58 S.M. Kleinendorst et al. 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. 9.3 Number of iterations needed before convergence within a set accuracy limit is reached for the Gauss-Newton 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 9.3.2 Trust-Region For the Trust-Region method the MATLAB optimization toolbox is used. The used Optimization Toolbox algorithm is trust-region-reflective in the fmincon solver. Trust-Region algorithms makes use of trust regions; a defined subdomain N in which the objective function f is approximated with a simpler function q which reflects the behavior of the true objective function. A trial steps is computed by minimizing over this trust region. When the sum of squares is lower for the trial x Cs, the current point is updated. Otherwise, the current point remains and the trust region N is shrunk. The approximated function q is often the Taylor approximation of f .x/. Similar to Gauss-Newton, the Trust-Region algorithm requires an objective gradient and the Hessian Matrix of the posed function, which tends to be slower but more robust to local minima. Using a Trust Region further increases robustness. Figure 9.4 shows the amount of iterations required to reach the convergence criterium, with 3 (i.e. NDOF C1) FE simulations per iteration due to the calculation of the gradient and Hessian. The Trust-Region method shows less robustness for low initial guesses, but convergence performance is higher than the custom Gauss-Newton method. 9.3.3 Nelder-Mead The Nelder-Mead algorithm is a derivative-free method, i.e. no Hessian matrix or gradient is required. The Nelder-Mead method is simplex-based. A simplex S inRn is defined as the convex hull of nC1vertices x 0; : : : ; xn 2 Rn. One iteration of the Nelder-Mead method consists of the following three steps: 1. Ordering: Determine the indices h; s; l of the worst, second worst and the best vertex, respectively, in the current working simplex S. 2. Centroid: Calculate the centroid c of the best side; this is the one opposite the worst vertex xh. 3. Transformation: Compute the new working simplex from the current one. The transformation of the simplex is controlled by four parameters; ˛ for reflection, ˇ for contraction, for expansion and ı for shrinkage. Figure 9.5 shows resp. (1) reflection, expansion, contraction and (2) shrinking operations. Based on the function values of vertices x0; : : : ; xn, an appropriate transformation step is chosen. It depends on this choice how many FE evaluations are required per iteration.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==