Neural network based constitutive modeling of nonlinear viscoplastic structural response

Mechanics Research Communications 95 (2019) 85–88

Contents lists available at ScienceDirect

Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom

Neural network based constitutive modeling of nonlinear viscoplastic structural response Marcus Stoffel∗, Franz Bamer, Bernd Markert Institute of General Mechanics, RWTH Aachen University, Templergraben 64, Aachen D-52056, Germany

a r t i c l e

i n f o

Article history: Received 5 December 2018 Revised 10 January 2019 Accepted 10 January 2019 Available online 11 January 2019 MSC: 74K20 74-05

a b s t r a c t In the present study constitutive equations in finite element simulations are replaced by means of an artificial neural network (ANN). Following this approach, a physically nonlinear stress-strain behavior with strain rate dependency is substituted by an algebraic system of equations. Implementing this mathematical approximation of a constitutive law into a finite element code, a so-called intelligent element is created. This approach leads to a significant reduction of computing time, because a complex material model is treated numerically by matrix multiplications as in the case of elasticity. Here, a viscoplastic material analysis by means of an ANN is proposed and applied to nonlinear structural behavior. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Artificial neural network Structural mechanics Intelligent finite element

1. Introduction Artificial neural networks have already been applied to engineering problems as an alternative approach compared to classical methods based on continuum mechanical modeling, e.g. steel structures [1], stability [2] or welding problems [3]. However, the modeling of the entire structure with neural networks limits the application to one particular problem. But if only the material law is replaced by an ANN, then the neural network based constitutive model can be applied to different engineering problems wherein the same material behavior is considered [4]. For this reason, in the present study, an ANN is developed to mimic nonlinear viscoplastic material behavior. Studies about the substitution of a constitutive model by means of an ANN have been published in [5]. A beam element, based on a neural network, is proposed in [6] and leads to less computing time than a classical finite element approach. This advantage is in multiscale approaches even more visible [7]. Several neural network constitutive models (NNCM) were discussed in [8]. However, it was pointed out that the correct choice of the training data is essential for a reliable intelligent finite element [5]. In this study, a neural network based constitutive viscoplastic model with nonlinear hardening is proposed and implemented into a finite element code. As an example, this intelligent element code is applied to structural deformations caused by shock wave

∗

Corresponding author. E-mail address: [email protected] (M. Stoffel).

https://doi.org/10.1016/j.mechrescom.2019.01.004 0093-6413/© 2019 Elsevier Ltd. All rights reserved.

loadings. A comparative study is then conducted between measurements of plate deformations in a shock tube, finite element simulations, and intelligent element predictions.

2. Experiment In order to obtain measurements of structural deformations and corresponding loads, experiments in shock tubes are carried out [9]. In Fig. 1, the shock tube is shown for subjecting aluminum plates of 553 mm diameter and 2 mm thickness to impulsive loadings. The tube consists of a high pressure chamber (HPC) and a low pressure chamber (LPC), separated from each other by an aluminum membrane. Due to the pressure difference between HPC and LPC, the membrane bursts, generating a shock wave which is moving through the LPC and loading the plate specimen at the end of the tube. Consequently, a high-density impulse during milliseconds is caused, leading to inelastic deformations of the plate sample. The mid-point deflection of the plate is measured during microseconds by a capacitive sensor, developed for this purpose [9]. The pressure acting on the plate is measured during the impulse period by means of piezoelectric pressure sensors. The pressure evolution during the shock wave loading can be varied by using different gases in the HPC. If a light gas, such as helium, is used in the HPC, then a faster shock wave and, hence, a higher pressure on the plate specimen can be caused, than in the case of nitrogen.

86

M. Stoffel, F. Bamer and B. Markert / Mechanics Research Communications 95 (2019) 85–88

of all weighted normalized input values in the propagation function N 

Xj =

xni wi j

(5)

i=1

with N as the number of input neurons and j for the jth neuron in the hidden layer. Bias terms are neglected, because they were not necessary in the present study. For training the neural network, a matrix A with Z input data rows is necessary. Each row denotes the input vector xTn leading to a matrix multiplication

B = A w,

(6) XT

see Fig. 2. The vector is obtained Z-times, with components Xj representing the values which are important for the activation of a neuron in the hidden layer. Hence, they are the arguments of the sigmoid activation function

1 1 + e−X j

Fj =

for each neuron in the hidden layer. Calculating forwardly to the output layer, the next propagation function reads

Fig. 1. Picture of the shock tube.

Yk =

3. Artificial neural network based viscoplastic model



ε



σi j − Xi j 3 = p˙ with p˙ = 2 J2 (σrs − Xrs ) 

σv = J2 (σi j

− X

ij

) − k and

 σ n v

(1)

K

quently, the output values are defined as plastic strain rate ε˙ i j and backstress rate X˙ i j . In order to obtain better convergence and numerical stability, all input and output values are normalized [11,12]. Here, this is carried out for all stress, strain, strain rate, backstress, and backstress rate values, denoted by xi , in the form p

ximax − ximin

(3)

leading to unified values xni and with ximin and ximax as minimum and maximum values. The index i stands for the ith input neuron. Following this approach, an input vector with normalized components, abbreviated with the index n, is obtained in the form

xTn =



pn pn pn n n n n σ12 σ13 σ23 ε11 ε22 ε12 σ11n σ22  pn pn n n n n n ε13 ε23 X11 X22 X12 X13 X23 .

(8)

with K as the number of output neurons, k for the kth neuron in the output layer, and new weights w∗jk between hidden layer and output layer. With Z vectors F, each in a row of matrix C in Fig. 2, a second matrix multiplication is performed by

D = C w∗ .

(2)

Here, σ ij , Xij , k, p, σ v denote plastic strain tensor, stress tensor, backstress tensor, yield limit, equivalent plastic strain, overstress, and a, s, n, K are material parameters. The deviatoric part of  ˙ indicates the material time derivaa tensor is denoted by () , () tive. During the numerical simulation at a time t, the values of stresses, strains, and backstresses are known and can be regarded as input values for the ANN. With this information in the current configuration the ANN must predict the strain and backstress inp crements εi j and Xij , respectively, for a time step t. Conse-

 x −x  i imin

Fj w∗jk

(9)

The obtained components Yk in Z rows, one row for each vector YT (Fig. 2), are inserted into the activation function

2 X˙ i j = aε˙ ipj − sXi j p˙ . 3

εipj ,

xni = 0.1 + 0.8 ·

K  i=1

The starting point for developing a material model based on an ANN is the viscoplastic law from [10] with kinematic hardening. It is expressed by p˙ ij

(7)

(4)

In the present study, a feed forward neural network is developed, see Fig. 2. This type of ANN is well established in literature [13]. All components of the input vector have to be multiplied with weights wij between input layer and hidden layer, leading to a sum

Gk =

1 1 + e−Yk

(10)

leading to normalized components Gnk of the output vector

GTn =



 pn pn pn pn pn ˙ n ˙ n n ˙n ˙n ε˙ 11 ε˙ 22 ε˙ 12 ε˙ 13 ε˙ 23 X11 X22 X˙ 12 X13 X23 .

(11)

Transforming these normalized values by means of Eq. (3) back to physical ones, the strain rate and hardening rate tensors are obtained for the simulation time t + t. As an example, the neurons in the upper row in Fig. 2 are marked with their mathematical meaning. In order to identify the weight matrices wij and w∗jk , the ANN has to be trained by given input and output values with Z input and output vectors. The training data is obtained from the constitutive law in the above mentioned ranges (Eq. (3)) expected during the simulation. The ANN is implemented in a separate python code. In this algorithm a loop is generated over the entire training data together with the back-propagation method. This gradient descent algorithm is applied in order to minimize the least square error between calculated and provided output data. The topology of the ANN has to be determined by the user. However, the use of an ANN represents mathematically a function approximation, in which the user has to determine the number of neurons in the hidden layer iteratively. This process could be continued by including further internal variables, such as additional hidden layers or biases, if necessary. Nevertheless, all variables are obtained by applying an optimum criterion. In the present study, the determined topology of the ANN, shown in Fig. 2, lead to the smallest error between input and output values during the training procedure. Changes around this optimized topology, e.g. changes in the number of neurons in the hidden layer, would impair the accuracy the output results. Consequently, seven neurons in the hidden layer have been identified iteratively to lead to the minimal least square error.

M. Stoffel, F. Bamer and B. Markert / Mechanics Research Communications 95 (2019) 85–88

87

Fig. 2. Artificial neural network with arithmetic operations for replacing viscoplastic constitutive equations.

Fig. 3. Comparative study between measured and simulated plate deflections using FEM and ANN.

For detailed descriptions of activation and propagation functions in ANNs it is referred to related textbooks [14]. Extended studies about backpropagation methods are reported in [15]. Finally, two optimized weight matrices wij and w∗jk are obtained, which are needed to replace the constitutive law in the finite element simulation by means of the ANN. In the finite element code [9], the material law in Eqs. (1) and (2) is substituted by Eqs. (3)–(11), leading to a new material description in form of matrix multiplications, see Fig. 2. Then, one input vector in matrix A denotes the input data in one Gaussian point. This mathematical way is comparable to pure elastic constitutive equations. Thus, no treatment of history variables representing the inelastic evolution

of the material is required in the present study, which is essential for reducing computing time. The structural part in this finite element code is based on a geometrically nonlinear first-order shear deformation shell theory as reported in [9]. 4. Results and discussion In Fig. 3 middle point deflections of two aluminum plates and the pressures acting on them during the impulse duration are shown. Two experiments with peak pressures pp = 7 bar and pp = 3.5 bar using helium (He) and nitrogen (N2 ) in the HPC, see

88

M. Stoffel, F. Bamer and B. Markert / Mechanics Research Communications 95 (2019) 85–88

Fig. 1, are presented together with numerical predictions based on FEM and ANN. In the case of the ANN result the finite element simulation includes the so-called intelligent element based on the developed neural network. Due to the fact, that the ANN simulation is trained with the constitutive data, which is also used for the FEM simulation, the ANN simulation can only tend towards the finite element result. However, the ANN simulation can simulate the plate deflection faster than the FEM solution, which is the advantage of the present approach. In this study, the computing time using the ANN was up to 50% lower, than in the case of the FEM simulation. However, the accuracy of the ANN depends strongly on the trained data. Interpolations between provided training data could be a reason between differences of the ANN and FEM solutions. 5. Conclusions The developed intelligent element based on an artificial neural network leads in the present study to lower computing times and very similar simulation results compared to classical finite element analyses. The ANN was able to account for nonlinear hardening and strain rate dependency by using an algebraic system of equations. Two weight matrices of the mathematical system were obtained by an optimized feed-forward neural network python algorithm. Finally, the combination of a neural network based constitutive model with a geometrically nonlinear shell theory within a finite element was successful. References [1] A.-A. Chojaczyk, A.-P. Teixeira, C. Luìs, J.-B. Cardosa, C.-G. Soares, Review and application of artificial neural networks models in reliability analysis of steel structures, Struct. Saf. 52 (A) (2015) 78–89.

[2] Z.-R. Tahir, P. Mandal, Artificial neural networks prediction of buckling load of thin cylindrical shells under axial compression, Eng. Struct. 152 (2017) 843–855. [3] D. Zhao, D. Ren, K. Zhao, S. Pan, X. Guo, Effect of welding parameters on tensile strength of ultrasonic spot welded joints of aluminum to steel by experimentation and artificial neural network, J. Manuf. Process. 30 (2017) 63–74. [4] M. Stoffel, F. Bamer, B. Markert, Artificial neural networks and intelligent finite elements in non-linear structural mechanics, Thin-Walled Struct. 131 (2018) 102–106. [5] A.-A. Javadi, M. Mehravar, A. Faramarzi, A. Ahangar-Asr, An artificial intelligence based finite element method, ISAST Trans. Comput. Intell. Syst. 1 (2) (2009) 1–7. [6] V. Papadopoulos, G. Soimiris, D.-G. Giovanis, M. Papadrakakis, A neural network-based surrogate model for carbon nanotubes with geometric nonlinearities, Comput. Methods Appl. Mech. Eng. 328 (2018) 411–430. [7] M. Lefik, D.-P. Boso, B.-A. Schrefler, Artificial neural networks in numerical modeling of composites, Comput. Methods Appl. Mech. Eng. 198 (2009) 1785–1804. [8] H. Shin, G. Pande, On self-learning finite element code based on monitored response of structures, Comput. Geotech. 27 (3) (20 0 0) 161–178. [9] M. Stoffel, Evolution of plastic zones in dynamically loaded plates using different elastic-viscoplastic laws, Int. J. Solids Struct. 41 (2004) 6813–6830. [10] J. Lemaitre, J. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1994. [11] M. Shakiba, N. Parson, X.-G. Chen, Modeling the effects of cu content and deformation variables on the hight-temperature flow behavior of dilute Al-Fe-Si alloys using an artificial neural network, Materials 9 (536) (2016) 1–13. [12] N. Kiliç, E. Bülent, S. Hartomaciog˘ lu, Determination of penetration depth at high velocity impact using finite element method and artificial neural network tools, Defence Technol. 11 (2015) 110–122. [13] V.-K. Ojha, A. Abraham, V. Snášel, Metaheuristic design of feedforward naural networks: a review of two decades of research, Eng. Appl. Artif. Intell. 60 (2017) 97–116. [14] A. Engelbrecht, Computational Intelligence, An Introduction, John Wiley & Sons, Ltd, 2007. [15] M. Huk, Backpropagation generalized delta rule for the selective attention sigma-if artificial neural network, Int. J. Appl. Math. Comput. Sci. 22 (2) (2012) 449–459.