NASA Contractor Report 1_5

ICASE Report No. 91--48

ICAS

HOMOGENIZATION TEC_S AND ESTIMATION OF

MATERIAL PARAMETERS _TRIBUTED STRUCTURES

_:'2U2-

E:A::.Z -

H. T. Banks

D. Cioranescu

A. Das

R. Miller

D.A. Rebnord .......

Contract No. NAS1-18605 :.__ .=_.=

June 1991 ;_-!

Institute for Computer Applications in _ce and Engineering

NASA Langley Research Center ,r-r-_ ....

Hampton, Virginia 23665-5225

Operated by the Universities Space Re_h Association

NI SA

National Aeronautics and

Space Administration

L_ngley Re_esrch Center

Hamplon, Virginia 23665-5225

_i_ •

:i_iiiiiii!ii

L

HOMOGENIZATION TECHNIQUES AND ESTIMATION

OF MATERIAL PARAMETERS IN DISTRIBUTED STRUCTURES*

H.T. Banks t

D. Cioranescu*

A. Das §

R. Miller¶

D.A. Rebnordll

ABSTRACT

We discuss the use of homogenization techniques to derive approximate models with simple geometry for physical models of grids and trusses which have a complex geometry that

gives rise to computational difficulties. Our presentation is in the context of inverse or parameter estimation problems for composite material structures with unknown characteristics

such as stiffness and internal damping. We present the necessary theoretical foundations for

this approach and discuss comparison of modal properties of the resulting homogenization

model for a two-dimensional grid structure with modal properties observed in experiments

with this grid.

*This research was supported in part by the Air Force Office of Scientific Research under grant AFOSR90-0091, in part by the National Science Foundation under grant DMS-8818530, and in part by the National

Aeronautics and Space Administration under NASA grant NAG-l-Ill6. Parts of this research were carried

out while the first author was a visitor at the Institute for Computer Applications in Science and Engineering

(ICASE), which is supported under NASA Contract No. NAS1-18605.

tCenter for Applied Mathematical Sciences, University of Southern California, LA, CA 90089-1113.

_Laboratoire d'Analyse Num_rique, Universit_ Pierre et Marie Curie, 75252 Paris, France.

§Phillips Laboratory, Edwards Air Force Base, CA, 93523.

_lCenter for Applied Mathematical Sciences, University of Southern California, LA, CA 90089-1113.

IIDepartment of Mathematics, Syracuse University, Syracuse, NY 13244.

1. Introduction

We report here on a part of our continuing efforts on the development

of high fidelity dynamic models for composite material structures. The focus of our investigations has been on models to be used in estimation and

control of large flexible structures mainly intended for use in space (e.g., antennas, platforms, solar panels, experimental arrays, etc.). At present there

is a reasonably adequate understanding of the dynamics of beams and plates

made from known materials such as aluminum alloys. Our recent efforts (see

[Bin] and the references therein) involved models and methods to determine

material parameters in composite material structures with simple geometry

(beams with attached solid bodies or solid plates). These result in inverse

problems that are computationally tractable as long as the physical geometry

is relatively simple. However, substantial difficulties arise in cases involving

more complex geometries such as grids (which may be viewed as plates with

many holes) and trusses (solid columns from which most of the material is

removed in some periodic, regular fashion). In these cases the difficulties

associated with unknown composite material characteristics such as stiffness

and internal damping are combined with severe difficulties related to computational grid selection for a domain that is mostly holes or perforations.

The purpose of this paper is to outline one possible methodology for

dealing with these structures of complex geometry and to report on some

of our initial investigations in this regard. This methodology is based on

ideas from homogenization and requires that the structures be highly periodic

(many perforations repeated in a regular pattern) and sparse in material (i.e.,

a grid or truss with thin members).

We formulate our ideas in the context of an inverse problem approximation framework that has been developed for problems involving simple

geometries. A summary of this theoretical framework is presented in Section

2. In Section 3 we describe a particular grid structure which we have used in

our experiments and indicate how to formulate a direct physical model for

this grid within the framework of Section 2. This model leads to computational methods that are inherently intractable for estimation of parameters
(and also for the ultimately desired control investigations) due to the underlying computational domain.

In Section 4 we present a summary of results for a homogenization procedure that approximates (in a nonstandard way) the original direct physical

model on a perforated domain by a homogenized model on a domain that is

very simple (the perforated domain with all of the perforations filled in). The

resulting approximate model is also in a form to fit into the inverse problem

framework of Section 2. Finally, in Section 5 we report on our initial efforts

on validation of this approximate model by comparing experimentally observed modal properties of the grid with those possessed by the homogenized

model.
2. Review of Theory for Second Order Systems

In this section we give a brief summary of the theoretical background

necessary for a rigorous discussion of estimation problems for second order

systems. Detailed discussions can be found in [BI], [BR1], [BR2], [BW], [B],

[BK].

Let V and H be complex Hilbert spaces with V continuously and densely

embedded in H. We may then formulate a Gelfand triple V _ H _-_ V* with

duality pairing <, )y*,y (e.g., see [W]), which we shall denote by (,) when

no confusion results. Let (Q,d) be a compact metric space of admissible

parameters q. We consider the parameter dependent abstract second order

system

(2.1) ii(t) + A2(q)it(t) + Al(q)u(t) = f(t,q) t > 0

= = vo,

where the operators Ai(q) E £(V, V*),i = 1,2, arise from parameter dependent sesquilinear forms _r_(q) : V × V _ C which represent generalized

stiffness (al) and damping (_2), respectively. More precisely, we consider the

equation

(2.2) (ii(t), ¢) + a_(q)(it(t), ¢) + a1(q)(u(t), ¢) = (f(t, q), ¢}

for all ¢ E V, where ol is symmetric and or1, a2 satisfy the following conditions

(a theory for damping forms a2 that satisfy weaker conditions than (B),

namely H-semiellipticity, can be found in [BI]):

(A) Parameter Continuity: For each ¢ G V there is a function

c(., .; ¢) on Q x Q satisfying _(q,p; ¢) _ 0 as q --* p in Q such that

for each _ C V we have

la(q)(¢, _b) - a(p)(¢, ¢)1 -< e(q,p; ¢)l_b]v.

(B) V-ellipticity: There exists cl > 0 such that for all q G Q and all

¢ E V we have

Re ¢) >__c1[¢#.
(c) Boundedness: There exists c2 > 0 such that for all q 6 Q and all

¢,¢26V we have

]_(q)(¢,_)[ _ c2[¢]v[¢lv.

Under these conditions on al and (r2, there exist operators Ai(q)

£:(V, V*) such that

ai(q)(¢,%b) = (Ai(q)¢,¢) ¢,¢ 6 V.

6

Hence equation (2.1), or equivalently, (2.2), is to be interpreted as an equation

in V*. Following standard practice, we may rewrite (2.1) in first order vector

form in the coordinates w = (u, ti) and use semigroup considerations when

discussing solutions. To this end, we define 7-/ = V x H and ]2 = V × V

and note that 12 _-+ 7"( _ 12" also forms a Gelfand triple. We also define the

sesquilinear form a(q) : Y x Y ---+C by

(2.3) a(q)((¢,¢),(rl,_))=-(¢,rl)v+al(q)(¢,_)+a2(q)(¢,_)

so that (2.2) may be written

(tb(t),x) + ar(q)(w(t),x) = (F(t,q),x)

(2.4)

w(0)= (u0,v0)

for w(t) = (u(t),i_(t)) and X = (¢,%b)in 12 with F(t,q)= (O,f(t,q)). Equivalently, we may write the equation in ]2" as

(2.5) tb(t) = .A(q)w(t)+ F(t,q)

where a(q)(x, () = (-.4(q)x, () with

Io i] .4(q)=

-A,(q) -A2(q)

Since a, and a2 satisfy (B) and (C), it is readily established that A(q) is

the infinitesimal generator of an analytic semigroup T(t; q) on ]2, 7f and ]2".
Moreover, the unique solution w E L2(O, T;'H) of (2.5) for w0 = (u0, v0) C

= V x H and f E L2(O,T;H) is given by

/0' (2.6) w(t;q)=T(t;q)wo+ T(t-s;q)F(s,q)ds.

For computational purposes, one must consider approximation schemes

for (2.6). Let H N be a family of finite dimensional subspaces of H satisfying

H 2v C V and the condition

(el) For each _ E V, th6re ezist8 _N _ O N ,_uch that I_-- oNI V --+ 0 a8 X _ oo,

Let 7-/N = H N × H N and let pU be the orthogonal projection of 7-/onto 7-/u.

Moreover, let AN(q) 6 £;(7"/x) denote the operator obtained by restricting

(7(q) of (2.3) to _N x _N; i.e., for X, _ C _N,

c_(q)(x, _) = <--AN(q)x,_).

We denote by TN(t; q) = e AN(q)t the corresponding analytic semigroups on

7-/N. The approximating systems for (2.6) are then defined by

/0' (2.7) wN(t;q)= TN(t;q)pNwo+ TN(t-s;q)pNF(s,q)ds.

From an application of the Trotter-Kato theorem on convergence of semigroups, one can readily obtain that under conditions (C1), (A), (B), (C),

we have for each X E 7-/, TN(t; qN)PN X _ T(t; q)x in 7-/, uniformly in t on

compact intervals, whenever {qN} is an arbitrarily chosen sequence in Q with

qN _+ q. Moreover, using the theory of analytic semigroups, one can argue

that for each X C N and each positive integer k, AN(qN)kTN(t; q_C)PN X

A(q)kT(t; q)x in 7_, again uniformly in t on compact intervals.

It follows immediately that wN(t; qN) of (2.7) converges in 7ff to w(t; q) of

(2.6), with the convergence being uniform in t on compact intervals. Furthermore, for f sufficiently smooth (e.g., f e Cl([0, T],/4) sumces),we find that

tbN(t; q_V) ___.tb(t; q) in _ as N --_ oc, uniformly in t on compact intervals.

Thus, we see that the first component uN(t; qN) converges to ut(t; q) in V

while the approximate acceleration uN(t; qN) converges in H to utt(t; q). With

more smoothness on I (e.g., I e C2([0, T], H)), we obtain that u_ (t; qN) converges in V to uct(f; q) as N _ oo.
If V embeds continuously in C(fl) (this is the case, for example, if

V C H2(f/) where fl C R 1 or _ C R2), then the above results lead to

pointwise convergence (in the spatial variables as well as time) of approximations to the displacement, velocity or acceleration in dynamic problems

involving structures such as beams and plates. These approximation results

are precisely those needed to treat certain questions arising in inverse or

parameter estimation problems for such structures.

To be more specific, suppose we are given a set of observations z in

the observation space Z along with an observation map F(q) from _) C

C([0, T], H) to Z for the system (2.1). Thus, z is a set of observations or

data for Fu(q) where u(q) is the solution to (2.1). A least squares estimation

problem then consists of finding _ E Q which minimizes over Q the criterion

(2.8) ¢(q) --IVu(q)- zl_ ,

subject to (2.1). Typical examples included in this setting are problems

for beams and plates where acceleration measurements (accelerometers) are

available. In the case of a beam of length a, we might have observations zjk

of the acceleration utt(tj, xk; q) at several times tj,j = 1,2,..., J, in [0, T]

and at several locations xk E (0, a], k = 1,2,..., K, along the beam. Then

Y C H2(0, a) in the usual formulation and 7) -- C2([0, T], V), Z = R JK with

Pu(q) = {utt(tj,zk;q): j = 1,...,Y, k = 1,...,K} so that

o(q) = lu (t3,xk;q)- zj l

j,k

For a rectangular plate occupying the region (x, y) E Ft = (0, a) × (0, b), use of

accelerometers at locations (xk, yk), k = 1,..., K, would lead to observations

{zjk} for {utt(tj,xk, yt,;q)}. With Y C g2(fl),:D = C2([O,T],V) and Z =

R ag we have Fu(q)= {utt(tj, xk, yk;q): j = 1,...,J, k= 1,...,K,} and

O(q)= E q) - 2

j,k

The corresponding approximate problems are formulated using the criterion

(2.9) ON(q)-_-trIAN(q)- Zl2Z
for solutions wN(q) = (uN(q),i_N(q)) of (2.7). One then seeks a parameter

qN E Q that minimizes ON(q) over q E Q subject to (2.7).

Among the important questions related to such approximate problems

are those pertaining to parameter convergence; i.e., for a given fixed set of

data z C Z, do optimal parameters _N for (2.9) converge in some sense to

an optimal parameter for (2.8)? More generally, one can also incorporate

continuous dependence of the optimal estimates on the data by employing

the concept of method stability. If one is given a sequence {z TM} of data

sets that converge in Z to a data set z °, and one denotes by _tN(z m) and

_(z °) the optimal parameters (in general, these are sets) for (2.9) and (2.8)

corresponding to z m and z °, respectively, then method stability requires that

(TN(z m) converges (in some appropriate setwise sense) to _](z °) as N, m _ oc.

These issues are carefully discussed in [B], [BK], where it is shown that to

insure both parameter convergence and method stability, it suffices to argue

that FuN(q N) --_ Fu(q) in Z for any sequence {qN} in Q with qN _+ q. From

the discussions above one thus has that the conditions (14), (B), (C) on 0" 1

and a2 and (CI) along with a compact parameter space (Q, d) are sufficient

to treat these questions in the case of accelerometer data for beams and

plates.
3. The AFAL Grid Structure

We consider a rectangular plate perforated with rectangular holes as depicted in Figure 3.1. A similar plate has been the subject of numerous

experimental investigations at the Air Force Astronautics Lab (now a part of

the Phillips Lab) at Edwards Air Force Base. Using the Love-Kirchhoff plate

theory (the 2-dimensional analogue of the Euler-Bernoulli theory for beams),

one can model such a grid structure as a second order system of the form

(2.1) so that conditions (A),(B),(C)of the previous section are satisfied.

We summarize here previous findings and refer the reader to [Ball, [_a21,

[R] and the references therein for more detailed discussions.

b

X2

Figure 3.1

We assume that the plate is cantilevered along the xl axis (at x2 = 0) as

depicted in Figure 3.1 and is free on the other edges as well as the edges of the

holes (perforations). If we use the Love-Kirchhoff theory and assume KelvinVoigt damping as well as viscous (air) damping, the transverse displacements

8
u(t, xl,x2) at time t and location x = (xa,x2) in the perforated domain

aper = (0, a) X (0, b)-{ perforations } satisfy a second order (in time) system:

. 02U OU 02M 1 0s/_112 OSM s

(3.1) ph-0-_+7-b-7+ Oz--T+ 2 0_I O_ + Ox _ - f

t > O, (Zl, ZS) _ _per,

where p is the mass density, h is the thickness of the plate, and M 1, M 2, and

M is are the bending moment about the xa axis, the bending moment about

the za axis and the twisting moment, respectively. If we include Kelvin-Voigt

damping in these moments, we find

1-u s -_z_ + v oz_ J + 1- u 2 oz_ot + u o--@2ot ;

(3.2) M 2 _ 1-uEI 2 { _02u +u_(,OSu} + 1-uc°I 20x_Ot { Oau +u_ Oau _

E1 O2u CDI C3aU

M 12 = +

1 -F u OxlOx2 1 + u OxlOx20t

where E1 and CDI are the usual stiffness and Kelvin-Voigt damping coefficients and u is Poisson's ratio. The plate is clamped along the xa axis where

we must have the (essential) boundary conditions

OU

(3.3) u - Ox_ - O, for x2 = 0, 0 <_ 271 < a, t :> 0.

Since the plate is free on its other edges and the hole edges, it must satisfy zero

moment and zero shear conditions on these edges. These result in natural

boundary conditions given by

(3.4) M x =0, OM1 +20Mls-O

Oxx 0272

on edges parallel to the x_ axis, and

(3.5) M s=O, OMS +20Mls-O

027s Oxl

on edges parallel to the 27x axis.
The system (3.1) - (3.5) can be readily formulated in the sesquilinear

framework of Section 2 so that the results given there are applicable. To

that end, define

and

Hper = L2(_per).

o¢

= 0 on x_ = O}

Ox2

Then Vner _-* //per _ Vp'_r forms a Gelfand triple. Moreover, we may define

sesquilinear forms rr_(q) : Vper X Vper --* C, i = 1,2 by

(3.6)

e)=

{qi(¢,11 _,11 q-¢,22 _/3,22 ) -}- qiq3(¢,11 ¢,22 +¢,22 ¢,11 )

per

+2qi(1 - q3)¢,12 ¢,x2 }dx,

E1 CDI 02¢ It is not

where q = (ql, q2, q3, q4) = ( 1 - u 2' 1 --u _' u, 3') and ¢,ij = OxiOx.i"

difficult to verify that cryPt(q) satisfies conditions (A), (B), (C) whenever Q is

a compact subset of C(_2pe_, R 4) N {q : q_ :> c > 0, i = l, 2, 3; q4 _> 0, q3 < 1 }.

The system may then be written for u(t) E Vp_ as

(3.7) <phutt, ¢) + a_(q)(u(t), ¢) + aPer(q)(it(t), ¢) + (q4z2(t), ¢) = <f, ¢)

for all _b e Yper. Given initial data u(0) = u0,/_(0) = v0 with u0 E Vpe_, v0 E

Hpe_, the methods outlined in Section 2 can be used to develop direct numerical methods as discussed in detail in [BR2]. While the theoretical aspects

of approximation in parameter estimation problems are relatively straightforward conceptually given the theory of Section 2, the problems are computationally quite demanding. This is true even though the computational

domain f_pe_ is among the simplest and most regular of those arising in grid

and truss structures.

The computational difficulties alluded to here motivate one to develop

alternative methods for modeling of grid and truss structures for the purposes

of identification and control.

10
4. Homogenization for Grid Structures

Given the difficulties associated with the model for the grid described in

the previous section, it is of interest to approximate grid-like and truss-like

structures with a model that retains physical fidelity but is computationally

more tractable than a direct physical model. One such approximate model

can be formulated using homogenization techniques which were originally

developed to model composite material structures. Here we outline results

for a grid structure similar to that of Section 3. Details for this grid and

other lattice/truss derivations can be found in [BCR], [R], [BLP], [CS], [CD],

and [SP].

The basic idea in our use of homogenization is to approximate a model

such as (3.1) - (3.5) or, equivalently, (3.7), by a homogenized model (HE) on

a domain in which the "holes" or perforations have been "filled in". That

is, instead of a perforated domain _p,r aS described in Section 3, we wish to

compute on the domain fl = (0, a) × (0, b). The model associated with this

domain, even if it is plate-like in form, will, of course, be nonphysical. The

coefficients in the distributed system will be nonphysical parameters.

The homogenization procedures we outline here require periodicity in the

structure (many regularly placed "holes") as well as sparseness of material

in the structure. Thus we consider a grid structure like that in Figure 3.1,

but with many "holes" or bays. We

ll
----_X 1

t" "1

I I

I 1

w

t. .... .J

--[ II--

z=(Xl,X2)

Y2

_2

x

y = -g

P

Figure 4.1

T

Y = [0, 11× [0,e:]

= Y*UT

_y*

consider a typical cell as depicted in Figure 4.1. We map this typical cell to

a fixed size cell Y = [0, _1] x [0, e2] via a mapping y = z/¢ so that the original

cell has dimension ¢_1 by 5e2. As _ _ 0, periodicity in the original fixed

domain flper increases. As # ---* 0, the thickness of the members in the grid

decreases. Our approximate equation on fl will be obtained by taking limits

as _ ---* 0, # _ 0. Of course, the approximate limit model can be expected to

be a better approximation to the actual grid structure if the grid has a large

number of bays and thin members.

To facilitate our discussions, we parameterize the grid structure, denoting

the perforated region by _/,,,, and consider a model of which (3.7) is a special

case. (For convenience, we drop the viscous damping term for the discussions

in this section.) We define H_,, = L2(fl_,,) and V_,, = H_(fl_,,) = {¢ C

H2(12_,,) : ¢ = 0¢_ = 0 on x2 = 0}. The generalized stiffness and damping (0x2

sesquilinear forms are defined as

(4.1) 0"1'#(¢_ ¢) aijkh t x )9,kh (X)_),ij (x)dx

c,t, i,j,k,h

12
(4.2) a_'"(¢, ¢) -- fa bi_h(X)¢,kh ¢,ij (x)dx

e,u i,j,k,h

for ¢, ¢ E V_,.. Here and throughout we adopt the notation ¢,ij for --

We assume that the functions defined by

aijkh -- aijkh(X ) and bijkh = bijkh(X ) for x E f/_,.

are Y-periodic and that initial data

u0_,_ E V_,., V;,_ E H.,.

are given. We then consider the system for u_'U(t) E

(4.3)

V_,. satisfying for

t E (0, T),

=_ u0, = V0 .

We wish to take limits as ¢ --, 0,# _ 0 in this system. Note however that as ¢ and # change so does the domain f_,_,; hence limit taking

is not a straightforward process. We need to first extend functions such as

u_'"(t), u_'", f, etc. to the entire domain f_. We can do this by assigning the

value zero to functions in the region f_ - f_,,. We shall denote by _0this extension by zero of any function g E L2(f_,,). This results in _ E L2(f_). Such

an extension is not so useful if we wish to deal with functions g E H2(E_,,)

(such as u_'_'(t)) for which the H 2 smoothness is to be preserved. For such

an extension, special extension operators must be formulated.

We shall proceed in two steps. We fix # at some value and take limits

as e ---* 0, obtaining an intermediate "homogenized" solution u" satisfying a

certain intermediate equation. We then take the limit as # _ 0, obtaining

the desired homogenized system (HE).

Let # > 0 be fixed and let u _ = u _'" be the solution to (4.3) where we

shall in our notation temporarily suppress the dependence on the fixed value

of #. Then we have

Lemrna 4.1. There exists an extension operator P_

P_ E L(L_(0, T; V_,u),L_(O,T;H_(a)))

13
and a function u = u _' such that for some sequence e,_ _ 0 we have

P_"u _" ---+ u in L°°(O,T;H_(_)) weak*

(P_"u'")t = P_"u_" ---, ut in L°°(O,T;L2(_)) weak*

With additional assumptions on the initial data in (4.3), we can determine

a limit equation that u = u u of Lemma 4.1 satisfies. Define 0 = IY*I/IYI,

where IYI denotes the measure (area) of Y. Then we have

Theorem 4.1. Let u _ = u _'u be the solution of (4.3) and suppose that

f • L2(_,,u) so that ] • L2(_). Moreover, suppose there exist Uo • H:(_)

and Vo • L2(12) such that the initial data for (_,.3) satisfy

o "-+ Uo as ¢ _ O, weakly in L2(_)

_ e11.$

o _ Vo as _ --* O, weakly in L2(_).

Then the convergence of Lemma 4.1 holds with the limit function u(t) =

uU(t) • H_(_) satisfying the homogenized system

(HE _)

(Ophutt(t),¢)+a(t)(u(.),¢)=(O],¢) forte(O,T) andallC• H_(12)

u(o)= _olO, _,(o)= volO,

corresponding to the homogenized hysteresis sesquilinear form

(4.4) _(t)(u(),¢)=/. ]o'E p,_kh(t-,)u,kh(,,_)_,,j(_)d,dx

i,j,k,h

The coefficients p_jkh(t) = L-l[ib;jkh(s)] are the inverse Laplace transforms of

the functions

(4.5)

_,jkh(s)= L[p;j,_](s)------

.{aqkh(y) + sbqkh(y)- _(aemkh(Y) + sbtmkh(Y))X,_ (s,y)}dy

g,rn

14
where the functions x_J(s,y) are Y-periodic solutions of

fy. (a mkh(Y)+ ¢,kh(y)dy = 0

g,m,k,h

1

for all Y-periodic _ e H2(Y*). Here pij(y) - gYiYj.

We note that the general homogenized plate equation (for # fixed) for

a grid with damping involves a time hysteresis functional of the solution.

It is instructive to consider the special case of the grid with no damping;

i.e., bijkh -_ O. In this case we find that Pljkh(S) -_ L[Pijkh](S) is actually a

constant, say f)ijkh(S) = a_kh with

1

/y.{aijkh(y) - _ at,_ku(y)x,_ j (y) }dy (4.6) =

where X ;j is the solution of

fy. _ a,mkh(Y)(X'J(y) -- PiJ(Y)),em _b,kh (y)dy = 0

£,m,k,h

for all Y-periodic ¢ C H2(Y*). Recalling that the inverse Laplace transform

of a constant is that constant times the Dirac delta function (i.e., L -1 [_jkh] =

aijkh6), we find the sesquilinear form (4.4) reduces to

a(t)(u(.),¢) =/_ _ a,_khU,kh (t,x)¢,,j(x)dx,

i,j,k,h

or

i-

(4.7) a(¢,¢) = L _, _,jkh¢,kh(X)¢,,j(x)dx.

i,j,k,h

Thus, the hysteresis in the model is a result of nontrivial damping terms in

the grid model. Moreover, we point out that the coefficients in (4.7), and

hence the coefficients in (HE ") for grids without damping, are not simple

averages of the original coefficients a_jkh of (4.3). Rather, the homogenized

coefficients are averages of the originM coefficients over the structure plus

some correction terms as shown in (4.6).

We proceed to the next step by letting # _ 0 in (HE ") and the associated

equations given in Theorem 4.1. To facilitate our discussions, we restrict

15
our considerationsto the constant coefficientcase;i.e., aijkh and bijkh are

constants. Moreover, we take fl = f2 = 1; hence Y = [0,1] x [0,1] and

IYI = 1, IY*I = #(2 - it) so that 0 = #(2 - it). It follows that

0

- -, 2 as it ---*0.

it

One can establish the following.

Theorem 4.2. For the constant coefficient problem with 21 = 22 = 1, there

exists a sequence #,_ _ 0 such that:

(i) The coefficients Pijkh "" of (4.5) converge in the sense that

1

itnps_kh *

up "-_ Pijkh

where

(4.8)

(aHtt + sbijtt)(a_tkh + sbeekh)

ihi*jkh(s) = L_i_kh](S ) = 2(aijkh + sb, jkh)-

£=1 atilt' + S btftl

(ii) The solutions u"" of (HE "") satisfy

u"" _ u" in L°o(0, T; Hi(a)) weak"

where u* is the solution to

(HE)

(2ph_;_(t), ¢) + _*(t)(_'(.),_)= (2],¢)

u*(0) = lim it uo/2

tt ---*0 _U0

# v0/2, u;(o)= _ _o =

_.(t)(u.(.),¢)= £ £

with

(4.9)

t e (O,T) and all g, C H_(f_),

pi*jkh(t -- v )u,*kh (v, x )¢ ,ij ( x )dT dx.

i,j,k,h

16
We return to the AFAL grid structure of Section 3 corresponding to a direct physical model using the Love-Kirchhoff theory with Kelvin-Voigt damping which is given by equation (3.7) with the sesquilinear forms given by (3.6).

This was the abstract variational form of (3.1) - (3.5). We specialize the results of this section to that example. We have the coemcients aijkh = aijkh

given by

EI

allll _ a2222 -- I - u 2 - ql

EI

a1122 = a2211 = v I _ u2 - qlq3

E1

a1212 = a2121 = a1221 = a2112 - 2(1 + u)

with all other aijkh zero. The bijkh are given by

CDI

blm=b2222- 1-u 2 -q2

1

-- _ql(1 -- q3)

CDI

b1122 ----b2211 -_ u 1 _ u2 -q2q3

CDI

lq2(1 -- q3)

b1212 = b2121 = b1221 = b2112 - 2(1 + u) - 2

with all other bijkh zero. In this case (3.6) and (4.1), (4.2) agree. Moreover,

using (4.8) of Theorem 4.2, we find

P;111(8) = i0_222(S) _-- E[ -_-SCDI = ql(1 -- q_) + sq2(1 -- q_)

^, ^, ^, ^, EI +s cnI _ q1(1--qa)+sq2(1--q3)

P1212(S) = P2121(S) ---- P2112(S) ---- P1221(S) -- 1 -t- _ 1 + u

with all other 15i*jkh zero. It follows that a* from (4.9) can be written as the

sum of two sesquilinear forms a_' and G_ given by

fa{qi(1 - _,11 -t-¢,22 ¢,22 ) + 4qi(1 q3)_,12_,12}dx. _ri(¢,0)=* q32)(¢,11 --

We observe that ar,a_ also satisfy the conditions (A), (B) and (C) of Section

2 so that the convergence results summarized in that section are readily applicable to the homogenized model in this case. If we write the homogenized

17
model system in strong form analogous to (3.1) - (3.5), we obtain (we include

the viscous damping term so as to compare with (3.1))

02?2" On* O2M J,1 O2M._12 OM',2 2

ph--_-i-+_/---_- + Oz----_l -l- 2OXlOX2 -{- 0x22--] t e (O,T),x e f_,

(4.11)

u'(o) = no/2, u;(o) = v0/2

where fi_)'_' _ no, _5_'u ---* Vo weakly in L2(f_) as e ---* 0 as stated in Theorem

4.1 and the moments M, 1j , M,,2 M, 12j are given by

EI 02u * CDI 03U *

M 1 - +

20x_ 20x_Ot

E1 02u * CDI OaU *

(4.12) M, 2 - +

20x 20x Ot

M1, _ =_

E1 02u * CDI 03U *

1 + u OXlOX 2 .A[_1 + U OXlOX2Ot"

We note that M, _2 is the same as M 12 of (3.2) while the form of equation

(4.11) is the same as that of (3.1), with only the moments being different.

Of course u* must also satisfy the clamped boundary condition (3.3) along

the edge x2 = 0 of f_ and the free boundary conditions (3.4) (with M, _, M, x2)

along Xl = 0 and Xl = a and (3.5) (with M, 2, M, x2) along x2 = b.

18
5. Comparison with Experimental Results for the AFAL

Grid

The homogenized model (4.11) - (4.12) for the APAL Grid is, of course,

a type of approximation to the physical model (3.1) - (3.5). Unlike many approximate models, we cannot increase the order of approximation by changing some parameters or mesh sizes. Even though (HE) is the result of a

limiting procedure e,# --+ 0, the periodicity of bays and the thickness of

members in a given grid are fixed and hence e and # are fixed. We can expect the homogenized model to be a better approximation for structures with

more bays and thinner members than for structures with a small number of

bays and relatively thick members.

Before using the model (4.11) - (4.12) for estimation and control problems for the AFAL Grid, we performed some initial experiments to test how

well the homogenized model described quantitative properties of vibrational

characteristics of the grid. One such investigation involved vibrational experiments (the grid was displaced initially from equilibrium and free release

vibrations were observed; the data consisted of accelerometer measurements

at several locations on the grid). Analysis of the data included experimental

determination of the fundamental frequencies (which, of course, depend on

the internal damping) for the grid. We then compared these values with

those predicted by the analysis of the homogenization model (4.11) - (4.12).

The AFAL Grid used in our experiments was a 5ft. square (a = b =

5) with 16 = 4 x 4 square bays measuring 12" x 12". This corresponds to

a calculated _ __ 1.2" with ¢gi _- 14.4". The grid was constructed from

aluminum alloy 6061-T6 for which hand book values for stiffness, etc., were

available. We used ph = .0550132slugs/ft _, u = 1⁄2, E1 = 135.641b • ft, cD :

1.728 × 1051b • sec/ft 2 and 7 = .02slugs/ft 2 sec in our calculations with the

model (4.11) - (4.12). We calculated the first eight frequencies w_' for the

homogenized model using two different approaches: an approximate mode

shape technique based on textbook approximations [G], ILl and an eigenvalue

analysis (MATLAB) for a finite element (bicubic B-splines modified to satisfy

the essential boundary conditions) Galerkin approximation to (4.11) - (4.12).

19
Weobtained essentiallythe sameresultswith both approaches.Thesevalues

are listed along with the experimentally obtained frequenciesw_ xP in the

table below. We observe that for i = 1, 2, 3, there is good agreement while

w_ is approximately the average of w ExP and w ExP. There is reasonably good

agreement between the pairs ca7 and w ExP i+l for i = 5,6, 7. This comparison

suggests that use of the homogenized model is a reasonable approximation

at least for vibrations involving modes with frequencies less than 16hz.

Mode i w___k _ 03/EXP

1 .785 .781

2 2.52 2.15

3 4.93 4.69

4 6.84 6.35

5 8.44 7.23

6 13.8 8.11

7 15.1 13.67

8 15.5 15.5

Table 4.1

2O
6. References

In]

[BCR]

[Bin]

[BI]

[BK]

[BLP]

[BRI]

[BR2]

[BW]

H.T. Banks, On a variational approach to some parameter estimation

problems, Distributed Parameter Systems, Springer Lee. Notes in Control and Info. Sci., 75 (1985), p. 1-23.

H.T. Banks, D. Cioranescu, and D.A. Rebnord, Homogenization models for 2-D grid structures, to appear.

H.T. Banks and D.J. Inman, On damping mechanisms in beams, CAMS

Rep. No. 89-3 (USC, Sept, 1989); ASME Journal of Applied Mechanics, to appear.

H.T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control-Theory and

Adv. Tech., 4 (1988), p. 73-90.

H.T. Banks and K. Kunisch, Estimation Techniques for Distributed

Parameter Systems, Birkh£user Boston, 1989.

A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis

for Periodic Structures, North Holland, Amsterdam, 1978.

H.T. Banks and D.A. Rebnord, Analytic semigroups: applications to

inverse problems for flexible structures, CAMS Rep. No. 90-3, Proc.

Intl. Conf. in Differential Eqns. and Applications, (Retzhof, June

18-24, 1989), Lec. Notes in Pure and Applied Mathematics, Marcel

Dekker, to appear.

H.T. Banks and D.A. Rebnord, Estimation of material parameters for

grid structures, J. Math. Systems, Estimation, and Control, 1 (1991),

p. 107-130.

H.T. Banks and J.G. Wade, A convergence framework for approximation methods for DPS estimation problems, Proc. Intl. Symp. on

Inverse Problems in Engineering Sciences, (Osaka, Aug. 19-20, 1990),

to appear.

21
[CD] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures et Appl., 68 (1989), p. 185-213.

[CS] D. Cioranescu and J. Saint Jean Paulin, Asymptotic analysis of elastic

wireworks, preprint.

[G] D.J. Gorman, Free Vibration Analysis of Rectangular Plates, Elsevier,

1982.

[L] A.W. Leissa, Vibration of Plates, NASA SP-160, 1969.

[R] D.A. Rebnord, Parameter Estimation for Two-Dimensional Grid Structures, PhD. Thesis, Brown University, 1989.

[SP] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory,

Lecture Notes in Physics, 127, Springer-Verlag, New York, 1980.

[W] J. Wloka, Partial Differential Equations, Cambridge, 1987.

22
l

Report Documentation Page !

1, Report No i 2 Government Accession No 3 Recipient's Catalog No

I

NASA CR- 187585 !

ICASE Report No. 91-48 i

4 Title and Subtttle

HOMOGENIZATION TECHNIQUES AND ESTIMATION OF MATERIAL

PARAMETERS IN DISTRIBUTED STRUCTURES

7. Author(s)

H. T. Banks

D. Cioranescu

A. Das

R. Miller

D. A. Rebnord

9. Pedormtng Organization Name and Address

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

5 Report Date

June 1991

6 Performing Organization Code

8. Performing OrgamzaDon Report No

91-48

10 Work Unit No.

505-90-52-01

11. Contract or Grant No

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supplementaw Notes

Langley Technical Monitor:

Michael F. Card

Final Report

To appear in Proc. 2nd Conf. on

Computational Control, Birkhauser,

(August 1991)

16, Abstract

We discuss the use of homogenization techniques to derive approximate models

with simple geometry for physical models of grids and trusses which have a complex

geometry that gives rise to computational difficulties. Our presentation is in the

context of inverse or parameter estimation problems for composite material structures

with unknown characteristics such as stiffness and internal damping. We present the

necessary theoretical foundations for this approach and discuss comparison of modal

properties of the resulting homogenization model for a two-dimensional grid structure

with modal properties observed in experiments with this grid.

17, Key Words(Suggestedby Author(s))

estimation, material parameter, grids,

homogenization

19. SecuriW Cla_if. (of this repot)

Unclassified

18. Distribution Statement

64 - Numerical Analysis

66 - Systems Analysis

Unclassified - Unlimited

_. SecuriW Cla_if. (of this pa_l 21. No. of pa_s

Unclassified 24

22. Price

A03

NASA FORM 1626 OCT 86

NASA-Langley, 1991

_,,cCc-D,N_ PAGE BLANK NOT F!LMED