911 DASSIOS, G. ; RIGOU, Z.: Reconstruction of a Rigid Body in Elasticity

ZAMM . Z. angew. Math. Mech. 77 (1997) 12, 911-923

DASSIOS, G. ; RIGOU, Z

On the Reconstruction of a Rigid Body in the Theory of Elasticity

A method is proposed fo r the solution of the inverse scattering problem associated with the shape reconstruction of a 3-0, star shaped, rigid scatterer in the theory of elasticity. The inversion procedure is based on the use of elastic Herglotz functions. A key point of the method is a basic connection formula associating the scattered field and the Herglotz function on the surface of the scatterer, with the corresponding scattering amplitudes and Herglotz kernels on the unit sphere. Analytical difficulties caused b y the complexity of the spectral Navier operator were overpassed by embedding the vector elastic scattering problem into a dyadic scattering problem, which absorbs the dependence of the incident field upon the transverse polarization by considering a complete incident dyadic field. The actual elastic problem is then obtained by projecting the dyadic scattering problem into the particular polarization of the assumed incident held. The method, which for the scalar case has been developed by COLTON and MONK, leads to an optimization scheme which is similar, but much more complicated, to the corresponding scheme in acoustics.

MSC (1991): 73D25, 73D50, 9OC90, 35Q72, 35R30

1. Introduction

The classical direct scattering problem deals with the solution of an exterior boundary value problem that shows how a physically and geometrically prescribed obstacle effects the propagation of a known time-harmonic incident wave [3]. All the properties of the obstacle that enter this interactive scattering process are mapped onto specific functions of directions far away from the obstacle, which are known as scattering amplitudes or far field patterns.

Inverse scattering problems deal with the recovery of the physical or the geometrical characteristics of the obsta- cle, once the incident field and the far field patterns are given. Science and modern technology is heavily dependent on inverse scattering methods. Techniques connected to nondestructive evaluation, geophysical exploration, medical imag- ing, remote sensing, and sea environment research furnish some examples in this direction.

Within the last fifteen years intense efforts have been directed towards the solution of the inverse scattering problem associated with shape reconstruction of an unknown obstacle when the boundary conditions on its surface are known. This is a typical problem where the physics is known but the geometry is unknown. References [4, 51 provide some basic background questions for inverse scattering methods. The book by COLTON and KRESS [9] accumulates a large portion of these efforts for scattering by acoustic (scalar) and electromagnetic (vector) waves. Much less work has been done in the area of elastic scattering theory, and this was not without reason. Elastic waves are considered to be much more difficult than electromagnetic waves for two main remons. First, in contrast to the electric and the mag- netic wave which propagate at the same speed, the longitudinal and the transverse waves that an elastic medium allows to propagate, are travelling with different speed, resulting in a continuous change of their relative phases. A second, but more important difference between electromagnetic and elastic waves, is connected with the fact that the two elastic waves are independent of each other, and we can not recover any one of them once the other is known. This independence does not hold in electromagnetism, where if any one of the electric or the magnetic field is known, the other is readily obtained through Maxwell’s equations.

The present work constitutes an attempt to solve the inverse scattering problem for an unknown star shaped rigid scatterer that is excited by time harmonic elastic waves. The question of uniqueness for this problem has been investigated in 118, 221. The proposed inversion method is based on Herglotz functions [15], and it forms an extension of the Colton-Monk method for acoustics [6-81. The basic theory of elastic Herglotz functions is developed in [14] and the necessary completeness and density results are included in [12].

One of the difficulties of the elastic case concerns the dependence of the general elastic wave on two orthogonal vectors, the direction of propagation and the direction of polarization of the incident transverse wave. In order to avoid this difficulty we had to raise the rank of the tensorial character of the fields involved by one. Hence, we embedded the (elastic) vector scattering problem into a dyadic scattering problem. Then the incident dyadic field is dependent only on one vector, the direction of propagation, while the transverse polarization vector is now replaced by the 2-D orthogonal complement of the direction of propagation. This technique was first used in electromagnetic scattering by TWERSKY [20], who eliminated the effects of incident polarization by considering all of them at once. Obviously, the particular elastic problem at hand can be recovered from the general dyadic scattering problem by contracting the dyadic solu- tion with the particular direction of polarization. The need for this extension from vector to dyadic scattering demands a corresponding extension of all the known elastic results to their dyadic counterparts. This has been done in a sys- tematic way throughout this work. It involves extensions to dyadic fields of Herglotz functions, of representation theo- rems, of the basic Colton-Monk connection formula, of the completeness arguments, of eigenvector expansions and so on. For the theory of dyadics we refer to [l], while for results on dyadic scattering we refer to [13].

912 ZAMM Z . angew. Math. Mech. 77 (1997) 12

Section 2 provides a short exposition of scattering theory for dyadic fields, while Section 3 contains the basic elements of Herglotz dyadics. The fundamental integral identity that connects the interaction between total fields and Herglotz functions on the surface of the scatterer, with the corresponding interaction between far field patterns and Herglotz kernels on the unit sphere, is developed and discussed in detail in Section 4. This connection formula is the key relation for the development of an orthogonal decomposition of the space of square integrable functions, i.e. a projection theorem on which the inversion method is founded. Section 5 contains the characterization of the scattering regions to Herglotz and to non Herglotz domains, the above projection theorem, and a related approximate projection theorem. The optimization problem that defines the actual inversion scheme is discussed in Section 6, while the final Section 7 contains a discussion based on intuitive and physical characteristics of the proposed method of inversion.

2. Dyadic scattering

Let V - be an open and bounded domain in lR3 with a smooth boundary S. The complement V of v-, which will be referred to as the medium of propagation, is characterized as a linearized, homogeneous, and isotropic elastic medium with mass density e, and Lam6 constants il and p. The behaviour of the scatterer is that of a rigid body, i.e. a body that allows no local deformations, and therefore the displacement field should vanish on its boundary. We work on the frequency domain, assuming the time dependence to enter through the harmonic spectral component e-iwt, where w denotes the angular frequency. The origin of our coordinate system is taken in the interior V- of the scatterer.

The scatterer is excited by a complete dyadic field propagating in the direction k which is decomposed into a plane longitudinal and a plane transverse wave. Such an incident field assumes the general form [13]

(1) &(r; k) = A P ~ @ e ik&. r + AS(^ - @ 1;) eik3k.r

where kP, k, are the wave numbers, and A!', As are the corresponding amplitudes for the longitudinal (P) and the transverse (S) waves. The dyadic I stands for the identity. Note that the general form of the incident dyadic field (1) eliminates from the scattering process the complications that come from any particular transverse polarization, by incorporating all possible polarizations into the transverse dyadic (1 - k @ k) which is the orthogonal complement of the longitudinal dyadic k 8 k.

Then, a vectorial longitudinal incidence is obtained by the contraction

'p"(r; k, k) = i; . &(r; k) = &(r; 1;) .k = A P ~ eikDk.r, ( 2 )

cpS(r; ic, fi) = i, . &(r; k) = &(r; k) . i, = ~ ' 6 . (I - k B k) eikak'r = AS$ . (6, @ 6, + +k @ i j k ) eikb'.r, (3) where 6,, i j k denote the tangential unit vectors for the spherical system with radial vector along k.

Note that the dyadic field &(r; k) is dependent only on the direction of incidence k, while in addition to k the vector fields @'(I-; k, k) and 'ps(r; k, p) are dependent on the polarization vectors k and p, respectively, as well. This observation reflects the generality of the dyadic field & as it compares to the vector fields 'pp, and ' p s , and it forms a basic element of our work below.

The dyadic field (1) can be interpreted as a dyadic superposition of three vector fields which appear as the first vectors of the dyads, while the second vectors of the dyads are provided by the incident orthogonal base {k, 6 k , Qk}. In other words,

while a.ny vectorial transverse incidence polarized along p is recovered by the contraction

&(r; L) = cpp(r; L, 6) B L + cps(r; L, 6,) B 6, + cps(r; k, Q ~ ) B i p k , r E ~ 3 , (4)

r E R 3 , (5)

qs(r; k, 6,) = eikq'.r , rElR3, (6)

$(r; k, +J = AS+^ ez'a'.r, (7)

U(r; L) = uP(r; k, k) g L+ us(r; L, 6 k ) B 6, + us((.; h,

where

rpP(r; 1;, h) = ~ ~ 1 ; e W . r

r E ~3 .

The dyadic character of the incident field & is inherited in the scattered field U it generates, preserving the order of the dyads as in the following form,

B Q,, r E V , (8) where the first vectors of the dyads correspond to the vector scattered fields generated by the first vectors of the dyads of &, respectively.

Then the total dyadic field

Y(r; k) = &(r; 1;) + U(r; 1;) (9)

913 DASSIOS, G.; RIGOU, Z.: Reconstruction of a Rigid Body in Elasticity

solves the dyadic spectral Navier equation

denote the longitudinal and the transverse phase velocities, respectively. The scatterer is considered to be rigid, which demands that the total field vanishes on its surface, i.e.

C ( r ; k ) = I s , r E S . (13) Furthermore, each displacement field appearing as a first vector of each dyad in (8) has to satisfy the Kupradze radia- tion conditions [17] at infinity, which are equivalent to the leading terms of the complete exterior expansion in powers of inverse distance from the scatterer Ill].

It can be shown [13] that the solution of the above problem has the integral representation

- , . V(r; k) = &(r; k) - - f r(r, r') . TflY(r'; k) ds(r') , r E V , (14) 4Jte

where

Tr = 2pn. V + A n div +pn x rot

denotes the surface traction operator, n indicates the exterior unit normal on S , and

is the Kupradze fundamental dyadic (10, 171 which solves the equation

[c iur + (c; - c;) Or B Vr + w2I] . F(r, r') = -4nd(r - r'> I

The representation (14) can be used [13] to obtain the uniform over S2 asymptotic form

u(r; k) = Gr(t; k) h(k,r) + Gt(t; k) h(k,r) + o ( I / T ~ ) ,

h(x) = h, (1) ( x ) = eix/ix

T ---f +cc , (18)

(19)

where

is the dimensionless fundamental solution for the scalar Helmholtz operator while the radial and the tangential far field patterns assume the representations

ik, t t . { e-ikpf, r' TrtY(r'; k) ds(r') = - k, Fr(f; k) , GI(?; k) = -- 4.76ec; 2QZCZ

S

and

respectively. The patterns G have units of length, and the patterns F have units of force. The reason for introducing the patterns F will be clear in Theorem 4.

Note that, in accord with known results from elastic scattering [lo, 131, the radial pattern G,, or Fr, is a purely longitudinal spherical wave propagating along t, and the tangential pattern Gt, or Ft, is a purely shear spherical wave propagating along f and polarized orthogonally to f .

The connection between the dyadic patterns G,, Gt, and the nine spherical patterns [lo] generated by the three first vectors of the dyads in (4) are provided by

where the third argument in g denotes the corresponding polarization of the incident wave.

60 Z. angrw blath. Mech., Bd. 77.11. 12

914 ZAMhl. Z. angew. Math. Mech. 77 (1997) 12

The inverse scattering problem we want to solve in the present work concerns the recovery of the unknown surface S of the rigid scatterer when the incident field 6 and the far field patterns G, and Gt, or equivalently F, and Ft, are known for a fixed frequency.

3. Herglotz dyadics

An elastic Herglotz function [14] is defined to be an entire solution u of the spectral Navier equation (11) which satisfies the asymptotic condition

B(0. r )

where B(0; r ) denotes the ball centered at the origin with radius r. The need for a scattering theory of dyadic fields demands the extension of our previous work on elastic Herglotz

functions [14] to Herglotz dyadics. This is a straightforward extension guided by the tensorial product (4). If v,, i = 1, 2, 3, are three elastic Herglotz functions [14], then we define the following Herglotz dyadic,

V(r) = v1 (r) 8 X I + vz(r) @ xi2 + vg(r) @ x 3 , r E IR~ . (24) Then obviously. V is an entire solution of (11) which satisfies the Herglotz boundedness condition

1 lini -

I - L X I' J where

\lV(r'))I2 dv(r ') < +m,

In [14] one can find the proof of the following

the unit sph,ere S2, such that Theorem 1: If v is an elastic Herglotz function then there are square integrable functions L and T , defined o n

L(k) eZkp" ds(k) + T(k) ezkb.r ds(k) , r E IR3, 2n

where k,, = kPk, und k, = k,k. Conversely, ifv is given by (27) with L longitudinal and T transverse square integrable functions over S', th,en v is a n elastic Herglotz function.

The functions L and T define the longitudinal and the transverse Herglotz kernels, respectively. The representa- tion (27) actua,lly effects a decomposition of v into longitudinal and transverse plane waves with amplitude distribu- tions over S' specified by the corresponding Herglotz kernels L and T.

Tkieoreni 1 implies immediately the following

Corollary 1: I f v i , i = 1, 2 , 3, are elastic Herglotz functions, then there are square integrable dyadics L and T defzned o n th,e unit sphere S2, such that the Herglotz dyadic V, given b y (24), enjoys the representation

ds(k) , r E I R ~ 2n

5 2

L(k) eikp'r ds(k) + S2

Conversely, the entire dyadic field V given by (as), where the longitudinal and the transverse Herglotz kernels L and T are square integrable dyadics over S2, is a Herglotz dyadic, i.e. it i s a n entire dyadic solution of (11) which satisfies condition (25).

of V assume the Orthogonality arguments show that the longitudinal part Vp, and the transverse part individual representations

DASSIOS, G.; RIGOU, Z.: Reconstruction of a Rigid Body in Elasticity 915

with

V(r) = Vp(r) + v"(r) , r E I R ~ , (31)

where all the first vectors of the dyads of Vp are irrotational while all the first vectors of the dyads of are sole- noidal.

COLTON and MONK [6] proved that, under certain conditions, the set of all scalar Herglotz functions in two dimensions forms a dense set within the set of all solutions of the Helmholtz equation in an interior domain. This result has to be extended to three dimensional Herglotz dyadics. We do that in two steps: first we prove it for the scalar 3-D case and then we show how it extends to the full dyadic field.

For the definitions of the Holder spaces C",* with n = 0, 1, 2 , . . . , and a E (0, I] we refer to the well known books [3, 91.

Theorem 2: Let V - be a bounded and star shaped domain with a C2,a, a E (0, 11, boundary S. Then, for every Y- E C2,a(v- ) which is a solution of the Helmholtz equation in V - , and for every E > 0 there exists a Herglotz fun,ction PI

suck that

max IYu-(r) - u(r)l 5 E . r E V -

P r o o f : The proof imitates the corresponding 2-D proof in [6]. Using results from [21] and straightforward manipulations we show that any two functions u0 E ker A and u E ker(A + k 2 ) in 3-D are connected by the transform pair

and

where 51 is the Bessel function of the first order. From [19] we know that, given any E > 0, there exists a star shaped domain 4- 3 V- and a harmonic function h0 E C2ia($-) such that

max Iho(r) - uo(r)l 5 E . (35) r E V -

Runge's approximation property [a] implies that h0 is uniformly approximated in v- by a finite linear combina- tion of harmonic polynomials, and then (33), (35) furnish a uniform approximation of u in v- by a finite linear combination of the spherical eigenfunctions { jn(kr ) Y,"(i.)}, where j , denote spherical Bessel functions of the first kind and Y," are the spherical harmonics. Finally, the representation formula

implies that such combinations of eigenfunctions are Herglotz functions, and the density argument follows. The above density theorem is next extended to the dyadic case.

Theorem 3: Let V - be as in Theorem 2 and suppose that Y- is a dyadic field which belongs to C2ia(v/-) and solves (11). Then for every E > 0 there exists a Herglotz dyadic V such that

rnv / /@-(r ) - V(r)// 5 e . (37) r e v -

P roof : The dyadic field @- has the Cartesian representation 3 3

2 = 1 3=1 Y-(r) = c C Y;(r) XJ B Xz

which by combining the first vectors of the dyads can be written as 3

2 = 1

W ( r ) = C Y;(r)BXz,

where the vector fields

(39)

60*

916 ZARIWI . Z. angew. Math. Mech. 77 (1997) 12

solve the spectre1 Navier equation

c:AYc(r) + (ci - cz) v B V ,YL(r) +w2Y;(rj = 0,

Y;(r) = YZ;’(r) + YFS(r), r E V - , (42)

r E V - , i = 1, 2, 3. (411

In view of the Helmholtz decomposition theorem each elastic solution YL can be decomposed into an irrotational and a solenoidal part as follows,

for every i = 1, 2, 3. Then the classical theory of elasticity shows that every component of Y;’ and every component of Yis for i = 1, 2 , 3 satisfies the 3-D Helmholtz equation. Therefore, Theorem 2 can be applied componentwise to furnish

max JY;’(rj - wyi(r)I 5 E I , (43) rev-

for every 2, J = 1, 2, 3 and ~1 > 0, where u,”z and uil denote the Herglotz functions that approximate the Helmholtz fields Yy3/” and ‘Pi’, respectively.

Define the dyadic 3 3

V(r) = C C (vyL(r) + w;$(r)) xJ x, , r E I R ~ . (45) L = l J = 1 “

Then, V is a Hcrglotz dyadic for which

rriax llW(r) - V(r)II 5 m e

and inequality (37) follows by choosing EI

r E l . - rev-

= ej18.

4. The basic connection formula

In scattering t,heory there is a one-to-one mapping between any scattered field and the far field patterns it generates. In fact, given the scattered field, the far field pattern is obtained through the asymptotic expansion of the scattered field in thP radiation zone, and conversely, given the far field pattern the scattered field is recovered via Atkinson’s expansion theorem. Such a mapping also exists between a Herglotz function and the corresponding Herglotz kernels. It is provided by the integral representation formula and its inversion [15]. The first mapping connects an exterior do- main with a function on the unit sphere (a function of directions), while the second mapping connects a full space field with a funct,ioii also defined on the unit sphere.

If we can connect the Herglotz function with the scattered field, then the Herglotz kernel and the far field pattern will also be connected and vice versa,. Such a connection formula will then furnish a means to transfer informa- tion, about the interior and the exterior domains, from the surface of the scatterer to the surface of the unite sphere. This was first obtained by COLTON and MONK [6] for the case of scalar acoustic waves, and it is based on the plane wave decomposition that a Herglotz function enjoys via its representation theorem. The electromagnetic analogue of this formula can be found in [8, 91. In the present work we need the following generalization of this connection to dyadic fields, from which all cases of physical interest, acoustic, electromagnetic, and elastic of any polarization, follow as special cases.

In what follows the double dyadic contraction is defined as

( a @ b ) : ( c @ d ) = ( a . d ) ( b . c ) . (47) Theorem 4: Le t Y be the dyadic solution of the scattering problem (11), (13), and F,, F, are the corresponding

dyadic f a r field patterns given by (20), (21). If V is a Herglotz dyadic with Herglotz kernels L and T, then

[TrY(r; k)IT : V * ( r ) ds(r) = [F:(i; k) : L*(i) +- F l ( F ; k) : T*(F)] d s ( i ) , f S s2

where 9 is given by (as), ‘7“ indicates transposition, “*” indicates conjugation, S i s the surface of the scatterer, and S’ is the m i t sphere.

P r o o f : Consider the incident orthonormal system

(k, k’, k 3 ) = (k h k , %I, (49)

DASSIOS, G.; RIGOU, Z. : Reconstruction of a Rigid Body in Elasticity 917

and the corresponding decompositions 3 3

3 = 1 j=1 W(r; 1;) = c Y3(r; k) 8 k3 ,

L(i) = c L 3 ( q @ k,, T(i) = T,(f) kl.

T ~ W ( ~ ; k) = c (Trvl(r; 1;)) 8 k, ,

3 3

3 = 1 3 = 1

Then, in view of (28) and (50)-(53) the left hand side of (48) is written as

f [TrV(r; k)IT : V*(r) ds(r) S

As it is shown in [14] the longitudinal kernel L, in (27), is a radial vector field and the transverse kernel T, in (27) , is a tangential vector field, i.e. for every j = I, 2, 3 we have that

L,(P) = P c3 p ' Lj(P) 1

Tj(p) = (I - i, p) . T j ( p ) . (56)

(55)

and

Using these properties as well as (20) and (21) into (54) we arrive at

[TrY(r; k)IT : V*(r) ds(r) S

*

S *

f 3 C p @ p . (TrYJ(r; k)) 8 k, ePkpP ds(r) L ( p ) c3 km] ds(p)

1

1 3 (I - p @ p) . # (TrYJ(r; k)) 8 kj T,(p) 8 km] ds(p)

S

= FT(p; k) : L*(p) ds(p) + SZ S2

Fl(p; k) : T*(p) ds(p)

which is the right hand side of (48). Hence the basic connection formula is established.

Corollary 2: Under the hypotheses of Theorem 4 the radaal connection formula

f [TrY(r; k)IT : [Vp(r)]* ds(r) = FT(f; k) : L*(i) ds ( i ) , S 52

I

# and the tangential connection formula

# [TrY(r; k)lT : [V"(r)]* ds(r) = F:(i; k) : T*(i) ds(f) S 52

hold h e .

P r o o f : Similar to the proof of (48).

Formula (58) recovers to the acoustic connection formula [6] if we assume

*(r; k) = Y(r; k) I

(57)

(59)

918 ZAMM . Z. angew. Math. Mech. 77 (1997) 12

and Vp(r) = v( r) n B n , r E s ,

where Y solves the scalar scattering problem and ZI is a scalar Herglotz function. Similarly, the electromagnetic connection formula [8] is obtained as a special case of (59) if we set

@(r; k) = E(r; k) @ k,

V(r) = (I - ii 63 fi) . EH(r) B k , and

r E S , where EH is a vector Herglotz function, and E solves the perfect conductor scattering problem with the standard approximation that a,,E and n are linearly dependent on S [16].

5. Herglotz domains and orthogonal decomposition

The value -a2 is called an interior rigid eigenvalue if a nontrivial dyadic field V exists in V- such that

c ? n V ( r ) + (c: - c,”) v g v . V(r) + w2V(r) = 0 , (64)

V ( r ) = O , r E S . (65)

r E V- , and

Any such V is called a rigid eigendyadic.

for the rigid elastic case [17]. A smooth domain V; is called a Herglotz domain if:

dyadic, and

The existence of such eigenvalues is secured componentwise, from the existence of the corresponding eigenvalues

(i) whenever -u2 is an interior rigid eigenvalue, at least one of the corresponding rigid eigendyadics is a Herglotz

(ii) for -w2 not a rigid eigenvalue the unique solution of the boundary value problem

c:AV(r) + (c: - c,”) v @ v . V(r) + a2V(r) = O f

V(r) = F*(o, r) , r E S , (67)

r E V< , (66)

is a Herglotz dyadic, where

where the origin of the coordinate system lies within V-. As we will see in the sequel, the choice of the boundary values of V is dictaded by the form of

representations needed in the proof of the projection theorem. The following theorem provides a decomposition of the space of square integrable dyadics defined

sphere into two orthogonal components one of which is generated by the far field dyadic patterns and generated by the Herglotz dyadic kernels.

Theorem 5: L e t V< be a Herglotz domain. Then , (i) i f -a2 is a n interior rigid eigenvalue, t hen

[L’((s~)]’ = span {F, + Ft) B span {L + TI, k € S Hcrglutz

the integral

on the unit the other is

(69)

where “span” denotes the subspace generated by the Herglotz kernels of the rigid Herglotz eigendyadics corresponding to

-a2, and Herglotz

(ii) if -w2 is n,ot a n interior rigid eigenvalue and k1 E S2, t h e n

1

where the superindex 1 corresponds t o the direction of excitation k1, “span” denotes the subspace generated by the Herglotz kernels of the unique solution of (66), (67). 1

P r o o f : (i) Let -w2 be an interior rigid eigenvalue. Since V i is a Herglotz domain there exists a Herglotz eigendyadic V

corresponding to -a2. The dyadic V vanishes on S , and the connection formula (48) implies that

f F,(1; k)T : L*(i.) ds(r) + Ft(l; k)T : T*(i.) ds(1) = 0 , S’ S2

f

DASSIOS, G. ; RIGOU, Z. : Reconstruction of a Rigid Body in Elasticity 919

which by orthogonality arguments [13, 141 is written as

f [F,(i.; k) + F,(f; k)IT : [L(f) + T(f)]* ds(r) = 0 , 5

and furnishes the inclusion

We will show that the dual inclusion of (73) is also true. So, let i

{ L o + T o } E { span{F, + F t > } , L o E span{F,)I , TO E

Then (72) holds, and formula (48) implies

(72)

(73)

(74)

f [TrY(r; k)IT : V;(r) ds(r) = 0 , (75) S

where Y is the dyadic field associated with F,, F,, and VO is connected to L o , TO via (28). Using (50), (51) and related results from [12] we can show that, since (75) holds true for every dyadic traction field Try, their density over S confirms that

Vo(r)=O, re^. (76)

Hence V o vanishes on S , and VO is a Herglotz eigendyadic. This proves the case where -w2 is an eigenvalue. (ii) Since Vg is a Herglotz domain and -w2 is not an interior rigid eigenvalue, there exists a Herglotz dyadic V

which solves the associated interior problem (66)-(67). If L and T denote the Herglotz kernels of V, then (48), in view of the boundary conditions (13) and (67), provides

f F,(f; k)T : L*(t) ds(i) + Ft(f; k)T : T*(f) ds(f) f S2 52

= f [Tr+(r; k)IT : r(0, r) ds(r) = [ r (O, r) : T,@(r; k) - Y(r; k) : T$(O, r)] ds(r) f S S

= f [ r (O, r) : T,fJ(r; k) - U(r; k) : T$(O, r)] ds(r) + f [ r ( O , r ) : T,&(r; k) - &(I; k) : Trr(O, r)] ds(r) S S

= 4 n ~ ( t r & ( O , k)) = 4ng(AP + 2AS), (77) where the vanishing of the integral representation for the scattered dyadic U at the origin has been used, and AP, As are the amplitudes of the incident plane waves given in (2), (3). Since the right hand side of (77) is a constant, orthogonality arguments lead to

- 1 ^. 1; f [ ( F r ( f ; k) - Fr(r, I ) ) ~ + ( F t ( f ; k) - Fi(f; k~))~] : [L(f) + T(f)]* ds(i) = 0 S2

which implies that 1

span{L + T I c { span{(Fr - F:) + (Ft - F:))} .

In order to show the dual inclusion we consider an arbitrary pair LO and T o from the right hand side of (79) in the following sense:

L o E spa@, - F,1) , T o E span(Ft - F;) , I

( L o + T o ) E {span(F, - F:) + span(Ft - FI)) . (80) Then

1 [F,(i; k) - F:(f; k1)IT : Lo*(?) ds(f) + [Ft(f; k) - Fi(f; k,)lT : Ti(?) ds(f) = 0 , S2 SZ

f and the connection formula (48) implies

f [Tr*(r; k) - TrY1(r; k1)IT : V;(r) ds(r) = 0 . S

920 ZAMM . Z. anEew. Math. Mech. 77 (1997) 12

On the other hand, from (77) we secure the relation

f [Try(.; k) - TrY1(r; k1)IT : r(0; r) ds(r) = 0 S

for every dyadic solution Y, which confirms that

Relations (82) and (83) imply that

V;(r) E span(f(0; r)} , (85) where the Herglotz kernels Lo, T o generate, via (28), the Herglotz dyadic V, which solves the associate interior prob- lem (66), (67). That shows the dual inclusion of (79) and proves part (ii) of the theorem.

We remark here that if -02 is an interior rigid eigenvalue, then a necessary and sufficient condition for the span of the far field patterns to be dense in [L2(S2)]9 is that the scatterer is not a Herglotz domain. Similarly, when -w2 is not an interior rigid eigenvalue then density of the far field patterns in [L2(S2)]9 implies that the scatterer is not a Herglotz domain.

The need for raising the rank of the fields from vectors to dyadics is dictated by relation (77). In fact, if we only use vector fields then the value of the integral in (77) will depend on the direction of polarization, which does not have to be the same for k and k 1 incidence, in order to furnish the orthogonality relation (78). But, if we use dyadic fields, which incorporate all possible polarizations and perform double contraction to eliminate all of the directional depen- dence of (77), then we arrive at a scalar value independent of propagation, or polarization vectors. In other words, we take advantage of the higher symmetry that the dyadic fields enjoy, which in our case appears as the ability of the dyadic incident field to absorb all possible polarizations.

Lack of a priori information about the unknown domain leads us to consider the following theorem which allows an approximate decomposition of the space of square integrable dyadics.

Theorem 6: Let V - be a bounded, star shaped region with a C2,a, a E (0 , 11, boundary S, which is defined by the radial function Q : S2 -+ IR. The region V- is not a Herglotz domain. Th,en for every E > 0 there exists a constant A4 independent of E and a , and Herglotz dyadics V 1 and VTZ with Herglotz kernels L1, T I , and L 2 , T 2 , such that

(i) if -02 is an interior rigid eigenvalue and F,, Ft are far field dyadic patterns, then

(c) f C : [L1(k) + Tl(k)]* ds(k) = constant, (88) s2

where C is a constant dyadic, and

to the k, and to the k 1 incidence, respectively, then (ii) if -a2 is not an interior rigid eigenvalue, and Fr, F t , and F:, Fi are far field dyadic patterns corresponding

(c) Re $ C : [Lz(k) + Tz(k)]* ds(k) = constant, (91) S2

where is a constant dyadic.

P r o o f : (i) Since the domain V- is not Herglotz any interior eigendyadic Yl is not Herglotz either. By the density

Theorem 2 there exists a Herglotz dyadic V 0 1 such that

r t V - max II*;(r) - v o l ( r ) I I I E

Ll(i ) = L,l(f) , (93)

(92)

for some E > 0. If Lo1 and T o 1 are the Herglotz kernels representing V,, we choose the Herglotz dyadic Vl(r) which is generated by the kernels

DASSIOS, G.; RIGOU, Z. : Reconstruction of a Rigid Body in Elasticity 921

and

T I ( ? ) = T,,(i.). (94) Then, in view of the connection formula, the estimate (92) and the vanishing of Y, on S , the left hand side of (86) furnishes

where Y denotes the total field corresponding to the patterns F,, Ft, and M is an upper bound of the norm of the total traction field over S. This shows inequality (86).

satis- fies, i.e.

Inequality (87) is a simple consequence of the restriction of (92) on S and the boundary condition that

IIVl(@(i.) f ) l l 5 yp; IlWe(2) i., k) - V'ol(@(f) f ) I I I E . (96)

Finally, relation (88) is obtained from the representation (28) as follows,

f C : [Ll(k) + Tl(k)]* ds(k) = C : f [Ll(k) eik~k.o]* ds(k) + C : [Tl(k) eikEk"]* ds(k) SZ sz 4

f = C : [2nV1(0)]* = constant, (97)

since the dyadic V, is real.

Herglotz dyadic, but the density property (ii) Since V - is not a Herglotz domain, the unique solution Y i of the associated problem (66), (67) is not a

max II*;(r) - V02( r )1 I I E (98) r E V -

still holds true for some Herglotz dyadic V 0 2 . Then, the connection formula (48), applied to the scattered fields Y and Y', corresponding to the far field patterns Fr, Ft and F:, F;, and to the Herglotz dyadic V 0 , which is constructed by the kernels

L 2 ( f ) = L 0 2 ( t ) , (99)

T a p ) = T o z ( i ) (100) and

provides the following estimates for the left hand side of (89),

LHS 5 + l[TrY(r; k) - TrY1(r; k1)II 11Y;(r) - V02(r)ll ds(r) S

-t [Tp*(r; k) - TrY1(r; kl)lT : Yi*(r) ds(r) If + [Tr@(r; k)IT : f'*(O; r) ds(r) - [TrY1(r; kl)lT : r*(O; r) ds(r) It S

5 E M + ~ z Q / ~ ~ & ( o ; L)-trcD(o; k1)1 = E M ,

where M estimates the norm of the total surface traction, and relation (77) has been also used. Inequality (90) is a simple consequence of (98) and of the boundary condition (67), i.e.

IlVz(e(i.) i.1- f'*(o; Q(i.1 i.)ll 5 ~t;llVm(~(f) i.1 -*,(e(i.) i.111 I E .

Finally, the proof of (91) imitates the proof of (88) where the solution V z of (66), (67) is not real anymore. This completes the proof of the theorem.

6. The inversion scheme

All the previous information is now incorporated into the following non linear optimization problem which furnishes the inversion scheme for the reconstruction of the unknown rigid surface S. We excite the scatterer from the N differ- ent directions kl, kz,. . . , k ~ , and let F y , F:, n = 1, 2 , . . . , N , be the far field patterns they establish. In order t o

922 ZAMM . Z. angew. Math. Mech. 77 (1997) 12

secure existence for the solution of the minimization problem we utilize some a priori information about the shape of the scatterer to rest,rict the set of possible radial functions e(f) into a compact subset U of C2,a . Similarly, we allow for the Herglotz kernels L and T to vary within a compact subset W of [L2(SZ)]’.

Then we consider the functional

+ llV(e(i’) i’) - cF”(0; e(i’) r)l12 ds(i ) , f S2

where V is a Herglotz dyadic generated by L and T, the constant c is equal to zero whenever -02 is an interior rigid eigenvalue, and c is equal to one whenever it is not. The minimization problem we have to solve is

m i n { E ( e , L , T ) J g E U , L E W , T E W , c=0 ,1} , (104)

which provides the best approximation of the unknown surface S through a minimization radial function eo = eo(f). Obviously, the solution is not unique as long as the compact set U is not convex. The optimization scheme (104) is similar to the one proposed in [6, 7, 91 but as it is expected it has many more terms, and a numerical implementation will demand much more elaborate algorithms. It will be of interest to examine the possibility of a numerical treatment of (104) using parallel algorithms for the components of the dyadic fields involved.

7. Discussion and physical considerations

The proposed inversion method is based on the theory of Herglotz functions and their dyadic generalizations. A Her- glotz dyadic is an entire solution of the spectral Navier equation which satisfies a certain boundness condition at infinity. It is actually a superposition [14,15] of an outgoing (scattered) wave and an incoming wave which emanates at infinity, and it is such that it wipes out the singularities of the scattered field which are located within the scatter- ing region. Therefore, besides the outgoing wave, the incoming wave, too, incorporates information about the scattered wave through its singularities. This means that if a Herglotz dyadic is associated with some scattering problem then it is possible to transfer to infinity information about the actual interaction between the incident field and the obstacle. But not all scattering problems are associated with some Herglotz dyadic. Only those where the scattering region is a Herglotz domain enjoy the affiliation with such a Herglotz dyadic. A Herglotz domain is a domain for which a Herglotz dyadic is attached. If -w2 is an interior eigenvalue the Herglotz dyadic is provided by an eigendyadic of -a2, while if -a2 is not an interior eigeiivalue the Herglotz dyadic is furnished as the solution of the associated interior problem (66), (67). The boundary condition (67) that the later Herglotz dyadic has to satisfy is dictated by the integral repre- sentation (14) and the form of the fundamental dyadic r. In fact, (67) has to be chosen in such a way that the integral (77) provides the representation of the first scalar invariant of the incident field at the origin. This is exactly the point where the use of dyadic fields became necessary, since a vector field would leave us with the dependence of the value of the incident field on the polarization vector. But considering all possible polarizations and taking the first scalar invari- ant of the incident field at the origin we end up with a constant. This property is needed for relation (77) to hold and therefore to establish the orthogonality implied by (78).

Coming back to the Herglotz dyadics we remark that they are specified as solutions of interior boundary value problems, but being Herglotz, it means that they are extensible to the full 3-D space. In other words, they can “reach” infinity, where they are uniquely identifiable by their kernels. In that sense, Herglotz dyadics carry information about the interior of the scatterer all the way to infinity and this information is recorded on the corresponding Herglotz kernels there.

On the other hand, the far field is also the place where information about the geometry of the scatterer arrives through the scattered wave and this information is codified on the scattering amplitudes, which depend only on the directions of incidence.

The inversion method we discuss utilizes the fact that a Herglotz domain splits the space of square integrable dyadics over the unit sphere into two orthogonal complements, one spanned by the set of all scattering amplitudes, and the other spanned by all Herglotz kernels generated by the corresponding Herglotz dyadic which characterizes the Herglotz domain. The importance of the inversion method lies in the fact that the information about the shape of the scatterer, which comes from the interior, through the Herglotz kernels, and the shape information which comes from the exterior, through the scattering amplitudes, arrive at the far field in the two orthogonal subspaces mentioned above. The information from the scattering problem arrives in the subspace of far field patterns, and the information from the interior problem arrives in the subspace of Herglotz kernels. The subspace spanned by the Herglotz kernels is much “smaller” than its orthogonal complement. Therefore, it makes a lot of sense to work on the “small” space of Herglotz kernels instead of the “large” space of far field patterns, since the corresponding computational schemes are expected to be much more efficient.

DASSIOS, G. ; RIGOU, Z.: Reconstruction of a Rigid Body in Elasticity 923

The mathematical mechanism that provides the connection between the values of the exterior and the interior solutions on the surface of the unknown scatterer, and their far field patterns on the unit sphere, is furnished by the key formula (48). This formula identifies an inner product on the surface of the scatterer with another inner product on the unit sphere. It actually indicates the possibility to let the information about the scattering region from the inside and from the outside interact, either on the surface of the scatterer, or at the far field.

More than ten years ago, when the Colton-Monk method was first introduced, the physical meaning behind the mathematical ideas was not obvious, and that justifies the remark “. . . this method defies physical intuition” written in [6, page 10411. In view of the above comments, it is our understanding that there is clear physical justification and intuitive explanation associated with the Colton-Monk inversion method.

Acknowledgements

The authors want to express their thanks to Dr. ANTONIOS CHARALAMBOPOULOS for long and helpful discussions during the prepara- tion of the present work. The first author acknowledges support from the Scientific and Environmental Affairs Division of NATO for a joint project with the Technical University of Sofia as well as from the Greek Secretariat General for Research and Technology.

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Received April 9, 1996 at one of the members of the Editorial Board, and May 3, 1996 at the Editorial Office of ZAMM, accepted July 26, 1996

Addresses: Prof. Dr. GEORGE DASSIOS, Division of Applied Mathematics, Department of Chemical Engineering, University of Patras; Dr. ZAFIRIA RIGOU, Institute of Chemical Engineering and High Temperature Chemical Processes, GR-26500 Patras, Greece