REDUCED ENRICHED BASIS TO REPRESENT
AN OBSERVED BEHAVIOR
V. Feuardent, M. Reynier
Laboratoire de Mecanique et Technologie
ENS Cachan I CNRS I Universite Paris VI
61, av.du President Wilson, 94235 CACHAN ex FRANCE
ABSTRACT. The improvement of finite element model by
m,n
updating procedures uses experimentally obtained dynamics
dimension of the truncated modal basis
of the sub-structure 1, 2
data. But the mechanical quantities are only partially
(!)
measured on the structure, and their extension to the whole
{ }
column vector
structure needs to be appropriatedly build. to allow the
[l
matrix
correction of the modelling errors .. As the models of complex
[ ] t{ } t transpose of a matrix, vector
structures have commonly a great dimension, the numerical
<1>
implementation is easier to carry out if the mechanical
quantities are described on a reduced basis. A simply
u
v
truncated modal basis associated with the initial finite
natural frequency for the whole structure
free mode of the truncated modal basis
displacement field on the whole structure
Vectors belonging to
V,
dynamics contribution to the
displacement field U
space of displacement fields
element model appear to be ill suited to describe the
mechanical quantities we look for. Using a modified Mac
space generated by the truncated modal basis of the
Neal's approach, we propose to enriche the truncated modal
sub-structure i
basis by static
contributions, taking the location of the
v
space generated by the truncated modal basis of
shakers and sensors. into account It appears to be well-
both the two sub-structures and verifying the
suited to the description of stress and displacement
constraint of the displacement gap.
w
quantities on the whole structure.
statics contribution of the displacement field U. Its
kinetic energy is negligible regarding the strain one.
w
NOMENCLATURE
space generated by the statics contributions and
verifying the constraint of the displacement gap and
subscript denoting the sub-structure
the orthogonality constraints between Wand V.
504
T
interface stress distribution
[~'
2.1 Mac Neal's substructuring technique
O]
K2
matrix where K1 (respectively K2) is the
Let Q be a bounded domain corresponding to the structure
stiffness matrix of the sub-structure 1 (resp. 2)
with the boundary
an .
Let us specify the boundary
a, Q
H
Hooke's operator of elasticity
conditions with the two complementary subsets
< ,>
scalar product related to the strain energy
o Q. Let the displacement field be given on a, Q. And let us
( ,)
scalar product related to the kimetic energy
introduce :
and
2
11 = {!l ,!1' regular, !1' la
Q
1
= 0} ..
The behavior is assumed to be elastic and we do not take the
1.
INTRODUCTION
damping into account. Eigen frequencies and associated
eigen modes of the structure are obtained by solving the
If we try to update finite element models with a great number
problem:
of degrees of freedom using test results, we have to describe
mechanical quantities such as displacement and stress
To find ( ffi, U) on Q, where ffi is a natural circular
frequency and !1 E 11 an associated modal displacement
fields. A priori this description does not require the use of the
(U
-:f.
0 ),
in such a way that they make
finite element model dimension. But the truncated basis of the
(!)2
,!J.') = t(!J.' ,!J.')-2(1!' ,_!::!')
initially given finite element model is often too far from the
\lf(W'
eigen basis of the model we look for : in previous works, we
stationary, with :
proposed to describe stress and strain using a truncated
(!:!, !:!') =
eigen basis [1]. We have shown that this choice can
ftr[ HE(!:!)E(!:!') ]dn
(1)
(!:!·!:!')=I P!:!·!:!' dQ(2)
n
Q
constitute a filter, consequently a loss of accuracy can
( tr[ ) : trace operator, £ is the strain operator in the
occur.
Classical strategies which use a truncated eigen basis
completed with static modes [2], [3], [4]. [5] have proposed to
assumption of small perturbation s. If we divide the Q
domain into two complementary sub-domains
enrich the truncated eigen basis with statics vectors. Here,
we choose a strategy based on the test. The method comes
from a sub-structuring technique close to Mac Neal's strategy
where
r 12
n,
and Q2
defines the interface. We name : !.!.1 the
displacement on
n,, and 1!2 the displacement on
Q2.
[6] in which we introduce a stiffness interface.
(3)
We use Mac Neal's method modified as proposed in [7] and
developped in
[1 0] To build a reduced basis for a finite
The Lagrange multiplyer
element model, one of the sub-structures is reduced to a
lfl( m' ' U' 'r ) =
point. We show that this strategy enables us to obtain an
1
m' 2
{I}
introduces an interface force,
2(U' ,U')- 2(U' ,U')-
efficient static contribution for each measurement point. We
fr (u
(4)
1
-U' 2)ds
rl2
test it for the numerical implementation of updating problems.
(!J., !t) =
2.THE
CHOICE
OF
A
Jtr[ H£(!11)E(!ld )dn + Jtr[ H£(!1 2)£(!1 2')]dn,
n,
SUB-STRUCTURING
n2
(!J.,!J.')= fp!ll!ll'dQ+ fp!12!12'dQ.
n,
n,
APPROACH
505
For each
we look for a displacement field
{!)
U such that :
U=V+W
- where
We are going to see that
(5)
y_ is built using the
modal modes of the sub-structures
=0
is related to an eigen problem,
and \jf w is related to a static problem with an imposed
displacement.
and W is the statics contributions.
( U 1 - U 2 )\r"
\If v
==>
- the statics problem:
(W,- W2l, = (1C2 -I,lz
=A
To find
which introduces
WE W such that it makes:
ttl w (w ,r ,a;,aD =
the gap A , a given displacement.
We choose Mac Neal's sub-structuring approach ; first, a
1
truncated eigen basis with a free interface is computed for
2
each sub-structure 1 12 :
-on Ql' the modes
(~: '····~~)
generates a
~m
- on Q2' the modes
(~~ ' ... ' ~~)
generates a
v;
space,
f
(w, w)- r(w,- W 2 -d)ds (8}
r1z
stationary
- the eigen problem :
To find ( ro, V),
V
E
V, such that they make station nary :
space.
We denote the spaces :
V
Yz
= {.YoF-0 and regular on Q,
E o/2n on n2,
y 1 E~m
on
Q 1,
(YI- Yz)ln" = -£H
and we are interested in finding statics contributions having
the following orthogonality properties:
W
{WoF-0
and
regular
on
Q,
I is a stress distribution on r 12 , and it is a linear function of
Wlan=O,
I
the displacement discontinuity A ; and the use of \jf v leads to
solve an eigen problem.
'v'ism,(w 2 .~Uin
2 =0 'v'jSn},
We impose the orthogonality constraint using Lagrange 's
multipliers (
A is a given displacement defined on 1 12 .
111'
(w· r
al, a~) , and obtain :
'I' w - • - ,
As
!:[ 1 =.Y 1 + W 1 on Q 1
,
!:[ 2
=y 2 + W 2
ai.ai
)=l(w
I· 2
2 - 'w)-
on Q 2 , the
Jr(w
-A)cts
- -1 -W
-2
-
nz
orthogonality properties give:
(y,w) =O
The value of the multiplyer
(6)
I is directly connected to the
choice of the truncated eigen basis. I and !l show an
The following partition of
'II= 'lfv + 'l'w
'1'( ffi' , .!:!:' ,!') is possible :
additional stiffness on
r 12 .
contribution of the high modes.
(7)
506
W
corresponds
to
the
where [P] is a projection matrix on the boundary
2.2 The Finite Element discretization
r 12 . I
is a
linear function of 8.
The functions \jf v and \jf w are written :
'V' w =
!{:~}[:' :J{:~}-[[P]{:~}- {A}J{T}
+
+
3.1 A basis associated to the tests
n
{ ¢Ii}t[MI
L.a{
~]{~}
i=l
0
0
L.a~
m
j=l
{
o}t[o
j
¢2
0
3. THE ENRICHMENT TECHNIQUE
If we reduce one of the sub-domains ( n 1 or Q 2 ) in such a
(11)
~J{~J
way that it becomes a sample point where we impose a given
displacement, we obtain a reduced basis associated with this
point. We
build a reduced basis associated with the
excitation points. W becomes the statics responses of an
elastic problem where the displacement is imposed on the
~,]{~}
excitation points. '1... is built using the truncated basis of the
whole structure.
~,]{~}
3.2 Example : a resulting enriched basis
(12)
On figure 1 we have an embedded beam discretized
into 70 finite elements of beam.
Equations (11) and (12) lead to solve :
[d :J:,}-1t {T(d)}
1
+
~0
1
~a~[~ ~]{~}
___ I
-figure 1-Beam structure (140 degrees of freedom)
(13)
The first three .U.-vectors are shown on figure 2. Ten eigen
modes are used to build theY-vectors.
[[~' ~J{~}-~{T(A)jl
-m 2
[M,0 ~,]{~} ={~}
[PJ{::}={A}
(16)
[PJ{~:}=-{A}
w
v
(14)
-U=V+W
--
(17)
507
11
35
em=0,2
23
48
ek=-O,I
70
elements
- figure 3 - structure
On the figures 4, 5, 6, we show the results of the location
stage obtained by the local error measure on the constitutive
relation (8].
-.
-1.!5
,'
0.8
0
elements
0.6
lli
0.4
'
0.2
,·.
0
'
,.
··'
0
0
10
20
30
40
50
elelllfnts
60
-figure 4- location stage ( complete basis )
The use of the complete modal basis enable us on figure 4 to
elements
verify the visibility of the errors by using only the first five
-figure 2- description of the vectors of the enriched basis
"experimental" modes.
On figure 5, the location of the
stiffness error is approximative when the computation uses
3.3
Updating problem
validation of the use of
only a truncated modal basis of ten computed modes, and
el
the enriched basis
T
1
The model is described on figure 3,. The structure is
0.B
Base modale tronqlltt
(10 premiers vecteurs propres)
discretized into 70 finite elements of beams. A 20% simulated
error
0.6
is introduced on the elementary mass of the 11th
is introduced on the
o.4
elementary mass of the 48 th element, The "experimental"
0. 2
element,
A
10% simulated error
point is the 23 th degree of freedom. Only five modes are
10
assumed to be measured.
20
_rfh
30
AlbJ
40
u50 ~
elements
-figure 5- location stage (truncated modal basis)
508
«)
the location of the mass error is loss. If the truncated modal
basis is enriched with one static modes, the location of the
[2): Dhatt G., Talbot M.
"La technique d'espace reduit pour /'analyse dynamique des
modelling errors becomes very good.
El
structures" Calcul des structures et intelligence artificielle,
T
Pluralis (Paris), vol. 3, 17-29, 1990.
0.8
Base vecteurs U
(MacNeal)
[3]: Destuynder Ph.
0.6
Remarks on Dynamic substructuring", European Journal of
0.4
Mechanics, NSolids, vol.8, W3, 1989.
[4]: Leger P.
20
30
40
50
60
elements
-figure 6- location stage (enriched basis)
Methodes de superposition modale pour !'analyse dynamique
des structures non lineaires
Revue europeenne des
elements finis, vol. 1, W2, 115-135, 1992.
4. CONCLUSION
[5]: Bricout
We have here proposed a strategy to build a basis allowing to
reduce the dimension of updating problems. efficiently A
J.
N.,
Mercier
F.,
Ladeveze
P.,
Reynier M.Different approaches of LMT Cachan FEM
updating method, ESA, CNES, Paris, Juin 1993.
modified Mac Neal's approach has been used, it assumes a
stiffness on the boundarie between the sub-domains. The
[6]: Rozenblum G.
enriched basis we have proposed is strongly associated to
Modal
the tests. The great advantage of this choice is to obtain a
method. Theoretical basis", 48, 139-154, North-Holland,1985.
synthesis
:generalization
of
Mac
Neal's
description of the mechanical quantities well suited with the
visibility coming from the tests condition. This enrichment
[8] Ladeveze P., Reynier M. and Maia N.
Error on the constitutive relation in dynamics : theory and
strategy is now used by the CNES, the French National Space
application for model updating", 1ST Lisbonne, Proc. Second
Agency, Examples of these spatial applications can be found
Int. Symposium on Inverse Problems in Engineering
in [9].
Mechanics ISIP'94, Paris, pp 251-256, 2-4 Nov. 1994.
5.
ACKNOWLEDGEMENTS
[9) : Feuardent V.
The authors are grateful to the CNES (French National Space
PhD thesis, Amelioration des
Agency ) for providing the financial support.
application aux structures spatiales, These de I'Ecole
modeles par recalage :
Normale Superieure de Cachan, Oct. 1997.
6. REFERENCES
[10] : Feuardent V., M. Reynier
[1]: Ladeveze P., Nedjar D., Reynier M.
Reduced Basis for structure dynamic model updating, Proc.
Updating of Finite Element Models Using Vibration Tests",
of the 2nd Int. Cont. on Inverse Problems in Engineering:
AIAA Journal, vol. 32, number 7, 1485, Jul. 1994.
theory and Practice.9-14 Port aux Roes,- Le croisic,
Engineering Foundation Ed., June1996.
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