Available online at ScienceDirect. Procedia Engineering 161 (2016 )

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1 Available online at ScienceDirect Procedia Engineering 161 (2016 ) World Multidisciplinary Civil Engineering-Architecture-Urban Planning Symposium 2016, WMCAUS 2016 Parameter Identification for a Multivariable Nonlinear Constitutive Model inside ANSYS Workbench Filip Hokeš a, *, Jií Kala b, Martin Hušek c, Petr Král d a,b,c,d Brno University of Technology, Faculty of Civil Engineering, Vevei 331/ Brno, Czech Republic Abstract This contribution aims to describe the process of the inverse identification of the parameters of a nonlinear material model from experimentally obtained data. This process takes place with the aid of approaches utilizing optimization procedures based on population methods which are currently implemented in the ANSYS Workbench environment. The input data for the described numerical procedure took the form of the points of a load-displacement curve which was measured during the performance of a three-point bending test on a concrete beam. This experiment was numerically simulated via the finite element method with the use of the Menétrey-Willam nonlinear material model. Great attention is paid to the description of the sensitivity analysis and the parameter correlation performed with the utilization of a programmed script that enables the correct understanding of the used material model. Emphasis is therefore placed on the analysis of individual parameters whose understanding and correct setting have a significant influence on the convergence of the nonlinear solution. The basic principle of the identification by optimization is the minimization of the difference between experimentally and numerically obtained load-displacement curves. However, the problem is how to formulate this difference as precise as possible because the right choice of objective function is crucial for achieving the optimum. One possible way is to use the root-mean-squared error that is often used for evaluation of accuracy of economy or weather mathematical models. The text also deals with the possibility of a reduction in the design vector according to the results of sensitivity analysis and shows how this reduction affects the accuracy of the sought parameters. The contribution provides another view on the utilization of optimization algorithms in the area of the design of safe and effective structures The Authors. Published by Elsevier by Elsevier Ltd. This Ltd. is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing committee of WMCAUS Peer-review under responsibility of the organizing committee of WMCAUS 2016 Keywords: Optimization; ANSYS Workbench; sensitivity; nonlinear material model; root-mean-squared error; identification; * Corresponding author. Tel.: address: hokes.f@fce.vutbr.cz The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing committee of WMCAUS 2016 doi: /j.proeng

2 Filip Hokeš et al. / Procedia Engineering 161 ( 2016 ) Introduction In civil engineering, optimization is commonly used when seeking an optimum geometrical configuration and a material with suitable mechanical-physical characteristics. The aim of such optimization is to ensure low investment and operational costs. However, the execution of correct optimization is not a simple task as it requires detailed knowledge of the analyzed problem and the selection of a suitable optimization algorithm. Many such algorithms are incorporated in computational systems, and their efficiency depends on the nature of the investigated problem [1]. Apart from the application mentioned above, optimization can also be used to identify the parameters of complex nonlinear constitutive relationships. The effort to design efficient and safe structures has led to the growing use of materially nonlinear simulations which aim to provide the most exact description possible of the real behaviour of structures [2]. With regard to these issues, special attention is paid to structures made from concrete, which are some of the most widely used thanks to the technical and structural advantages of this material [3]. However, the derivation of a correct constitutive relationship for concrete appears to be problematic. Problems arise as a result of the difference in tensile and compressive strengths and also due to the existence of irreversible plastic deformations, non-elastic volume changes and the influence of the decrease in stiffness that occurs due to the initiation of cracks. The occurrence of such phenomena has led to the creation of many nonlinear material models based on pure plasticity or damage theory, but as Cicekli et al. [3] and Grassl and Jirásek [4] suggest, the use of only one of these approaches individually is insufficient. As a result, they are more frequently used in mutual combination, or other tools are used that are based on nonlinear fracture mechanics. A representative of this concept is the multiplas material model database [5], which is used for nonlinear calculations in the ANSYS system [6]. However, the described concept based on the combination of various theories in one material model with the aid of this library leads to a complication in the form of the existence of an extensive class of mechanical-physical and fracture-mechanical parameters that are unknown in advance. These parameters can be identified relatively easily from experimental measurements, particularly if the help of neural networks [7] or an optimization [8] is enlisted. Optimization algorithms are currently an inseparable part of the ANSYS computational system, and thus the nonlinear calculation of a structure itself, including the identification of unknown parameters. They can be carried out inside this system with no other external optimization applications required. However, the determination of the objective function, which is minimized, is problematic. Thankfully, a solution can be easily found using a script programmed in the Matlab language. Nomenclature d Vertical displacement measured during testing the concrete specimen at the mid span on the lower surface E Young s modulus of elasticity f t Ultimate tensile strength f c Ultimate compressive strength G fc, Specific fracture energy in compression G ft Specific fracture energy in tension h Height of the specimen k Ratio between biaxial compressive strength and uniaxial compressive strength L Load measured on the testing machine l Length of the specimen n Number of observations (samples) RMSE Root-mean-squared error measure w Width of the specimen X Full design vector (all material parameters included) Xred Reduced design vector * y i Value of the force calculated within the framework of the numerical simulation y i Value of the force gained from the experimental L-d curve Plastic strain corresponding to the maximum load ml

3 894 Filip Hokeš et al. / Procedia Engineering 161 ( 2016 ) ci cr tr Poisson s ratio Dilatancy angle (friction angle) Relative stress level at the start of nonlinear hardening in compression Residual relative stress level in compression Residual relative stress level in tension 2. Analysis of input data For the purpose of inverse identification, data measured during a fracture experiment published in the contribution [9] were used. The test specimen for this experiment was made from concrete, class C25/30. The final dimensions of the specimen were: length l = 360 mm, height h = 120 mm and width w = 58 mm. A 40 mm high notch was made in the sample at midspan. The specimen was tested in a 3-point bending test configuration in which the supports were 300 mm apart. The result of the experiment was a diagram showing the dependence of loading force L and vertical displacement d measured at midspan. The coordinates of the points of this L-d diagram were used as points for the reference curve entering the optimization process. 3. Identification of parameters in the ANSYS Workbench environment The process of the inverse identification of the parameters of the chosen (Menétrey-Willam) nonlinear material model from the multiplas database was carried out in the ANSYS Workbench environment via the repeated numerical calculation of the experiment described above. Because the setting of the material model was carried out via a programmed batch, it was possible to perform the whole calculation in this way. An advantage of this procedure was the simplicity of calling an external script for the evaluation of the similarity between the reference curve and the calculated L-d curve. The programmed batch thus carried out the assembly of the geometry of the computational model, the definition of the material model, the launch of the nonlinear calculations, the creation of the resultant L-d curve and the calling of the external script which enabled the calculation of the objective function value Geometry of the computational model For the creation of the computational model of a notched concrete beam, PLANE182 4-node planar elements were used. With the aid of this type of element, the experiment was carried out as a plane stress task with a pre-defined element thickness w = 58 mm. The created finite element mesh was regular, the edge length of each element being 6 mm. The stated simplifications were absolutely justified according to [10], as that publication shows that various finite element mesh sizes produced the same results in the case of the elements SOLID185 and PLANE182. It was necessary for the simulation to be repeated many times, but due to the described simplifications there was a significant decrease in computational time requirements Material model An elasto-plastic material model from the multiplas database was used to ensure the computational model exhibited nonlinear behaviour as far as the material was concerned. Due to its suitability, proven in several studies published by the authors Hokes et al. [11] and Hokes [12], use was made of a constitutive relationship based on the plasticity surface derived by Willam and Warnke [13] and modified by Klisinský [14] and Menétrey and Warnke [15]. This model ranks among the group of models with non-associated flow rule that consider invariants of stress tensors as well as invariants of deviatory stress tensors. From the point of view of the use of the FEM, the chosen material model utilizes the smeared crack concept [16]. The given problem was solved with the aid of a softening function based on the dissipation of specific fracture energy G ft. With regard to the need to remove the negative dependence of the solution on the size of the mesh of finite elements, the nonlinear Menétrey-Willam model makes use of Bazant's Crack Band concept [17]. The described nonlinear material model was only assigned to elements from the area of the notch, namely in a 240

4 Filip Hokeš et al. / Procedia Engineering 161 ( 2016 ) mm wide strip. Due to the occurrence of local stress peaks, only a linear isotropic material defined with the help of elasticity modulus E and Poisson s ratio was selected for use above the supports The objective function and its calculation For the evaluation of the similarity between the reference and numerically calculated L-d curves, the RMSE (rootmean-square error) measure was used. It is frequently used for measuring the difference between the values generated by a mathematical model and observed values [18]. This measure is mainly used in the fields of meteorology, economics and demography. The expression for the calculation of the RMSE had the following form: RMSE n * 2 ( y y ) i i i1 (1) n where yi * is the value of the force in the i-th point of the experimental L-d curve, and y i is the value of the force calculated using a nonlinear material model for the i-th point of the curve. However, the distribution of points on the horizontal axis was not identical to the distribution of points on the reference curve due to the different behaviour of the solver, and therefore it was not possible to calculate the RMSE immediately after solving the calculation task. For this reason, a script programmed in the Matlab language was used which carried out a mapping of the position of such points according to the reference curve with the help of linear interpolation Sensitivity analysis and correlation of parameters The identification of the material model was preceded by a sensitivity analysis. The main aim of this part of the analysis was to map the area of design variables and determine the level of sensitivity of the individual material parameters to the value of the objective function. 250 simulations were carried out in total, during which the uniformity of the coverage of the design space was ensured using the LHS method. The sensitivity level of the individual material parameters to the RMSE values was expressed using Spearman s correlation coefficient. The sensitivity analysis showed that the elasticity modulus E, uniaxial tensile strength f t and specific tensile fracture energy G ft had the highest sensitivity. The results of the sensitivity analysis are shown in Fig. 1. Fig. 1. Results of the sensitivity analysis.

5 896 Filip Hokeš et al. / Procedia Engineering 161 ( 2016 ) Identification with the aid of an optimization algorithm Identification of the unknown parameters of the selected material model was carried out via direct optimization using a genetic algorithm. The calculation involved the introduction of a programmed batch in which material parameter values were re-written using a genetic optimization algorithm. The identification of the parameters was carried out twice, and for different design vectors. In the first case, all the parameters of the design vector were sought: t c ml fc cr ci ft T tr X E,, f, f, k,,, G,,, G, (2) In the second case, identification was carried out on the basis of the sensitivity analysis in the design area of three variables. The form of the reduced designed vector was thus: 4. Results T X red E, f t, G ft (3) The use of the optimization algorithms based on genetic and population theories proved to be the right choice, as the algorithm converged both for the parameter identification in the reduced design vector and (mainly) for the identification of a full design vector. In the case of the full design vector, the optimum (RMSE = ) was reached in the 209 th design point, with the following parameter values: E = GPa, f t = MPa, G ft = Nm/m 2. In the case of the reduced design factor, the optimum (RMSE = ) was achieved in the 33 rd iteration, with the identified parameter values: E = MPa, f t = MPa, G ft = Nm/m 2. A comparison of the development of the objective function for both identification processes and a comparison of the resultant curves is shown in Fig Conclusion Fig. 2. Objective function timecourse (left) and comparison of L-d curves (right). The aim of the described study was to carry out the inverse identification of the parameters of a nonlinear material model in the ANSYS Workbench. As the task was conducted using a material in tension during flexure, only the elasticity module E, tensile strength f t and specific tensile fracture energy G ft can be considered to be realistically identified parameter values. The submitted study shows that even though ANSYS does not offer its own calculator enabling the evaluation of the objective function, the appropriate results can be achieved using an external application. It can also be stated that when the design vector is reduced, there is a significant saving in the computational time needed for the execution of a complete identification.

6 Filip Hokeš et al. / Procedia Engineering 161 ( 2016 ) Acknowledgements This contribution was created with the financial aid of project GACR S Aspects of the use of complex nonlinear material models provided by the Czech Science Foundation and also with support of the project FAST-J "Implementation of material models of concrete in system ANSYS and their experimental verification" of the specific university research of Brno University of Technology. References [1] F. Fedorik, J. Kala, A. Haapala, M. Malaska, Use of design optimization techniques in solving typical structural engineering related design optimization problems, Structural Engineering and Mechanics, vol. 55, no. 6, pp [2] F. Hokes, The Current State-of-the-Art in the Field of Material Models of Concrete and other Cementitious Composites, in: K. Wongseedakaew, Q. Luo (Eds.), Applied Mechanics and Materials, Trans Tech Publications, Switzerland, vol. 729, pp [3] U. Cicekli, G. Z. Voyiadjis, R. K. Abu Al-Rub, A plasticity and anisotropic damage model for plain concrete, International Journal of Plasticity. 23, (10-11), pp [4] P. Grassl, M. Jirásek, Damage-plastic model for concrete failure, International Journal of Solids and Structures, vol. 43, no , pp [5] Dynardo GmbH, Multiplas: User s Manual Release for Weimar. [6] ANSYS Inc, ANSYS Mechanical Theory Reference: Release Canonsburg PA, USA. [7] D. Novák, D. Lehký, ANN inverse analysis based on stochastic small-sample training set simulation, Engineering Applications of Artificial Intelligence, vol. 19, no. 7, pp , [8] M. Vaz, Cardoso, E. L., Muñoz-Rojas, P. A., Carniel, T. A., Luersen, M. A., Tomiyama, M., da Silva, J. O., Stahlschmidt, J., and Trentin, R. G., Identification of constitutive parameters - optimization strategies and applications, Materialwissenschaft und Werkstofftechnik, vol. 46, no. 4-5, pp [9] T. Zimmermann, A. Strauss, D. Lehky, D. Novak and Z. Kersner, Stochastic fracture-mechanical characteristics of concrete based on experiments and inverse analysis, Construction and Building Materials, vol. 73, pp [10] F. Hokeš, Selected Aspects of Modelling of Non-Linear Behaviour of Concrete During Tensile Test Using Multiplas Library, Transactions of the VŠB Technical University of Ostrava, Civil Engineering Series, vol. 15, no. 2. [11] F. Hokes, J. Kala, O. Krnavek, Optimization as a Tool for the Inverse Identification of Parameters of Nonlinear Material Models, in: A. Bulucea (Eds.) Proceedings of the 9th International Conference on Continuum Mechanics, Rome, Italy, pp [12] F. Hokes, Comparison of suitability of selected material models of concrete for inverse identification of parameters with aid of optimization algorithms, in: Extended Abstracts of 31th conference with international participation Computational Mechanics Plze, Czech Republic, University of West Bohemia, pp [13] K. J. William, E. P. Warnke, Constitutive model for the triaxial behavior of concrete, International Association of Bridge and Structural Engineers, vol. 19, pp [14] M. Klisinsky, Degradation and plastic deformation of concrete, Polish Academy of Sciences, Iftr report 38. [15] P. Menetrey, K. J. Willam, Triaxial failure criterion for concrete and its generalization, ACI Structural Journal, vol. 92, no. 3, pp [16] R. Pölling, Eine praxisnahe, schädigungsorienten Materialbeschreibung von Stahlbeton für Strukturanalysen, PhD Thesis, Bochum. [17] Z. P. Bažant, Oh, B. H., Crack band theory for fracture of concrete, Matériaux et Constructions, vol. 16, no. 3, pp [18] R. J. Hyndman, A. B. Koehler, A. B., Another look at measures of forecast accuracy. International Journal of Forecasting, vol. 22, no. 4, pp