Boundary shape identi cation in two-dimensional ... - CiteSeerX

Boundary shape identi cation in two-dimensional electrostatic problems using SQUIDs H.T. BANKS 1 and F. KOJIMA 2

Abstract | This paper is concerned with a quantitative nondestructive evaluation of conduc-

tors using superconducting quantum interference devices ( SQUIDs ). A measurement system is described for an electrical potential problem with an unknown boundary. Domain identi cation is discussed within the theoretical framework of a parameter estimation problem for the electrostatic eld analysis. Applying the method of mappings to the problem considered here, we present computational methods, including theoretical convergence results for the associated nite dimensional problem identi cation techniques.

1. INTRODUCTION Recently, demand has grown for assessing the structural integrity of materials used in nuclear power plants using advanced sensor technologies. An important eort on such problems entails quantitative nondestructive evaluation methods in magnetic ux imaging techniques. These methods involve an attempt to characterize structural aws or defects which may not be detectable by visual inspection. SQUIDs ( superconducting quantum interference devices ) have the potential to detect material defects in conductors due to their extremely high magnetic ux sensitivity [1]. In this paper, we propose a computational method for recovering defect shapes with magnetic ux density data from high critical temperature SQUIDs (HTc-SQUIDs). In the proposed nondestructive test, a stationary current density is applied to the conductor inspected. The magnetic ux density can be measured from a SQUID sensor located near the conductor. Figure 1 illustrates the inspection process using an HTc-SQUID. As shown in Fig. 1, the existence of a defect ( represented by a nonconducting volume ) corrupts the current ows inside the conductor and, as a result, this material defect can be Center for Research in Scienti c Computation, North Carolina State University, Raleigh, NC 276958205, USA. E-mail:[email protected] 2 Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1, Ohmiya, Asahi-ku, Osaka 535-8585 JAPAN. E-mail:[email protected] 1

1

detected as the perturbation of magnetic ux data. It is well-known that a mathematical model for this nondestructive test can be derived from Maxwell's equations. By Faraday's law and Gauss's law, the electrical potential E satis es

rE = 0 r
E = 

(1) (2)

where  and  denote the permittivity and the charge density of the sample specimen, respectively. We introduce the electrical scalar potential  such that

E = ?r: Pick-up coil of SQUID Material flaw G S0

C Sample

S1

Fig. 1 Inspection process using HTc-SQUID Assuming that there is no charge density inside the material and that the conductor is a homogeneous plate, we can rewrite Eqs. (1)-(2) as the Laplace equation

r2  = 0

in G3  R3 ;

(3)

with non-homogeneous boundary conditions

 = v on @GD3 ; r
n^ = 0 on @GN3 [ @GC3 ; where G3 and @G3 = @GD3 [@GN3 [@GC3 denote, respectively, the domain and the boundary of the conductor with a aw. More speci cally, the boundary @GD3 [ @GN3 denotes the exterior boundary of the conductor and @GC3 characterizes the defect shape. In the above equations, v denotes the prescribed electrical potentials on @GD3 . The measurement of magnetic ux density by HTc-SQUIDs is described by Biot-Savart's law [2], for x = (x1; x2; x3) 2 R3 ? G3, x0) dx0; B (x) = ? 400 rj x?(xx? 0 j3 G3 Z

2

where 0 and 0 denote the electrical conductivity and magnetic permeability of the conductor, respectively. Such measurements can be obtained by detecting voltage changes in SQUID magnetmetors. We assume that the measurements are only made on the vertical direction (x3-axis). Since the voltage change can be measured through the pickup coil, we can only take the average of the magnetic ux density at the sensor location. Suppose that xp 2 R3 ? G3 is the location of sensor and Sp denotes the region of the pickup coil with the center xp. The problem considered here is that of identifying, from input and output data fv; B (xp)g, the geometrical shape of the defect. If we assume that the conductor is a thin plate and that the defect to be identi ed is uniform in the x3 direction, the corresponding electrical potential becomes @ ; @ ; 0 : r  @x (4) 1 @x2 Then the observation at xp can be approximated by !

Bx3 (xp)  Yx = ? 40jS0 j p

Z

"

Z

@ (x2 ? x02) @x 0

(

@ ? (x1 ? x01) @x 0

)

jx ? x0j?3dx0

#

dx; (5) where jSpj denotes the crosssectional area of the HTc-SQUID. From our assumptions, the identi cation problem is well approximated by using a 2-D spatial domain. Thus, in this paper, our attention is restricted to a domain identi cation problem in two dimensions. p

Sp

G3

1

2

This paper is organized as follows. In Section 2, the mathematical model for the system is described by the Laplace equation in a two-dimensional spatial domain. The measurements are derived from Biot-Savart's law. Then these problems are treated as domain identi cation problems in electrostatic eld analysis. In Section 2, we also formulate this problem in an abstract setting in a Hilbert space. The ideas proposed in Banks et. al., [3][4] using the \method of mappings" are adopted to the problem considered here. In Section 3, for computational purposes, we approximate the Hilbert space by nite dimensional subspaces and we discuss convergence analysis for the approximate identi cation problems. A practical computational algorithm is proposed brie y in the last section.

2. PROBLEM FORMULATION AND BASIC ASSUMPTIONS Let G be the sectional plane of the sample specimen G3 as shown in Fig. 1. Let q be a constant vector which characterizes an unknown defect domain where q belongs to an admissible parameter set Q such that

(H-0) The admissible parameter set Q is a compact subset of RM . We consider a bounded domain Cq ( G  R2) which is parametrized by unknown values

of q. As depicted in Fig. 2, the bounded domain Gq is considered as the system domain 3

such that Gq = G ? Cq . Corresponding to the boundary @GD3 , @GN3 and @GC3 in three dimensions, we assume the boundary of Gq consists of the following components: @Gq = [4i=1 @Gi @Cq We further assume that meas(Gq )   > 0, uniformly for q 2 Q, for some positive . [

Cq

Fig. 2 Spatial domain G in two dimensions Thus, from Eqs.(3) and (4), the system can be rewritten by r2 = 0 in Gq (6)  = vi on @Gi; i = 1; 3; (7) r
n^ = 0 on @Gi [ @Cq ; i = 2; 4: (8) Suppose that the SQUID sensor is scanned on the x1 ? x2 plane and measurements are made at fxip = (xi1p; xi2p)gmi=1. Let Spi be the region of the pick-up coil whose center is located at xip. We also assume that the distance between the surface of the plate and SQUID sensor is taken as h. Then, from (5), the observation in case of two-dimensions is described by @ ? (x ? x0 ) @ Ypi = ? 40jS0
i j (x2 ? x02) @x 1 1 @x0 0 1 2 p S G ? 3 = 2  (x1 ? x01)2 + (x2 ? x02)2 + h2 dx01dx02 dx1dx2; for i = 1; 2;


; m; (9) "Z

Z

i p

)

(

q

n

o

4



where
denotes the thickness of the plate. Let  be an arbitrary smooth function on Gq satisfying  = vi in @Gi; i = 1; 3: Then the system (6)-(8) is equivalent to the following boundary value problem :

 = u +  where u is the solution of

r2u = ?r2

in Gq u = 0 on @Gi; i = 1; 3; ru
n^ = ?r
n^ on @Gi [ @Cq ; i = 2; 4:

(10) (11) (12)

Let Vq be the closed subset of functions in H 1(Gq ) satisfying the homogeneous boundary conditions on @Gi; i = 1; 3, i.e.,

Vq = f j 2 H 1(Gq ); = 0 on @Gi; i = 1; 3 g: endowed with the norm j jV def = G jr j2dx: Z

q

q

(13)

Lemma 1. Suppose that (C-1)  is an element of Vqv , the inhomogeneous analogue of Vq de ned by Vqv = f j 2 H 1(Gq ); = vi on @Gi; i = 1; 3 g: Then, for every q 2 Q, there exists a unique solution (q) 2 Vqv of (6)-(8) in the sense that u =  ?   2 Vq satis es (10)-(12). The solution u of (10)-(12) is in the weak sense, i. e.,

(q)(u; ) = ?(q)(; ) for 2 Vq :

(14)

Here (q)(
;
) denotes a sesquilinear form on Vq  Vq de ned by

(q)('; Moreover, we have

) def =

Z

Gq

r'
r dx:

ju(q)jV  K1jjV

where K1 is a constant independent of q.

q

5

q

(15)

Remark. We note that even though  is in Vqv and not in Vq , the value of (q)(; ) for 2 Vq can be de ned as in (15) and the value of jjV is well de ned as in (13). We q

tacitly assume in the statement of Lemma 1 and in subsequent discussions that such an interpretation is understood. Proof. From (15), for arbitrary ' 2 Vq , (q) is V-elliptic with constant C1 = 1 which is independent of q since (16) (q)('; ') = jr'j2dx = jr'j2L2(G ) = j'j2V : Z

Gq

Since we have

q

q

j(q)('; )j  j'jV j jV : q

the linear functional ?(q)(;
) is bounded

q

on Vq , i. e., j ? (q)(; ')j  j'jV jjV for ' 2 Vq : (17) From (16) and (17), we can apply the Lax-Milgram lemma, taking the Hilbert space Vq , sesquilinear form (q)(
;
), and linear functional ?(q)(;
), respectively. Hence, for each q 2 Q, there exists a unique solution u 2 Vq in the sense of (14). Similarly, from (14), (16) and (17), we have juj2V  j ? (q)(; u)j  jjV jujV from which follows jujV  jjV ; thus completing the proof. For convenience of theoretical developments, we use the polar coordinate system x = (r; ) instead of the Cartesian coordinate x = (x1; x2) in the sequel. To this end, we introduce the unknown defect function c(; q) and the prescribed outer boundary function l(). These functions are assumed to be de ned on (0; 2) and are 2-periodic, respectively. As depicted in Fig. 3, it is further assumed that the system domain Gq can be described by Gq = f (r; ) j 0 <  < 2; c(; q) < r < l() g; and hence  ! c(; q) is a parameterized function which is assumed to characterize the unknown defect shape. The boundary of Gq is also de ned by @Gi = f (r; ) j i?1 <  < i; r = l() g for i = 1; 2; 3; 4; @Cq = f (r; ) j 0 <  < 2; r = c(; q) g; where 0 = 0 and 4 = 2, respectively. The sesquilinear form (15) can be rewritten as 2 l() @' @ @ drd: r @r @r + 1r @' (q)('; ) = (18) @ @ 0 c(;q) q

q

q

q

q

Z

q

q

!

Z

6

l (θ )

G (q)

c (θ , q ) O

Fig. 3 The spatial domain G and its boundary using the polar coordinate system For the discussions here, we restrict the geometrical structure of the boundary @Gq by imposing the following hypotheses: (H-1) For each q 2 Q, c(
; q) 2 W11 (0; 2) and l 2 W11 (0; 2). (H-2) There are constants 1 and 2 such that, for q 2 Q, 0 < 1  c(; q)  2 < l() < 1 for  in (0; 2):

(H-3) There exists a function d : Q  Q ! R1 with d(q; q~) ! 0 as jq ? q~j ! 0 such that jc(
; q) ? c(
; q~)j1;1  d(q; q~) for q; q~ 2 Q; where j
j1;1 denotes the norm of W11 .

Following standard procedures in the method of mapping techniques ([3],[4],[9]), we introduce the ane mapping x = T (q)~x where x = (r; ) and x~ = (~r; ~). The precise form of T (q) in this paper is given by r = (q; r~; ~) = fc(~; q) ? l(~)gfc ? l(~)g?1fr~ ? l(~)g + l(~) : (19)  = ~ Let G~ be the reference domain given by G~ = f x~ = (~r; ~) j 0 < ~ < 2; c < r~ < l(~) g which is independent of the parameter q as depicted in Fig. 4. (

7

l (θ )

~ G -c O

Fig. 4 The reference domain G~ Then, under this coordinate change T (q), the reference domain G~ is transformed into the unknown domain Gq . Let us de ne the Hilbert space de ned on G~ , V~ = f j 2 H 1(G~ ); = 0 on @Gi; i = 1; 3 g endowed with the norm 2 l() @ 2 @ 2 def ~ (20) j jV~ = 0 c @ r~ + @ ~ dr~d: Z

Noting that

we have

Z

8 < :





9 = ;

r~ @ =@ ~ rT (q) = @ =@ 0 1 jrT (q)j = @ @ r~ ; "

#

@' = @ ?1 @ '~ @r @ r~ @ r~ @' = ? @ ?1 @ @ '~ + @ '~ @ @ r~ @ ~ @ r~ @ ~ ~ drd = @ @ r~ dr~d: 8 !

!

Hence (18) can be transformed into the sesquilinear form on V~  V~ as follows:

~ (q)('; )

2 Z l() ( @ a (q) @' 1 @ r~ @ r~ 0 c

Z

def =

+a3(q) @'~ @ ~ dr~d~ @ @ )

@ + @' @ + a2(q) @' @ r~ @ ~ @ ~ @ r~

!

(21)

where

a1(q) def = a2(q) def = a3(q) def =

?1 2 2 + @ (q ) ( q )

(q)?1 @ ; @ r~ @ ~ ? (q)?1 @ ~(q); @ @ ? 1

(q) @ r~ (q): )

(

2

(

4

) 3 5

Lemma 2. With the hypotheses (H-0) to (H-3), there exist positive constants ; ; K2 , and K3 such that, for q; q~ 2 Q, the sesquilinear form ~ (q)(
;
) satis es the following inequalities for all '; 2 V~ : ~ (q)( ; )  j j2V~ (22) j~ (q)('; )j  K2j'jV~ j jV~ (23) j~ (q)('; ) ? ~ (~q)('; )j  K3d(q; q~)j'jV~ j jV~ (24) where

d(q; q~) ! 0 as jq ? q~j ! 0: Furthermore, , K2, and K3 can be chosen as constants which are independent of the parameter vector q.

Proof. From (21), the associated quadratic form is a1(q)12 + 2a2(q)12 + a3(q)22 ?1 2 2 @ @ @ @ @ 2 2 ? 1

+ ~ 1 ? 2 @ r~ ~ 12 + @ r~ 22 : = @ r~ @ @ !

28 < 4 :

!

9 = ;

!

3 5

For any quadratic form QF = a12 ? 2b12 + c22 with a > 0; c > 0, completion of the square arguments yield QF  12(ac ? b2)=c and QF  22 (ac ? b2)=a so that 2 b2  2 : 2QF  ac ?c b 12 + ac ? a 2

9

Choosing

a = 2 +

we easily nd that

@ 2 ; b = @ @ and c = @ 2 ; @ r~ @ ~ @ r~ @ ~ !

!

a1(q)12 + 2a2(q)12 + a3(q)22 2 ?1 ?1 2 @ @ 1 @ 2 2 1 + @ r~ + ~ 22 :  2 @ r~ @ !

2 6 4

! 8