0Inverse - HY

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

Copyright 2011 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 23032317

23

03

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].

23

04


I. CIMRK

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)


23

05

I. CIMRK

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//.

23

06


I. CIMRK

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.


23

07

I. CIMRK

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

08


I. CIMRK

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.


23

09

I. CIMRK

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).

23

10


I. CIMRK

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.


23

11

I. CIMRK

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.

23

12


I. CIMRK

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


23

13

I. CIMRK

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.

23

14


I. CIMRK

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,


23

15

I. CIMRK

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).

23

16


I. CIMRK

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.


23

17

Documents

0Inverse - HY