Christiansen, Niels Hørbye

Høgsberg, Jan Becker

Winther, Ole

It is shown how artificial neural networks can be trained to predict dynamic response of a simple nonlinear structure. Data generated using a nonlinear finite element model of a simplified wind turbine is used to train a one layer artificial neural network. When trained properly the network is able to perform accurate response prediction much faster than the corresponding finite element model. Initial result indicate a reduction in cpu time by two orders of magnitude

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Downloaded from orbit.dtu.dk on: Dec 19, 2017Artificial Neural Networks for Nonlinear Dynamic Response Simulation in MechanicalSystemsChristiansen, Niels Hørbye; Høgsberg, Jan Becker; Winther, OlePublished in:Proceedings of the 24th Nordic Seminar on Computational MechanicsPublication date:2011Link back to DTU OrbitCitation (APA):Christiansen, N. H., Høgsberg, J. B., & Winther, O. (2011). Artificial Neural Networks for Nonlinear DynamicResponse Simulation in Mechanical Systems. In Proceedings of the 24th Nordic Seminar on ComputationalMechanicsProceedings of the 24th Nordic Seminar on Computational MechanicsJ. Freund and R. Kouhia (Eds.)c©Aalto University, 2011Artificial Neural Networks for Nonlinear Dynamic ResponseSimulation in Mechanical SystemsNiels H. Christiansen1,2, Jan Høgsberg1, and Ole Winther3(1)Department of Mechanical Engineering, Technical University of Denmark, Kgs. Lyngby, Den-mark, e-mail: Niels.Christiansen@dnv.com(2)DNV Denmark, Hellerup, Denmark(3)Department of Informatics and Mathematical Modeling, Technical University of Denmark,Kgs. Lyngby, DenmarkSummary. It is shown how artificial neural networks can be trained to predict dynamic response of asimple nonlinear structure. Data generated using a nonlinear finite element model of a simplified windturbine is used to train a one layer artificial neural network. When trained properly the network is ableto perform accurate response prediction much faster than the corresponding finite element model. Initialresult indicate a reduction in cpu time by two orders of magnitude.Key words: Nonlinear structural dynamics, Artificial neural networks.IntroductionTime domain simulation of nonlinear systems using finite element method (FEM) analysis canbe computationally very expensive - especially in fatigue calculations where long response his-tories are needed in order to obtain reliable time series statistics. The use of artificial neuralnetworks (ANN) combined with classical methods have shown promising results in reducing thiscomputational cost [1]. This paper presents a hybrid method for simulation of dynamic responseof a simple nonlinear structure. It is shown how an ANN can be trained to predict dynamicresponse of a simplified model of a wind turbine. FEM models of nonlinear structures oftenrequire fine element mesh discretization, small time step size and iterative procedures in orderto obtain equilibrium between internal and external forces. For large complex models this can bevery time consuming. The ANN’s ability to perform nonlinear mapping between a given inputand a system output makes it capable of response prediction without equilibrium iterations.Hence, a properly trained ANN can save a lot of computational effort in response prediction.Artificial Neural NetworkThe architecture of a typical one layer artificial neural network is shown in Figure 1. The ANNconsists of an input layer, a hidden layer and an output layer. Each connection between twoneurons in two neighboring layers has a weight. Training of an ANN is optimization of theseweights for a given data training set.Following [2] the ANN set up and training procedure can be written as follows. The ANNoutput is calculated byy = W>o z, z = tanh(W>i x), x0 ≡ z0 ≡ 1, (1)1x1x2x0z0y1y2xd−1xdz1z2zdWiWoBiasBiasInputHiddenOutputTime shiftFigure 1. Sketch of artificial neural network.where x is input vector, y is output vector and Wi and Wo are neuron connection weights be-tween input and hidden layer and hidden and output layer, respectively. The tangent hyperbolicis used as activation function between input and hidden layer.The error function which is minimized during training can be written asE (W) =12N∑n=1c∑k=1{y(xn;W)k − tnk}2 +12αW2, (2)where y is the ANN output, t is the target value and α is the weight decay that controls thevalue of the weights and prevents the ANN from overfiting to noise in the training data. Optimalweights are found with an iterative procedure stepping in weight space towards minimal error.The weight update is done by gradient decent going the opposite direction of the cost functionsgradient as in (3).Wnew = Wold +∆W, ∆W = −η∂E (W)∂W, (3)where η is the learning step size parameter. This parameter can either be constant or updatedduring the training of the ANN. For this application the dynamic learning step size parameteris adjusted for each iteration so that it is increased if the training error is decreased comparedto previous iteration step and reduced if the training error increases.Structural ModelTo illustrate the hybrid method a simplified model of a wind turbine is set up, see Figure 2.The height of the wind turbine is 100 m. The diameter of the turbine steel tower is 4 m with awall thickness of 0.1 m. At the top of the tower a 100 · 103 kg mass is placed to represent thenacelle and turbine blades. The lowest eigenfrequency of the structure is 0.3 Hz. The load f(t)applied to the structure corresponds to the horizontal load on a 100 m diameter rotor in a 15m/s wind. The mean wind load is 284 kN with a standard deviation of 44 kN.The FEM model of the structure uses co-rotational beam element formulation as describedin [3]. This formulation separates beam motion into two parts: a rigid body motion associatedwith a local frame of reference, and a deformation of the beam within this frame. The localdeformation of the beam element implies geometrical stiffness contributions which depend onthe deformation. Thus, we have a nonlinear model which can handle large deformations, whenthe governing equation is given as2Mr¨+Cr˙+K(r)r = f(t). (4)The lumped masses and beam stiffness contributions are assembled in a mass matrix M anda stiffness matrix K, respectively. The beam is Rayleigh damped through damping matrix Cintroducing 3 % damping to the lowest vibration mode of the wind turbine. The force vector fconsists of external force components at each degree of freedom (DOF) for each time step andthe vector r contains the degrees of freedom (DOF) of the structure. Note that the componentsin the stiffness matrix K dependent on the deflection r of the structure. In the FEM model theturbine tower is divided into 10 elements.The response of the structure is calculated by Newmark’s method of direct integration. TheNewton-Raphson method is used to achieve force equilibrium in each time step and the updateof the stiffness matrix follows the procedures described in [3]. The time step size is 0.1 s andequilibrium is assumed when force and displacement residuals are below 1 %.0 5 10 15 20200250300  [kN]f(t)tpqrrMkkkkkkkkkkmmmmmmmmmf(t)f(t)Figure 2. a) Harmonic wind load, b) Sketch of wind turbine, c) Simple turbine FEM model.Based on the response history data generated by the FEM model an ANN is trained topredict the future dynamic response of the wind turbine.SimulationThe ANN is designed and trained to predict the horizontal deflection of the wind turbine atthe location of the mass M, see Figure 2. Note that the ANN output only gives the horizontalresponse of top and hence, as oppose to the FEM model, not the response of all model DOFs.The load together with the state space variables (r, r˙) of previous time steps are used as ANNinputxt = [f(t) . . . f(t− d) r(t− 1) r˙(t− 1) . . . r(t− d) r˙(t− d)]T , (5)where d is number of previous time steps included in the input i.e. the model memory. ANNoutput is the current deflection and velocityyt = [r(t) r˙(t)]T (6)Since the ANN in this case simulate a theoretical model of a physical system and thereforereplicate an exact solution to the system equations of motion there is no noise in the targetoutput data. Therefore the weight decay in (2) is set to zero (α = 0) in this example.The wind turbine response history generated by the FEM model (4) is divided into a trainingand a test set as shown on Figure 3. Out of the 100 s response history the first 80 s is used30 10 20 30 40 50 60 70 80 90 100−2−1012  Training Testtr(t)Figure 3. Training and test data.500 510 520 530 540 550 560 570 580 590 600−2−1012  FEM ANN PeaksFEM PeaksANNtr(t)Figure 4. Exact FEM solution together with ANN simulation.for training the ANN and the last 20 s is used for testing. With a time step size of 0.1 s and adominating vibration frequency of 0.3 Hz, this gives a training set of 800 points covering about24 vibration cycles. Parameter investigations not included in this paper show that an accurate,compact and robust ANN is obtained with 50 units in the hidden layer and 10 previous timesteps. After training the ANN is able to predict the wind turbine response to a similar windload history as the one used to generate the training data. The results of simulations using FEMmodel and ANN is shown in Figure 4. It is seen that the ANN captures the dynamics of thesystem very well and predicts the deflection quite accurately.Concluding remarksIn the example presented in this paper the reduction in cpu time spend on a simulation of 10 min.wind is about a factor of 100 when using the ANN compared to the FEM - that is when the ANNis set up and trained. This factor is obviously dependent on the structure, the loading, modelconfiguration etc. However, the examples indicates that the method holds a great potential.Furthermore, the reduction in calculation cost may be increased even further if it is possible toskip a few post possessing steps so that, instead of just predicting structural response, the ANNcan be trained to predict material stresses or structural damage directly based on force input.References[1] R. Guarize, N.A.F. Matos, L.V.S. Sagrilo, and E.C.P. Lima. Neural networks in dynamic re-sponse analysis of slender marine structures. Applied Ocean Research, (29):191–198, January2008.[2] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, October 2006.[3] Steen Krenk. Non-linear Modeling and Analysis of Solids and Structures. Cambridge Uni-versity Press, August 2009.4

Artificial Neural Networks for Nonlinear Dynamic Response Simulation in Mechanical Systems

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Artificial Neural Networks for Nonlinear Dynamic Response Simulation in Mechanical Systems

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