56 S.M. Kleinendorst et al. After all, the residual converges to zero if the FE simulation matches the true, experimental, behavior of the specimen and the back-deformed image calculated with this correct displacement field correlates to the undeformed image. Different approaches exist to solve this minimization problem. Most commonly used in DIC algorithms is a (modified) Gauss-Newton method. This method is relatively fast in the proximity of the solution, but gives robustness issues if the initial guess is farther from the solution. Other methods exist which have their own advantages and disadvantages. A key issue in integrated DIC is the computational cost consumed by the number of FE simulations performed. Every iteration in the optimization method requires at least one FE evaluation. Some methods, however, require the derivative of the objective function, i.e., the residual, towards the degrees of freedom, i.e., the model parameters. This derivative is not analytically available and therefore, often a finite difference scheme is used to approximate the derivative. This calculation requires the number of degrees of freedom amount of FE evaluations, thereby increasing calculation cost and CPU time immensely. Therefore, the different optimization methods are compared based on the number of FE simulations needed to converge. The lay-out of this paper is as follows. First the virtual experiment used to evaluate the performance of the different optimization methods is introduced in Sect. 9.2. Then, in Sect. 9.3 three different optimization methods are presented and their performance is compared. Finally, conclusions are drawn in Sect. 9.4. 9.2 Virtual Test Case The performance of the different optimization methods is analyzed based on a virtual experiment. This virtual experiment concerns a tensile experiment with a dogbone tensile specimen, see Fig. 9.1, on which a load is prescribed on both edges. In a virtual experiment no real experimental images are used, but a finite element (FE) simulation is executed and the resulting displacement fields are used to artificially deform a reference image, see Fig. 9.2. These images form the input for the integrated DIC method. The objective is to correlate these images in order to find the model parameters, in this case the Young’s modulus E and Poisson ratio , that were used in the elastic isotropic FE model used to create the images. The parameters used to create the virtual experiment are Ep D1:3 10 5 Pa and p D0:28. 9.3 Optimization Methods Three different iterative optimization methods are tested. The first is a Gauss-Newton method that is often implemented in (integrated) digital image correlation approaches. This is a custom coded method. The other two methods are used from the MATLAB optimization toolbox. The first of these methods is a Trust-Region method with a Newton approach. The last Fig. 9.1 Finite element simulation for the virtual experiment concerning a dogbone tensile sample. A load is prescribed on the left andright edge. A quarter of the structure is modeled, the red lines indicate symmetry lines. The region of interest is indicated by the green rectangle
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