AP04-Bittnar1.vp 1 Introduction Utilization of an appropriate material model for realistic modeling of concrete is essential in order to capture both experimental results and real structures. Generally, the more sophisticated the model we deal with, the greater the number of model parameters that have to be considered. In better cases, basic parameters such as compressive strength, modu- lus of elasticity, etc. are known. In worse cases, practically nothing is known. Some parameters can be estimated using recommended formulas from the literature, but in most cases these formulas can be used only as a first approximation of the parameters. If an experimental load-deflection curve is to be captured, e.g., by. a nonlinear fracture mechanics model, the first calculation using an initial set of material model parameters usually deviates from desired set. Then it is ne- cessary to make some correction of the parameters using a trial-and-error method. The parameters are changed step by step, the numerical calculations have to repeated many times and the numerical simulation results are compared with the experimental results. Such a classical approach is not very efficient, especially if a complex material model with many parameters is used. That is why different alternatives of iden- tification algorithms have been proposed in the literature, and they are now becoming increasingly attractive (e.g. [1], [2], [3]). The aim of this paper is to present a new approach for identifying material model parameters. The proposed ap- proach is based on coupling stochastic nonlinear fracture mechanics analysis and an artificial neural network. The identification parameters play the role of basic random vari- ables, with the scatter reflecting the physical range of possible values. The efficient Monte Carlo type simulation method Latin Hypercube Sampling (LHS) is used. The statistical sim- ulation provides the set of data, “a bundle” of numerically simulated load-deflection curves. The generated basic ran- dom variables and the subsequently calculated load-deflec- tion curves are used for training a suitable type of neural network. Once the network is trained it represents an approx- imation which can be utilized in a reverse way: for a given experimental load-deflection curve to provide the best possi- ble set of material model parameters. Several software tools had to be combined in order to make the identification possible. First, ATENA [4] nonline- ar fracture mechanics softwaand FREET [5] probabilistic soft- ware package – these can be combined under the SARA soft- ware shell [6], [7]. Then DLNNET, new neural network soft- ware was developed [8]. The results of the approach were recently presented [9]. The similar concept of using Latin Hypercube Sampling sta- tistical simulation for stochastic training of a neural network was used to estimate microplane model parameters [10]. The methodology is demonstrated on selected numerical examples of identifyinf a material model for concrete, called SBETA (the classical often used model available in ATENA software): a notched specimen under three-point bending of high-strength concrete used for railway sleepers and shear wall experiments. 2 Fundamental difficulty of nonlinear fracture mechanics modeling For realistic modeling of structural failure from quasi- brittle materials, an advanced computational analysis should utilize nonlinear fracture mechanics. ATENA software [4] is an efficient tool for analysis of concrete, reinforced con- crete and prestressed structures. The software employs a set of advanced material models for realistic calculation of the structural response. The well-known SBETA material model was verified during long development of the software, and it reflects all important aspects of concrete behaviour in both tension and compression. A fundamental difficulty naturally exists when an experimental load-deflection curve is to be captured by numerical simulation. Such a virtual experiment needs good material data in order to reproduce the load-de- flection curve properly. In the case of the SBETA material model the main parameters are: modulus of elasticity E, ten- sile strength ft, compressive strength fc, fracture energy GF, critical compressive displacement wd, compressive strain in the uniaxial compressive test �c. A typical situation is that a set of preliminary parameters is first used for modeling, and in most cases only poor agreement with the experimental 110 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004 Neural Network Based Identification of Material Model Parameters to Capture Experimental Load-deflection Curve D. Novák, D. Lehký A new approach is presented for identifying material model parameters. The approach is based on coupling stochastic nonlinear analysis and an artificial neural network. The model parameters play the role of random variables. The Monte Carlo type simulation method is used for training the neural network. The feasibility of the presented approach is demonstrated using examples of high performance concrete for prestressed railway sleepers and an example of a shear wall failure. Keywords: Neural network, nonlinear fracture mechanics, Latin Hypercube Sampling, identification. data is achieved. A heuristic user-based iteration has to be performed. For illustration, Fig. 1 shows an experimental load-deflec- tion curve and a result of numerical simulation with the first set of preliminary parameters of a notched beam. This exam- ple will be discussed later in this paper. Although most of the material parameters were known (the relevant experiments had been done), the disagreement between the two curves is significant. It is clear that the parameters were not de- termined satisfactorily (errors, assumptions, contaminated experiments, influence of size effect, etc.), and the parameters need to be modified. 3 Identification of material model parameters 3.1 General concept The new identification technique is based on a combina- tion of statistical simulation and training the neural network. Several software tools had to be combined to make the iden- tification possible. The whole procedure can be itemized as follows (the software relevant to the individual steps is referenced): 1. First, a computational model has to be developed using the appropriate FEM software, which enables modeling of both pre-peak and post-peak behaviour. The initial cal- culation uses a set of initial material model parameters. Software: ATENA [4]. 2. The parameters of the material model to be identified are considered as random variables described by a probability distribution. Rectangular distribution is a “natural choice” as the lower and upper limits represent the bounded range of physical existence. However distributions can also be used, e.g. Gaussian (in spite of the fact that it is not bounded). These parameters are simulated random- ly based on a Monte Carlo type simulation, and LHS small-sample simulation is recommended. The statistical correlation between some parameters can be taken into account. Software: FREET [5]. 3. A multiple calculation of a deterministic computational model using random realizations of the material model parameters is performed, resulting in “a bundle” of load- -deflection curves (usually overlapping experimental curve). Software: SARA [6], [7]. 4. The random load-deflection curves serve as the basis for training an appropriate neural network. Such training can be called stochastic training, due to the stochastic origin of the load-deflection curves. After training, the neural network is ready to answer the referse task: to select the material model parameters which can capture the ex- perimental load-deflection curve as closely as possible. Software: DLNNET [8]. 5. The final calculation using the identified material model parameters should verify how well the parameters were identified (ATENA). The complex program communication and the necessary interfaces are schematically shown in Fig. 2. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 111 Acta Polytechnica Vol. 44 No. 5 – 6/2004 0 0. 0 5. 1 0. 1 5. 2.0 0 00. 0 05. 0 10. 0 15. 0 20. deformation [mm] fo rc e [k N ] experiment model - param. from exp. Fig. 1: Load-deflection curves – experiment and initial numerical simulation SARA ATENA FREET DLNNET random material parametersrandom L-D curves experimental L-D curve identified material parameters A D A P T IV E P H A S E -- A C T IV E P H A S E -- T R A IN IN G Fig. 2: Identification software communication scheme 3.2 Statistical simulation In order to prepare the set of random load-deflection curves for training the neural network, a proper efficient Monte Carlo type simulation has to be performed. The SARA system originally developed for statistical and reliability anal- ysis of concrete structures consists of two major parts – the FREET statistical and reliability package and the ATENA nonlinear finite element simulation. The stochastic part of the SARA system is the FREET – Feasible Reliability Engineering Efficient Tool – probabilistic program. This probabilistic software for statistical, sensitivity and reliability analysis of engineering problems was designed with its focus on computationally intensive problems, which do not allow thousands of samples to be performed [5], [11]. A special type of numerical probabilistic simulation, LHS, makes it possible to use only a small number of Monte Carlo simulations for a good estimation of the first and second moments of the response function. LHS uses stratification of the theoretical cumulative probability distribution function (CPDF) of the input random variables. CPDFs for all ran- dom variables are divided into N equivalent no overlapping intervals, where N is the number of simulations. The repre- sentative parameters of the variables are selected randomly on the basis of random permutations of integers 1, 2, …, j, …, N. Every interval of each variable must be used only once during the simulation. ATENA nonlinear finite element software is well-estab- lished for realistic computer simulation of damage and failure of concrete and reinforced concrete structures in a determin- istic way [12], [13]. The constitutive relation at a material point (constitutive model) plays the most crucial role in the finite element analysis and decides how the structural model represents reality. Since concrete is a complex material with a strongly nonlinear response even under service load con- ditions, special constitutive models for the finite element analysis of concrete structures are employed [4]. The SARA system can easily be used for stochastic training of a neural network. The parameters for identification are simulated as random variables with prescribed variability. The resulting random load-deflection curves with random realiza- tions of the parameters serve for training the network. Such a “bundle” of curves is shown in Fig. 5. 3.3 Artificial neural networks The basic idea of an artificial neural network was to pro- vide a numerical model of processes in the brain. Nowadays this approach is used in various fields of technical practice, mainly for classification problems [14]. In our identification technique a multilayered neural network is used. All neurons in one layer are connected with all neurons in the following layer. The connecting paths among the neurons are weighted, which models their conductivity. At the level of the neuron, the bias is added to the sum of the weighted impulses from each neuron of the previous layer. Then the transfer function is applied. Three types of transfer functions can be used: hard-limit, linear and nonlinear (e.g. sigmoid) transfer func- tions. The synaptic weights, biases and transfer functions determine the behavior of neurons and the whole neural network. The output from a single neuron (Fig. 3a) can be calcu- lated as: y f x f w p bk k k � � � � � � � � � �( ) ( ) , (1) where: k – number of input impulses (1, …, K), wk – synaptic weight of the connecting path from the k-th neuron of the previous layer, pk – impulses from the k-th neuron of the previous layer, b – bias of the neuron and f – transfer function of the neuron. If the output vector of the whole neural network is re- quired, the output vectors have to be calculated layer by layer from the input layer to the output layer of the network. Out- put of the u-th layer of the network is: y f w y bk u u kj u j u k u j J � � � � � � � � � � � �( )1 1 , (2) where: k – number of components in the output vector in the u-th layer (1, …, K � number of neurons in the u-th layer), j – number of components in the output vector in the (u�1)-th layer (1, …, J � number of neurons in the (u�1)-th layer), yk u – one component of the output vector, wkj u – synaptic weight – this connects the k-th neuron of the u-th layer with the j-th in the (u�1)-th layer, y j u�1 – one component of the output vector in the previous layer, bk u – bias of the k-th neuron in the u-th layer and f u – transfer function of the neurons in the u-th layer. If u is the number of the last layer, then y u is the output vector of the network. An artificial neural network works in two phases – active and adaptive. In the active phase the signal passes through the connecting paths from the input layer to the output layer of the network. To obtain correct results of that process, the weights and biases must have appropriate values. To assign these values, an adaptive phase must be used. This process is called training the neural network. For network training a set of training parameters is needed. This set consists of ordered pairs [pi, yi], where yi are the expected output vectors (in our case random realizations of the model parameters selected for identification), which are yielded by simulation of the net- work with input vectors pi (in our case points on the load-de- flection curves). The main aim during training is to minimize the following criterion: E y yik ik k K i N � � �� �� 1 2 2 11 ( )*� , (3) 112 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004 � p p p � � � � w w � w � y b a) b �� p � p �� w � b b � �� w �� w �� w � b � b � � y y 1 2b) Fig. 3: a) Scheme of single neuron, b) scheme of neural network with two layers where: N – number of ordered pairs input – output in the training set, yik * – required output value of the k-th output neuron at the i-th input and yik � – real output value (at the same input). In order to minimize criterion E some optimization tech- nique is used. A description of these techniques can be found in, e.g. [15], [8]. 4 Numerical examples 4.1 Notched specimen of high-strength concrete used for railway sleepers A classical experiment involving three-point bending of a notched plane concrete beam was performed in order to determine fracture parameters of concrete for mass produc- tion of railway sleepers. Six specimens 80×80×480 mm were tested with a notch 25 mm in depth. The fracture-mechanical parameters were determined on the basis of the recom- mendation of RILEM [16] and improvements according to Elices [17], Stibor [18] and Veselý [19]. Based on this ex- periment, the following parameters were obtained: modulus of elasticity Ec � 32.4 GPa, fracture energy GF � 188.7 N/m, compressive strength fc � 75.0 MPa and estimation of tensile strength ft � 4.0 MPa. The mean values of the parameters used for stochastic simulation and consequent identification are as follows: mod- ulus of elasticity Ec � 60 GPa, tensile strength ft � 5 MPa, com- pressive strength fc � 75 MPa, fracture energy GF � 170 N/m © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 113 Acta Polytechnica Vol. 44 No. 5 – 6/2004 F x 2L x wa L = 200 mm w = 80 mm a = 25 mm b = 80 mm b Fig. 4: Notched beam under three-point bending 0 0. 0 5. 1 0. 1 5. 2 0. 2.5 0 00. 0 05. 0 10. 0 15. 0 20. deformation [mm] fo rc e [k N ] Fig. 5: Random load-deflection curve realizations – 20 simulations of LHS 0 0. 0 5. 1 0. 1 5. 2.0 0 00. 0 05. 0 10. 0 15. 0 20. deformation [mm] fo rc e [k N ] experiment model - 5 parameters model - 3 parameters Fig. 6: Load deflection curves – experiment and numerical simulation using identified parameters and compressive strain at compressive strength in the uniaxial compressive test �c � 0.003. For stochastic training, randomness was introduced using the same coefficient of variation 0.15 and rectangular proba- bility distribution for all random variables. Twenty simula- tions of LHS resulted in the load-deflection curves presented in Fig. 5. Note that none of these random curves captured the experiment well. This input-output information serves for training the selected neural network: a network with 20 in- puts (20 points on the load-deflection curve for each simula- tion is utilized for training), one hidden layer consisting of 15 neurons with a nonlinear transfer function and one output layer of 5 neurons with a linear transfer function. Instead of 5 neurons, a second alternative with only 3 output neurons was also used. The original number of parameters (5) could be de- creased (to 3) as the sensitivity analysis showed the dominat- ing and nondominating random variables. The trained neu- ral network provided the material model parameters (for 3 or 5 considered parameters): Ec � 70.3 and 73.2 GPa, ft � 5.5 and 5.3 MPa, fc � 75 (mean value) and 103.22 MPa, GF � 128 and 141 N/m, �c � 0.003 (mean value) and 0.004. The final calculation using ATENA resulted in a very good agreement with the experimental load-deflection curve for both alternatives, Fig. 6. 4.2 Shear wall failure The shear wall shown in Fig. 7 was tested by Maier and Thürliman [20]. The square panel was orthogonally rein- forced and provided with stiffening flanges. Loading by a ver- tical force was first applied representing a dead load. Then a horizontal force was applied and increased to failure. During the experiment there was extensive diagonal cracking prior to failure, followed by explosive crushing of the concrete under maximum load. The experimental failure pattern is shown in Fig. 7(a). The analysis was done by ATENA using plane-stress iso- parametric finite elements with the composite reinforced concrete material. All 10 shear wall parameters of the mate- rial models (both concrete and steel reinforcement) were identified here. The mean values of the parameters used for stochastic simulation and consequent identification are as follows: for concrete (SBETA model) – modulus of elas- ticity Ec � 30 GPa, compressive strength fc � 35 MPa, tensile strength ft � 2.5 MPa, fracture energy GF � 75 N/m, compres- sive strain in the uniaxial compressive test �c � 0.0025, critical compressive displacement wd � 0.003 m; for steel (bilinear law) – yield strain x1 � 0.0027, yield stress fx1 � 574 MPa, ulti- mate strain x2 � 0.015 and ultimate stress fx2 � 764 MPa. For stochastic training, randomness was intuitively intro- duced using coefficient of variation 0.10 for Ec, ft and fc, 0.2 for GF and �c, 0.3 for wd and 0.1 for all steel parameters. A rectangular probability distribution for all random variables is used. The experimental load-deflection curve and 20 simu- lations of LHS are presented in Fig. 8. This input-output information serves for training the selected neural network: network with 24 inputs (24 points on load-deflection curve for every simulation is utilized for training), two hidden layer consisting of 12 and 10 neurons with nonlinear transfer func- tions and one output layer of 10 neurons with a linear transfer 114 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004 0 200 400 600 800 1000 0 2 4 6 8 10 12 14 Horizontal deformation [mm] H o ri z o n ta l fo rc e [k N ] Fig. 8: Random load-deflection curve realizations – 20 simulations of LHS (a) (b) Fig. 7: Shear wall experimental failure and its ATENA simula- tion: (a) experimental failure, (b) failure simulated by ATENA function. The trained neural network provided the material model parameters: Ec � 33 GPa, fc � 35.3 MPa, ft � 2.47 MPa, GF � 77.85 N/m, �c � 0.0026, wd � 0.0031 m, x1 � 0.0028, fx1 � 570.7 MPa, x2 � 0.0147 and fx2 � 768.8 MPa. The final calculation using ATENA resulted in a very good agreement with experimental load-deflection curve, Fig. 9. 5 Conclusion Determining the material model parameters generally presents a great problem when using nonlinear analysis and sophisticated material constitutive laws. A methodology for efficient numerical identification of material model para- meters is suggested. This approach utilizes stochastic com- putational analysis in combination with an artificial neural network. Small-sample simulation techniques are employed, which enables analysis of computationally intensive problems of nonlinear fracture mechanics. The feasibility of the ap- proach is documented by numerical examples. Application of the concept to problems, where the experimental load-deflec- tion curve is known could result in much better identification of the material model parameters than with a heuristic “trial and error” approach. Such well-identified parameters based on experiment can then be used for calculating a real struc- ture. Very good results have been achieved, which indicates that efficient techniques have been combined at all three basic levels: deterministic nonlinear modeling, probabilistic strati- fied simulation and neural network approximation. References [1] Bažant Z. P., Becq-Giraudon E.: “Statistical prediction of fracture parameters of concrete and implications for choice of testing standard.” Cement and Concrete Research, Vol. 32 (2002), No. 4, p. 529–556. [2] Iacono C., Sluys L. J., van Mier J. G. 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[18] Stibor M.: „Problémy s určováním lomové energie ce- mentových kompozitů.“ In: Sborník semináře Problémy lomové mechaniky, Brno, 2001, p. 72–77, (in Czech). [19] Veselý V.: „Lomově-mechanické vlastnosti pražcového betonu.“ In: Sborník 4. odborné konference doktor- ského studia (CD-ROM), (in Czech), VUT FAST v Brně, Brno, 2002. [20] Maier J., Thürliman B.: “Bruchversuche an Stahlbeton- scheiben.” Institut für Baustatik und Konstruktion ETH Zürich, Bericht Nr. 8003-1, 1985. Prof. Ing. Drahomír Novák, DrSc. phone:+420 541 147 360 fax: +420 541 240 994 e-mail: novak.d@fce.vutbr.cz Ing. David Lehký phone: +420 541 147 376 fax: +420 541 240 994 e-mail: lehky.d@fce.vutbr.cz Institute of Structural Mechanics Brno University of Technology Faculty of Civil Engineering Veveří 95 602 00 Brno, Czech Republic 116 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004