Optimization as a tool for the inverse identification of parameters of nonlinear material models

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1 Optimization as a tool for the inverse identification of parameters of nonlinear material models FILIP HOKES, JIRI KALA, ONDREJ KRNAVEK Institute of Structural Mechanics Brno University of Technology, Faculty of Civil Engineering Veveri 331/95, Brno, CZECH REPUBLIC hokes.f@fce.vutbr.cz, kala.j@fce.vutbr.cz, krnavek.o@fce.vutbr.cz Abstract: - At present, optimization methods are finding application in many technical as well as nontechnical fields. One of the possible applications of optimization procedures is in the identification of formerly unknown parameters of nonlinear constitutive relationships of construction materials which are used in numerical simulations of the behaviour of built structures. In this respect, optimization constitutes a certain counterweight to other soft-computing methods which also include methods of identification based on the exercise of artificial neural networks. The topic of the following contribution is the description of the possible connection of a computational system based on the finite element method with an optimization tool, and an analysis of the identification of the mechanico-physical and fracture mechanical parameters of a material model for concrete based on the Willam-Warnke yield surface from an experimentally-measured loading curve. The individual steps of this inverse identification process are described in individual paragraphs, including a description of the model used, all with the aim of creating a comprehensive procedure which can be utilized in further research or in the practical design of efficient built structures. Key-Words: - FEA, nonlinear material model, inverse identification, optimization, concrete, ANSYS 1 Introduction The numerical solution of tasks in the area of the design of structures has been very popular for many years. This popularity is a result of both the already very good existing knowledge of numerical methods of mathematical modelling and, among other things, the fact that the numerical simulation of certain complex and atypical structural elements in computational systems such as ANSYS [1] or LS- Dyna [2] is significantly cheaper than the experimental testing of real specimens [3,4]. Despite the level of development of current computational tools, efforts are constantly underway to improve them still further, which is most true of all for the area of nonlinear mechanics. Research into the description of the nonlinear behaviour of materials and the testing of the usability of derived constitutive relationships in the calculations needed for specific civil engineering and mechanical engineering tasks is of high importance. With the correct use of an exact model, and when the given prerequisites are fulfilled, the closer approximation of a simulation to the real behaviour of a structure can be achieved with numerical calculations. One result of such an approximation may be the design of still more economical and safer buildings. An example of such a calculation can be found in [5]. With regard to the efforts mentioned earlier, a database of elastoplastic material models, multiplas, has been developed for nonlinear simulations in the ANSYS system. However, the use of this and similar tools 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, these parameters, which can be both mechanico-physical and fracture-mechanical in nature, may not be well known in advance. In the above-mentioned situation it is possible to make advantageous use of experimentally obtained load displacement curves measured during the testing of test specimens fabricated from selected materials and reverse identify needed parameters with the aid of inverse identification. The inverse identification process is currently executed by exercising artificial neural networks or with the help of optimization algorithms implemented in academic or commercial optimization software tools. In the case of inverse identification using optimization, the optimization task can be defined as the minimization of the ISBN:

2 difference between the experimental load displacement curve and a curve obtained as the output of a numerical simulation in the selected computational system. In such a task, the design vector is created by previously unknown parameters of the nonlinear material models while the delimiting conditions are derived from the prerequisites from which the used material model was derived, or from the requirement for the smooth convergence of the solution. An example of a similar task can be found in conference contribution [5]. The presented article deals with the abovementioned inverse identification of the parameters of a nonlinear material model from the multiplas library with the help of an optimization algorithm implemented in the optislang programme. These parameters were established from an L-d load displacement curve which was obtained from a three-point bending test on a notched concrete beam. The ANSYS computational system was selected for the numerical simulation of this task. The aim of the article is thus to describe the individual steps for the above-mentioned inverse identification method and provide a comprehensive procedure which is useable in further research and in the design of larger structural units. 2 Definition of the problem The inverse identification of parameters for the Menétrey-Willam material model from the multiplas library was performed on a loaddisplacement curve obtained from an experimental investigation published in [6]. This curve was the output of a three-point bending test carried out on a notched concrete beam with the dimensions 360 x 120 x 58 mm. This is one of several typical tests used to identify the material and fracture parameters of quasi-brittle materials [6]. In order to achieve savings with regard to the computation time needed for the nonlinear simulation, the complexity of the plane stress task was reduced from 3D to 2D. With regard to the simplification mentioned above, boundary conditions were also simplified. In real experiments, support and the application of load both take place using steel cylinders. When the numerical simulations were carried out, only zero vertical displacements were prescribed at the supports. The application of load was also realized in the form of vertical displacement: in order to maintain solvability and obtain the convergence of the solution, horizontal displacement was prevented at the load input location. The final appearance of the idealization of the solved task is depicted in Fig. 1. Fig.1 Idealization of the testing configuration 2.1 Analysis of the input data The cited publication [6] presented tests on four sets of concrete samples of strength class C25/30 according to [7]. Within these four sets, concrete mixtures with slump values of F45 and F70 were combined and then tested at the age of 28 and 170 days. Each set was composed of 5 test specimens. As previously mentioned, the concrete samples were subjected to three-point bending tests, the output of which was a diagram of the dependence of loading force L (measured on a testing machine) and displacement d (measured on the test specimen at midspan). For the purposes of the presented article, only one load-displacement curve was chosen for one of the test specimens with a slump value of F45 and an age of 28 days. The form of this L-d curve is clearly depicted in Fig. 2. A maximum deformation value d max of m was derived from this reference curve. The stated load value enabled the post-critical behaviour of the concrete sample (softening curve) to be expressed within the numerical simulation. Fig.2 Reference L-d curve from a three-point bending test ISBN:

3 3 Description of the process of inverse identification using optimization methods The whole process of inverse identification in the optislang programme was divided into two main parts: sensitivity analysis and global optimization. The individual numerical simulations were dealt with using the Newton-Raphson nonlinear equation solver implemented in ANSYS software. 3.1 Nonlinear numerical analysis A numerical analysis of the three-point bending test was conducted via the finite element method within the ANSYS 15.0 computational system. The nonlinear behaviour of the computational model used in the simulation was determined using the Menétrey-Willam material model from the multiplas library of elasto-plastic material models. Due to task automation requirements, the whole calculation was controlled by a programmed macro which set up the geometry of the computational model and the parameters of the chosen constitutive law, input the solver settings, solved the calculation and finally exported the file containing the points for the L-d diagrams The computational model A computational model of a concrete specimen was put together with previously-defined dimensions from a total of 4800 PLANE182 4-node planar elements with 3 mm long edges and a thickness of 58 mm. At the location of the notch, two parallel lines were created with a shared node at the top of the notch. With regard to the fact that the monitored area with strongly nonlinear behaviour was located above the top of the notch, and with regard to the occurrence of local peaks of tension in the area above the supports, a linear material model was assigned for the elements at both ends of the computational models within a 30 mm-wide area on both sides of the supports The final appearance of the computational model is depicted in Fig The material model A material model from the multiplas library called the Menétrey-Willam model was selected for the numerical solution of a three-point bending test simulation in the ANSYS system. This material model belongs to a group of material models of concrete which do not consider the influence of the strain rate on stress and in which the decomposition of the total plastic strain vector, ε tot, into an elastic component, ε el, and a plastic part, ε pl, is thus expected [8]. ε = ε + ε (1) tot el pl From the perspective of plastic flow, the model used ranks among the group of models with nonassociated plastic flow. The Menétrey-Willam material model [9] is based on the Willam-Warnke yield surface [10], which is, in contrast with the Drucker-Prager surface, a function not only of the first and second but also of the third invariant of the stress deviator (known as the Lode angle ). This modification achieves the softening of the corners of deviatoric planes of the yield surface which (in addition) do not lie at a constant distance from the hydrostatic axis in Haigh-Westergaard space. This aspect is the basic difference in relation to the modified Drucker- Prager model, and it thus expands the possibilities for the utilization of the plasticity model for other types of loading conditions. From the point of view of the use of FEM the chosen material model utilizes the smeared crack concept [11]. The given problem was solved with the aid of a softening function based on the dissipation of specific fracture energy G ft, which thus acts as one of the sought parameters. 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 Bažant s Crack Band concept. [12]. In order for this material model to function correctly, a further 11 parameters had to be defined in addition to the above-mentioned specific fracture energy. Basic description of these parameters can be found in Tab. 1. Fig.3 The computational model ISBN:

4 Table 1 Parameters of the selected material model Par. Unit Description E [Pa] Young s modulus of elasticity ν [-] Poisson s ratio f c [Pa] Uniaxial compression strength f t [Pa] Uniaxial tension strength k [-] Ratio between biaxial compressive strength and uniaxial compressive strength ψ [ ] Dilatancy angle (friction angle) ε ml [-] Plastic strain corresponding to the maximum load G fc [Nm/m 2 Specific fracture energy in ] compression Relative stress level at the start of Ω ci [-] nonlinear hardening in compression Ω cr [-] Residual relative stress level in compression G ft [Nm/m 2 ] Specific fracture energy in tension Ω tr [-] Residual relative stress level in tension Within the sensitivity analysis, a total of 300 random realizations of design vectors were executed. Sufficient coverage of the design space was ensured using the ALHS method [14]. The form of the boundary curves and their position with regard to the reference L-d curve is depicted in Fig. 4 along with sufficient coverage of the design space via random realizations. Sensitivity analysis proved that the following material model parameters were most used to find the corresponding form of the L-d curve: Young s elasticity modulus E, Poisson s ratio ν, specific fracture energy in tension G ft, and the relative value of residual tensile strength Ω tr. Another interim result of the sensitivity analysis was the first generation of design vectors used for the subsequent optimization process. These realizations were selected on the basis of the calculated value of the target function. 3.2 Inverse identification The inverse analysis itself was performed in the environment of the optislang optimization system in two stages. The aim of the sensitivity analysis was to reveal the level of sensitivity of the individual parameters to the resultant form of the L- d curve, and thus limit the number of parameters of the material model to the necessary minimum and possibly modify the intervals of the values of these parameters. Within the second stage, the optimization itself was performed with a reduced vector of the designed parameters using a suitable optimization algorithm Sensitivity analysis Sensitivity analysis is basically a task which seeks the level to which output data uncertainties are influenced by the variability of input data [13]. Before the beginning of the sensitivity analysis itself, a set of purely empirical testing simulations was peformed whose aim was to obtain boundary curves. The parameters which correspond to these curves were then used as limit boundaries of the design intervals for the sensitivity analysis. The secondary aim of these testing calculations was to prove the solvability of the calculations for the boundary values of parameters in the computational system, which also implied their solvability within the selected intervals as a result. Fig.4 Realizations of L-d curves in sensitivity analysis Global optimization According to [15], the term optimization can be defined as an effort to obtain the best possible result under a given set of conditions. Optimization is used for the minimization of needed effort or the maximization of an effect. The described inverse identification task can also be defined in the stated intentions. This is namely the effort to obtain an L-d curve from nonlinear numerical simulations that would best correspond to the sample curve obtained from the experiment. In other words, the aim of the executed optimization is to minimize the difference between the reference and calculated curve by varying the input parameters of the constitutive relation. The global optimization process was carried out using an algorithm which ranks among what are known as population methods. These are stochastic ISBN:

5 optimization methods based on the principle of the imitation of biological processes [16]. Two types of population method were implemented in the optislang program, (a) genetic algorithms (GA) and (b) evolution strategies (ES); a combination of these methods could be used within the given software for the solution of the described problem. The objective function for the given optimization problem had the following form: n i= 1 * i y i 2 ERROR = ( y ), (2) where y i * represented the value of the force originating from the numerical simulation and y i was the value of the force obtained from an experimentally obtained L-d curve. With regard to the results of the previous sensitivity analysis, the reduced design vector in the form given below was selected for the optimization task in order to reduce the time required for the calculation: red { E, ν, f, ψ,, Ω, G } T X = ε (3) t ml tr The initial population of design vectors for the optimization was composed of the 10 best sensitivity analysis realizations. In order to find an optimum solution, a total of 25 generations of the design vectors were created and thus a total of 250 numerical simulations were carried out in the ANSYS system. An optimum result was achieved with a final objective function value ERROR = A comparison of the curve for the final values of the parameters and the reference L-d curve is shown in Fig. 5. More accurate results can be achieved with more generations of design vectors and further improvement of the results can be increased by implementing a local optimization process, which is included in the future plans of the authors. Fig.5 L-d curve for optimized parameters ft 4 Results The form of the resultant curve for the identified parameters of the material model has already been presented in Fig. 5. The sizes of the individual values of the parameters of the material model are documented further in Tab. 2. Apart from the final values, the table also shows the values of the preoptimized parameters which originate from the sensitivity analysis, including the value of the target function for this set of parameters. Table 2 Results final values of parameters Par. Units Sensitivity Global optimization ERROR [N 2 ] E [Pa] ν [-] f c [Pa] f t [Pa] k [-] ψ [ ] 10 9 ε ml [-] G fc [Nm/m 2 ] Ω ci [-] Ω cr [-] G ft [Nm/m 2 ] Ω tr [-] Conclusion The results stated in the previous paragraphs document the usability of optimization methods in the area of the inverse identification of the parameters of nonlinear material models. It became apparent, though, that when solving the calculations for a specific task such as a simulated three-point bending test, the only parameters that can successfully be defined are those which are utilized in the given loading method. In order to completely obtain all the parameters of the material model, parameters also need to be identified from experimental data originating from different tests. An undeniable advantage of the described inverse identification method is the fact that every numerically obtained curve for a certain set of parameters represents a truly converged numerical solution. This demonstrates the functionality and wide range of applications of the selected material model. A possible refinement of the optimum can be achieved by increasing the number of design vector generations or via the use of other, different ISBN:

6 optimization procedures, the particularities involved in whose application options are currently the focus of attention of the authors of this contribution. 6 Acknowledgement 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 under the project No. LO1408 "AdMaS UP - Advanced Materials, Structures and Technologies", supported by Ministry of Education, Youth and Sports under the National Sustainability Programme I".. References: [1] ANSYS, Inc.: ANSYS Mechanical Theory Reference, Release 15.0, [2] Livermore Software Technology Corporation (LSTC). LS-DYNA Theory Manual, [3] F. Hokes, Different Approaches to Numerical Simulations of Prestressed Concrete Structural Elements, Applied Research in Materials and Mechanics Engineering, Vol. 621, 2014, ISSN: , pp [4] P. Hradil, J. Kala, Analysis of the Shear Failure of a Reinforced Concrete Wall, Applied Research in Materials and Mechanics Engineering, Vol. 621, 2014, ISSN: , pp [5] T. Most, Identification of the parameters of complex constitutive models: Least squares minimization vs. Bayesian updating, Reliability Conference in München, [6] A. Strauss, T. Zimmermann, D. Lehký, D. Novák, Z. Keršner, Stochastic fracturemechanical parameters for the performance based design of concrete structures. Structural Concrete, Vol. 15, 2014, pp [7] European Committee for Standardization: Concrete Part 1: Specification, performance, production and conformity, EN 206-1, [8] Dynardo, GmbH.: Multiplas, User s Manual Release for ANSYS 15.0, Weimar, [9] P. G. Menétrey, Numerical Analysis of Punching Failure in Reinforced Concrete Structures, PhD Thesis, École polytechnique fédérale de Lausane EPFL, Lausanne, [10] K. J. Willam, E. P. Warnke, Constitutive Models of Triaxial Behavior of Concrete. In Proceedings of the International Association for Bridge and Structural Engineering, Bergamo, vol. 19, pp [11] R. Pölling, Eine praxisnahe, schädigungsorientierte Materialbeschreibung von Stahlbeton für Strukturanalysen. PhD Thesis, Ruhr-Universität Bochum, Bochum, [12] Z. P. Bazant, B. H. Oh, Crack band theory for fracture of concrete. Material and Structures, Rilem, vol. 16, 1983, pp [13] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola, Global Sensitivity Analysis: The primer. Wiley, London, [14] D. E. Huntington, C. S. Lyrintzis, Improvement to limitations of Latin hypercube sampling. Probabilistic Engineering Mechanics, Elsevier, Amsterdam, vol. 13, 1998, pp [15] S. S. Rao, Engineering Optimization: Theory and Practice, 4th edn. Wiley, London, [16] Dynardo GmbH.: Method for multi-disciplinary optimization and robustness analysis. Germany, [17] Kala, J., Hradil, P., Bajer, M. Reinforced concrete wall under shear load Experimental and nonlinear simulation, International Journal of Mechanics, 9, pp , ISBN: