PSI - Issue 19

A. Halfpenny et al. / Procedia Structural Integrity 19 (2019) 150–167 Author name / Structural Integrity Procedia 00 (2019) 000–000

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In the case of experimental design it is usual to interpret this as meaning there should be exactly one sample taken from each row and column of the square. Fig. 5a. shows the Latin Square for a 2-factor design (i.e. 2 parameters). The columns pertain to parameter X and the rows to parameter Y. In Fig. 5a. each random variable is calculated based on 5 divisions using the approach illustrated in Fig. 4. The Latin Square can also be represented as a 2-column matrix as illustrated in Fig. 5b. Each division in X is paired with a randomly selected division in Y, such that all divisions are used exactly once. This is easily extrapolated to a 3-factor system (i.e. Latin Cube) in Fig. 5c., and the approach is scaled similarly to any n dimensional problem (Latin Hypercube).

Fig. 5. Latin Square and Latin Cube multivariate sampling

As the number of simulations is reduced it becomes more necessary to check that tests are sufficiently random in their distribution through the design space. Failure to do this could result in either: two or more tests having similar input values, therefore biasing the statistical results. For example, the random values selected in columns 3 and 4 of Fig. 5a are close together whereas other areas of the matrix are left under sampled. a high correlation between some of the tests so the design space represents points along a preferred line rather than evenly and randomly distributed. An additional optimization is applied to the Latin Hypercube design in order to maximize the minimum distance between random test values, whilst simultaneously minimizing the maximum cross-correlation. 2.3. Discrete distributions The method is not limited to continuous probability distributions. A discrete distribution is returned as a list of enumerations. Each enumeration is given a weighting factor. Enumerations are typically used to select options from a list, for example, surface finishes and treatments, or filenames pointing to alternative input load spectra. This allows for a statistical simulation of varying driver/road profiles or flight spectra to be considered in the stochastic design. An illustration of the approach is given in Fig. 6.