PSI - Issue 28

C. Mallor et al. / Procedia Structural Integrity 28 (2020) 619–626 C. Mallor et. al. / Structural Integrity Procedia 00 (2020) 000–000

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The most general propagation of uncertainty method for non-linear combinations, the FOSM, determines the stochastic moments of functions with random input variables by using Taylor expansions, provided that the moments of the input variables are known. Let � � � , � , … , � � � be a set of � functions which are a non-linear combination of � random input variables � � � , � , … , � � � as it is shown in the multi-input multi-output system of Fig. 1.

Fig. 1. Multi-variable vector-valued function. The first-order Taylor approximation of becomes Eq (1), where � is the mean value vector of the random variables vector , � � � is a compressed notation for the vector of the model response, that is, the vector-valued function � � evaluated at the mean value vector � and J � � � � is the Jacobian matrix of the vector-valued function � � evaluated at the mean value vector � . � � � � � � � J � � � �� � � � (1) The output expected value vector of the first-order Taylor approximation Eq (1) is given by Eq. (2). E� � � � � � � � (2) The output covariance �d � �d � � -matrix � of the first-order Taylor approximation Eq (1) is given by Eq. (3), where � is the symmetric input covariance �d � �d � � -matrix which contains all variances and covariances of . � � J � � � � � J � T � � � (3) The previous approximation is the most general equation for the propagation of uncertainty from one set of variables onto another using a first order Taylor approximation. In other words, it provides the effect of the input variables uncertainties on the uncertainty of functions based on them approximated though Taylor. In this case, the uncertainty is quantified in terms of the variances and covariances of the output random variables. The Eq. (3) can be interpreted as it is depicted in Fig. 2.

Fig. 2. Interpretation of the error propagation law in its matrix form. In the interpretation of the previous figure, the input uncertainties have a role in the covariance matrix of the input random variables, � , and also in the mean value vector for evaluation of the Jacobian matrix of the output functions J � � � � . Then the Jacobian matrix, that can be thought as the system, is used to transform the rows and columns of the covariance matrix of the input random variables, providing the covariance matrix of the output random variables, � , i.e. it is used to map the input randomness onto the output randomness. As mentioned, the complete mathematical derivation of the FSOA for the first to fourth moments of functions of random variables using summation notation is presented in [14,15]. They present the expressions involving tensors of different orders in a simple and comprehensible way. Notice that, the first to fourth moments are related by definition to the expected value (first raw moment), the variance (second central moment), the skewness and the kurtosis (third and fourth central standardized moments, respectively) of the random output variable.