FINITE ELEMENT MODEL UPDATING USING RESPONSE SURFACE METHOD
Tshilidzi Marwala
School of Electrical and Information Engineering
University of the Witwatersrand
P/Bag 3, Wits, 2050, South Africa
t.marwala@ee.wits.ac.za
This paper proposes the response surface method for finite element model updating. The response
surface method is implemented by approximating the finite element model surface response equation
by a multi-layer perceptron. The updated parameters of the finite element model were calculated using
genetic algorithm by optimizing the surface response equation. The proposed method was compared
to the existing methods that use simulated annealing or genetic algorithm together with a full finite
element model for finite element model updating. The proposed method was tested on an unsymmetrical H-shaped structure. It was observed that the proposed method gave the updated natural frequencies and mode shapes that were of the same order of accuracy as those given by simulated annealing
and genetic algorithm. Furthermore, it was observed that the response surface method achieved these
results at a computational speed that was more than 2.5 times as fast as the genetic algorithm and a full
finite element model and 24 times faster than the simulated annealing.
proximation model, which traditionally is a polynomial
and therefore is less expensive to evaluate. This makes
RSM very useful to FE model updating because optimizing the FE model to match measured data to FE model
generated data is a computationally expensive exercise.
Furthermore, the calculation of the gradients that are
essential when traditional optimization methods, such as
conjugate gradient methods, are used is computationally
expensive and often encounters numerical problems
such as ill-conditioning. RSM tends to be immune to
such problems when used for FE model updating. This
is largely because RSM solves a crude approximation of
the FE model rather than the full FE model which is of
high dimensional order. The multi-layer perceptron
(MLP)6 is used to approximate the response equation.
The RSM is particularly useful for optimizing systems
that are evolving as a function of time, a situation that is
prevalent in model-based fault diagnostics found in the
manufacturing sector. To date, RSM has been used extensively to optimize complex models and processes7,8.
In summary, the RSM is used because of the following
reasons: (1) the ease of implementation that includes low
computational time; (2) the suitability of the approach to
the manufacturing sector where model-based methods
are often used to monitor structures that evolve as a
function of time.
FE model updating has been used widely to detect
damage in structures9. When implementing FE updating
methods for damage identification, it is assumed that the
FE model is a true dynamic representation of the structure and this is achieved through FE model updating.
This means that changing any physical parameter of an
element in the FE model is equivalent to introducing
damage in that region. There are two approaches that
are used in FE updating: direct methods and iterative

Introduction
Finite element (FE) models are widely used to predict
the dynamic characteristics of aerospace structures.
These models often give results that differ from the
measured results and therefore need to be updated to
match the measured data. FE model updating entails
tuning the model so that it can better reflect the measured data from the physical structure being modeled1.
One fundamental characteristic of an FE model is that it
can never be a true reflection of the physical structure
but it will forever be an approximation. FE model updating fundamentally implies that we are identifying a
better approximation model of the physical structure
than the original model. The aim of this paper is to introduce updating of finite element models using Response Surface Method (RSM)2. Thus far, the RSM
method has not been used to solve the FE updating problem1. This new approach to FE model updating is compared to methods that use simulated annealing (SA) or
genetic algorithm (GA) together with full FE models for
FE model updating. FE model updating methods have
been implemented using different types of optimization
methods such as genetic algorithm and conjugate gradient methods3-5. Levin and Lieven5 proposed the use of
SA and GA for FE updating.
RSM is an approximate optimization method that
looks at various design variables and their responses and
identify the combination of design variables that give the
best response. The best response, in this paper, is defined as the one that gives the minimum distance between the measured data and the data predicted by the
FE model. RSM attempts to replace implicit functions
of the original design optimization problem with an ap*

Associate Professor

1

methods1. Direct methods, which use the modal properties, are computationally efficient to implement and reproduce the measured modal data exactly. Furthermore,
they do not take into account the physical parameters
that are updated. Consequently, even though the FE
model is able to predict measured quantities, the updated
model is limited in the following ways: it may lack the
connectivity of nodes - connectivity of nodes is a phenomenon that occurs naturally in finite element modeling because of the physical reality that the structure is
connected; the updated matrices are populated instead
of banded - the fact that structural elements are only
connected to their neighbors ensures that the mass and
stiffness matrices are diagonally dominated with few
couplings between elements that are far apart; and there
is a possible loss of symmetry of the systems matrices.
Iterative procedures use changes in physical parameters
to update FE models and produce models that are physically realistic. Iterative methods that use modal properties and the RSM for FE model updating are implemented in this paper. The FE models are updated so that
the measured modal properties match the FE model predicted modal properties. The proposed RSM updating
method is tested on an unsymmetrical H-shaped structure.

and stiffness matrices. The frequency response functions (FRFs) are defined as the ratio of the Fourier transformed response to the Fourier transformed force. The
FRFs may be expressed in receptance and inertance
form. On the one hand, receptance expression of the
FRF is defined as the ratio of the Fourier transformed
displacement to the Fourier transformed force. On the
other hand, inertance expression of the FRF is defined as
the ratio of the Fourier transformed acceleration to the
Fourier transformed force. The inertance FRF (H) may
be written in terms of the modal properties by using the
modal summation equation as follows10:
N
− ω 2φkiφli
H kl ( ω ) = ∑
(3)
2
2
i =1 − ω + 2ζ i ω i ωj + ω i
Equation 3 is an FRF due to excitation at position k and
response measurement at position l, ω is the frequency
point, ω i is the ith natural frequency, N is the number of
modes and ζi is the damping ratio of mode i. The excitation and response of the structure and Fourier transform method10 can be used to calculate the FRFs.
Through equation 3 and a technique called modal analysis10, the natural frequencies and mode shapes can be
indirectly calculated from the measured FRFs. The modal properties of a dynamic system depend on the mass
and stiffness matrices of the system as indicated by
equation 2. Therefore, the measured modal properties
can be reproduced by the FE model if the correct mass
and stiffness matrices are identified.
FE model updating is achieved by identifying the
correct mass and stiffness matrices. The correct mass
and stiffness matrices, in the light of the measured data,
can be obtained by identifying the correct moduli of
elasticity for various sections of the structure under consideration1. In this paper, to correctly identify the
moduli of elasticity of the structure, the following cost
function that measures the distance between measured
data and FE model calculated data, is minimized:

Mathematical Background
In this study, modal properties, i.e. natural frequencies
and mode shapes, are used as a basis for FE model updating. For this reason these parameters are described in
this section. Modal properties are related to the physical
properties of the structure. All elastic structures may be
described in terms of their distributed mass, damping
and stiffness matrices in the time domain through the
following expression10:

[ M ]{ X ' ' } + [ C ]{ X ' } + [ K ]}{ X } = { F }
(1)
where [M], [C] and [K] are the mass, damping and
stiffness matrices respectively, and {X}, {X′} and {X′′}
are the displacement, velocity and acceleration vectors
respectively while {F} is the applied force vector. If
equation 1 is transformed into the modal domain to form
an eigenvalue equation for the ith mode, then10:

 ω m − ω calc
E = ∑ γ i  i m i
ωi
i =i

N

N

(

2


 ...



+ β ∑ 1 − diag( MAC({ φ }icalc ,{ φ }im ))

(4)

)

i

2

( −ω i [ M ] + jω i [ C ] + [ K ]){ φ }i = { 0 }
(2)
th
where j = − 1 , ω i is the i complex eigenvalue, with
its imaginary part corresponding to the natural frequency
ωi, { 0 } is the null vector and { φ }i is the ith complex
mode shape vector with the real part corresponding to
the normalized mode shape {φ}i. From equation 2, it
may be deduced that the changes in the mass and stiffness matrices cause changes in the modal properties of
the structure. Therefore, the modal properties can be
identified through the identification of the correct mass

Here m is for measured, calc is for calculated, N is the
number of modes; γ i is the weighting factor that measures the relative distance between the initial estimated
natural frequencies for mode i and the target frequency
of the same mode; the parameter β is the weighting
function on the mode shapes; the MAC is the modal
assurance criterion11; and the diag(MAC)i stands for the
ith diagonal element of the MAC matrix. The MAC is a
measure of the correlation between two sets of mode
shapes of the same dimension. In equation 4 the first

2

part has a function of ensuring that the natural frequencies predicted by the FE model are as close to the measured ones as possible while the second term ensures that
the mode shapes between measurements and those predicted by the FE model are correlated. When two sets of
mode shapes are perfectly correlated then the MAC
matrix is an identity matrix. The updated model is
evaluated by comparing the natural frequencies and
mode shapes from the FE models before and after updating to the measured ones.

Initial Conditions
Updating
Parameters

Updating
Objective

Updating Space

Generation of surface
response data using
the FE model

Response Surface Method

Functional
approximation

RSM method is a procedure that operates by generating a response for a given input. The inputs are the parameters to be updated and the response is the error between the measured data and the FE model generated
data. Then an approximation model of the input parameters and the response, called a response surface
equation, is constructed. As a consequence of this, the
optimization method operates on the surface response.
This equation is usually simple and not computationally
intensive as opposed to a full FE model. RSM has other
advantages such as the ease of implementation through
parallel computation and the ease at which parameter
sensitivity can be calculated.
The proposed RSM consists of these essential components: (1) the response surface approximation equation; and (2) the optimization procedure. There are
many techniques that have been used for response surface approximation such as polynomial approximation12
and neural networks13. A multi-layer perceptron is used
as a response surface approximation equation6. Further
understanding of different approaches to response surface approximation may be found in the literature14-19.
In this paper, MLP is used because it has been successfully used to solve complicated regression problems.
The details of the MLP are described in the next section.
The second component of the RSM is the optimization
of the response surface. There are many types of optimization methods that can be used to optimize the response surface equation and these include the gradient
based methods20 and evolutionary computation methods21. The gradient based methods have a shortcoming
of identifying local optimum solutions while evolutionary computing methods are better able to identify global
optimum solution. As a result of the global optimum
advantage of evolutionary methods, in this study the GA
is used to optimize the response surface equation. The
manner in which the RSM is implemented is shown in
Figure 1.
In this figure it shown that the RSM is implemented
by following these steps:
1) Setting initial conditions which are: updating parameters, updating objective, which is in equation 4,

Global
optimization

Functional
evaluation at
the optimum
solution on the
FE model

Updating
criteria
satisfied?

N

Replace the worst
surface response
data with the
optimum point and
its FE response

Y
Stop

Figure 1. The flowchart of the RSM. Here N stands for no
and Y stands for yes.
and updating space.
2) The FE model is then used to generate sample response surface data
3) MLP is used to approximate the response surface
approximation equation from the data generated in
Step 2.
4) GA is used to find a global optimum solution.
5) The new optimum solution is used to evaluate the
response from the full FE model.
6) If the optimum solution does not satisfy the objective, then the new optimum and the corresponding
FE model calculated response replaces the candidate
with the worst response in data set generated in Step
2 and then steps 3 to 5 are repeated. If the objective
is satisfied then stop and the optimum solution becomes the ultimate solution.
Step 6 ensures that the simulation always operates in
the region of the optimum solution. The next section

3

sic dimensionality of the data. Models of this form can
approximate any continuous function to arbitrary accuracy if the number of hidden units M is sufficiently large.
The relationship between the output y, representing error
between the model and measured data, and input, x,
representing updating parameters may be written as
follows6:

describes an MLP, which is used for functional approximation.

Multi-layer Perceptron
Multi-layer perceptron is a type of neural networks
which used in the present study. This section gives the
over-view of the MLP in the context of functional approximation. The MLP is viewed in this paper as parameterized graphs that make probabilistic assumptions
about data. Learning algorithms are viewed as methods
for finding parameter values that look probable in the
light of the data. Supervised learning is used to identify
the mapping function between the updating parameters
(x) and the response (y). The response is calculated using equation 4. The reason why the MLP is used is because it provides a distributed representation with respect to the input space due to cross-coupling between
input, hidden and output layers. The MLP architecture
contains a hyperbolic tangent basis function in the hidden units and linear basis functions in the output units6.
A schematic illustration of the MLP is shown in Figure
2.
This network architecture contains hidden units and
output units and has one hidden layer. The bias parameters in the first layer are shown as weights from an extra

M

 d

y k =  ∑ wkj( 2 ) tanh ∑ w (ji1 ) x i + w (j 01 )  + wk( 02 ) 
(5)
 i =1

 j =1

Here, w (ji1 ) and w(ji2 ) indicate weights in the first and second layers, respectively, going from input i to hidden
unit j, M is the number of input units, d is the number of
output units while w (j 01 ) indicates the bias for the hidden
unit j. Training the neural network identifies the weights
in equations 5 and a cost function must be chosen to
identify these weights. A cost function is a mathematical
representation of the overall objective of the problem.
The main objective, this is used to construct a cost
function, is to identify a set of neural network weights
given updating parameters and the error between the FE
model and the measured data. If the training set
D = { x k , t k }kN=1 is used and assuming that the targets t
are sampled independently given the inputs xk and the
weight parameters, wkj, the cost function, E, may be
written as follows using the sum-of-square error func6
tion :

Output Units

N

zM

The sum-of-square error function is chosen because it
6
has been found to be suited to regression problems . In
equation 6, N is the number of training examples and K
is the number of output units. In this paper, N is equal to
150, while K is equal to 1.
Before the MLP is trained, the network architecture
needs to be constructed by choosing the number of hidden units, M. If M is too small, the MLP will be insufficiently flexible and will give poor generalization of the
data because of high bias. However, if M is too large,
the neural network will be unnecessarily flexible and
will give poor generalization due to a phenomenon
known as over-fitting caused by high variance. In this
study, we choose M such that the number of weights is at
most fewer than the number of response data. This is in
line with the basic mathematical principle which states
that in order to solve a set of equations with n variables
you need at least n independent data points. The next
section describes the GA, which is a method that is used
to solve for the optimum solution of the response surface
approximation equation.

Hidden Units

bias
xd

(6)

n =1 k =1

z0
z1

K

E = ∑ ∑ { t nk − y nk } 2

yc

y1

x1

x0

Input Units

Figure 2. Feed-forward network having two layers of
adaptive weights
input having a fixed value of x0=1. The bias parameters
in the second layer are shown as weights from an extra
hidden unit, with the activation fixed at z0=1. The model
in Figure 2 is able to take into account the intrinsic di-

Genetic Algorithms
GA was inspired by Darwin’s theory of natural evolution. Genetic algorithm is a simulation of natural evolu-

4

tion where the law of the survival of the fittest is applied
to a population of individuals. This natural optimization
method is used to optimize either the response surface
approximation equation or the error between the FE
model and the measured data. GA is implemented by
generating a population and creating a new population
by performing the following procedures: (1) crossover;
(2) mutation; (3) and reproduction. The details of these
procedures can be found in Holland21 and Goldberg22.
The crossover operator mixes genetic information in the
population by cutting pairs of chromosomes at random
points along their length and exchanging over the cut
sections. This has a potential of joining successful operators together. Arithmetic crossover technique22 is
used in this paper. Arithmetic crossover takes two parents and performs an interpolation along the line formed
by the two parents. For example if two parents p1 and
p2 undergo crossover, then a random number a which
lies in the interval [0,1] is generated and the new offsprings formed are p1(a-1) and pa.

The next section describes simulated annealing which
is used to update an FE model using a FE model.

Simulated Annealing
Simulated Annealing is a Monte Carlo method that is
used to investigate the equations of state and frozen
states of n degrees of freedom system23. SA was inspired by the process of annealing where objects, such as
metals, re-crystallize or liquids freeze. In the annealing
process the object is heated until it is molten, then it is
slowly cooled down such that the metal at any given
time is approximately in thermodynamic equilibrium.
As the temperature of the object is lowered, the system
becomes more ordered and approaches a frozen state at
T=0. If the cooling process is conducted insufficiently
or the initial temperature of the object is not sufficiently
high, the system may become quenched forming defects
or freezing out in metastable states. This indicates that
the system is trapped in a local minimum energy state.
The process that is followed to simulate the annealing
process was proposed by Metropolis et al.24 and it involves choosing the initial state with energy Eold (see
equation 4) and temperature T and holding T constant
and perturbing the initial configuration and computing
Enew at the new state. If Enew is lower than Eold, then accept the new state, otherwise if the opposite is the case
then accept this state with a probability of exp -(dE/T)
where dE is the change in energy. This process can be
mathematically represented as follows:

Mutation is a process that introduces to a population,
new information. Non-uniform mutation22 was used and
it changes one of the parameters of the parent based on a
non-uniform probability distribution. The Gaussian distribution starts with a high variance and narrows to a
point distribution as the current generation approaches
the maximum generation.
Reproduction takes successful chromosomes and reproduces them in accordance to their fitness functions.
In this study normalized geometric selection method was
used22. This method is a ranking selection function
which is based on the normalized geometric distribution.
Using this method the least fit members of the population are gradually driven out of the population. The basic genetic algorithm was implemented in this paper as
follows:

if E new < E old accept state E new
 E − E old  (7)

else accept E new with probability exp new
T


This processes is repeated such that the sampling statistics for the current temperature is adequate, and then
the temperature is decreased and the process is repeated
until a frozen state where T=0 is achieved.
SA was first applied to optimization problems by
Kirkpatrick, et al. 23. The current state is the current
updating solution, the energy equation is the objective
function in equation 4, and the ground state is the global
optimum solution.

1) Randomly create an initial population of a certain
size.
2) Evaluate all of the individuals in the population using the objective function in equation 4.
3) Use the normalized geometric selection method to
select a new population from the old population
based on the fitness of the individuals as given by the
objective function.

Example: Asymmetrical H-structure
An unsymmetrical H-shaped aluminum structure
shown in Figure 3 was used to validate the proposed
method. This structure was also used by Marwala and
Heyns4 as well as Marwala25. This structure had three
thin cuts of 1mm that went half-way through the crosssection of the beam. These cuts were introduced to elements 3, 4 and 5. The structure with these cuts was used
so that the initial FE model gives data that are far from
the measured data and, thereby test the proposed proce-

4) Apply some genetic operators, non-uniform mutation
and arithmetic crossover, to members of the population to create new solutions.
5) Repeat steps 2-6, which is termed one generation,
until a certain fixed number of generations has been
achieved

5

dure on a difficult FE model updating problem. The
structure was suspended using elastic rubber bands. The
structure was excited using an electromagnetic shaker
and the response was measured using an accelerometer.
The structure was divided into 12 elements. It was excited at a position indicated by double-arrows, in Figure
3, and acceleration was measured at 15 positions indicated by single-arrows in Figure 3. The structure was
tested freely-suspended, and a set of 15 frequency response functions were calculated. A roving accelerometer was used for the testing. The mass of the accelerometer was found to be negligible compared to the mass

was run 150 times to generate the data for functional
approximation. The MLP implemented had 12 input
variables corresponding to the 12 elements in the FE
model, 8 hidden units and one output unit corresponding
to the error in equation 4. As described before, the MLP
had a hyperbolic tangent activation function in the hidden layer and linear activation function in the output
layer. The RSM functional approximation via the MLP
was evaluated 10 times (iterations) each time using the
GA to calculate the optimum point and evaluating this
optimum point on the FE model and then storing the
previous optimum point in the data set for the current

A
200mm

400mm
B

600mm
y
Cross-section indicated by line AB
9.8mm
x
32.2mm

Figure 3. Irregular H-shaped structure
of the structure. The number of measured coordinates functional approximation. The scaled conjugate gradiwas 15.
ent method was used to train the MLP, primarily beThereafter, the finite element model was constructed cause of its computational efficiency27. The initial funcusing the Structural Dynamics Toolbox26. The FE tional approximation was obtained by training the MLP
model used Euler-Bernoulli beam elements. The FE for 150 training cycles and on a subsequent functional
model contained 12 elements. The moduli of elasticity of these elements were used as updating parameTable 1. Results showing measured frequencies, the initial freters. When the FE updating was implemented the
quencies and the frequencies obtained when the FE model is
moduli of elasticity was restricted to vary from
updated using the RSM, SA and GA
10
10
-2
6.00x10 to 8.00x10 N.m . The weighting factors,
Measured
Initial
Frequencies Frequencies Frequencies
in the first term in equation 4, were calculated for
Freq
Freq
from RSM
from SA
from GA
each mode as the square of the error between the
(Hz)
(Hz)
Updated
Updated
Updated
measured natural frequency and the natural frequency
Model
(Hz)
Model
(Hz)
Model (Hz)
calculated from the initial model and the weighting
53.9
56.2
52.2
54.0
53.9
function for the second term in equation 4 was set to
117.3
127.1
118.4
118.8
120.1
0.75. When the RSM, SA and GA were implemented
228.4
209.4
209.7
211.3
for model updating the results shown in Table 1 were 208.4
254.0
263.4
251.1
253.8
253.4
obtained.
445.1
452.4
432.7
435.8
438.6
On implementing the proposed RSM, the FE model

6

and that from the initial model was 8.4%. When the
RSM was used, this error was reduced to 0.9% while
using SA it was reduced to 1.3% and using the GA it
was reduced to 2.4%. The error of the third natural frequencies between the measured data and the initial FE
model was 9.6%. When the RSM was used, this error
was reduced to 0.5% while using SA reduced it to 0.6%
and using the GA and a full FE model reduced it to
1.4%. The error between the fourth measured natural
frequency and that from the initial model was 3.7%.
When the RSM was used for FE updating, this error was
reduced to 1.1% while using the SA reduced it to 0.1%
and using the GA and a full FE model reduced it to
0.2%. The error between the fifth measured natural frequency and that from the initial model was 1.6%. When
the RSM was used, this error was increased to 2.8%
while using SA increased it to 2.1% and using the GA
and a full FE model the error was reduced to 1.5%.
Overall, the SA gave the best results with an average
error, calculated over all the five natural frequencies, of
0.9% followed by the GA with an average error of 1.1%
and then RSM with an average error of 1.7%. All the
three methods on average improved when compared to
the average error between the initial FE model and the
measured data, which was 5.5%.
The updated FE models implemented were
9
also
validated on the mode shapes they pre8
dicted. To make this assessment possible the
7
MAC11 was used. The mean of the diagonal of
6
the MAC vector was used to compare the mode
5
shapes predicted by the updated and initial FE
models to the measured mode shapes. The av4
erage MAC calculated between the mode shapes
3
from an initial FE model and the measured
2
mode shapes was 0.8394. When the average
1
MAC was calculated between the measured data
0
and data obtained from the updated FE models,
it was observed that the RSM, SA and GA up1 2 3 4 5 6 7 8 9 10 11 12
dated FE model gave the improved average of
Element
the diagonal of the MAC matrix of 0.8413,
Initial
RSM
GA
SA
0.8430 and 0.8419, respectively. Therefore, the
Figure 4. Graph showing the initial moduli of elasticity and the SA gave the best MAC followed by the GA
moduli of elasticity obtained when the FE model is updated using which was followed by the RSM. However,
the RSM, GA and SA. Here e10 indicates 10 to the power 10 and these differences in accuracies of the MAC and
the units are Nm-2
natural frequencies were not significant.
The computational time taken to run the complete
RSM method was 46 CPU seconds, while
measured natural frequency and that from the initial FE
the
SA
and
a
full
FE model took 19 CPU minutes to run
model, which was obtained when the modulus of elasticand
the
GA
and
a
full FE model took 117 CPU seconds.
10
-2
ity of 7.00x10 N.m was assumed, was 4.3%. When
The
RSM
was
found
to be faster than the GA which was
the RSM was used for FE updating, this error was rein
turn
much
faster
than
the SA which was faster that the
duced to 3.1% while using SA it was reduced to 0.2%
GA.
On
implementing
the
RSM, 160 FE model evaluaand using the GA approach it was reduced to 0%. The
tions
were
made,
while
on
implementing the SA 19485
error between the second measured natural frequency
FE model evaluations were made and on implementing
Modulus of Elasticity (e10)

approximation, where the data set had the previous optimum solution added to it, used 5 training cycles. On
using the RSM, the MLP was only initialized once. The
GA was implemented on a population size of 50 and 200
generations. The normalized geometric distribution was
implemented with a probability of selecting the best
candidate set to 8%, mutation rate of 0.3% and crossover rate of 60%.
When SA and a full FE model was implemented for
FE updating, the scale of the cooling schedule was set to
4 and the number of individual annealing runs was set to
3. When the simulation was run, the first run involved
7008 FE model calculations, in the second run 6546 FE
model calculations and in the third run 5931 FE model
calculations were made.
On implementing the GA and a full FE model, the
same options as those that were used in the implementation of the RSM were used. The results showing the
moduli of elasticity of the initial FE model, RSM updated FE model, SA updated FE model and GA updated
FE model are shown in Figure 4. Table 1 shows the
measured natural frequencies, initial natural frequencies
and natural frequencies obtained by the RSM, SA and
GA updated FE models. The error between the first

7

the GA 10000 FE model calculations were made. In this
paper, a simple FE model with 39 degrees of freedom is
updated. It can, therefore, be concluded that if the FE
model had several thousand degrees of freedom, the
RSM will be substantially faster than the other methods.
This conclusion should be understood in the light of the
fact that FE models usually have many degrees of freedom.

changes in their vibration characteristics: a literature
review” Los Alamos National Laboratory Report LA13070-MS, 1996.
10
Ewins, D.J., Modal testing: theory and practice, Research Studies Press, Letchworth, U.K, 1995.
11
Allemang, R.J. and Brown, D.L., “A correlation coefficient for modal vector analysis” Proceedings of the 1st
International Modal Analysis Conference, 1-18, 1982.
12
Sacks, J., Welch, W.J., Mitchell, T.J., and Wynn,
H.P., “Design and analysis of computer experiments”
Statistical Science, 4(4), 1989, pp. 409-435.
13
Varaajan, S., Chen, W., and Pelka, C.J., “Robust
concept exploration of propulsion systems with enhanced model approximation” Engineering Optimization, 32(3), 2000, pp. 309-334.
14
Giunta, A. A., and Watson, L. T., “A Comparison of
Approximation Modeling Techniques: Polynomial Versus Interpolating Models,” AIAA-98-4758, American
Institute of Aeronautics and Astronautics, Inc., 1998,
pp.392-401.
15
Koch, P. N., Simpson, T. W., Allen, J. K., and Mistree, F., 1999, “Statistical Approximations for Multidisciplinary Design Optimization: The Problem of Size,”
Journal of Aircraft, 36(1), 1999, pp. 275-286.
16
Jin, R., Chen, W., and Simpson, T., 2000, “Comparative Studies of Metamodeling Techniques under
Multiple
Modeling
Criteria,”
8th
AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, September 6-8, 2000.
17
Lin, Y., Krishnapur, K., Allen, J. K., and Mistree,
F., 2000, “Robust Concept Exploration in Engineering
Design: Metamodeling Techniques and Goal Formulations,” Proceedings of the 2000 ASME Design Engineering Technical Conferences, DETC2000/DAC14283, September 10-14, 2000, Baltimore, Maryland.
18
Wang, G.G., 2003, “Adaptive response surface
method using inherited Latin hypercube design points,”
Transactions of the ASME, Journal of Mechanical Design,125, pp. 210-220.
19
Simpson, T. W., Peplinski, J. D., Koch, P. N., and
Allen, J. K., “Metamodels for Computer-based Engineering Design: Survey and Recommendations,” Engineering with Computers, 17, 2001, pp. 129-150.
20
Fletcher, R., Practical Methods of Optimization.
2nd edition, New York: Wiley, 1987.
21
Holland, J, Adaptation in Natural and Artificial Systems, Ann Arbor: University of Michigan Press, 1975.
22
Goldberg, D.E., Genetic algorithms in search, optimization and machine learning, Addison-Wesley, Reading, MA, 1989.
23
Metropolis, N, Rosenbluth, A.W., Rosenbluth,
M.N., Teller, A.H., and Teller, E., “Equations of state

Conclusion
In this study, RSM is proposed for FE model updating. The proposed RSM was implemented within the
framework of the MLP for functional approximation and
GA for optimization of the MLP response surface function. This procedure was compared to the GA and SA.
When these techniques were tested on the unsymmetrical
H-shaped structure, it was observed that the RSM was
faster than the SA and GA without much compromise on
the accuracy of the predicted modal properties.

Acknowledgment
The author would like to thank Stefan Heyns, the now
National Research Foundation as well as the AECI, Ltd
for their assistance in this work.

References
1

Friswell, M.I., and Mottershead, J.E., Finite element
model updating in structural dynamics, Kluwer Academic Publishers Group, Norwell, MA, 1995, pp. 1-286.
2
Montgomery, D.C., Design and analysis of experiments, 4th Edition, John Wiley and Sons, NY, 1995,
Chapter 14.
3
Marwala, T., “Finite element model updating using
wavelet data and genetic algorithm” Journal of Aircraft,
39(4), 2002, pp. 709-711.
4
Marwala, T., and Heyns, P.S., “A multiple criterion
method for detecting damage on structures” AIAA Journal, 195(2), 1998, pp. 1494-1501.
5
Levin, R.I. and Lieven, N.A.J, “Dynamic finite element model updating using simulated annealing and
genetic algorithms” Mechanical Systems and Signal
Processing, 12(1), pp. 91-120.
6
Bishop, C.M., Neural Networks for Pattern Recognition. Oxford: Clarendon, 1996.
7
Lee, S.H., Kim, H.Y., and Oh, S.I., “Cylindrical tube
optimization using response surface method based on
stochastic process” Journal of Materials Processing
Technology, 130-131(20), 2002, pp. 490-496.
8
Edwards, I.M., and Jutan, A., “Optimization and control using response surface methods” Computers &
Chemical Engineering, 21(4), 1997, pp. 441-453.
9
Doebling, S.W., Farrar, C.R., Prime, M.B., and
Shevitz, D.W., “Damage identification and health monitoring of structural and mechanical systems from

8

26

calculations by fast computing machines,” Journal of
Chemical Physics, 21, 1953, pp. 1087-1092.
24
Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P.,
“Optimization by simulated annealing,” Science, 220,
1983, pp. 671-680.
25
Marwala, T, A multiple criterion updating method
for damage detection on structures. University of Pretoria Masters Thesis, 1997.

Balmès, E., Structural Dynamics Toolbox User’s
Manual Version 2.1, Scientific Software Group, Sèvres,
France, 1997.
27
Møller, M. “A scaled conjugate gradient algorithm
for fast supervised learning” Neural Networks, vol. 6,
1993, pp. 525-533.

9

