روش مبتنی بر طبقه بندی موثر برای تجزیه و تحلیل حساسیت جهانی

New classification based methods for global sensitivity analysis of structural models are presented which do not require the full approximation of the model response for qualitatively good sensitivity measures. Instead, only the level sets of the model response are identified by partitioning it into a number of classes with a few available sample points. The average change in class memberships of simulated points on the model domain is considered as sensitivity measure. The new methods are realized using Support Vector Machines and their results are compared with existing methods by using analytical as well as practical industry examples.

Global sensitivity analysis is a procedure to analyze the full range of plausible values of (random) parameters and their interactions in a structural model in order to assess their impact on the model response. Global in a sense that the analysis is not performed locally or one-factor-at-a-time but considering the whole domain of each model parameter. Each individual model parameter X i, i = 1,2,…,n is compared with the other remaining parameters X 1, … X i−1, X i+1, … , X n in order to evaluate its influence on the model response Y . The objective of sensitivity analysis is to identify the most significant model parameters affecting a specific model response [18]. Significant parameters are those which have a large impact on the model response. Results of the sensitivity analysis are the sensitivity measures S i,i = 1,2,…,n, i.e., significance of a parameter X i is identified with reference to its sensitivity measure S i. The normalized sensitivity measure S i for a model parameter X i is given as equation(1) View the MathML sourceSi=S∼i∑j=1nS∼j, Turn MathJax on where View the MathML sourceS∼i represents the influence of Xi on Y according to a specific sensitivity measure. Sensitivity measures are used for the identification of the significant parameters before the optimization of a design structure [10], [14], [16] and [17]. Optimization of design structures is a computationally extensive process. During the optimization process, the objective function which is formulated on the basis of the model response is analyzed depending on the design parameters and constraints. An optimization model is often dependent in part on the number of design parameters. The complexity of the optimization problem can be reduced if the relationship between the design parameters and the model response is effectively identified and only the significant design parameters are then used. This relationship is captured by the methods of global sensitivity analysis. Different sensitivity measures have been proposed which are determined using variance based methods [18] and [19], sampling based methods [9], and derivative based methods [19]. In variance based methods, the unconditional variance of the model response is decomposed into terms due to individual factors and the terms due to the interaction among factors. Common variance based global sensitivity measures are for example ANOVA measures [18] and Sobol Indices [19]. Sobol Indices are popular because they capture the non-linearity in the models as well [16]. Sampling based methods include screening methods and correlation analysis, which is only applicable for linear problems. Derivative based methods use partial derivatives to determine the importance of the input parameters because they represent the instant slope of the underlying function for each value of the input parameter [19]. If the model is unknown, meta-model based global sensitivity methods can be used for sensitivity analysis. An approximate model is constructed for the available response values using meta-models and certain intrinsic properties of the meta-models are then used to derive sensitivity measures [22]. Based on neural networks as meta-models, different equations have been proposed which calculate the products of the weights of the network and then obtain the sum of the calculated products according to a certain criteria [7], [8] and [23]. These methods are broadly characterized as weight based sensitivity measures. Values stored in the static matrix of weights of the neuron connections in a neural network can be used to determine the relative influence of each input parameter on the network response. On the other hand, using the derivability property of meta-models, derivative based sensitivity measures can also be determined. In [12] a partial derivative based method using a single hidden layer neural network for determining sensitivity measures is presented. The problem with the above mentioned methods is that they require a large number of sample points for calculating sensitivity measures and thus are computationally expensive. Also, application of these methods using meta-models requires a large number of sample points for an accurate approximation of the model response which in turn influence the sensitivity measures [13]. In this paper we introduce a new class of meta-model based global sensitivity measures termed as classification based sensitivity measures. The difference between classification based methods and previously presented methods is the granularity of the approximation. The new approach does not requires full approximation of the model response but only the level sets of the model response which in turn requires relatively lesser sample points for qualitatively good sensitivity measures. These level sets are identified by partitioning the values of the model response into a set of disjoint classes with the help of Support Vector Machines [2] and [24] and then calculating the influence of a model parameter on the model response with the help of change of class on that parameter domain through Monte Carlo simulation. The next section introduces methods of global sensitivity analysis for non-linear models and highlights some meta-models particularly the Support Vector Machines and their use for the classification of the values of the model response. Section 3 explains the new classification based sensitivity measures in detail and defines new Vertical Class Jump Method, Horizontal Class Jump Method, and Boundary Method. Section 4 shows the results of these methods applied on industry relevant example problems and their comparison with existing global sensitivity methods. The conclusions are presented in Section 5.

In this paper, new methods for global sensitivity analysis are presented which use classification of the model response for calculating sensitivity measures. Three methods for classification based sensitivity analysis are presented and discussed together with some numerical aspects. These methods do not require the full approximation of the model response and therefore require relatively less sample points which are demonstrated with the help of two industry relevant crash test examples. The results of classification based methods are comparable to existing approaches, however, they require less sample points. The results also show that classification based methods provides relatively stable sensitivity measures for the increasing number of sample points as compared to existing methods.