Generation of controlled acoustic waves by optimal design of surface loads with constrained forms

Pergamon

Int. J. Engng Sci. Vol. 33, No. 6, pp. 907-920, 1995

0020-7225(94)00101-4

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00

GENERATION OF CONTROLLED ACOUSTIC WAVES BY OPTIMAL DESIGN OF SURFACE LOADS WITH CONSTRAINED FORMS YOUNG SIK KIM Department of Applied Science, Hong-IK University, Mapoku, Seoul 121, Korea

HERSCHEL RABITZ and MOHSEN T A D I t Department of Chemistry, Princeton University, Princeton, NJ 08544, U.S.A.

ATI'ILA ASKAR School of Sciences, Koc University, 80860 Istanbul, Turkey

JOHN B. McMANUS Aerodyne Research Inc., Billerica, MA 01821-3976, U.S.A.

Abstrael--Model calculations are presented for the optimal design of surface force patterns to generate acoustic waves that come to a focus within the bulk of a homogeneous elastic solid. The optimal design consists of achieving a high level of energy at the target at a prescribed time by applying a relatively minimal surface force while aiming for a minimal system disturbance away from the focal target. Such optimal designs were derived in an earlier paper, in which no restriction was imposed on the functional form of the applied stress. In this paper we examine the importance of the fine detail in the earlier derived forcing functions in achieving efficient acoustic focusing. We repeat the optimal design calculations with the surface stress constrained to be in the form of rings of variable radius, with cross sectional profiles made by the superposition of two Gaussians. The optimality conditions are secured via the conjugate gradient algorithm (CGA) and the mechanics of the elastic medium are treated by the finite element method along with using the half space Green's function matrix. We use a criterion for focusing efficiency of the ratio of acoustic energy in the target volume to the total work done on the surface, at a prescribed time. The calculations show the high levels of focusing efficiency derived in earlier work with unconstrained force patterns also can be achieved with constrained and simplified force patterns. This observation is encouraging in terms of the robustness of the optimal solution as well as the possibility of laboratory realizations of the designed force patterns for generating focused acoustic waves.

1. I N T R O D U C T I O N

There are a number of areas in the study of the properties, structure and dynamics of solid materials that would benefit by an ability to design and create specific acoustic wave fields. Non-destructive testing (NDT) may employ the generated acoustic focus within a material to probe mechanical properties in depth. The focusing of high amplitude acoustic waves also can be used to produce nonlinear effects that reveal yield effects in engineered structures. Very high amplitude pressure waves may be used to modify materials in a useful way. This range of effects, from NDT to materials modifications, might benefit from an ability to generate specific wave structures, particularly with a high amplitude focus. One approach to the problem of generating particular acoustic wave structures within solids is to act directly on the surface, rather than to use separate transducers, acoustic lenses and coupling media. Investigators have considered how best to produce specific acoustic waves by designing surface forces [1, 2]. In the low-strain linear-medium regime, the response of a solid to distributed surface forces can be described in terms of the sum of responses due to point forces on the surface, with the

t To whom all correspondence should be addressed. ss ~ , ~

907

908

Y . S . KIM et al.

response from each point source described by a dyadic Green's function [3]. The acoustic response due to a single point force is fairly complex, with the generation of shear, compressional and surface modes, all of which have distinct spatial and angular dependences. If one desires to design the surface forces to produce a specific acoustic state on the interior of a solid while also taking into account technical or practical constraints, then it would be quite difficult to account for all of this complexity without a systematic procedure. Optimal control theory [4] provides a natural means to manage the complexity of designing the best surface loading to generate a specified acoustic wave. Optimal control theory is based on the calculus of variations, and naturally takes into account the physical response as well as practical constraints in calculating the optimum application of forces which drive a physical system to a desired state. In addition, optimal control theory allows for the inclusion of practical constraints as cost functions in the calculation so one can, for example, derive the surface forcing function which produces the best interior acoustic focus with the minimum expenditure of work in moving the surface. Optimal control theory has been applied to a wide variety of problems, and we have recently applied these techniques to calculate the optimum temporal structure of the driving forces on a solid surface for generating a prescribed state of interior stress [1]. We have shown how to maximize the acoustic energy within a subsurface volume, while minimizing the expended energy and disturbance of the material away from the focus via careful designs of the surface driving forces. The resultant tailored acoustic design procedure could provide a new and possibly general tool for producing localized strain fields in solids. We envision these acoustic wave forms and fields to be potentially useful in a broad range of applications for internal materials diagnostics and possibly even modifications. Direct generation of acoustic waves on surfaces has been demonstrated using lasers or electron beams [5, 6], as well as with arrays of individually driven transducers. The technique of laser acoustics is especially interesting from this perspective, since laser beams can be controlled both spatially and temporally, and the energy can be delivered in a non-contact manner in the ambient environment. Thus, we believe that optimal design techniques will be particularly useful to the field of laser acoustics. In our previous work on the design of optimal surface loads to produce subsurface focusing [1], we allowed the designed forces to assume virtually any form, guided only by the optimization goals. To generate a subsurface acoustic wave focus at a specified time, the derived load forms were collapsing rings with surface velocities consistent with the simultaneous arrival of shear and compressional waves at the focus. We specified a minimum disturbance of the material away from the focus and a minimal total driving force, but otherwise did not limit the detail in the surface forces. As a result, the derived surface loads had subtle spatial and temporal features. An important question to ask is: how important are the various fine scale features of the driving force designs in producing an efficient acoustic focus? In the work reported here, we have performed optimal control calculations with constrained forms of surface driving forces to produce subsurface acoustic focusing. In recognition of the practical difficulties in producing load forms of arbitrary detail, we have limited the surface loads to have simple shapes with only a few time-dependent parameters. The parameters in these load forms have been adjusted by optimal design procedures to give the best acoustic focus while minimizing total expended energy and disturbance of the material away from the focus. In Section 2 we will present the optimal control formulation for the constrained load forms. This treatment is based on our earlier work [1], which contains more mathematical detail. In Section 3 we will give the results of numerical optimal control calculations with constrained driving forms. The results and their implications are discussed in Section 4.

Generation of controlled acousticwaves

909

2. OPTIMAL CONTROL PROBLEM The use of the optimal control theory to design forcing functions that can generate acoustic waves was described in extensive detail in our earlier publication [1], so we will present here only an outline of the process. Optimal control problems contain several elements, an objective to be met by means of control inputs, and a physical system that is subject to costs and constraints. The objective is a specified state that one tries to drive the system towards and the controls are parameters in functions that drive the system. The constraints include the physical laws governing the system and the costs are penalties to be avoided. The objective, constraints and costs are combined into a functional to be minimized, by varying the controls. The optimization typically does not try to achieve the objective state exactly, which is not possible in most realistic cases, but rather tries to find the best possible solution to meet the objective, in competition with the costs. There is considerable flexibility in the optimization, since the relative importance of various costs can be balanced through a set of weight parameters in the cost functional. The degree of satisfaction of the objective, i.e. the closeness of the final state to the desired state, will depend upon the physical accessibility of that state and upon the stringency of the cost functions. The derived forcing functions therefore typically are not unique, but depend upon the weights assigned to the various costs and the compatability of those costs with the physical system. We begin with the elasticity equation (1), which describes the response of the system to external forces. For simplicity, we consider an elastic, homogeneous and isotropic semi-infinite solid whose free surface is subjected to a boundary traction zi(x, t). The initial and boundary conditions, expressed in equations (2) and (3) are: zero initial displacements and velocities within the solid (domain V); and with a traction applied to the surface (S) at positive times: /xui,jj + (A +/z)uj,j~ = pUi, u~(x, O) = a;(x, O) = O,

Aujjni q- t.l~(l~i,j q- tlj, i)nj

= "~i,

(x, t) E V x T +

(2)

x ~ v

(X, t) ~ S

(1)

X T +.

(3)

Here, n i are the direction cosines of the unit vector normal to the surface S, A, and/x are the Lam6 coefficients and p is the mass density. Throughout the text, the standard index notation is adopted where summation is assumed over repeated indices, commas denote spatial differentiation and dots indicate differentiation with respect to time. Using the half-space dyadic Green's function G~+) and the initial conditions in equation (2), one may obtain the displacement ui(x, t) at any point in the half space as [3]:

u~(x, t) = fo'dto fsdSG!/-)zj(Xo, to).

(4)

In the present optimization problem, we want to efficiently maximize the acoustic energy in a subsurface target volume at a particular time, such that that the total energy of the system remains as small as possible elsewhere and by applying a minimum surface force by means of controlling the surface traction during the process. The physical objective functional may be written as:

J[u,, rl] = ¢(7") + L,

(5)

Y.S. KIM et al.

910 with

( ~ ( T ) = [ . dYe(x, T) - E , ) ,

L 1=

L2 =

fo ?f fo W1

dV

S

dS

(6)

dte(x, t) - W'l

dVe(x, T)],

(7)

c

dt~,

(8)

where the error function ~b(T) measues the degree of satisfaction of the objective of localizing a specified amount of energy, Ep, in the target (focal) volume Vc at the target time t = T. In competition with this goal are the penalty cost functions, L = L~ + L:, corresponding to the desire of minimizing the total energy of the system (except in the volume Vc at t = T) and the total surface force during the energy focusing process. The total energy density, e(x, t), may be expressed as a sum of the kinetic and strain energy densities as: e(x,t) ~- 2p UiUi . . -~ -2 A (Uk'k) e -'~ ~2 ui'J(l'~i'J +uj;). '

(9)

The choice of the weight factors (wl, w'l and w2) in equations (7) and (8) allows for flexibility in balancing the role of the ~ and L contributions to J. The numerical computations in Section 3 will illustrate some of the logic involved with specific results for different choices for the weights. The dynamic optimization of surface loading for the purpose of efficient focusing of acoustic energy within the solid interior may be stated as the problem of finding the optimal control function Ti from a set of admissible controls such that the objective functional J is minimized under the system constraints in equations (1)-(3). Such a constrained minimization problem can be treated as an unconstrained one by introducing Lagrange multiplier functions ~i [4]. Instead of minimizing J subject to the constraints in equations (1)-(3), one minimizes the modified objective functional, ]:

][ui, ~:i, ~i] = J -

dV

dt ~;[l~u;,jj + (A + tx)uH; - Ou~].

(10)

The necessary conditions for the stationarity of ] are obtained by utilizing the techniques of variational calculus for the arbitrary variations 6u;, 60; and 6~. The variation of Y with respect to the costate function qJ~ simply yields the dynamical equations in equation (1) which are the constraints for the optimization problem. Similarly the costate (the Lagrange multiplier) equations of motion are obtained for arbitrary variations of u; as: ~.£l~i,jj q- (1~ q- ~£ )ffl],ji = DffIi,

(X, t) e V × T -

(11)

where the costate boundary condition and final conditions at t = T are respectively A~j,jRi + I£(~bi,j + ~bj.i)rli = T c,

(X, t) e S × T +

(12)

r~ = w,[(T - t)'t, - v,], ~bi(x,T)={~WlW:-l-a)fli(x,T)

(bi(x,r)={(WlW'l±l)iii(x,T), '

(0,

(13) if x ~ V~; otherwise.

(14)

where ~ denotes the costate boundary traction. The equation for the costate function ~; is the same as that of the displacement field, as a consequence of the self-adjointness of the wave operator. The costate function can be written as follows in terms of the half-space dyadic

Generation of controlled acoustic waves

911

Green's function G ~ ) as:

q,i(x,t)=PfvdV{G,7)o £fs dSGb-)~(Xo, to).

o G~-) },0=r Ot---~oOJy(Xo,to) - ~bj(Xo, to) "~o

+

dt0

(15)

The advanced dyadic Green's function can be rewritten from the reciprocal relation as [7] G~7-)(x, t; Xo, to) = G}7)(x, - t ; Xo, -to) = G~-+)(x, to; Xo, t).

(16)

We consider here only vertical surface forces, i.e. only the normal component z~ is present. Also, due to the symmetry of the problem the traction is assumed to have an axially symmetric form, zz = Zz(r, t). For the axially symmetric problem, we employ a reformulation of the Green's function matrix for a symmetric ring load [1, 8]. In our preceeding paper [1], the form of the surface traction function ~'z was without any restriction in the level of derived detail. Here however, we are concerned with simplified forcing functions that will be practical for laboratory realization. We have chosen the example of a forcing function that is axially symmetric with a cross sectional profile made from a superposition of Gaussian forms. Two cases are treated: (1) with ~'~constrained to be positive, i.e. as an imposed pressure distribution which might correspond to laser driven surface ablation, and (2) a case where z~ may be positive or negative, i.e. surface "pull" is allowed which would correspond to driving by a piezoelectric transducer array. Hence we have: M

Case 1: Zz(r, t) = - ~

ot2(t)gi(r, t),

(17)

i M

Case 2: zz(r, t) • ~ oti(t)~i(r , t)gi(r, t),

(18)

i

with,

gi(r, t) = exp[-~:ie(r, t)],

/~i(r, t)

r-

),

(19)

mi

where M denotes the number of the pulses (M = 2 in the present calculations). These two cases are schematically shown in Figs l(b) and 1(c) for a particular value of/3, and Ai. The reason for selecting two pulses in each case is the desire for a simple design while allowing for the possibility of two different pulses travelling with different velocities. This structure is suggested by our previous results where the optimal solution, though not Gaussian, was obtained by the superposition of two pulses each corresponding to the generation of shear and compressional waves. The function ~,(r, t) serves to track the center of each load and to control its width through Ai. The derivative-like shape of Zz in equation (18) was suggested by the structure of the corresponding optimal load shape [1]. After the substitution of Zz into the above formulation, the variation of J with respect to the control parameter functions a, and/3i and Ai yields respectively the equations below. Case 1: 6J

f

- 2 J~ dS(r)[w,(T t)tiz + w2"fz - qtz]Oti(t)gi(r, t), 80t i( t ) 8] f_ _ ot2i(t) 8fli(t) = - 2 Js dS(r)[Wl( T - t)ti z + w2 ~:2 - ~bz] ~ ~i(r, t)gi(r, t), 8'--2 6k,

fsdS(r) £ dto{W, £ dv(tz(r, v)--~i "" c~/2(t°)

+ [WeZz(r, to) - qJz(r, to)l ~

(20) (21)

,Zi(r, v)gi(r , v)

~Zi(r, to)gi(r,

/o)}.

(22)

912

Y. S. KIM et al.

Case 2:

a/

fs

dS(r)[wl(T

6cti(t)

6ill(t)

m ~ 6A i

-

d S ( r ) [ W l ( T - t)ft z + wavz - qJz] ~

rr

(23)

- t)ti z + wavz - t~z]~i(r, t)gi(r, t),

f

dS(r) Jo dt°'tw

70

Jo dvC, (r,

+ [wave(r, to) - ~Oz(r, to)] ~

Ai

{2~,z(r, t) - 1}gi(r, t),

ai(v)

(24)

1]gi(r,

~i(r, to)[2~2(r, to) - 1]gi(r, to)}.

(25)

The optimization problem consists of iteratively solving the integral equations for the state function [equation (4)] and the costate function [equation (15)] until specified convergence criteria are satisfied. With an initial guess on the form for vi on S, the state problem is solved for ui from equation (4) under the boundary condition in equation (3). With this ui, the costate problem equation (15) is consequently solved for qJ~ with equations (12)-(14). Better values for the control function parameters O/i, /~i and A~ can be found via the conjugate gradient algorithm [4] by using the objective functional J [equation (10)] and its gradients [equations (20)-(25)]. For the numerical solution, the continuous space-time optimal control problem posed above must be discretized in space and time [9, 10]. However, by using the half-space dyadic Green's function we avoid discretizing the whole space domain, needing only to discretize the boundary surface and target volume. The boundary surface is discretized as a series of axial-symmetric ring segments by the Boundary Element Method and the target volume (Vc) as a series of cylindrical segments by the Finite Element Method. Special care is taken on the choice of time intervals and boundary discretization in order to avoid violating the causality property of the Green's function [7].

3. N U M E R I C A L R E S U L T S AND D I S C U S S I O N

We have performed two sets of calculations, for the cases described above where the surface load is, or is not, constrained to be positive. Furthermore, for each case two different numerical calculations are carried out, with different weights in the cost functions to represent different system optimization choices. First we use the weight pair, wl = 0 and w2 = 1, where wl = 0 indicates that there is no cost penalty imposed that would tend to keep the system disturbances to a minimum and at the same time, we = 1.0 tends to keep the applied surface loads to a minimum (surface sensitive). The weight w'~ is chosen as the dimensionless time step size throughout. Next we have wl = 0.15 and w2 = 0, where wl = 0.15 indicates there is some penalty for disturbing the solid away from the target (interior sensitive). These two cost function choices represent somewhat different practical tradeoffs while maximizing the acoustic energy in the target volume. In the first option we minimize the surface forces and ignore the energy outside the target volume, where as in the second option we minimize the energy outside the target volume at the target time and allow high peak surface forces. In Table 1 these two cases are refered to as s u r f a c e s e n s i t i v e and i n t e r i o r sensitive.

Generation of controlled acoustic waves

913

Table 1. Energy yields of target energy versus work done on the boundary Load case

Weighting options

(+ ) (+ ) ( + ) and ( - ) ( + ) and ( - )

Surf. Inte. Surf. Inte.

Weights

sensitive sensitive sensitive sensitive

Pulse amp.'t

Pulse widthsi"

wI

wE

d

s

d

s

0.0 0.15 0.0 0.15

1.0 0.0 1.0 0.0

1.8 1.2 1.5 6.0

2.2 3.0 2.5 3.5

0.098 0.106 0.120 0.122

0.049 0.061 0.089 0.095

Yields % Res$ Unres§ 10.7 26.0 25.0 31.0

13 65 27 68

t d and s refer to dilatational and shear wave generating loads. :~ Restricted form of the surface load in the present paper. § Using an unrestricted form of the surface load derived in [1].

The numerical results below are based on the example of aluminum for concreteness. The relevant physical properties [1] are: the mass density p = 2.77gm 3, the Lam6 constants A = 0.546 × 1012 g / c m 2,/x = 0.257 × 10 ]2 g / c m s2, and the longitudinal and shear wave velocities Ca = 0.618 × 10 6 c m / s , c s = 0.304 × 10 6 c m / s . The longitudinal and shear wave velocities ca and cs are related to the Lam6 constants by c 2 = (A + 2 ~ ) / p and c 2 = t z / p . We convert all of the expressions to a dimensionless form, with a hat indicating the original physical variables. The dimensionless displacement, surface tractions, energy density, Lam6's constants, mass density, space coordinates, time, and G r e e n ' s functions are defined in terms of their dimensional counterparts by: u = f~lZo,

,~ = X l ~ c ~ , x = i/Zo,

r = elect,

e = ~l~c~

p = t~/fi = 1

~ = fLI~c~, t = cd'i/Zo,

(26)

G = ~caZ~o~.,

where Z0 is the length scale chosen here as the plate thickness. For the axially symmetric problem depicted in Fig. 1, we assume the m a x i m u m radius of the surface load to be

T(=j,t ) (a)

l

_t_1 "TI

I I

III

I1"1 III Ill III

"1 I I

r

z

Fig. ](a). Caption overleaf

Y. S. KIM et al.

914

°~

0.5

0

X

5 ,-i

o~ t~

0

.o u

d-5

Fig. 1. (a) Finite-element mesh for the half-space under the axial-symmetric surface load r:(r, t). The shaded part denotes the target volume V~ which is centered at (r, z) = (0.0, 0.8); (b) functional form of a typical surface loading with instantaneous radius, /3, and width, ~ (Case 1: no negative loading allowed); (c) functional form of a typical surface loading (Case 2: negative loading allowed).

2

P,l~p

0

1

~

-"

8 t/"l" 0.4

0.2

0.6

0.6

r/Ro

1

o.

(b) 3.0

2.0

1.0

/

/ / / 0.0

,

I

I

0.2

0.4

0.6

0.8

J

i

.....

0.0

1.0

t/T (c) 2.0

,

i

1.6

1.2

....................................

O

~-" o . 8 0.4

, ...........

0.0 0.0

0.2

0.4

0.6

0.8

1.0

t/T Fig. 2. (a) T h e positive constrained optimal surface normal load for the weights w 1 = 0 and w2 = 1, corresponding to minimal surface load. T h e normalized load, Pz/ep, is plotted as a function of normalized radius and time. T h e acoustic pulse arrives at the subsurface target at tiT = 1; (b) the amplitudes of the two Gaussian pulses. T h e solid curve is for the load stream generating a longitudinal wave while the dashed curve is for the load stream generating shear wave; (c) radius as a function of time for the two Gaussian pulses in Fig. 2(a). T h e solid curve is for the longitudinal wave while the dashed curve is for the shear wave. 915

916

Y.S. KIM et al.

R0 = 2.5Z0, which is divided into 25 axisymmetric ring segments with Ar = 0.1. Similarly, the depth from the surface z is discretized with the same mesh refinement, AZ = 0.1. The target volume Vc is a cylinder of radius r = 0.2 and height 0.4 centered at Zc = 0.8. The target volume is discretized as eight axial symmetric ring elements with each element having nine nodal points. Pz stands for the optimal normal surface loading which is related to the surface traction rz as Pz = - r z at z = 0. Also, the target time is taken as T = 7.0, in order to allow sufficient time for all possible control mechanisms to act. In order to preserve the causality property of the Green's function, a time step At = 0.046 is adopted. The optimizations result in amplitudes and radii as a function of time for Gaussian forms that are assumed to make up the load streams. The results are summarized in Figs 2-5 and in Table 1. For each of the cases we show the derived optimal surface loads as a function of radial coordinate and time, as well as the peak intensities and center positions as a function of time for the Gaussian forms. Figures 2 and 3 present the results for the positively constrained surface load, for cost functions corresponding to minimal surface load (Fig. 2: wl = 0, w2 = 1) and minimal system disturbance (Fig. 3: wl = 0.15, w2 = 0). Figures 4 and 5 are the counterparts of the preceding two figures where the surface traction can have both positive and negative values. All of the cases show similar overall features. The two concentric load streams [with the earlier and later streams denoted as Ps (shear) and Pd (dilatation)] shrink to the center of the surface with velocities that are asymptotically equal (at large radius) to the shear and compressional wave speed, respectively. Therefore, we associate the earlier and later load streams to the generation of shear and compressional waves that arrive simultaneously at the target volume at the target time. The velocities of the two force streams are similar in all cases. A comparison of Figs 2 and 3 shows that the optimal surface load in the former case (surface sensitive) mainly consist of the broad but small amplitudes of P~ pulse load and in the latter case (interior sensitive) mainly consist of the sharp but high amplitudes Pd pulse load. Also as seen in the Table 1, the yield of the controlled energy focusing in the former case is 10.7% along with an efficient 26% yield in the latter case. It follows that for the surface sensitive case, most of the energy is transmitted from the shear load Ps whereas, for the interior sensitive case, most of the energy is transmitted from the compressional load Pd. It is noted from Fig. 3(a) that to achieve acoustic focusing at the target with minimum system disturbance the optimal loading is mostly a Pd load directly above the target volume. Comparing Figs 2(b) and 4(b) it is noted that although the amplitudes of the optimal surface loads are similar for both cases, the percentage of the energy focusing at the target for the bipolar surface force, shown in Fig. 4(a), is more than twice as large as the positively constrained case shown in Fig. 2(a). This difference was also observed when the surface load forms were not restricted [1]. As it is noted in Table 1 the percentage of energy yield at the target for restricted load forms are very close to that of the unrestricted forms presented earlier [1]. It follows that under the minimum surface force option (surface sensitive), the percentage of energy yield at the target does not depend heavily on the fine details in the structure of surface loading. Such robustness is highly desirable in the sense of laboratory realization of these loading patterns. Figures 3 and 5 show the results for the interior sensitive case, when the surface loadings are chosen to have positively constrained form and bipolar form, respectively. It is noted that although they both focus about the same percentage of input energy at the target ( ~ 3 0 % ) , the amplitude of the load stream corresponding to the longitudinal wave Pd decreases by a factor of two, when going from a positively constrained loading form to bipolar form. It is also noted from Table 1 that, for the interior sensitive case, although restricting the loading patterns on the surface results in the loss of energy yields at the target, it is still possible to focus a high percentage of the input energy at the target using simplified loading patterns. It follows that for the interior sensitive case the fine details in the loading patterns on the surface are important

15 10 /',/~p 5 0

0 4 0"if/~

1 t/T

0.2

o.

1

(b) 12.0 __

8,0

4.0

0.0

0.2

0.4

0.6

0.8

1.0

0.8

t,0

t/T (c) 2.0

s

.

.

.

.

.

.

.

.

i

.........

1.6

1.2

.............

r~

~" 0.8 0.4

0.0 0.0

0.2

0.4

0.6

t/T Fig. 3. (a) The positive constrained optimal surface normal load for the weights w~ = 0.15 and w~ = 0, corresponding to minimal system disturbances. The normalized load, P~[ep, is plotted as a function of normalized radius and time. The acoustic pulse arrives at the subsurface target at t i t - 1; (b) the amplitudes of the two Gaussian pulses. The solid curve is for the load stream generating a longitudinal wave while the dashed curve is for the load stream generating shear wave; (c) radius as a function of time for the two Gaussian pulses in Fig. 3(a). The solid curve is for the longitudinal wave while the dashed curve is for the shear wave. 917

P,#P

0

-1

10 8 ~ 0 ~ , 6 0.1

3.0

0.4 r/Ro

o.

(b) i

i

r

0.6

0.8

.0

0.6

0.8

1,0

2.0

<

//

o~

// / /

8~

1.0

/ /

0.0

, ~:

0.0

I

I

0.2

0.4

%

t/T (c) 2.0

1.6

0

1.2

................

0.8

0.4

0.0 0.0

I

I

0.2

0.4

t/T Fig. 4. (a) T h e optimal surface normal load without the positive constrained for the weights wt = 0 and w2 = 1, corresponding to minimal surface load. The normalized load, Pz/ep, plotted as a function of normalized radius and time. T h e acoustic pulse arrives at the subsurface target at t i T = 1; (b) the amplitudes of the two Gaussian pulses. T h e solid curve is for the load stream generating a longitudinal wave while the dashed curve is for the load stream generating shear wave; (c) radius as a function of time for the two Gaussian pulses in Fig. 4(a). The solid curve is for the longitudinal wave while the dashed curve is for the shear wave. 918

I (a)

-2 ~ 1

08060"4~2

0,9

0,8

0,7

0,6

0.5 t/l"

0.4

0.3 0.2

0.1

o.

(b) 7.O_ 6.0 5.0 4.0

<

~,-, 3.o 2.0 1.0 0.0 0.0

0.2

0.4

0,6

0.8

1.O

0.6

0,8

1.0

t/T

(c) 2.0

1.6

o

1,2

'",,

..

0.8

0.4

0.0 0.0

0.2

0.4

t/T Fig. 5. (a) The optimal surface normal load without the positive constraned for the weights w~ = 0,15 and w2 = 0, corresponding to minimal system disturbance. The normalized load, Pz/em is plotted as a function of normalized radius and time. The acoustic pulse arrives at the subsurface target at t / T = 1; (b) the amplitudes of the two Gaussian pulses, The solid curve is for the load stream generating a longitudinal wave while the dashed curve is for the load stream generating shear wave; (e) radius as a function of time for the two Gaussian pulses in Fig. 5(a), The solid curve is for the longitudinal wave while the dashed curve is for the shear wave, 919

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in achieving such high percentages of energy focusing at the target where as, for the surface sensitive case such details were not significant. A summary of the essential results in the figures is presented in Table 1. The table shows the amplitudes and the widths of the pulses along with the energy focusing efficiencies. In interpreting the results in the table, wl = 0.0 indicates the case where there is no penalty for disturbing the system away from the target and w2 = 0.0 indicates the case where there is no penalty imposed for keeping the surface traction to a minimum. Thus, the pulse amplitudes are larger in the cases where w2 = 0. The table indicates that the optimal widths do not vary from case to case and the widths of the Po pulses are larger than those of the Ps pulses. This is expected because the widths of the pulses are largely controlled by the target size and the velocity of the Pd load stream is faster than that of the Ps load stream.

4. C O N C L U S I O N S

Two important observations are derived from this paper. The first observation concerns the issue of the robustness of optimal designs for subsurface focusing. In a previous paper we derived optimal surface forcing functions that produced efficient subsurface focusing while minimizing excitations elsewhere in the solid. The derived forcing functions had a high degree of detail in their spatial and temporal structure. Here, we have examined the use of constrained, simplified forms for the forcing functions to test the importance of the fine detail. We have found that the designs are indeed robust, preserving the high efficiency focusing of the earlier work while constraining the forces to forms based on Gaussians. This is important in terms of practical applications in the laboratory, where the pulse shapes are limited to simple forms. The second observation is of a general nature and confirms the finding in the earlier paper. The optimal dynamic surface loads which lead to successfully achieved objectives with high yield are not easily arrived at by a simple intuitive design process. Although some intuition into the operative physical processes can be gained with hindsight, subtleties arise due to the acoustic wave interference nature of the control process. Thus, without the use of a systematic design process, one would at most expect only a small percentage of input energy to be guided into the target volume. Acknowledgements--The authors acknowledge support for this work from the Department of Energy. Y. S. K. acknowledges support by the Korea Science and Engineering Foundation under Contract Nos 931-0300-002-2 and 91-08-00-05.

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(Revision received 11 March 1994; accepted 8 September 1994)