Issue 39

P. Král et alii, Frattura ed Integrità Strutturale, 39 (2017) 38-46; DOI: 10.3221/IGF-ESIS.39.05 44 Optimization The aim of the optimization process consisted in finding such identified parameter values at which the value of the objective function was minimal. The objective function utilized within the entire inverse analysis process was defined as a sum of squares by the formula:   2* 1 n i i i LSM y y     (6) where, for y i , we substituted the loading force values obtained from the numerically-simulated load-displacement curve at the given deformations, and y i * was substituted with the loading force values obtained from the experimentally-measured load-displacement curve at the same deformations. As already mentioned, the objective function was minimized during the optimization process ( LSM  min), meaning that we sought such material parameter values at which the load- displacement curve obtained from the numerical simulation exhibited the smallest possible deflection from the reference load-displacement curve produced by the experiment; in other words, we sought such values at which the sum of squares was minimal. It then follows from this description that the inverse analysis was based on minimizing the sum of squares (the Least Squares Minimization) [15]. As pointed out above, the optimization involved only those material parameters that were part of the reduced design vector. The remaining parameters were defined by the constant values from their initial range of the variability. The optimization of the parameters was performed using the optimization procedure known as Evolutionary Algorithm (EA) [14], an optimization approach that exploits processes inspired by biological evolution, including, for example, reproduction, mutation, and recombination. More concretely, we utilized in this context the Evolutionary Algorithm used for global optimization, with the 10 best realizations acquired within the sensitivity analysis serving as the starting point for the discussed algorithm. The values of the identified material parameters provided by the best generation of the Evolutionary Algorithm are, together with the relevant minimum value of the objective function, presented in Tab. 2. Fig. 4 below compares the load-displacement curve obtained via the numerical simulation, in which we applied the optimized parameter values of the K & C Concrete model from the Evolutionary Algorithm (EA), with the experimentally-measured load-displacement curve. It is then obvious from the representation that the parameters of the selected material model were identified very accurately because the results of the numerical simulation ensure a very good approximation of the experimental data. Figure 4 : The load-displacement curve for the optimized parameter values compared with the experimental load-displacement curve. C ONCLUSION his paper was based on performing an inverse analysis to identify the material parameters of the Karagozian & Case Concrete constitutive model. The inverse analysis was carried out utilizing the load-displacement curve experimentally measured during the triaxial compression strength testing of concrete cylinders. The actual results T