9 On the Boundary Conditions and Optimization Methods in Integrated Digital Image Correlation 59 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x log(ν/ν p ) log(E/E p ) −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 # FE simulations Fig. 9.4 Number of iterations needed before convergence within a set accuracy limit is reached for the Trust-Region method for a range of initial guesses. On the x-axis the initial guess in Poisson ratio , relative to the reference value p, is plotted. On the y-axis the initial guess in Young’s modulus E, with respect to the reference value Ep Fig. 9.5 Operations in the Nelder-Mead algorithm The following results are for the MATLAB Optimization Toolbox using the Nelder-Mead algorithm. The Nelder-Mead algorithm shows significant convergence towards a minimum for all given initial guesses, which proves robustness, but takes many FE simulations to accurately converge. For all cases in Fig. 9.4, the algorithm was converging, as can be seen in Fig. 9.6. 9.3.4 Comparison of the Methods To compare the methods more visually, a convergence plot for each method is shown for one specific initial guess combination: log. = p/ D 0:1and log.E=Ep/ D 0:3. Nelder- Mead takes significantly more simulations to converge and cannot rely on a derivative for the search direction, but takes multiple iterations to determine the direction. Gauss-Newton is the fastest method, followed by Trust-Region (Fig. 9.7).
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