Issue 39

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02 8 behaviour of plain concrete can be found in the work of authors in [4], Willam and Warnke [5], Bazant [6], Dragon and Mroz [7], Schreyer [8], Chen and Buykozturk [9], Onate [10], Pramono and Willam [11], Etse and Willam [12], Menetrey and Willam [13], and Grassl [14]. The use of pure plasticity theory is not sufficient due to the gradual decrease in the stiffness of concrete due to the occurrence of cracks [1]. This problem can be removed when damage theory is used, i.e. by using an adequate damage model. However, as Grassl claims [15], independent damage models are not sufficient when the description of irreversible deformations and the inelastic volumetric expansion of concrete is required. Despite the above-mentioned limitations of both approaches, there are advantages to using both of them in mutual combination, and they can be combined further with other approaches formulated within the framework of nonlinear fracture mechanics. However, the use of combined material models results in a problem in real life in the form of the large amount of parameters which need to be known for the selected material model before the launch of the numerical simulation itself. Unfortunately, not all data regarding these parameters, which can be both mechanico-physical and fracture-mechanical in nature, may be available in advance. This problems can be resolved with use of other advanced mathematical approaches like optimization analysis. The varied spectrum of possibilities characterizing the use of optimization methods includes, among other options, the above mentioned inverse identification of the unknown parameters of the nonlinear material models utilized in numerical analyses performed with the finite element method. The optimization algorithms exploited in the inverse analysis of unknown material parameters, described within references [16, 17], constitute a counterpart to methods based on the training of artificial neural networks as discussed by Novak and Lehky [18]. However, both in cases where optimization is applied to the identification problem and during any classic use of the optimization methods presented in [19], the decisive factor to support a successful optimization process consists in selecting the appropriate algorithm and correctly formulating the relevant objective function. The actual need of such a function becomes even more prominent in identification using optimization modules implemented within ANSYS Workbench [20], where the definition and computation of the objective function value have to be performed with an external program or script. The task embodying the inverse identification of unknown material parameters consists in utilizing the experimentally measured curves that characterize the relationship between the load L and the deformation d ( L-d curves). During the actual identification, this reference pattern is compared with the L-d curves produced by the nonlinear numerical simulations within the corresponding experiment. The basis of the comparison then rests in calculating the similarity ratio, which is represented by one or more numerical values and also prescribes the objective function. With respect to the formulation of the objective function, the optimization task is, in a given case, defined as the minimization of the similarity ratio. For the discussed purpose, it appears advantageous to employ the RMSE (Root-Mean-Square Error) ratio, an instrument that, according to [21], enables us to compare the differences between the values measured and those generated via a mathematical model; in this context, the authors of reference [22] analyze the application of the RMSE ratio within disciplines such as meteorology, economics, and demography. Considering the shape of the L-d curves, it is then possible, as shown in study [23], to exploit them in comparing the value of the surface below the loading curve with the maximum load value. Importantly, if the second one of the described variants is used, we also have to select a correct and robust algorithm to facilitate the optimization including multiple objective functions; this problem can be further encountered in the computation of more RMSE ratios for partial sections of the curve, whose positive impact is embodied in the analysis of the individual parameters‘ sensitivity to specific sections of the loading curve. With respect to the above-outlined conditions, the present article examines the effect exerted by different formulations of the objective function in a given identification task. For the identification proper, we chose the L-d curve measured during a three point bending test on a notched concrete beam, according to [24]. The numerical simulation of the experiment was performed with ANSYS via a nonlinear, multi-parametric material model of concrete adopted from the multiPlas library [25]. Generally, the paper aims to describe the applicability of the above-mentioned possibilities of formulating the objective function; the computation of the individual options was enabled by scripts created in Python. I NPUT D ATA n order to analyze the suitability of the selected objective functions, we chose one L-d curve associated with the set of fracture tests published by Zimmermann et al. [24]. The specimen, a notched concrete beam manufactured from class C25/30 concrete and having the length l equal to 360 mm, height h of 120 mm, width w corresponding to 58 mm, and notch height of 40 mm, was configured for three point bending test; the vertical deformation d was measured in the middle of the span of 300 mm at the bottom side of the specimen. The cited article presented experimental and numerical research where the identification procedure based on utilization of neural network was used. The identification I