In science and engineering, numerical simulation is a powerful tool that predicts the behaviour of physical systems. Despite the advancement of the computing
capabilities such as power and speed, the numerical prediction of the dynamic behaviour of structure via the finite element method is still incapable of representing the behaviour of physical structure accurately. Model updating techniques have been used to improve the finite element models to closely characterise the physical behaviour of the actual structure. Most of the model updating methods is based on the minimisation of structural parameters by minimising the error function between the measured data and numerical model.
Teughels et al. (2003) revealed that the success of the application of the finite element model updating method depends on the numerical finite element model, quality of the measured data, definition of the optimisation problem and the mathematical capabilities of the optimisation algorithm. Zingg et al. (2008) showed that the computation of the optimisation problem is typically proportional to the number of design variables and constraints because a large number of iterations of the respective system had to be computed in order to find an optimum value. This is because most of the local optimisation algorithms are based on the iterative method and they are widely used for solving a variety of optimisation problem such as model updating, because these methods are easy to perform, fast and robust (Zabel and Brehm, 2009). However, the gradient based method highly depends on the given starting point and if the objective function has several local minimums, the search algorithm may get stuck in the local minimum rather than the global optimum (Ren et al., 2011). The sensitivity based finite model updating methods are determined on the construction of sensitivity matrices which can affect the optimisation process due to large computational efforts.
Alternatively, there are many alternatives methods that have been developed through which the finite element models of structures are adjusted by varying the parameters of numerical models to fit the experimental data (Venter, 2010). Recently, the alternative methods such as the evolutionary algorithms (genetic algorithms (GA) and simulated annealing (SA)) and model replacement designs (RSM) and statistical method (Monte Carlo) have attracted the attention of the engineering communities. Response surface method (RSM) is based on the replacement model of the finite element model of the system/structure which requires less computational efforts. Meanwhile, the evolutionary algorithms such
as genetic algorithms (GA) and simulated annealing (SA) are gained attention by researcher because its capability in finding globally optimum results to complicated optimisation problems (Khan and Prasad, 1997; Baumal et al., 1998 and Correia et al., 2005).
Meanwhile, Mares et al. (2004) and Mottershead et al. (2006) presented a stochastic model updating method using inverse Monte-Carlo propagation of actual structure variability and model uncertainty together with multivariate multiple regression for optimisation by the gradient method. Abu Husain et al. (2012) demonstrated the stochastic model updating using perturbation method to update flat plates and hat shape structures.
2.5.1 Response surface method
The RSM that was originally developed by Box and Wilson (1951) is the combination of mathematical and statistical technique. The RSM approach is to create the response surface by replacing the expensive computer analyses as the approximated model by utilising the generated numerical sample (Myers and Montgomery, 2002 and Carlo et al., 2002) found that the RSM has become popular and been widely used because RSM is insensitive to numerical noise such as round-off errors and can be efficiently used with other computer programme.
The RSM has been widely used in different applications such as engineering, biological and food science (Myers et al., 1989). For instance, Giunta et al. (1997) applied RSM to the analysis and design of aircraft. He used stepwise regression to obtain the optimal model. Meanwhile, Stewart et al. (2002) applied the RSM for the development of aerospace simulations. The response surface was used to attain a real time and useable accurate response for complex aerospace component simulations. Nicolai et al. (2004) used the automated setting of RSM in optimisation exercise when there was a little information about the objective function and they used the stochastic objective functions with unknown variance and objective function that were very time consuming to evaluate for every solution. On the other hand, the RSM also has been used for damage
identification. In civil engineering (Fang and Perera, 2009) used an RSM to predict damage on a beam that made of reinforced concrete (RC) and the full scale bride structure.
2.5.2 Numerical sampling
The main components of the RSM are normally coupled to design of experiment (DOE) for the computer analysis and the response surface analysis. A set of numerical sampling data is generated based on appropriate DOE. These numerical samples will be used to generate the response surface based on polynomial approximation functions. The goal of the numerical sampling is to compute the values of design variables which are considered in the design constraints. The quality of numerical sampling is essential in order to obtain an accurate model of the function and also to reduce computational effort (Chaloner and Verdinelli, 1995; Helton and Davis, 2003).
A design optimal distribution can be created using the space filling sampling, in the sense that all areas of the parameter space are randomly sampled. Several space-filling methods requiring only information on the domain are available in the literature, such as Monte Carlo (Metropolis and Ulam, 1949), Latin hypercube sampling (LHS) (Stein, 1987) and Sobol (Kocis and Whiten, 1997). On top of that, Simpson et al. (2001a) and Rutherford et al. (2006) provided a comprehensive overview of a few types of space filling design and sampling methods. The space-filling method such as LHS is used to generate numerical sampling in order to find the output features. This method which is the most ambitious is about ensuring a good coverage of the random parameter space (McKay, 1992).
Figure 2.4: Latin hypercube sampling (Stein, 1987)
Figure 2.4 illustrates how the LHS can be used to draw samples from this bi- dimensional parameter space. As shown in the figure, the samples are taken randomly from each subspace and they are distributed uniformly on the parameter space.
2.5.3 Evolutionary algorithm
The generated surface of the response can be complex and it is also very difficult to obtain the optimal value due to many local optima points. Therefore, it is difficult to apply traditional optimisation methods such as the steepest ascent for searching the global optimum due to many local optima (Alvarez et al., 2009). The optimisation can be performed on the response surface by applying gradient based method or evolutionary algorithms such as genetic algorithm (GA) and simulated annealing (SA) (Raphael and Smith, 2000). The basic GA was introduced by Holland (1975). GA works on the principle of genetic and natural selection based on Darwin’s survival fitness strategy where the dominant members of population will compete with each other to survive and reproduce successfully. As a result, the combination of dominant genes is likely to across the populations. GA has a higher probability of identifying a global optimum solution than the gradient based approach especially finite element model updating which produces a non-smooth objective function that makes the process of searching a global minimum become extremely difficult.
GA has been widely used in mechanical and civil engineering, (Coley, 1999; Chambers, 2001 and Kwak and Kim, 2009). For example, Kim et al. (2002) applied GA on RSM to determine the optimal conditions of welding processes. Akula and Ganguli (2003) utilised the finite element model updating based on GA to update the helicopter rotor blade design. Canyurt et al. (2008) used GA to estimate the strength of laser hybrid welded joint. Meanwhile, Perera and Ruiz (2008) successfully applied a GA based on the finite element model updating for damage identification of a bridge.
Despite having several attractive features, GA algorithm also have a several weaknesses and drawbacks like slow convergence at which solution is arrived at, making incredibly difficult to implement for a structure that contained many choices and parameters (Marwala, 2010). Levin and Lieven (1998) used a combination of GA and SA for the finite element model updating and it was revealed that the GA had led to high computational efforts and to slow rate of convergence, especially near an optimum making it incredibly difficult to implement for a large scale structure.