PSI - Issue 19

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

158

9

Fig. 7. 2-level factorial design performed at the edges of a design space

Fig. 7. considers the case where there are only 2 possible values for each variable. These are usually at the edges of the distribution. The number of permutations m is given by Equation (4): � (4) Where n is the number of factors (or variables). A more general case of the 2-level case is where there are k possible outcomes for each variable n. For example, this might consider mid-points along the edges, the surfaces, or within a volume. The number of permutations is then given as Equation (5): ∏ � ��� � (5) These approaches also extend to discrete probability distributions. For example, to determine the most conservative setting for a calculation method, such as: Mean stress correction method = ‘None’, ‘Morrow’ or ‘Smith-Watson Topper’? The number of permutations considered in the factorial approach can become prohibitive and so there are a number of methods available to reduce these. These are known as fractional-factorial designs and include methods such as Plackett and Burman [PLA 46] and Taguchi’s Orthogonal Arrays [TAG 05]. These methods are beyond the scope of this paper and are discussed in [REL 15a], chapter 9. Response surface A response surface is used to interpolate between the simulation points in the design matrix. The response surface is multidimensional. It is usual to limit the response surface to either a simple linear surface or a second-order as illustrated in Fig. 8. The surface does not need to be accurate as it is used as part of an iterative solution process to converge on the desired solutions. Further information is given in [REL 15a], chapter 10.

Fig. 8. Types of second order response surfaces