FIELD TESTING AND FINITE ELEMENT MODEL UPDATING
5.7 Finite Element Model Updating
5.7.1 Response Surface Method (RSM)
Response surface method (RSM) consists of a group of mathematical and statistical techniques used in the development of an adequate functional relationship between response of interest (output variable) and several independent variables which influence output of the system response (input variables). This method uses the response surface function (RSF) to approximate the actual state function, which is usually implicit and difficult to express. The basic steps involve in the response surface based finite element updating are discussed below:
5.7.1.1 Design of Experiments (DoE)
An important aspect of response surface method is the design of experiments (Box and Draper, 1987), usually abbreviated as DoE. These strategies were originally developed for the
model fitting in physical experiments, but can also be applied to numerical experiments. The objective of DoE is the selection of the points where the response should be evaluated. The term experiment herein refers to either physical experiments or computer experiments. DoE plays an important role in constructing a response surface. A detailed description of the design of experiments theory can be found in Box and Draper (1987), Myers and Montgomery (1995) and Montgomery (2000). There are many designs available for response surface function. The most popular method is the central composite design (CCD) introduced by Box and Wilson (1951). This design method gives most reliable response surface function especially for civil engineering structures (Ren and Chen, 2010).
The central composite design consists of factorial points, central points, and axial points.
CCD was often developed through a sequential experimentation. The number of center points at the origin and the distance of the axial runs from the design center are two parameters in the CCD design. The center runs contain information about the curvature of the surface. If the curvature is significant, the additional axial points allow ones to obtain an efficient estimation of the quadratic terms. There are many possible way of combining these three points depending on the number of factors; some of the combinations may be insignificant for a particular problem. In view of the practical aspect of bridge model updating, Deng and Cai (2009) listed out CCD for different number of factors given in Table 5.2. Two-factor factorial CCD is shown in Fig. 5.14. In the figure factorial points is represented as small circle, star represent axial points and a small circle located at the center denotes central point. X1 and X2
are the independent variables.
Table 5.2 Central Composite Design for different number of factors (Deng and Cai, 2009) Number of factors
2 3 4 5 6 7
Factorial points 22 23 24 25-1 26-1 27-1
Axial points 4 6 8 10 12 14
Center point 1 1 1 1 1 1
1.4141 1.6818 2.0 2.0 2.3784 2.8284
Total number of trials 12 18 28 30 48 82
Fig. 5.14 Two factor Central Composite Design
A three-factor CCD adopted by Mayers et.al (1989) is shown in Fig 5.15 and their corresponding values are given in Table-5.3, which has been used in the present finite element model updating of the tested bridge. In the figure, X1, X2 and X3 are independent variables. It should be noted that the given CCD design is a standard design form for a case with three factors. The center point is repeated four times since the information at the center point is more important to the response surface compared with the other points in the three factors central composite design. Here, has been taken as 1.6818 (Mayers et.al, 1989).
Fig. 5.15 Three factor Central Composite Design
Table 5.3 Three factor Central Composite Design (Mayers et.al, 1989) Experimental
trial
Factor Level Setting
X1 X2 X3
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 0
9 - 0 0
10 0 0
11 0 - 0
12 0 -
13 0 0
14 0 0 0
15 0 0 0
16 0 0 0
17 0 0 0
18 0 0 0
5.7.1.2 Construction of Response Surface Function
In general, the low-order polynomial model is used to describe the response surface for conceptual as well as computational reasons. A polynomial model is usually a sufficient approximation in a small region of the response surface. Therefore, depending on the approximation of unknown function, either first-order or second-order models are employed.
If the response can be defined by a linear function of independent variables, then the approximating function is a “first-order model”. If there is a curvature in the response surface, then a higher degree polynomial should be used (Douglas, 2005). A two degree polynomial is termed as “second-order model”. The two models can be given as
=
+
= n
j jXj
Y
1
0 β
β (5.1)
= =
=
=
+ +
+
= n
i n
j ij i j
n
j jj j
n
j jXj X X X
Y
1 1
1 2 1
0 β β β
β (5.2) Eqs. (5.1) and (5.2) represents first-order model and second-order model respectively. n is a number of independent variable. Yis the response and X denotes the independent variables or
system parameters to be updated, 0i, j, ij, are the regression coefficient to be estimated from experimental design data. The Least squares method has been applied to estimate the coefficients in the second order model.
The relationship between the response variable Y and independent variables X is usually unknown. Therefore, depending on the approximation of unknown function f, either first- order or second-order models are employed. The second-order model is flexible, because it can take a variety of functional forms and approximates the response surface locally Therefore, this model is usually a good estimation of the true response surface (Douglas, 2005). Due to this reason, second-order model has been adopted in the present study for the construction of response surface function.
5.7.1.3 Development of Objective Function
In a response surface based finite element updating method, a deviation between actual measured response and the virtual response obtained from the experimental trial using response surface function has to be minimized by means of any suitable optimization method.
In the present study, a global search optimization method, called Genetic Algorithm has been employed for minimizing this deviation. The objective function can be written based on the least-squares method as shown below (Deng and Cai, 2009)
2 1
1
)2
( −
=
= n
k Yk Zk
ε (5.3)
where, is a deviation of measured and predicted response. Yk is the predicted response from the response surface function whereas Zk denotes measured response The number of system response considered for model updating is ‘n’.