9 On the Boundary Conditions and Optimization Methods in Integrated Digital Image Correlation 57 Fig. 9.2 Reference image f and artificially deformed image g. The reference image is a generated speckle pattern with both coarse and fine features, which is suitable for digital image correlation analysis. (a) Reference image f . (b) Deformed image g optimization method is the Nelder-Mead algorithm. The performance of the IDIC algorithm with the different optimization approaches is evaluated by comparing the number iterations needed to converge within a set accuracy. The computational cost is mainly determined by the number of FE evaluations executed and. This number per iterations differs for each method and therefore the total number of FE evaluations for convergence is also compared for the considered methods. The method are tested for different initial guesses on both parameters Eand . The initial guesses are given relative to the true material property values used to create the virtual experiment, Ep and jnup, and are chosen on a logarithmic scale. The set of initial guess values for Eranges from a 10 times lower value to a 10 times higher value. For the range also start with values 10 times lower than the reference value, but it can not increase to a 10 times higher value, since the Poisson’s ratio can not exceed 0.5 physically. Since the iterations are computationally costly, especially for the methods that require derivative information, a maximum of 20 iterations is set. If a initial guess combination for a certain method crosses this limit, but is in the process of converging to the correct solution, this is indicated by a blue color in the graphs. If the method is diverging, an ‘x’-symbol is depicted in addition to the blue color. 9.3.1 Gauss-Newton The Gauss-Newton method is a modification of Newton’s method for finding the objective’s minimum. Given the set of m image residuals r D.r1; ::; rm/ and the set of n variables D 1; : : : ; n/ and the initial guess .0/, the method proceeds by the iterations s described in Equation 9.1 using Jacobian Jr [5]. .sC1/ D .s/ .J T r Jr/ 1JT r r. .s/ /: (9.1) This algorithm is often used in digital images correlation methods. Usually JT r Jr is denoted as M, the optical correlation matrix, and JT r r. .s// as b, the right-hand side vector. To prevent the method from iterating towards non-physical negative values for the model parameters , an update limit it introduced. This limit states that the iterative update in can not exceed a factor ˛ < 1 times the current value of the variable, .s/. Here a factor ˛ D0:9is used. The results for a set of initial guesses is shown in Fig. 9.3. Since this method requires the derivative, each iteration requires .NDOF C1/ number of FE evaluations. The method converges for all initial guesses tried, however, for low values of the Young’s modulus converges becomes slow. The method is concluded to be relatively robust, but slow if far from the solution.
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