Mesh Density Effects on Finite Element Model Updating
M. Imregun and D. J. Ewins
Imperial College of Science, Technology and Medicine
Mechanical Engineering Department
London SW1 2BX
United Kingdom
ABSTRACT.
This paper investigates the
feasibility of updating finite element models with
meshes and examines the
coarser-than-usual
consequences of using such meshesfrom a model
updating viewpoint. A channel-section structure
with relatively simple geometry was modelled
using three different mesh densities and the
resulting analytical models were correlated with
the experimental
model
obtained from
measurements on an actual test structure. All
three finite element models were then updated
using the inverse eigensensitivity method and the
success of the updated models was investigated
with particular
emphasis on mesh density.
Although the model with the highest mesh density
produced the best updated model, because of the
good initial agreement between the analytical
and experimental modal models, it was concluded
that the coarse mesh route was worth pursuing,
especially in the light of the prohibitive
computational
requirements associated with
large finite element models.
NOMENCLATURE
[A]
[AM]
[M]
[K]
[AK]
[S]
: Modal parameter variation vector
: Mass correction matrix
: Mass matrix
: Stiffness correction matrix
: Stiffness matrix
: Sensitivity matrix
A
e
Ne
P
X
: Analytical
: Element
: Number of individual finite elements
: Correction factor
: Experimental
1. INTRODUCTION
Growing demands for quality and reliability in all
types of structure and machine generate a need
for the ability to predict the dynamic
characteristics of engineering structures. This
need can only be satisfied by the availability of
suitable mathematical representations of the
structures under study and state-of-the-art finite
element technology is addressing the problem
with varying levels of success trl. Although
structural dynamics models resulting from such
representations need to be validated against
measured data, this is not an end in itself since an
indication of poor (or even reasonable)
performance of the mathematical model is not of
much use to the design engineer who needs to
know the dynamic response of the structure with
good accuracy. Hence, the primary requirement
is the correction of the finite element model in
the light of measured data, a process known as
model updating [al.
One of the prerequisites in model updating is an
initial closeness of the analytical and experimental
models. In finite element modelling, the model
1372
quality is very often associated with the mesh
density, an axiom which remains unchallenged so
far. However, when the objective is to correct or
to update the initial finite element model of a
structure, there are enormous computational
advantages in keeping this initial model small
while retaining a basic correlation with the
experimental model. Indeed, large finite element
models are often vety difficult to update because
of the relative insignificance of the individual
elements which must be corrected one by one.
This is especially true for both response and
sensitivity-based methods which assign and
compute correction factors for each elemental
matrix. One is then confronted with the dilemma
of (i) updating a large (and hopefully fullyconverged)
model
and
accepting
the
computational consequences, or (ii) starting with
a small (and probably not fully-converged) model
and reducing the computational requirements by
several orders of magnitude. If both approaches
above lead to comparable updated models in
terms of representing the dynamic behaviour of
the actual structure, then there will be a very good
argument in favour of choosing route (ii) because
of its obvious computational advantages. In any
case, if the finite element model is going to be
corrected in a global sense by multiplying its
individual elements by arbitrary correction factors,
the initial discretization, determined by the mesh
size, becomes of secondary importance since the
updated model will not have all the discretization
properties of the initial one. Therefore, the main
purpose of this paper is to explore route (ii) and
to investigate the possibility of updating finite
element models with relatively coarse mesheswith
the eventual aim of tackling very large-size
problems.
element sense and the investigation will focus on
whether it is possible to update them in such a
way such that their performance is comparable to
that of the updated model obtained from the third
finite element model.
To make the exercise more realistic, it is also
proposed to use true measured FRF data, rather
than simulated test data; a feature which makes
any updating attempt much more taxing. Finally, it
was decided to use a sensitivity-based updating
algorithm rather than a response-based one but
this choice is rather arbitrary and work is on the
way to repeat the same updating exercise using
the latter technique.
2. BASIC THEORY
The Inverse Eigensensitivity Method (IESM) has
been the subject of numerous research papers on
model updating 131and only a summary will be
given here.
Let [MA] and [KA] be the mass and stiffness
matrices of the FE model, and [Mx] and [Kx]
those of the real structure. One can now write
[Mxl = [MAI + [AMI
(1)
Kxl = [&I + [AK1
where matrices [AM] and [AK] represent the
mass and stiffness error matrices between the two
models. These can be defined as follows:
To this end, it is proposed to have three different
finite element models of increasing mesh density
to represent a channel-like structure and to try to
update each of these models using the same
updating algorithm. Unlike the third one, the first
two models are not fully-converged in the finite
1373
[AM]=
2 pMc[M,]
e=,
[AK1= 2 P& [K, 1
e=,
(2)
Ne is the number of (super) elements in the FE
model, and pMe, pDe and pKe are the element
mass and stiffness correction factors for
individual finite elements. It is also possible to
include damping in a similar fashion.
being estimated to be E=206 GN/m* and p =7860
kg/m3. The structure is not continuous but is
made of several (sometimes overlapping) plates
joined together by numerous nominally-identical
spot welds. (Fig. 1).
By defining a number of design variables and
computing the sensitivity of the eigensolution to
changes in these parameters together with an
objective function to minimise the differences
between the current and target eigensolutions, it
is possible to write a matrix sensitivity equation
of the form:
The structure was tested in a free-free condition
using impact testing. Various support conditions
were tried to minimise the suspension and it was
eventually decided to suspend the test piece from
one comer by drilling a very small hole. A total
of 120 FRFs were measured along the channel
length in both cross-section axes directions in the
O-1600 Hz frequency range with a 2 Hz
frequency increment.
(3)
the iterative solution of which yields the required
correction factors.
One problem associated with equation (3) is the
inherent ill-conditioning of the sensitivity matrix
since the magnitudes of the eigenvector
derivatives are usually much smaller than those
of the eigenvalue derivatives. Therefore the
sensitivity matrix must be balanced prior to the
solution, a procedure which may well lead to
diminish the effect of the eigenvalues which are
suppressed numerically. In addition, some
introduce constraint
sensitivity
algorithms
equations of the form:
Cti pi = 0
(4)
where CXiis the penalty coefficient which
expressesthe level of confidence in the i*
element variable pi.
3. MODAL ANALYSIS
CORRELATION
The measured FRFs were analysed using a global
curve-fitter and the mode shapes were identified
directly. Most of the modes were found to be real
but some exhibited highly complex behaviour,
probably due to several spot welds joining the
various plates together (Fig. 2). The structure
was found to be lightly damped, the structural
damping values ranging between 0.5 and 1.7%
for the first ten modes.
3.2 Finite Element Modelling
The channel structure was modelled using the
finite element program ANSYS and 4-node
quadratic plate elements (STIF63) were used
throughout, The joint model was restricted to the
overlap of the two plates which were assumed to
be connected in a perfectly rigid fashion. The
discontinuity in the structure was also included in
the model by two overlapping plates.
Table 1
Details of the FE models used
AND
3.1 Modal Test and Analysis
The overall dimensions of the channel structure
are 570 x 90 x 1 mm, the material properties
1374
3.3 Correlation of Finite Element and Modal
Test Results
Table 2
The predicted natural frequencies (Hz) of the
M 1 Description
1 Small
1 MAC
1 Medium
1 MAC
I
Law?
1
The degree of correlation between the
experimental model and the three finite element
models was investigated in several stages:
(i) Comparison of natural frequencies (Table 3),
(ii) MAC calculations (Fig. 3),
(iii) 45O mode shape plots (Fig. 4), and
(iv) FRF overlays (Figs. 5).
I
10 I
I
As one would expect, the agreement between the
FE and experimental models increases with
increasing mesh density and the small FE model is
not a particularly good representation of the test
structure.
I
-
1999.3
16,
After experimenting with different mesh sizes, it
was decided to use three different meshes with an
increasing number of nodes and elements (Table
1). To avoid errors due to matrix reduction, the
individual finite element mass and stiffness
matrices were read from ANSYS and a full
eigensolution was performed by reassembling
these matrices, The results are summarised in
Table 2 which also includes MAC values for the
large-medium and the medium-small FE model
pairs.
4. UPDATING
THE FE MODELS
4. 1 Numerical Considerations
As can be seen from Table 2, the natural
frequency agreement between the models is not
very good although the basic vibratory behaviour
is predicted reasonably well in the sense that the
first two close modes are followed by a cluster of
again close modes with higher natural
frequencies. This behaviour is mainly due to the
lack of symmetry because of the discontinuity
along the channels length which is modelled by
two overlapping plates. It should also be pointed
out that almost all mode shapes involve a large
amount of torsional motion and hence the natural
frequencies are very sensitive to changes in the
cross-sectional properties.
Program MODULATE 141was used for updating
the three finite element models described above.
Individual element mass and stiffness matrices,
together with the connectivity information, are
read into the program and the global mass and
stiffness matrices are assembled internally at each
iteration step. In this particular study, the
correction factors were computed using both
eigenvalue and eigenvector sensitivities and it
was noted that the IESM updated model was not
unique since the results dependedon:
(i) the number of modes kept in the sensitivity
analysis,
(ii) the balancing of the sensitivity matrix,
(iii) the selection of elements to be included in the
formulation, and
(iv) the use of penalty functions.
The convergence of the algorithm for all three
cases was found to be dependent on the number
1375
Once an acceptable combination of parameters (i)
to (iv) was determined, it was possible to
compute iteratively a set of modification factors
which minim&d the difference between the
experimental and theoretical models. The main
parameters are summarised in Table 4.
of modes kept in the analysis. Generally speaking,
the convergence was found to be faster and more
stable when a small number of modes was used.
In all cases considered, it was found necessaryto
balance the sensitivity matrix as this contains
elements spanning several orders of magnitude.
However, as there are many ways to balance a
matrix (by columns, by rows or by using a
combination) there will be at least as many
updated models, each corresponding to a
particular balancing case. Here, it is proposed to
focus attention on the best results only since
matrix balancing is very case-dependent and
general conclusions cannot easily be drawn.
It was noted that some of the elements
contributed little to the variation of the correction
factors and so these were excluded from the
analysis for numerical reasons.However, it should
be noted that there is little physical or modelling
basis for adopting this approach since there is no
reason why modelling errors should not exist at
insensitive parts of the structure. In any case, the
convergence of the sensitivity algorithm was
found to be very dependent on the number of
elements kept in the analysis and, in all the cases
studied, no convergence was achieved when all
the elements were kept in the analysis.
Table 4
Key parameters for IESM updating
Modes used
Balancing
Elements kept
Penalty func.
Also, the modification factors were constrained to
be as small as possible by using a penalty function
technique, a safeguard which prevents them from
becoming unrealistically large and which reflects
the degree of confidence in the original FE model:
in general terms, the more weight that is given to
the penalty constraints, the more trust is placed in
the FE model. In cases where the very first
iteration yields excessively large modification
factors and the resulting system matrices are not
positive-definite, the use of penalty constraints
may shift the minimisation process to a different
solution path with faster convergence, an
acceptable course of action since the solution is
not expected to be unique.
1376
small
Model
10
Column
Medium
Model
10
cohlmn
Large
Model
10
70%
60%
60%
YES
NO
NO
Cd&
row
4.2 Comparing the Updated Models
The success of the updated models was assessed
by defining and comparing two parameters.
Global frequency error.
M0dC.S
IIAfll = [ z((Target fi - Input fi)/Target fj )2] O.5
:-I
Global eigenvector error
Modes Co-ords
IIA~II = [ X
i=l
C (Target @ij - Input ~ij
j=l
)
2
]
0.5
The percentage improvement of the updated
model was defined by yf and j’$, where
y = (Ilh II initial - IIA II updated) / llA ‘1 initial x 100
and the results are listed in Table 5.
The natural frequencies corresponding to all three
cases above are listed in Table 6. The MAC
values between the experimental model and the
three updated models are given in Table 7
As the final and most taxing comparison, the
measured, initially-predicted and updated FRFs
are plotted in Fig. 6 for all three finite element
models. At this stage it must be remembered that
the IESM actually updates the synthesised FRF
which is regenerated using identified modal
parameters rather than measured (raw) data and
much depends on the quality of modal analysis
which produces the reference (or target) set of
modal parameters.
4.3 Computational
Requirements
The updating exercise was conducted on a variety
of computers ranging from 486.based PCs to a
Cray Y-MP8, and including DEC 5100 and IBM
RS/6000
workstations.
The
main
two
requirements of in-core memory and dedicated
CPU time are summarised below in Table 8
where the latter has been normalised with respect
to the IBM RS/6000 model 530 processor. Four
iterations were needed for the small and medium
models while five iterations were used for the
large model. It should also be noted that Table 8
does not include the CPU effort required for the
trial runs in order to determine the key parameters
such as the number of modes, the number of
elements kept in the analysis and the balancing
method to be used.
Table 8
Computational requirements
1 Small
1 Medium 1 Large
1
1377
5. CONCLUDING
REMARKS
(i) An inspection of the updated FRFs reveals that
the large FE model produces the best updating
results, closely followed by the medium model.
Also, The small model results are not
disappointing at all and they certainly exhibit a
very marked improvement over those obtained
from the initial FE model. The immediate
conclusion is that it is possible to update any FE
model, including those with coarser meshes,
provided there is a basic initial agreement.
(ii) When both eigenvalue and eigenvector
sensitivities are used, the computational
requirement rapidly becomes prohibitive with
increasing model size. There are many instances
for which it is not possible to update existing
models unless the mesh size is reduced
significantly. The present feasibility study
originates from the need to explore such an
approach.
(iii) An immediate application of (ii) can be found
in the common practice of using the same finite
element model for both static and dynamic
calculations. In the former type of analysis it is
important to capture stress concentrations and
hence particular emphasis is placed on sudden
geometry changes. However, these features are
not all that important in a dynamic analysis where
the overall stiffness and mass play a much more
important role and hence there is a case for
simplifying the static analysis model
(iv) The question of which model to choose as the
starting point invites a much more fundamental
question: the purpose for which the updated
model is going to be used. As there are so many
different applications, it is not possible to
generalise or to make recommendations,
However, the coarse mesh route may well be an
attractive option for a number of cases.
6. ACKNOWLEDGEMENTS
2. Imregun M., and Visser W. J.
A Review of Model Updating Techniques,
Shock and Vibration Digest., Vol. 23, No 1,
~~9-20, Jan 1991
The authors gratefully acknowledge the
contribution of the Nissan Motor Co. for
sponsoring some of the work and for providing
the test specimen. The authors also thank Dr. A.
S. Nobari and Mr. N. Imamovic for providing
most of the numerical data presented in this paper.
3. Zhang, Q. and Lallement, G.
A Complete Procedure for the Adjustment of a
Mathematical Model From Identified Complex
Modes,
Proc. IMAC 5, April 1987.
7. REFERENCES
4. MODULATE
Users Guide to Version 1.0,
Imperial College Analysis,
Software, September 1993
1. Ewins, D. J. and Imregun, M.
State-of-the-Art Assessment of Structural
Dynamic ResponseAnalysis Methods (DYNAS),
Shock and Vibration Bulletin, Vol. 56 (I), 1986
Testing
and
Table 3
Measured and predicted natural frequencies (Hz) of the channel structure
Mode
Exp.
1
2
3
4
c
J
393.2
397.9
752.2
736.2
OAA
1
0”U.I
51; :.tl 49c..5.0
1044.4
1 962.6
6
7.
8
9
10
828.9
841.4
919.9
960.9
479
r;
884.3
-
I
Small
Mode,
898.5
Error
m
7”
,
l-30.4
l-24.7
--.~
I-38.8
(-30.7
-12.2
-6.7
1
I,
1
(
485.3
493.1
848.3
858.3
861.7
860.7
973.2
1114.6
992.3
I
Medium
\“^rl^l
I”I”UGI
24.2%
I
,
Error
I%
7”
j-23.4
l-23.9
I-12.8
I-16.6
-7.7
-3.8
-15.7
-21.2
-: 3.3
14.3%
,
Large
l”,.-l..,
I”I”“szI
,
Error
0%
I”
) 398.3
I 404.5
) -1.3
I -1.7
( 819.0
1 -8.9
1 830.9
808.3
846.~4
926.6
999.2
(1030.3
I
-
j-12.9
( -1.0
-2.1
-10.1
1 -8.6
1 -3.3
I
I
5.5%
I
Table 5
Updating parameters for the three finite element models
Stlltdl
IlAfll
Initial FE
0.85
Updated
0.40
l1Al$ll
26.2
26.3
III
Medium
y
53%
0%
Initial FE
0.54
21.1
Updated
0.01
21.2
1378
Large
y
98%
0%
Initial FE
0.13
updated
0.0
y
99%
26.2
22.0
14%
Table 6
Natural frequencies (Hz) predicted by the updated models
Exper.
Mode
I
10
Error
Small
960.9
-
I
Medium
4.4
I 1.9%
(14.3%)
-
/ 979.5
Average error
(Initial error)
Error
5.1%
(24.2%)
Large
Error
Yb0.l
978.9
I 0.U
0.0%
(2.9%)
Table 7
MAC values (%) between experimental and updated models
(The last row is the average value for that column. The correlating mode pairsareshown in brackets)
36
Large
MC:dium
Small
39
I
--
\.,.,
,
41
42
(8,ll)
(-~,t(,-’
51
146
,’ 4b
*I
(8,9)
(9,8)
’I43~~
j 72
50
I
Fig 1. The test structure
1379
i8.9,
(10.11)
59
(28
I65
I
I
I
(8,9)
(10.11)
-b-l
I
I
Fig 2. The experimentally-determined
1380
mode shapes
03)
(a)
(cl
Fig. 3 The MAC values for (a) small (h) medium and (c) large models
Fig. 5 Measured
and initially-predicted
1381
point inertances
”
1382