21]
where theauthors consider smooth domains and [5] for conical
domains. The FEM approximation of nonlinear interface problems have
beenstudied in [22, 24].
We formulate the properties of b1, b2 appearing in (1). These
properties are consistent with the thermal imaging application.
Fori D 1, 2
A1 There exist positive bmin, bmax such that bmin bi.s/ bmax ,A2
bi is differentiable.
The first assumption guarantees that the equation (4) does not
degenerate. Further, we assume that f .x/,rf .x/ 2 L1.D/.To show
the existence of the solutions to (2)(3), we make use of
theoretical results from [21] concerning the strong solution of
quasilinear diffraction problems. First finding from [21] is
that we have boundedness of u in L1.D/, that is for some positive M
2 Rwehave juj M. The main result from [21] is the following
theorem, adapted to our case.Theorem 1 (Theorem 2, [21])Assume that
for 1> > 0 the following smoothness conditions are valid
a1i .u,p/ :D b1.u/pi 2 C1,. M, MR/, a2i .u,p/ :D b2.u/pi 2 C1,.D
n M, MR/,where positive M is the L1 bound of u and i D 1, 2. Then
there exists a classical solution of (2)(3) satisfying
u 2 C 0,.D/,@u
@xi2 C0,./, @u
@xi2 C0,.D n/,
@2u
@xi@xj2 C0,./, @
2u
@xi@xj2 C0,.D n/, i, j D 1, 2.
The uniqueness is established in [25] for more general elliptic
operators in divergence form.
4. Material derivative
We refer to [23] and references therein for the overview of the
theory of material and shape derivative method. We let evolve in
timeintroducing a time variable t > 0. We denote byt the
evolved. The direct problem in the time instance t can be written
as
.bt.ut/rut ,r't/D .ft ,'t/, (6)with nonlinearity
bt.x, s/D
b2.s/ for x 2 D nt ,b1.s/ for x 2t . (7)
Symbol h.x/ stands for a velocity field. For non-negative t 2R
define the mapping Ft :R2 !R2 byFt.X/D X C th.X/, (8)
where h.X/D .h1.X/, h2.X//T 2 .C1,1.R2//2 and h D 0 on @D.
Further, we set h D 0 in the vicinity of !. This requirement
correspondsto the fact that the holes are not located in the area
of the measurements. For t sufficiently small, lett D Ft./ be the
image of thefixed domain. Because FtjtD0 D Id, we have0 D. We use
symbol X for the points inR2 whereR2 is considered as the
definitiondomain of Ft . We use x for points inR2 whereR2 is
considered as the range of Ft . Ft is considered as the mapping
from the fixed frameto the moving frame. The moving frame moves
under the velocity field h.
Symbol D in front of a vector function f, we understand the
matrix
Df D
0BB@@f1@x1
@f1@x2
@f2@x1
@f2@x2
1CCA
We denote M D .DFt/1, It D det.DFt/, At :D MMT It and A :D r hId
.DhT C Dh/. We list several important identities.
DFt D
0BB@1C t @h1
@x1t@h1@x2
t@h2@x1
1C t @h2@x2
1CCA , It D det.DFt/, M D DF1t D .DFt/1
At D MMT It D DF1t .DF1t /T It , A :D r hId .DhT C Dh/
(9)
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It 1t
jtD0 D r h, DFt Idt
tD0D
0BB@@h1@x1
@h1@x2
@h2@x1
@h2@x2
1CCA D Dh, AtjtD0 D Id (10)
MT Idt
tD0D .DF
1t /
T Idt
tD0D
0BB@
@h1@x1
@h2@x1
@h1@x2
@h2@x2
1CCA D DhT (11)
At Idt
tD0D
0BB@
@h2@x2
@h1@x1
@h1@x2
@h2@x1
@h1@x2
@h2@x1
@h2@x2
C @h1@x1
1CCA D r hId .DhT C Dh/D A (12)
We distinguish between the functions with domain in the fixed
frame from those having domain in themoving frame. The
functionsdepending on X (i.e. those with domain in the fixed frame)
will be marked by superscript t whereas functions depending on x
(i.e.,those with domain in moving frame) will be marked by
subscript t. Thus ut.X/D ut.x/D ut.Ft.X// or ut D ut Ft . Similarly
f t D ft Ft .
We have the following equivalences
rut D DFTt rut (13)
MT rut D rut . (14)The cost functional reads as
J.t/D 12
Z!
jut NKj2 (15)
where NK are the measurements.The material derivative Pu is
defined as
Pu D limt!0
ut ut
.
We derive the equation for Pu. Consider the direct problem (6)
for the positive time instance t > 0. After change of variables
x D Ft.X/we obtain
.b.ut/Atrut ,r't/D .Itf t ,'t/.
We introduce the notation wt D utut . We subtract the direct
problem for the time instance t D 0 from the previous equation,
andwe divide the resulting equation by t. The test functions will
be denoted simply by '. After somemanipulation we arrive at
b.ut/Atrwt ,r'
at1.wt ,'/
C
b.ut/At Id
tru,r'
bt1.'/
C
b.ut/ b.u/t
ru,r'
It 1t
f t ,'
bt2.'/
f t ft
,'
bt3.'/
D 0(16)
The first termon the left-hand side of the previous equality can
be considered as a bilinear form, andwedenote this formby at1.wt
,'/.The second term, the fourth term, and the fifth term can be
considered as linear functionals, andwedenote themby bt1.'/, b
t2.'/, b
t3.'/,
respectively.From Theorem 1, we know that the point-wise values
jutj and juj are bounded. From the differentiability of b, we
consequently
conclude that for each t and X there exists .X/ satisfying
min
ut.X/, u.X/ .X/ max ut.X/, u.X/
such that
b.ut.X// b.u.X//D b0..X//.ut.X/ u.X//.
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We plug the previous expression into the remaining integral on
the left-hand side of (16), and we get
b.ut/ b.u/
tru,r'
D b0./wtru,r'
at2.wt ,'/
. (17)
The right-hand side of the previous equation can be considered
as a bilinear form, and we denote it by at2.wt ,'/. The sum of at1,
a
t2 is
denoted by at , and the sum of bt1, bt2, b
t3 is denoted by b
t . Using the notations introduced above, we have
at.wt ,'/C bt.'/D 0. (18)
We would like to prove that for every positive t, there exists a
unique solution wt of (18). Here, we cannot use the classical
LaxMilgram theorem because at2.wt ,'/ is not coercive. However, we
do can use the results from [9] where the following
non-coercivelinear elliptic operators with Dirichlet boundary
conditions have been studied
r .Arw/ r .pw/C rw D L, in D,w D w0, on @D.
The strong formulation is not the same as in (2)(3), but the
authors have, anyway, studied the weak formulation that corresponds
to(18). We set
A D b.ut/At , p D b0./ru, r D 0.
Next, we verify the hypothesis from [9] on the coefficients
(i) A is a measurable matrix-valued function that is bounded and
coercive. This is true because from the L1 estimates of u, we
getthe continuity of at1. From 0< bmin b.s/, we have that at1.wt
, wt/ bmin=2krwtk22 that gives the coercivity of A.
(ii) p 2 L2C.D/. This is true, we can even obtain L1 estimate,
namely from the L1 estimates of ru and from boundedness of b0.(iii)
L 2 .W1,2.D//0. This is verified by showing the continuity of
bt.'/. From (12), we have that kt1.At Id/kL1.D/ C and
together with L1 boundedness of rut and ru we obtain the
boundedness (and thus continuity) of bt1. From (10), we havethat
kt1.It 1/kL1.D/ C and thus bt2 is bounded and continuous. From the
smoothness properties of ft , we can concludethat also bt3 is
bounded and continuous.
As a conclusion, we can use the existence and uniqueness result
from [9] and state that there exists a unique solution wt to
(18).Further, we need that the estimates of wt in H1.D/ are
independent of t. This is however true because change in t implies
change int ,and this does not influence the constant C appearing in
Theorem 2.1 from [9]. Thus, for the solution, the following
estimate holds
kwtkH1.D/ C, (19)
with C independent of t.In the following, we perform the
convergence analysis for t ! 0. From the previous estimate, we
directly have that krut ruk Ct
and therefore, we obtain that
ut ! u strongly in H1.D/.
From (11), we have MT ! Id in L1 and therefore we conclude that
MT rut ! ru strongly in L2.D/.If we now consider a sequence of
functions defined as wn D wtn , where tn ! 0, then we have the
boundedness of this sequence and
thus a weak convergence of a subsequence, still denoted wn in
H1.D/ to some element from H1.D/ that will be denoted as Pu.We are
going to derive an equation that is satisfied by Pu. We compute and
bound the following expressions:
A1 :D jatn1 .wn,'/ .b.u/r Pu,r'/jA2 :D jatn2 .wn,'/
.b0.u/Puru,r'/jB1 :D jbtn1 .'/ .b.u/Aru,r'/jB2 :D jbtn2 .'/ .fr
h,'/jB3 :D jbtn3 .'/ .rf h,'/j.
Remark 1We would like to emphasize the main difference between
the linear case and nonlinear case. In the linear case, b.u/ is
just a con-stant, independent of u. In this case, the expressions
A1, A2, B1, B2, B3 can be bounded directly. In nonlinear case, one
needs to carefullyconsider the properties of the functions, their
boundedness in various functional spaces, and the limit passes for
n ! 1.
Copyright 2011 John Wiley & Sons, Ltd. Math. Meth. Appl.
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Lemma 1Using the previous notations, we have the following
results
(a) b.utn/Atn ! b.u/Id strongly in L2.D/ for n ! 1.(b) b0./!
b0.u/ strongly in L2.D/ for n ! 1.
ProofTo prove the first statement, we begin with
b.utn/Atn b.u/
b.utn/.Atn Id/ C b.utn/ b.u/ .Using that b is Lipschitz
continuous, we obtain
b.utn/Atn b.u/2 kb.utn/k1k.Atn Id/k1 C C utn u2 .From (12), we
have that Atn ! Id in L1. Also utn ! u in H1.D/. This confirms the
first statement of the lemma.To show the second statement, we
recall a diagram on page 88 of [16] describing the mutual relations
between different types of
convergence. Of our interest is the relation between the
convergence in Lp.D/, and the existence of a subsequence that
convergesalmost everywhere. From this relation, we can state that
because un converges strongly to u in L2.D/, then there exists a
subsequencestill denoted by un such that for almost all X 2 D the
sequence un.X/! u.X/. But we know that
min
utn.X/, u.X/ .X/ max utn.X/, u.X/
that means that also .X/! u.X/ for almost all X 2 D. Using this
and the continuity of b0, we obtain that b0..X//! b0.u.X//.Finally,
from convergence almost everywhere, we conclude that b0./! b0.u/
strongly in H1.D/. We compute the limit for A1
limn!1 jA1j D limn!1
b.utn/Atn rwn,r'
.b.u/r Pu,r'/ lim
n!1b.utn/Atn b.u/rwn,r'
C b.u/.rwn r Pu/,r' .
The first limit is zero. This is true because rwn is L2 bounded,
and we suppose that ' 2 C1.D/. From Lemma 1 (a), we have
strongconvergence of the rest. The second limit is zero because wn
* Pu and b.jruj2/r' is fixed and bounded in L2.
The conclusion is that if ' 2 C1.D/ then limn!1 jA1j D 0.We
compute the limit of A2 for n ! 1
limn!1 jA2j D limn!1
b0./wnru b0.u/Puru,r'
limn!1
.b0./ b.u//wnru,r'
C b0.u/wn .rutn ru/,r'C b0.u/.wn Pu/ru,r' .
The last limit is zero because wn * Pu in L2.D/ and b0.u/ru r'
is fixed and bounded in L2. Next, for ' 2 C1.D/, we have
bounded-ness of r',ru in L1.D/. From (19) and from Lemma 1 (b), we
conclude that the first limit is zero, too. Next, we use the
boundednessof b0 and r' to end up with
limn!1 jA2j C limn!1 kw
nk2krutn ruk2
and because wn is bounded in L2.D/ and utn ! u in H1.D/,
strongly we conclude that for ' 2 C1.D/we have limn!1 jA2j D 0.For
the limits Bi , we have
B1 b.utn/At Idt ru b.u/Aru
2kr'k2
B2 It 1t f tn .fr h/
2k'k2
B3 f
tn ft
.rf h2k'k2.23
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The L2 norms for the expressions without ' can be bounded in
very similar manner that was done when computing the limits forA1,
A2 and for brevity, we skip the details. Eventually we get that
0 D limn!1 jB1j D limn!1 jB2j D limn!1 jB3j.
Remark 2The limits of A1 and A2 for n ! 1 are zero only under
the assumption that ' 2 C1.D/. Using the density argument in
further develop-ments leading to Lemma 2, we see that this
assumption is not restrictive. On the other hand, the limits of B1,
B2, and B3 for n ! 1 arezero for broader class of test functions,
namely for all ' 2 H1.D/. That means, for example, taking ' D wtn ,
we can obtain that
limn!1 jb
tn.wtn/ b.wtn/j D 0.
This observation will be crucial for later considerations about
the strong convergence of wt .
We are ready to derive the equation for Pu. We introduce the
bilinear form a and the functional b by
a.v,'/D .b.u/rv,r'/C .b0.u/vru,r'/, (20)
b.'/D .b.u/Aru,r'/C .r .f h/,'/. (21)
Similarly, as has been shown for at and bt , we can prove the
existence and uniqueness of the solution to a.w,'/C b.'/ D 0.
Usingthe density argument, we can prove that if the identity
a.w,'/C b.'/ D 0 is satisfied for all ' 2 C1.D/, then it is also
satisfied for all' 2 H1.D/. From the limits computed before, we
know that if wn * Pu in H1.D/ then Pu satisfies a.Pu,'/C b.'/D 0
for all ' 2 H1.D/. Butbecause the solution of a.v,'/C b.'/D 0 is
unique, we obtain that not only wn * Pu but also wt * Pu in
H1.D/.
To formalize this result, we state the following lemma.
Lemma 2wt * Pu in H1.D/ and the weak limit satisfies the
following equation
.b.u/r Pu,r'/C .b0.u/Puru,r'/C .b.u/Aru,r'/C .r .f h/,'/D 0
(22)
for all ' 2 H1.D/. Moreover, the solution Pu satisfies
@Pu@xi
2 C0,./, @Pu@xi
2 C0,.D n/. (23)
ProofThe first part of the lemma has just been proven. The
second part is a direct consequence of [17, Theorem 16.2]. To
fulfill the assump-tions of the theorem, one needs to guarantee
that the coefficients of the linear problem (22) belong to C0,./
and to C0,.D n/.Those coefficients are, however, the solutions of
(2)(3) and the required regularity is verified by Theorem 1.
Further, we would like to show the existence of strong material
derivative. As we see later, we do not succeed in this task
withoutadditional assumptions.
Lemma 3Using the previous notations, the following is valid
at.wt , wt/! a.Pu, Pu/ for t ! 0, (24)
at.Pu, Pu/! a.Pu, Pu/ for t ! 0, (25)
at.Pu, wt Pu/! 0 for t ! 0. (26)
ProofTo show the first statement set ' D wt in (18). We obtain
at.wt , wt/D bt.wt/. From Remark 2, we know that
limt!0 jb
t.wt/ b.Pu/j D limt!0 jb
t.wt/ b.wt/j C limt!0 jb.w
t/ b.Pu/j D 0,
which results in bt.wt/! b.Pu/. From (22), we have that a.Pu,
Pu/D b.Pu/ that proves the first statement.
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To show the second statement, we estimate
jat.Pu, Pu/ a.Pu, Pu/j b.ut/Atr Pu,r Pu .b.u/r Pu,r Pu/C
j.b0./Puru,r Pu/ .b0.u/Puru,r Pu/j
kb.ut/At b.u/k2kr Puk21C kb0./ b0.u/k2kPuk1krut C ruk1kr Puk1C
kb0.u/k1kPuk1krut ruk2kr Puk1
From Lemma 2, we know that r Pu is L1 bounded, and we also know
from Lemma 1 (a) that b.ut/At ! b.u/ in L2. From Lemma 1 (b),we
know that b0./ ! b.u/ in L2 and also rut and ru are L1 bounded.
Finally, because ut ! u strongly in H1, we can conclude thesecond
statement of the lemma.
To show the last statement of this lemma, we start with
limt!0 ja
t.Pu, wt Pu/j D limt!0 ja
t.Pu, wt/ at.Pu, Pu/j lim
t!0 jat.Pu, wt/ a.Pu, wt/j C lim
t!0 ja.Pu, wt Pu/j C lim
t!0 ja.Pu, Pu/ at.Pu, Pu/j
The second limit is zero because wt * Pu in H1.D/ and the third
limit is zero from (25). For the first limit, we estimatejat.Pu,
wt/ a.Pu, wt/j b.ut/Atr Pu,rwt .b.u/r Pu,rwt/
C j.b0./Puru,rwt/ .b0.u/Puru,rwt/j kb.ut/At b.u/k2kr
Puk1krwtk2
C kb0./ b0.u/k2kPuk1krut C ruk1krwtk2C kb0.u/k1kPuk1krut
ruk2krwtk2.
From Lemma 2, we know that r Pu is L1 bounded. Also krwtk2 C.
From Lemma 1 (a), we have that b.ut/At ! b.u/ in L2. Therefore,the
first term on the right-hand side tends to zero.
From Lemma 1 (b), we know that b0./ ! b.u/ in L2 and also rut
and ru are L1 bounded that means that also the second termtends to
zero.
Finally, because ut ! u strongly in H1, we can conclude that the
limit of at.Pu, wt/ a.Pu, wt/ is zero, and the statement of the
lemmais valid.
Remark 3In the proof of the previous lemma, we again see the
difference between our nonlinear case and the linear case. The
ingredients of ourproof are much more refined as in the linear
case, see, for example, [2].
Application of Lemma 3 gives
limt!0.a
t.wt , wt/ at.Pu, Pu//D limt!0.a
t.wt , wt/ a.Pu, Pu//C limt!0.a.Pu, Pu/ a
t.Pu, Pu//D 0.
Using this result together with (26), we end up with
limt!0 a
t.wt Pu, wt Pu/D limt!0.a
t.wt , wt/ at.Pu, Pu// 2 limt!0 a
t.wt Pu, Pu/D 0.
If the bilinear form at were coercive, then we would be able to
conclude that
at.wt Pu, wt Pu/ Ckwt PukH1.D/that together with the previous
result would give that wt ! Pu strongly in H1.D/.
However, at is not coercive and therefore for the original
problem (2)(3), we are not able to prove the existence of strong
materialderivative. The only lower estimate on at.v, v/ is the
Garding inequality
at.v, v/ krvk22 Cgkvk22with Cg dependent on D and bmin.
In the different scenario, when an internal heat source term is
introduced to the original problem, we will be able to prove
theexistence of the strong material derivative. Consider the
following elliptic problem with positive Cs
r .b.x, u/ru/C Csu D f .x/ in D, u D 0 on @D, (27)replacing the
original one (2).
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For this direct problem, the term Cs will change the Garding
inequality to
at.v, v/ krvk22 Cgkvk22 C Cskvk22.
It is thus sufficient to assume that Cs > Cg in order to get
the coercivity of at and subsequently to obtain an existence of the
strongmaterial derivative.
We formalize this result in the following theorem.
Theorem 2Assume that the internal heat source coefficient Cs
> Cg, where Cg is the Garding coefficient of at . Then there
exists a strong materialderivative Pu related to problem (27) for
which wt ! Pu strongly in H1.D/.
5. Shape derivative
The shape derivative will be denoted by u0 and defined by u0 D
Pu h ru. We derive an equation for u0. In this section, let us
supposethat u,' 2 W2,2./ and u,' 2 W2,2.D n/. Theorem 1 does not
guarantee this, therefore, we need to assume such regularity of u.
Letus compute the following expression denoted by R
R :D a.u0,'/C .r .b.u/ru/ ,h r'/D .b.u/ru0,r'/C .b0.u/u0ru,r'/C
.r .b.u/ru/ ,h r'/D .b.u/r Pu,r'/C .b0.u/Puru r'/ .b.u/r.h
ru/,r'/
.b0.u/h ruru,r'/C .b.u/u,h r'/C .r.b.u// ru,h r'/.
The sum of the first and the second term on the right-hand side
is equal to a.Pu,'/, and thus we can replace it by b.'/ from
Lemma2. We then regroup some terms to obtain
R D .b.u/Aru r.h ru/Cuh,r'/C .r .f h/,'/C .b0.u/jruj2h h
ruru,r'/.
Recall that the curl operator acting on a scalar function is
defined asrf D .fy ,fx/. It is straightforward, although a little
bit tedious,to verify that
Aru r.h ru/Cuh r' D r .h2ux h1uy/ r',jruj2h h ruru D r u.h2ux
h1uy/.
We can therefore use the previous findings to go on in the
computation of R
R D .b.u/r .h2ux h1uy/,r'/C .r .f h/,'/ .b0.u/r u.h2ux
h1uy/,r'/D .r b.u/.h2ux h1uy/,r'/C .r .f h/,'/. (28)
We use the Green theorem for a 2-dimensional region S
Z@S
rr' t D Z
Sr r r'. (29)
We are going to compute the first integral on the right-hand
side of (28). We split the integration domain D into two subdomains
and D n. We set r :D h2ux h1uy . Notice that space dependent
function r, as well as the function b.u/, both have a
discontinuityacross @. We use the superscripts C and to indicate
the limit values when approaching the boundary @ from outside of
andfrom inside of, respectively, that is
f C.x/D limxn!x f .xn/ for xn 2 D n, f
.x/D limxn!x f .xn/ for xn 2.
We perform integration by parts for two domains separately using
(29)
.r b.u/.h2ux h1uy/,r'/D .b.u/r,r' tD/@D .b.u/CrC,r' t/@ C
.b.u/r,r' t/@where t is defined as counter-clockwise unit
tangential vector to. Because of h D 0 on @D, we also have r D 0 on
@D and thus thefirst integral on the right-hand side vanishes.
We write h D hnn Chtt, the sum of its projections onto the
orthonormal system .n, t/. We know that t D .t1, t2/D .n2, n1/and
thus
r D h2ux h1uy D htn ru hnt ru.
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Therefore, we obtain
.r b.u/.h2ux h1uy/,r'/D .b.u/C.htn ruC hnt ruC/,r' t/@C
.b.u/.htn ru hnt ru/,r' t/@.
We can use the interface condition (3) to obtain
.r b.u/.h2ux h1uy/,r'/D .hn.b.u/CruC b.u/ru/ tt,r'/@.Now,
realize that .b.u/CruC b.u/ru/ tt is nothing else than the
projection of b.u/CruC b.u/ru onto t. But from
the interface condition, we know that b.u/CruC b.u/ru is
perpendicular to n and therefore.b.u/CruC b.u/ru/ tt D b.u/CruC
b.u/ru.
We put the obtained findings into (28) to obtain
a.u0,'/C .r b.u/ru,h r'/D .hn.b.u/CruC b.u/ru/,r'/@ C .r .f
h/,'/.From (2), we have that r b.u/ru D f and thus
a.u0,'/ .f ,h r'/ D .hn.b.u/CruC b.u/ru/,r'/@ C .r .f
h/,'/a.u0,'/ .f h nD,'/@D C .f h n,'/@ .f h n,'/@
D .hn.b.u/CruC b.u/ru/,r'/@.Because h D 0 on @D, we can
successfully conclude this section with the characterization of the
elliptic interface problem defined as
a.u0,'/D .h n.b.u/CruC b.u/ru/,r'/@. (30)that is satisfied by
the shape derivative u0.
6. Adjoint problem - shape derivative method
We differentiate the cost functional (15) with respect to t
DJ :D limt!0
J.t/ J./t
D limt!0
1
2t
Z!
jut NKtj2 ju NKj2.
From [15, 23], we have that
DJ :DZ
!.u NK/u0. (31)
We introduce an adjoint problem in order to explicitly compute
the derivative of the cost function J./. For the definition of
theadjoint problem, we use the bilinear form a, which has been
defined by (20). Denote by p a W1,2.D/ function such that puc 2
W1,20 .D/and
a.p, /DZ
!.u NK/ (32)
is satisfied for all 2 W1,20 .D/. Moreover, assume that p 2
W2,2./ and p 2 W1,2.D n/.Take the following test functions ' D p in
(30) and D u0 in (32). The left-hand sides of the resulting
equalities are equal and
therefore, we obtain
DJ DZ
!.u NK/u0 D .h n.b.u/CruC b.u/ru/,rp/@ (33)
Therefore, the steepest descend direction (denoted by hsd) for
the gradient-type algorithms minimizing J is given by
hsd D .b.u/CruC b.u/ru/ rpn, on @. (34)
7. Implementation
For the description of the geometry, we use the level set
method. We refer [15, 20] and the references therein for an
overview.The boundary of is represented by a zero level set of a
function . To minimize the shape functional J, we would like to
move the
interface @ in the steepest descent direction hsd. The level set
method allows us to do this by solving the HamiltonJacobi
equation
t C hsdjrj D 0, (35)where hsd :D jhsdj.
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7.1. Numerical algorithm
Given data on the domain !, the algorithm to identify the
unknown inside the domain D is outlined as follows.
(a) Set an initial level set function as an initial guess. For j
D 0, j D 1, : : : , do the following until the algorithm
converges(b) Solve (4) forj to obtain the solution of the direct
problem uj , where we use j to indicate quantities in the jth
step.(c) Solve (32) forj and uj to obtain the solution of the
adjoint problem pj .
(d) Evaluate the normal steepest descent direction hjsd from
(34).
(e) Update the level set function by solving jt C hjsdjr jj D
0.(f ) If the convergence is reached, then stop otherwise shift the
index j with the corresponding quantities and go to the (b) part
of
this algorithm.
We will use a finite element method for the finite dimensional
approximation of . For the approximation of H1.D/, we
chooseLagrange finite elements of the first order.
In part (b), we need to compute a nonlinear elliptic equation
(4). This equation can be considered as an operator equation G.u/ D
fwhere G is a mapping G : u 2 W1,2./! G.u/ 2 W1,2./ such that
b.jruj2ru,r' D G.u/,',
This operator equation is nonlinear, and therefore it will be
solved for all the numerical examples by the same iterative
algorithm.Starting from the initial guess u0, we use the
NewtonRaphson algorithm based on the following update
uiC1 D ui DG.ui/1.G.ui/ f /.
Notice that for each iteration one linear partial differential
equation (PDE) has to be solved. For the evaluation of hsd , we
need toproject .b.u/CruC b.u/ru/ rp onto space of Lagrange finite
elements. This is done by solving simple linear equation
[email protected]/CruC b.u/ru/ rp'dx D
ZD
hsd'dx. (36)
In Section 8, we discuss how we tackle the line integral on the
left-hand side.Part (e) involves the solution of the HamiltonJacobi
equation. We use simple approximation scheme
jC1 jt
C hjsdjr jj D 0.
The step size t is chosen dynamically. It is doubled if the
shape functional decreases, otherwise it is divided by 2 until we
obtainthe decrease in functional. For evaluation of jr jj,
different approaches can be used. For an overview of up-wind
schemes on triangu-lar meshes, we refer to [18] and the references
therein. The widely used ENO (Essentially non-oscillatory) and WENO
(Weighted ENO)schemes have been used in numerous applications. We
do not use any up-winding and still we obtain satisfactory results
without oscil-lations. The convergence in part (f ) is controlled
by checking if the shape functional J sufficiently drops. If
jJ.j/J.jC1/j< etr , whereetr is some small threshold, we stop
with the algorithm.
8. Numerics
We use a smeared out Heaviside function as recommended in [20].
The following smooth approximation of the Heaviside function
isused
Hk./D 0.5C 1arctan.k/. (37)
Real parameter k defines how steep is the approximation around
zero. For k ! 1, Hk./ converges pointwise to H./. Thecomputation of
line integrals becomes simpler, for example, instead of (36) we
have
ZD
H0k./.b2.u/ru b1.u/ru/ rp'dx DZ
Dhsd'dx.
Without any regularization, all simulations have oscillations of
the zero level set. To stabilize the optimization process, we
introducethe Tikhonov stabilizing term equal to the squared norm of
the gradient of the level set function. The coefficient controls
the weightof the regularization. The cost function J from (5) thus
obtains a new term
J./D 12
Z!
ju./ NKj2 C Z
jrj2dx.
The expression (36) for the evaluation of the normal steepest
descent direction hsd changes by adding the corresponding
derivativeof the regularization term to
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ZD
H0k./.b2.u/ru b1.u/ru/ rp'dx C 2Z
Dr r'dx D
ZD
hsd'dx. (38)
Throughout this section, we consider 2R2 to be a rectangle .1,
1/.0.25, 0.25/. All the linear problems are solved on the regu-lar
triangular mesh. Themeshwas obtained by splitting the longer side
into 60 line segments and the shorter one into 15 line
segmentsforming a grid with 90 small squares. Then, each square has
diagonally been split into two triangles resulting into a mesh with
1800triangular elements and with 976 nodes. Given the size of space
discretization, we choose k D 40 that follows the recommendationsin
[20].
The domain corresponds to a rectangular metal rod. Inside the
rod, two holes are located filled either with air or with water.
Werun tests for two different metals, for gold (Au) and zirconium
(Zr). With this choice of materials, we want to demonstrate that
ouralgorithm is robust enough to cover materials with qualitatively
different properties. The only dependence of the problem on a
mate-rial is expressed by the nonlinear thermal conductivity
function b2. Thermal conductivities of Au and Zr differ
significantly, namelyconductivity of Au is a decreasing function
whereas conductivity of Zr is an increasing function in the working
range of temperatures.
The explicit expressions for thermal conductivities of air and
water have been obtained from the Chemistry WebBook [19],
providedby the National Institute of Standards and Technology , see
also Figure 1
bO21 .s/D 8.38 109s2 C 8.68 105s C 1.66 103
bH2O1 .s/D 9.54 106s2 C 7.35 103s 0.737 for s 373.15
3.81 108s2 C 6.39 105s 5.25 103 for s > 373.15.
Thermal conductivity of water is discontinuous because water
changes the phase at 373.15K. Further, we use a formula
forconductivities of Zr from [12] and of Au from [8], see also
Figure 2
bZr2 .s/D 2.53 106s2 C 7.08 103s C 8.85C 2.99 103s1bAu2 .s/D 8
108s3 C 2 104s2 0.2s C 3.55 102.
We choose three different settings. In all of them, the rod has
two rectangular holes in the interior. Given these holes, we
generatesynthetic data to replace the real measurements. The
measurements of the temperature are available only to a specific
depth. SeeFigure 3 for the geometry. The holes are not reachable
with measurements. In the iterative algorithm as a first
approximation of theholes or of the domain, we choose two large
circles encircling the actual exact holes.
0.0250.03
0.0350.04
0.0450.05
0.0550.06
0.0650.07
0.075
300 400 500 600 700 800 900the
rmal
con
duct
ivity
(WK-1
m-1 )
ther
mal
con
duct
ivity
(WK-1
m-1 )
temperature (K)
oxygen
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
300 350 400 450 500temperature (K)
water
Figure 1. Thermal conductivities of O2 and H2O.
18.5
19
19.5
20
20.5
21
21.5
300 400 500 600 700 800 900 300 400 500 600 700 800 900ther
mal
con
duct
ivity
(WK-1
m-1 )
ther
mal
con
duct
ivity
(WK-1
m-1 )
temperature (K)
zirconium
275280285290295300305310315
temperature (K)
gold
Figure 2. Thermal conductivities of Zr and Au.
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solid,measurements
solid,no measurements
water oroxygen0.1
0.4
Figure 3. Geometry of the problem. Large rectangle is the rod.
The area with dense slanted lines represents the area of
measurements. Two white smaller
rectangles are two holes. Two circles depict the initial
approximation in the iterative algorithm.
(b)
(c)
(a)
Figure 4. Au and O2 (a) The exact shape is depicted by two small
rectangles, the measurement area is bounded by large white
rectangle and the borders of the
figure. The initial shape consists of two white circles. The
color palette ranging from blue (300K) to red (753K) shows the
distribution of the temperature over the
whole domain for the exact shape. (b) Approximating shapes for
the 5th iteration (yellow line), for 15th iteration (light green
line) and for 20th iteration (dark
green line). (c) Approximating shapes for the 32th iteration
(turquoise line), for 425th iteration (blue line) and for the final
615th iteration (black line).
8.1. Au and O2
Here, we consider the rod made of gold and the holes inside the
rod are filled with oxygen. The surface of the rod has been kept
at300K. The heat source term f has been set to f D 4.0106. The
temperature inside the rod has raised during the simulation up to
753K.
Here, the conductivities differ by three orders of magnitude,
and the regularization parameter has been set to D 1.0.
8.2. Zr and O2
Here, we consider the rod made of zirconium and the holes inside
the rod are filled with oxygen. The surface of the rod has been
keptat 673K that means that the Dirichlet boundary conditions u D
673K have been used in (2). The heat source term f has been set tof
D 105. The temperature inside the rod has raised during the
simulation up to 850K.
Here, the conductivities differ by two orders of magnitude, and
the regularization parameter has been set to D 0.25.
8.3. Zr and H2O
The last combination is a zirconium rod filled with water.
During this simulation, the temperature range is between 340K and
380K. Thisworking range is interesting because at 373.15K the water
inside the holes changes its phase from liquid to gas. The source
term f isset to f D 2.3 104 that is a value for which the isoline T
D 373.15 crosses both holes, and thus the water is in two phases.
The thermalconductivity of water is depicted in Figure 1, and we
see that in the range of temperatures 340K380K the conductivity is
discontinu-ous. The theory from Sections 36 is thus in fact not
valid because the hypothesis A2 is not fulfilled. Anyway, we wanted
to show thatthe algorithm works fine even in this case.
Another reason for choosing the combination of Zr and H2O was
that the conductivities differ only by one order of magnitude.
Inthis case, there is weaker distinction between the solid and the
liquid. The algorithm must thus be more sensitive. On the other
hand,
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because there are no different length scales involved,
thematrices are better conditioned. Also for this case, the
simulations show goodresults. The regularization parameter has been
set to D 0.05.
Discussion
From Figures 46, we can see that the evolution of the
approximating shapes is similar in all cases. The evolution of the
curves can besplit roughly in two parts:
First, the approximating shapes shrink in the vertical direction
only. This can be explained by the fact that the available data are
closerto the actual hole in the vertical direction than in the
horizontal direction. So the gradients are much higher on the upper
and lowersides of the approximation shapes then on the left and
right sides. Therefore, they push the circles from up and from down
towards theexact holes.
(b)
(c)
(a)
Figure 5. Zr and O2. (a) Similar objects as in Figure 4. The
color palette ranges from blue (673K) to red (850K). (b) The color
palette is reduced to the area of the
available measurements. Approximating shapes for the 10th
iteration (yellow line), for 20th iteration (light green line), and
for 85th iteration (dark green line).
(c) Approximating shapes for the 205th iteration (turquoise
line), for 400th iteration (blue line), and for the final 685th
iteration (black line).
(b)
(c)
(a)
Figure 6. Zr and HO2 . (a) Similar objects as in Figure 5. The
color palette ranges from blue (340K) to red (380K). The black line
indicates the isoline u = 373.15K at
which the phase of water changes. (b) Approximating shapes for
the 15th iteration (yellow line), for 22th iteration (light green
line), and for 33th iteration (dark
green line). (c) Approximating shapes for the 60th iteration
(turquoise line), for 200th iteration (blue line), and for the
final 275th iteration (black line).
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Table I. Regularization parameter versus conductivityratio.
AuO2 ZrO2 ZrH2O
Regularization 1.0 0.25 0.05Ratio b2=b1 6 103 4 102 3 101
Second, the approximating shapes shrink in both vertical and
horizontal direction towards the exact holes. In a specific time,
thegradients on the upper and lower sides of the approximating
shapes become comparable with gradients on the right and left
sides.Therefore, the curves are pushed also from left and right
side.
Interesting observation is that the regularization parameter is
proportional to the order of magnitude by which the
conductivitiesof solid and gas/liquid differ. Indeed in Table I, we
see this dependence. Greater the order ofmagnitude, greater was
needed to obtaina good solution. Moreover, the proportionality is
linear. When the ratio b2=b1 is divided by 10, the regularization
parameter is dividedby 4. This dependence is linked with the
condition number of the matrices arisen from solving linear
problems.
References1. Ammari H, Iakovleva E, Kang H, Kim K. Direct
alghorithms for thermal imaging of small inclusions. Multiscale
Modelling and Simulation 2005;
4(4):11161136.2. Ammari H, Kang H, Lee H. Layer potential
techniques in spectral analysis. In Mathematical Surveys and
Monographs, Vol. 153. AmericanMathematical
Society: Providence, 2009.3. Ammari H, Kang H. Polarization and
moment tensors: with applications to inverse problems and effective
medium theory. In Applied Mathematical
Sciences Series, Vol. 162. Springer-Verlag: New York, 2007.4.
Babuka I. The finite element method for elliptic equations with
discontinuous coefficients. Computing 1970; 5:207213.5. Borsuk M.
The transmission problem for quasi-linear elliptic second order
equations in a conical domain. i, ii. Nonlinear Analysis: Theory,
Methods &
Applications 2009; 71(10):50325083.6. Cantwell WJ, Morton J. The
significance of damage and defects and their detection in composite
materials: A review. The Journal of Strain Analysis
for Engineering Design 1992; 27(1):2942.7. Cimrk I, Van Keer R.
Level set method for the inverse elliptic problem in nonlinear
electromagnetism. Journal of Computational Physics 2010;
229(24):9269-9283.8. Cliver J, Hoang C. Thermal conductivity of
solids.
http://www.owlnet.rice.edu/.ceng402/proj03/choang/ceng402/ceng402.html.9.
Droniou J. Non-coercive linear elliptic problems. Potential
Analysis 2002; 17(2):181203.
10. Durand S, Cimrk I, Sergeant P. Adjoint variable method for
time-harmonic Maxwells equations. COMPEL: The International Journal
for Computationand Mathematics in Electrical and Electronic
Engineering 2009; 28(5):12021215.
11. Durand S, Cimrk I, Sergeant P, Abdallh A. Analysis of a
Non-destructive Evaluation Technique for Defect Characterization in
Magnetic MaterialsUsing Local Magnetic Measurements. Mathematical
Problems in Engineering 2010; 2010:574153.
12. Fink JK, Leibowitz L. Thermal conductivity of zirconium.
Journal of Nuclear Materials 1995; 226(12):4450.13. Frehse J. On
the boundedness of weak solutions of higher order nonlinear
elliptic partial differential equations. Bollettino della Unione
Matematica
Italiana 1970; 3:607627.14. Abel IR (ed.). Printed circuit board
fault detection and isolation using thermal imaging techniques,
Society of Photo-Optical Instrumentation Engineers
(SPIE) Conference Series, Vol. 636, January 1986.15. Ito K,
Kunisch K, Li Z. Level-set function approach to an inverse
interface problem. Inverse Problems 2001; 17(5):12251242.16. Kufner
A, John O, Fucik S. Function Spaces. In Monographs and Textbooks on
Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff
International Publishing: Leyden; Academia, Prague, 1977.17.
Ladyzhenskaya OA, Uraltseva NN. Linear and Quasilinear Elliptic
Equations, Mathematics in science and engineering, Vol. 46.
Academic press: New
York (N.Y.), 1968.18. Levy D, Nayak S, Shu C, Zhang Y. Central
WENO schemes for Hamilton-Jacobi equations on triangular meshes.
SIAM Journal on Scientific Computing
2006; 28(6):22292247. (electronic).19. National Institute of
Standards and Technology. NIST Chemistry WebBook.
http://webbook.nist.gov/.20. Osher S, Fedkiw R. Level Set Methods
and Dynamic Implicit Surfaces, Applied Mathematical Sciences, Vol.
153. Springer-Verlag: New York, 2003.21. Rivkind VY, Uraltseva NN.
Classical solvability and linear schemes for the approximate
solution of the diffraction problem for quasilinear equations
of parabolic and elliptic type. Journal of Mathematical Sciences
1973; 1(2):235264.22. Sinha RK, Deka B. Finite element methods for
semilinear elliptic and parabolic interface problems. Applied
Numerical Mathematics 2009; 59(8):
18701883.23. Sokolowski J, Zolesio JP. Introduction to shape
optimization, Springer Series in Computational Mathematics, Vol.
16. Springer-Verlag: Berlin, 1992.
Shape sensitivity analysis.24. enek A. Nonlinear Elliptic and
Evolution Problems and Their Finite Element Approximations.
Academic Press: (London, San Diego, CA), 1990. ISBN10
0-1277-9560-X.25. Zhang X. Uniqueness of weak solution for
nonlinear elliptic equations in divergence form. International
Journal of Mathematics and Mathematical
Sciences 2000; 23(5):313318.26. Zolsio JP. The material
derivative (or speed) method for shape optimization. In
Optimization of distributed parameter structures, Vol. II (Iowa
City,
Iowa, 1980), Vol. 50, NATO Adv. Study Inst. Ser. E: Appl. Sci,
1981; 10891151.
Copyright 2011 John Wiley & Sons, Ltd. Math. Meth. Appl.
Sci. 2011, 34 23032317
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