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Structural Damage Identification Using Co-Evolution and Frequency Response Functions
1
Structural Damage Identification Using Co-Evolution and Frequency
Response Functions

B. Kouchmeshky and W. Aquino
1

Cornell University, School of Civil and Environmental Engineering, Ithaca, NY 14853


Abstract

Structural damage identification can be mathematically described as an inverse problem. Data
sparsity in these applications results in inverse problems that are commonly ill-posed (e.g. non-
unique solutions and instability of solutions). Although multiple tests can be used to alleviate the
ill-posedness of the problem, it is not trivial how to select physical tests a priori, given the
constraint that structural tests are often expensive. This work describes a new co-evolutionary
algorithm that interactively searches for damage scenarios and optimal physical tests. The
methodology is presented in the context of steady state dynamics, in which changes of frequency
response functions are used as indicators of structural damage. The feasibility of the
methodology is demonstrated using a bridge truss example.

Keywords
Genetic algorithms, damage identification, co-evolution, structural health monitoring.
1. Introduction
Structural health monitoring is concerned with the identification and quantification of damage in
aerospace, civil, and mechanical infrastructure [1], and has emerged as a modern area of research
due to its key role in structural safety, maintenance, serviceability, and estimation of expected

1
Corresponding author, Cornell University, 313 Hollister Hall, Ithaca, NY 14853, Email:
wa27@cornell.edu
, Fax:
607-255-9004, Phone: 607-255-3294.
2
service life. Structural health monitoring systems use a combination of sensing, actuation,
simulation, and reasoning components to infer the location and extent of damage in a structure.
One of the main challenges in damage identification is to obtain sufficient information through
sensing so that reasoning systems can uniquely and unambiguously characterize damage.

The identification of damaged elements is inherently an inverse problem in which the state of the
structure is determined based on its response. In situations where sufficient data are not available
or the noise-to-signal ratio is high, the inverse problem can be ill-posed in the sense that more
than one solution can exist and/or the solution may be very sensitive to errors in the input data.
To address the non-uniqueness of the solution, multiple tests can be performed to increase
information about the damage state and constrain the number of possible solutions. Ghaboussi
and Chou [2] and, Hjelmstad and Shin [3] used multiple simulated static tests in damage
identification problems, recognizing that a simple test does not yield the necessary information
for solution uniqueness. However, it is not trivial to know a priori which tests to perform on a
structure so that its damage state can be unambiguously identified. Tests are expensive and
carrying out random experiments on a structure can result in unacceptable costs.

A co-evolutionary strategy is presented in this work that addresses the problem of optimizing the
number of tests needed for damage identification. Co-evolution is a biological process where
populations of individuals interact with each other while trying to evolve in response to the
evolution of the other populations. Co-evolutionary strategies have recently been proposed for
nonlinear system identification and structural damage detection [4, 5]. The methodology
proposed in this paper, called The Estimation-Exploration Algorithm (EEA) hereafter,
3
provides an intelligent approach for selecting multiple tests in damage identification problems so
that information contained in the measured response is maximized. In the co-evolution process,
models evolve over generations to predict current tests, while current tests evolve to create
discrepancy among models. This work currently concentrates on structures excited using
harmonic dynamic loading.

The paper is organized as follows. The formulation of the problem is presented first, followed by
a description of the proposed co-evolutionary algorithm. Then, the feasibility of the methodology
is demonstrated through numerical examples. Finally, some preliminary results, discussion, and
conclusions are presented.

2. Formulation of the problem
The dynamic behavior of a linear system with viscous damping can be written as

( )
( )
( )
( )
t
t
t
t
+
+
=
Mx
Cx
Kx
f
&&
&
, (1)
where M ,
C
and K are the mass, damping ,and stiffness matrices of the structure, x is the
displacement vector, and
f is force vector acting on the structure.

Assuming harmonic excitation, the force acting on the structure can be expressed in terms of its
angular frequency, , and a complex forcing amplitude vector,
f
, as

( )
i t
t
e =
f
f
. (2)

In addition, the steady state response of the structure can be expressed as

( )
i t
t
e =
x
x
, (3)
4
where x is the frequency component of the displacement. Substituting Equations (2) and (3) into
Equation (1) yields

( ) =
x H
f
, (4)
where

( )
(
)
1
2
i
= +
H
K
M
C
. (5)

( ) H
is called the frequency response function (FRF) matrix of the system or, more specifically,
the receptance matrix. The
th
aj member of an FRF matrix represents the response (displacement,
velocity or acceleration) of the
th
a degree of freedom subjected to a harmonic force applied at
the
th
j DOF.

The solution of Equation (4) is computationally expensive for large systems. A more efficient
approach is to use the spectral decomposition of the receptance matrix ( ) H
to compute the
frequency response at selected degrees of freedom. The spectral decomposition of the receptance
matrix for a proportionally damped viscous dynamic system can be expressed as [6]

2
2
1
( )
(
)
T
j
diag = H
. (6)

It is usually assumed that damage is manifested as change in the stiffness of the structure.
Considering one damage parameter per structural element, the updated local stiffness matrix of
an element can be described as

5

(1
)
ee
ee
l
l
l =
o
K
K
, (7)
where
ee
l
o
K
is the undamaged stiffness matrix of element
l
,
l is the damage ratio or index, and
ee
l
K is the updated element stiffness matrix. The global stiffness matrix is assembled from
individual element contributions as
( )
ee
elements
= K
K . (8)

Structural damage identification is usually cast as an optimization problem in which model
parameters are updated in order to minimize the discrepancy between a mathematical model and
the sensed behavior of the actual structure [1-3, 6-10] . As a result of the above formulation the
receptance function will depend on the damage parameter vector, , and on the frequency,
.
Damage parameters are obtained by formulating an optimization problem in which the error
between the computed and measured receptance functions is minimized. The error function for a
given structural test,
q
, as used in this paper, is given as

(
)
( )
( )
(
)
1
1 ,
( ) max
R
M
ak
p
ak
p
q
a
p
ak
p
H
H
E
H =
= =

, (9)
where
a
indexes the measured degrees of freedom,
k
is the excitation degree of freedom,
p

indexes the excitation frequencies,
R
is the total number of sensors in the structure,
M
is the
total number of excitation frequencies, and ak
H
is the measured receptance. The damage
parameters are then obtained by solving

1
1
( )
0
1
N
q
l
q
Minimize
E
Subject to
N =
<
<
, (10)

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where
N
is the number of physical tests performed on the structure.

2.1 Genetic Algorithms (GA)
Genetic algorithms are attractive for complex optimization problems because of their inherent
advantages such as parallelism, convergence to global optima, adaptation, and the lack of need
for the gradient of the objective function. Because of these advantages, genetic algorithms have
been successfully used in structural damage identification problems [2, 5, 6, 11-16]. Inspired by
Darwins theory of survival of the fittest, GA mimic the process of the evolution of an organism
and can be used to solve a wide variety of problems in engineering and science [17, 18].

The information describing each solution (i.e. individual) is encoded in a string that is called a
chromosome. The individual entries that form a chromosome are called genes. For instance, in
damage identification problems, damage parameters are encoded in real-number genes, which
form the chromosome of an individual solution. The algorithm starts with a set of solutions
(initial population) selected randomly. Then, the fitness of each individual is calculated based on
how well the response of the structure is predicted when the encoded set of damage parameters is
used in a finite element analysis. For instance, the error norm defined in Equation (9) can be used
as a measure of fitness.

In producing the next generation, pairs of individuals from the current population are selected
based on their fitness to serve as parents. Operators such as crossover and mutation are crucial in
genetic algorithms and are used to produce new individuals. The crossover operation relates to
th