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This section shows a brief survey about metamodelling techniques, which have become very important in the pursue of studying complex engineering problems. Despite the high advances in the computational power in the last decades it is still very costly to carry out certain types of engineering simulations, such as Computational Fluid Dynamics (CFD), Fluid-Structure Interaction (FSI), crashworthiness or post-buckling analysis. To cut down the computational cost, simplified but efficient representations, entitled surrogate models or metamodels, are built and used instead of performing sequentially FE simulations. These metamodels require a sampling scheme for their construction, which should be performed with High Performance Computing (HPC) in order to take profit of the parallelization possibilities inherent to these techniques.

2.2.5.1 Polynomial Regressions

Polynomial Regressions have been widely used in the design of complex engineering systems, as stated in Venter et al [203] or Chen et al [34]. They tend to represent the system responses recognizing the significance of different design factors by identifying the coefficients of the polynomial regression. They have the main advantage of behav- ing properly and allowing convergence in noisy functions thanks to their smoothing capability (Jin et al [107]). A typical example of a second-order polynomial model is expressed as: ˆ y = b0+ k X i=1 bixi + k X i=1 biix2i + X i X j bijxixj (2.18)

The Polynomial Chaos Expansion (PCE) is one of these metamodelling techniques that apart from the aforementioned advantages allows to quantify the uncertainty of a structural system, as exposed in Choi et al [38]. In this sense, PCE benefits from

special polynomials whose orthogonal weighting functions are the Probability Density Functions (PDF) of well-known random distributions.

2.2.5.2 Kriging Method

Kriging method (KM), whose detailed description can be found in Liu and Batilly [133], creates a combination of known fixed functions fj(x) as exposed in Equation 2.19:

ˆ y = k X j=1 λjfj(x) + Z(x) (2.19)

where fj(x) are k known regression models with their corresponding λj ponderation parameters and Z(x) is a sample of a stochastic process with mean zero and spatial correlation function given by:

cov[Z(xi), Z(xj)] = σ2R(xi, xj) (2.20)

where σ2 is the process variance and R is the correlation function, which can be selected by several means (Simpson et al [184]) although the most frequently used is the Gaussian one proposed in Sacks et al [177]. The main advantages of Kriging are that it is very flexible due to the high range of correlation functions and that the same data can be used for screening and building the predictor model, while the main drawbacks are that constructing the metamodel can be very expensive computationally and that the correlation matrix can be singular if some of the sample points are very close to others.

2.2.5.3 Multivariate Adaptive Regression Splines

Multivariate Adaptive Regression Splines (MARS, Friedman [72]) are non-parametric regression techniques that adaptively select a set of basis functions for approximating the real response through an iterative approach. The MARS approximation can be written as: ˆ f (x) = M X m=1 amBm(x) (2.21)

where am are the coefficients of the regression and Bm are the basis functions, which can take one of the following three forms:

2. A hinge function, which has the form max(0, x − C) or max(x − C, 0), where x is the variable and C is a constant location on each of the variables, denoted as knot.

3. A product of two or more hinge functions.

Hence the model consists of a sum of the M basis functions weigthed with the regression coefficients. MARS show the advantage of its accuracy and major reduction in the computational burden associated to the metamodel building compared to the kriging method, but in exchange its performance deteriorates when small or scarce sample sets are used (Jin et al [107]).

2.2.5.4 Artificial Neural Network

Artificial Neural Networks (ANN, see Swingler [197] or Waszczyszyn and Ziemianski [209]) are global approximation techniques inspired by biological neural networks that aim to estimate functions which depend on a large number of unknown inputs. ANN possess the ability to obtain an output response from external input data based on their training and past experiences. ANN are typically organized in layers, which are constituted by a number of computing units called nodes that are interconnected. There are usually three types of layers: input layer (which contains the input data information), output layer (which provides the answer or the response) and hidden layers (where the actual processing is done through a system of weighted connections). Figure 2.8 presents a graphic with the general scheme of an ANN.

Input data

Output response

Input layer Hidden layer Output layer

Figure 2.8: General scheme of an ANN.

Most ANN contain “learning rules” which adapt the interconnections and modify their weights according to the input data received, training the Network in order to minimize the differences between the computed outputs and the real ones.

2.2.5.5 Radial Basis Functions

Radial Basis Functions (RBF), which were developed for scattered multivariate data in- terpolation (Dyn et al [61]), are metamodelling techniques that describe the behaviour of non-linear functions from a set of N samples. The approximation is provided by a linear combination of univariate and radially symmetric functions, whose values at m points of a n dimensional Euclidean space are known. The general expression of a RBF is: s(x) = m X j=1 λj· φ(rj) rj = kx − xjk (2.22)

where the approximation function s(x) is represented as a sum of m radial basis func- tions φ(rj), each depending on the distance between the generic point x and the sample point xj used to build the metamodel. RBF provide accurate results in both determin- istic and stochastic response functions (Powell [162]), and show advantages such as the easy design, good generalization or strong tolerance to noisy input data. However, the condition number of the system that describes the metamodel worsens rapidly when the dimension or the number of data values to be fitted increase (Hussain et al [96]).