Issue 37

R. Sepe et alii, Frattura ed Integrità Strutturale, 37 (2016) 369-381; DOI: 10.3221/IGF-ESIS.37.48 374 Figure 1 : Elementary cell FE model. T HE SDI PROCESS set of techniques was introduced in order to obtain a robust design, i.e. one whose behavior is rather insensitive to all variations of the main variables, or, what is the same thing, a design whose statistics are characterized by the smallest standard deviation, as a function of the statistics of input. This approach can be also linked to another very relevant question; the result of an experimental test carried on an assigned structure is the consequence of the particular and real values of all design variables, whose density functions are supposed to be known: when we try to correlate the test results to a numerical simulation procedure, we want, in effect to assess all other aspects stated, which are the values that the design variables had in the real structure tested in that particular experiment. Both these problems can be effectively dealt with by an SDI (Stochastic Design Improvement) [12] process, which is carried out by means of several MC series of trials (runs) as well as of the analysis of the intermediate results. In fact, input – i.e. design variables x – and output – i.e. target y – of a mechanical system can be connected by means of a functional relation of the type   y F x  (11) Which, in the largest part of the applications, cannot be defined analytically, but only ideally deduced because of its complex nature; in practice, it can be obtained by considering a sample x i and examining the response y i , which can be carried out by means of a simulation procedure and, first of all, by means of one of M-C techniques, as recalled above. Considering a whole set of M-C samples, the output can be expressed by a linearized Taylor expansion centered about the mean values of the control variables, as       i x i x y i x d F y F μ x μ μ G x μ d x       (12) where  i represents the vector of mean values of input/output variables, and where the gradient matrix G can be obtained numerically, carrying out a multivariate regression of y on the x sets obtained by M-C sampling. If y 0 is the required target, we could find the new x 0 values by inverting the relation above, i.e. by   1 0 0 x y x μ G y μ     (13) as we are dealing with probabilities, the real target is the mean value of the output, which we compared with the mean value of the input, and considering that, as we shall illustrate below, the procedure will evolve by an iterative procedure, it can be stated that the relation above has to be modified as follows, considering the update between the k-th and the (k+1)-th step: A