Sensitivity of the solution to c, N and M

6.2. Sensitivity of the solution to c, N and M

After a parametric study using both pile-up (E/σy = 148, n = 0.15) and sink-in (E/σy = 1555, n = 0.4) materials, it was observed that both the error and the number of iterations to converge to the target solution are influenced by the contribution of the second objective function, as illustrated in Figure 6.1. The value of c that produced the best fit and fastest convergence was found to be 0.25 for a P-h curve and pile-up profile defined by N and M ~ 2.7N data points. The fastest convergence occurred at the limiting values c = 0 and c = 1 as a result of the less restrictive single-optimisation model in these cases. Given that the information provided by the residual imprint is strongly linked to the plastic behaviour of the material, the solution was significantly improved as a compromise between the contributions of each objective function was reached. As seen in Figure 6.1, a scaling coefficient c = 0.25 resulted in a reduction of the error in the predicted values of σy and n by more than 9 and 24% respectively, compared to the model neglecting the contribution of the second objective function, i.e. c = 1. Conversely, the Young’s modulus is related to the contact stiffness of the unloading curve (Equation 2.3) and hence the solution was relatively insensitive to the participation of the multiple objective functions. The information used to construct the graph in Figure 6.1 is detailed in Table 6.1; an extra column has been included with the result of the minimisation of the multi-objective function disregarding the scale coefficients, i.e. the components of each objective function are multiplied by a coefficient c1 = c2 = 1.

Figure 6.1. Effects of scaling coefficients on the optimised parameters.

Chapter 6- Development of a multi-objective function optimisation model

109

Table 6.1. Relative error [%] between optimised and target parameters and number of iteration.

Parameter c = 0 c = 0.25 c = 0.5 c = 0.75 c = 1 c1 = c2 =1 function containing the scaled components of each objective function was returned to the optimisation algorithm (the lsqnonlin solver), which implicitly computed the sum of squares of the components (referred to as the squared 2-norm of the residual) of f1(x) and f2(x) (See Equation 3.7). Therefore, the components of both f1(x) and f2(x) were ‘artificially’ reduced by c and 1 - c, respectively, with the purpose of equilibrating the global contribution of each objective function. Consequently, the global contribution of each of these functions was not uniquely influenced by the magnitude of the scaling coefficients but also by the magnitude of the residual computed at each sample point and the number of sample points. This could be explained by studying the minimisation problem without scaling the respective components. During the early iterations (i ≤ 3 in the current analysis), the normalised sum of squares of the unscaled components of the objective functions, i.e.

were of the same order of magnitude as can be inferred from the dashed line in Figure 6.2a. Note Ai and Bi, the normalised error computed between the experimental and predicted P-h curve and pile-up profile at iteration i, respectively, were not scaled so as to assess how well the optimised parameters in x fit the experimental data. However,

Chapter 6- Development of a multi-objective function optimisation model

110 the global contribution of f2(x) to f(x) was weaker as Bi was a few times smaller than Ai, or in other words, the predicted pile-up profile at the current point xi provided a better fit to the respective experimental data than Ai. Otherwise, the global contribution of f1(x) diminishes. It can be seen in Figure 6.2a, that the information provided by the P-h curve drove most of the optimisation procedure in a ratio of approximately 3:1 and hence, the purpose of the scaling coefficients is to equilibrate the unbalanced global contribution of each objective function. The same trend was observed starting at a different initial guess, e.g. x = [200000, 1300, 0.1]. Figure 6.2b and c, illustrate the outcome of the same assessment carried out in the optimisation procedures using c = 0.25 and c = 0.75. Accordingly, the global contribution of f1(x) and f2(x) to f(x) was more sensitive to c during the early iterations and thus, a scaling coefficient of c = 0.25 increased the global contribution of f2(x) in excess of 65% unlike c = 0.75, that diminished it to below 3%. However, as the iterative procedure progressed beyond i >

3, Ai and Bi remained in the same order of magnitude when c = 0.25 as illustrated in Figure 6.2b. Given that the magnitude of c in this case reduced the computed error between the experimental and predicted P-h curves ‘artificially’ to a greater extent, the optimisation algorithm tried to improve the minimisation of f(x) by finding a vector x that improves the fitting between the experimental and predicted geometry of the pile-up. For this reason, Ai was between 4 and 5.25 times higher than Bi. In contrast, when c = 0.75, Ai became nearly an order of magnitude smaller than Bi at i > 3, as it was now the error between the experimental and predicted pile-up pattern which was being reduced artificially to a greater extent. As the solution approached convergence, the minimisation problem scaled using c = 0.25 reached a better compromise between the global contribution of each objective function and ratio Ai/Bi and hence, the information provided by both the P-h curve and the geometry of the pile-up was better exploited.

Chapter 6- Development of a multi-objective function optimisation model

111 a)

b)

c)

Figure 6.2. Global contribution of each objective function along the optimisation procedure set up with a scaling coefficient of a) c1 = c2 = 1, b) c = 0.75 and c) c = 0.25.

In order to better understand the influence of the number of data points N and M on the solution, two additional optimisation analyses were run as follows: For the first analysis, N was kept constant at 82 while for the second analysis, M was kept constant at 31; the respective counterpart, M and N, was computed accordingly to set a ratio N/M ~ 5. Similarly, the global contribution of each objective function was greatly driven by c only during the early iterations, when the normalised squared 2-norm of the residual of both f1(x) and f2(x) were of the same order of magnitude. Interestingly, during the early iterations, the global contribution of f1(x) and f2(x) as well as the ratio

Chapter 6- Development of a multi-objective function optimisation model

112 Ai/Bi were equivalent provided that the ratio N/M was maintained constant. Additional pairs of optimisation analyses with ratios N/M ~ 2, 3, 4 and 7.5 were run in order to support the latter observation. Throughout the rest of the iterations, the global contribution of f1(x) and f2(x) varies so as to improve the minimisation of f(x) as detailed previously. Notwithstanding, as illustrated in Figure 6.3, this study concluded that the influence of the number of data points is negligible on the solution as both simulations using target curves defined by N/M ~ 5 data points reached convergence to within a difference of less than 0.5% in relation with the solution provided by the reference model, i.e. N = 81 and M = 31 (N/M ~ 2.7).

Chapter 6- Development of a multi-objective function optimisation model

113 a)

b)

c)

Figure 6.3. Iteration history showing the negligible sensitivity of the optimisation parameter a) E, b) σy and c) n to the number of

data points M and N along the P-h curve and pile-up profile, respectively.