Finite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique

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1 Finite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique I. Boulkaibet a, L. Mthembu a, T. Marwala a, M. I. Friswell b, S. Adhikari b a The Centre For Intelligent System Modelling (CISM), Electrical and Electronic Engineering Department, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa. b College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom. ABSTRACT The use of Bayesian techniques in Finite Element Model (FEM) updating has recently increased. These techniques have the ability to quantify and characterize the uncertainties of dynamic structures. In order to update a FEM, the Bayesian formulation requires the evaluation of the posterior distribution function. For large systems, this functions is either difficult (or not available) to solve in an analytical way. In such cases using sampling techniques can provide good approximations of the Bayesian posterior distribution function. The Hybrid Monte Carlo (HMC) method is a powerful sampling method for solving higher-dimensional complex problems. The HMC uses the molecular dynamics (MD) as a global Monte Carlo (MC) move to reach areas of high probability. However, the acceptance rate of HMC is sensitive to the system size as well as the time step used to evaluate MD trajectory. To overcome this, we propose the use of the Separable Shadow hybrid Monte Carlo (S2HMC) method. This method generates samples from a separable shadow Hamiltonian. The accuracy and the efficiency of this sampling method is tested on the updating of a GARTEUR SM-AG19 structure. Keywords: Bayesian, Sampling, Finite Element Model updating, Markov Chain Monte Carlo, Hybrid Monte Carlo method, Shadow Hybrid Monte Carlo. 1. Introduction Finite element model (FEM) is a numerical method used to model complex engineering problems [1, 2]. FEM is often used to compute displacements, stresses and strains in complex structures under a given set of loads. Due to the uncertainties (among other approximations) associated with the process of constructing a finite element model of a structure the analytical results are different from those obtained from experimental measurements [3, 4]. Thus for practical purposes the FE model needs to be updated. In recent years the use of the Bayesian framework to build model updating techniques has shown promising results in this system identification problem [4, 6, 7, 8]. This approach allows system modelling uncertainties to be expressed in terms of probability. This can be done by representing the parameters that need to be updated as random vectors with a joint probability distribution function (pdf). This distribution function is known as the posterior distribution function. For sufficiently complex problems this pdf is not available in analytical form. This is the case for the FEM updating problem where the parameter search space is non linear and of high dimension. When an analytical solution is not available sampling methods, such as the Markov Chain Monte Carlo (MCMC), offer the only practical solution to estimating the desired posterior distribution function [4, 7, 8]. One improvement on the classic MCMC is the Hybrid Monte Carlo (HMC) sampling technique. This algorithm is able to deal with an updating vector of a large size. In the HMC the derivative of the target log-density probability is used to guide the Monte Carlo trajectory and leads towards areas of high probability [5, 7, 13, 25]. An auxiliary variable, called the momentum vector is introduced and the updated vector is treated as a system displacement. The total system energy called the Hamiltonian function- is evaluated using the Störmer-Verlet (also called leapfrog) algorithm. The leapfrog algorithm requires the log-density derivative, which can be seen as a guide used to deliver global moves with a

2 higher acceptance probability. The Hamiltonian function is numerically evaluated using the popular Störmer- Verlet integrator [25]. This integrator does not conserve the energy especially when the time step used by the leapfrog algorithm or/and the system size is considered large. To overcome this limitation an algorithm called the Shadow Hybrid Monte Carlo (SHMC) has been proposed [9, 23]. The SHMC uses a modified Hamiltonian function for sampling and a reweighting to improve the acceptance rate of HMC [9, 23].However the SHMC uses a non-separable Hamiltonian which generates the momenta in a computationally expensive way. Furthermore this method requires an extra tuning parameter to balance the cost of rejection of momenta and positions [9, 23, 24]. In this paper the Separable Shadow Hybrid Monte Carlo (S2HMC) [24] is implemented. The S2HMC is able to sample the posterior distribution function of FEM updating parameters by using a separable shadow Hamiltonian function and without involving any extra parameters. This method is tested on updating a GARTEUR SM-AG19 aeroplane structure. The efficiency, reliability and limitations of the S2HMC technique are investigated when a Bayesian approach is implemented on an FEM updating problem. In the next section, the finite element model background is presented. In Section 3, an introduction to the Bayesian framework is introduced where the posterior distribution of the uncertain parameters of the FEM is presented. Section 4 introduces the HMC techniques. Section 5 introduces the Shadow Hamiltonian function. Section 6 introduces the S2HMC technique which is used to predict the posterior distribution. Section 8 presents an implementation on a GARTEUR SM-AG19 aeroplane structure. Finally, the Section 9 concludes the paper. 2 Finite Element Model Background In finite element modelling, an N degree of freedom dynamic structure can be described by the second order equation of motion [8, 14, 23]: ( ) ( ) ( ) ( ) (1) where and are the mass, damping and stiffness matrices of size, ( ) is the vector of N degrees of freedom and ( ) is the vector of loads applied to the structure. In the case that no external forces are applied to the structure and if the damping terms are neglected ( ), the dynamic equation may be written in the modal domain (natural frequencies and mode shapes) : [ ( ) ] (2) is the measured natural frequency, is the measured mode shape vector and is the error vector. In Eq. (2), the error vector is equal to if the system matrices and correspond to the modal properties ( and ). However, is a non-zero vector if the system matrices obtained analytically from the finite element model do not match the measured modal properties and. 3 Bayesian Inferences In this work the Bayesian method is used to solve the FEM updating problem in the modal domain. Bayesian approaches are governed by Bayes rule [4, 5, 8, 23]: ( ) ( ) ( ) (3) where represent the vector of updating parameters and the mass and stiffness matrices are functions of the updating parameters. The quantity ( ), known as the prior probability distribution, is a function of the updating parameters in the absence of the data. is the measured modal properties; the natural frequencies and mode shapes. The quantity ( ) is the posterior probability distribution function of the parameters in the presence of the data. ( ) is the likelihood probability distribution function [4,5,17]. The likelihood distribution can be seen as the probability of the modal measurements in the presence of uncertain parameters [8]. This function can be defined as the normalized exponent of the error function that represents the differences between the measured and the analytic frequencies.

3 It can be written as: ( ) ( ) ( ( ) ) (4) where is a constant, is the number of measured modes and is the analytical frequency. The prior density function represents the prior knowledge about the updating parameters and quantifies the uncertainty of the parameters [8]. This knowledge can be facts like some parameters need to be updated more intensely than others. For example in structural systems parameters next to joints should be updated more intensely than for those corresponding to smooth surface areas far from joints. Here the prior probability distribution function for parameters is assumed to be Gaussian and is given by [17, 18, 23] : ( ) ( ) ( ) (5) where is the number of groups of parameters to be updated, and is the coefficient of the prior density function for the group of updating parameters. The notation denotes the Euclidean norm of. In Eq. (5), if is constant for all of the updating parameters then the updated parameters will be of the same order of magnitude. Eq. (5) is chosen to be Gaussian because many natural processes tend to have a Gaussian distribution. The posterior distribution function of the parameters given the observed data is denoted as ( ) and is obtained by applying Bayes theorem as represented in Eq. (3). The distribution ( ) is calculated by substituting Eq. (4) and (5) into Eq. (3) to give ( ) ( ) ( ( ) ) (6) where ( ) ( ) ( ) (7) In FEM updating the analytical form of the posterior distribution function solution is not available. As discussed sampling techniques simplify the Bayesian inference by providing a set of random samples from posterior distribution [5, 7, 8, 17, 20]. In the case that is the observation of certain parameters at different discrete time instants the total Probability theorem provides probabilistic information for the prediction of the future responses at different time instants. Consider the following integral: ( ) ( ) ( ) [ ] (8) Eq. (8) depends on the posterior distribution function. The dimension of the updating parameters makes it very difficult to obtain an analytical solution. Therefore, sampling techniques, such as Markov chain Monte Carlo (MCMC) methods are employed to predict the updating parameter distribution and subsequently to predict the modal properties. Given a set of random parameter vector drawn from ( ), the expectation value of any observed function can be easily estimated. The integral in Eq. (8) can be solved using sampling algorithms [5, 7, 12, 13]. These algorithms are used to generate a sequence of vectors { } where is the number of samples and these vectors can be used to form a Markov chain. This generated vector is then used to predict the form of the posterior distribution function ( ). The integral in Eq. (8) can be approximated as ( ) (9) where is a function that depends on the updated parameters. As an example, if then becomes the expected value of. Generally, is the vector that contains the modal properties and is the number of retained states. In this paper, the SHMC method is used to sample from the posterior distribution function.

4 4 The Hybrid Monte Carlo Method The Hybrid Monte Carlo method, also known as the Hamiltonian Markov Chain method, is a sampling method for solving higher-dimensional complex problems [5, 7, 13, 23, 25]. The HMC combines a Molecular Dynamic (MD) trajectory with a Monte Carlo (MC) rejection step [8, 17]. In HMC, a dynamical system is considered in which auxiliary variables, called momentum are introduced. The updated parameters in the posterior distribution are treated as displacements. The total energy (Hamiltonian function) of the new dynamical system is defined by ( ) ( ) ( ), where the potential energy is ( ) ( ( )) and the kinetic energy is ( ). The kinetic energy depends only on and some chosen positive definite matrix. The joint distribution derived from the Hamiltonian function can be written in the following form: ( ) ( ( )) where is normalization constant. It is easy to see that ( ) can be written as ( ) ( ( ) ( ( )) or ( ) ( ) ( ). Sampling from the posterior distribution can be obtained by sampling ( ) from the joint distribution ( ). Also, the vectors and are independent according to ( ). The evolution of ( ) through time and time step is given by the following Störmer-Verlet algorithm [7, 8] ( ) ( ) [ ( )] (10) ( ) ( ) ( ) (11) ( ) ( ) [ ( )] (12) where is obtained numerically by finite difference as ( ) ( ) (13) [ ] is the perturbation vector and is a scalar which dictates the size of the perturbation of. After each iteration of Eqs. (10) - (12), the resulting candidate state is accepted or rejected according to the Metropolis criterion based on the value of the Hamiltonian ( ). Thus if ( ) is the initial state and ( ) is the state after the above equations have been updated then this candidate state is accepted with probability ( { ( ) ( )}). The obtained vector will be used for the next iteration and the algorithm stopping criterion is defined by the number of samples ( ). Theoretically, these moves preserve the total energy ( ) where the value of the total energy is constant. This can make the acceptance rate since the term { ( ) ( )}. However, the Hamiltonian dynamics is a discretised problem where the Störmer-Verlet is used to evaluate the pair ( ) through time. This integrator does not achieve the exact energy conservation. In this case, the time step needs to be small enough to reduce the error caused by the Störmer-Verlet integrator. The HMC algorithm can be summarized as follows: 1) An initial value is used to initiate the algorithm. 2) Initiate such that ( ) 3) Initiate the leapfrog algorithm with ( ) and run the algorithm for time steps to obtain ( ) 4) Update the FEM to obtain the new analytic frequencies and then compute ( ). 5) Accept ( ) with probability ( { ( ) ( )}). 6) Repeat steps (3-5) to get samples. 5. The Separable Shadow Hamiltonian function The S2HMC improves the sampling by changing the configuration spaces. This accelerates the convergence of averages computed with the method [24]. As a result the S2HMC improves the acceptance rate of HMC at a comparatively negligible computational cost. The S2HMC uses a processed velocity Verlet (VV) integrator instead of Verlet. The goal of a processing integrator is to increase the effective order of accuracy by using preprocessing and post-processing steps [24].

5 The rationale for increasing the effective order of accuracy is that a more accurate integrator has better acceptance rate in HMC. The S2HMC also uses a modified potential energy function, which is conserved to ( ) by the processed method instead of just ( ) by the unprocessed method. Moreover the S2HMC requires a reweighting step to compensate for modification of the potential energy. The shadow Hamiltonian function used in S2HMC is separable and fourth order [24]: ( ) ( ) ( ) (14) is the derivative of the potential energy with respect to. The modified or shadow Hamiltonian is a result of applying backward error analysis to numerical integrators [24]. In the analysis of numerical integrators for Hamiltonian systems, the shadow Hamiltonian has quantities that are better conserved than the true Hamiltonian. In particular, a fourth order shadow Hamiltonian is conserved within ( ) where is the discretization time step. For symplectic integrators one can construct shadow Hamiltonians of arbitrarily high order. The pre-processing step is given by: ( ( ) ( )) (15) ( ( ) ( )) (16) Equations (15) and (16) require an iterative solution for and a direct computation for. The post-processing step is given by: ( ( ) ( )) (17) ( ( ) ( )) (18) Equations (17) and (18) require an iterative solution for and a direct computation for. Finally, in order to calculate balanced values of the mean, the results must be reweighted. The average of an observable is giving by [24]:, where ( ( )) ( ( )) (19) The S2HMC algorithm can be summarized as follows [24]: 1) An initial value is used to initiate the algorithm. 2) Initiate such that ( ). 3) Compute the initial shadow energy ( ) using Eq. (14). 4) Pre-processing: Starting from ( ), solve iteratively for and a directly compute using Eqs. (15) and (16). 5) Initiate the leapfrog algorithm with ( ) and run the algorithm for time steps to obtain ( ) 6) Post-processing: Starting from ( ), solve iteratively for and a directly compute using Eqs. (17) and (18). 7) Update the FEM to obtain the new analytic frequencies and then compute ( ). 8) Accept ( ) with probability ( { ( ) ( )}). 9) Repeat steps (3-5) to get samples. 10) Compute weight: To compute the averages of a quantity ( ) using the S2HMC, reweighting of the sequence of is needed (Eq. 19).

6 The Modelled Structure and FE model All the finite element modeling was simulated using version 6.2 of the Structural Dynamics Toolbox (SDT ) under the MATLAB environment. In this paper, a GARTEUR SM-AG19 aeroplane structure is used to investigate the optimization capability of the four algorithms. The GARTEUR SM-AG19 structure was used as a benchmark study by 12 members of the GARTEUR Structures and Materials Action Group 19 [26, 27, 28, 29, 30]. One of the aims of the study was to compare the S2HMC and HMC methods with different time steps [23]. The benchmark study also allowed participants to test a single representative structure using their own test equipment. The experimental test data used in our analysis is data obtained from DLR Data, Göttingen, Germany. The above aeroplane has a length of 1.5 m and a width of 3m. The depth of the fuselage is 15cm with a thickness of 5cm. Figure 1 shows the FE model of the aeroplane. In our models all element materials are considered standard isotropic. The model elements are Euler Bernoulli beam elements. The measured natural frequency (Hz) data is: 6.38, 16.10, 33.13, 33.53, 35.65, 48.38, 49.43, 55.08, 63.04, Hz. Figure 1 FEM Garteur Structure The parameters to be updated are the right wing stiffnesses ( ), the left wing stiffnesses ( ), vertical tail statiffness ( ) and the overall structure s density( ). The constant of the posterior distribution is set equal 100, and all coefficients are set equal to, where is the variance of the parameter. The vector of is defined as [ ]. The initial position vector =[, ] and its bounds are given in Tables 1 and 2 where - Vertical Tail Plane, -Right and - Left. The time step is and the number of samples is. Table 1 The parameter vector and the mean values Parameter Parameter ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

7 Table 2 The Max/Min bounds of the updated vector Max_ position Min_ position Table 3 presents the initial value (the mean material or geometric value) of the update vector, as well as the updated values obtained by HMC and S2HMC methods for two different time step scenarios ( and ). In the first scenario ( ), the updated parameters obtained by the S2HMC algorithm are closer to the mean values i.e. they are physically realistic. There is a noticeable difference between the final updated values obtained by the HMC and S2HMC. The time step used for simulations in both methods,, provides a very good acceptance sampling rate - - for both methods. In the second scenario ( ) the updated parameters using the S2HMC method are much closer to the mean value. The reason is that the time step is large enough to allow significant jumps of the algorithm during the searching process. This also will lead to better results (see Table 4). In this setting the HMC method gives poor updating parameters (the same initial values) not shown in Table 3. This can be explained because the time step does not conserve the Hamiltonian function. This time step causes significant numerical errors of the integrator used (VV). In this case, the Hamiltonian function decreases with time which causes a sudden decrease of the acceptance rate (the acceptance rate decreases to less than 1% when the time step is ). The acceptance rate for the S2HMC is 71%, which is an acceptable rate compared to that for the HMC method. Table 3 Initial and updated parameter values Initial E Method Method Method Table 4 shows the modal results and output errors for the different sampling algorithms. The results show that the updated FEM natural frequencies are better than the initial FEM for all methods. The S2HMC provides a smaller final sum error compared to the HMC for both time steps. In the case, the error between the first measured natural frequency and that of the initial model is 10.47%. With the HMC method this error is reduced to 3.84% and by implementing the S2HMC it was further reduced to 3.73%. A similar observation can be made for the fourth, fifth, sixth, eighth and ninth natural frequencies. The total initial error was % but after using the HMC and S2HMC methods it reduce to % and 15.00% respectively. Both methods converge fast and they almost have the same convergence rate (the algorithms start converging in the first iterations).

8 Changing the time step for both methods gives different results. In the case where the time step is increased ( ), the S2HMC method improves the most. This can be seen in Table 4 where the total error is reduced to % with an acceptance rate of 71%. However, this is not the case for HMC where the acceptance rate decreases to less than 1%. Using this time step, the updated vector obtained from the HMC does not improve the FEM results. Table 4 Modal results and errors for S2HMC and HMC at two different time steps. Mode Measured Frequency (Hz) Initial FEM Frequencies (Hz) Error (%) Frequencies HMC Method (Hz) Error (%) Frequencies S2HMC Method (Hz) Error (%) Frequencies S2HMC Method (Hz) Error (%) Total errors The time step,, provides a good acceptance sampling rate for both methods: HMC and S2HMC ( ). Choosing a different time step may reduce the acceptance sampling rate for the HMC method which can significantly affect the results obtained as well as the convergence rate. At the same time, it may provide a good convergence rate for the S2HMC method since the S2HMC provides samples when the time step is large. Fig. 2 shows the acceptance rate when the time step varies between to. The acceptance rate for both methods is 99.9% when the time step is. The acceptance rate starts decreasing when the time step increases for both methods but this decrease is faster and more significant in the case of the HMC method. When the time step, the acceptance rate for the HMC method decreases slightly to 98.7% and stays the same for the S2HMC methods (99.9%). When the time step used is, the S2HMC acceptance rate reduces slightly to 97.8%. However, it reduces significantly to 53.2% in the case of the HMC method. Finally, when the time step reaches, the S2HMC acceptance rate reduces to 71.3% which is an acceptable rate comparing to that obtained by the HMC method (less than 1%). Figure 2: The acceptance rate obtained for different time steps using HMC and S2HMC methods.

9 9. Conclusion In this paper Bayesian FEM methods are used to update a GARTEUR SM-AG19 aeroplane structure. To evaluate the posterior distribution function, two Markov Chain Monte Carlo (MCMC) sampling techniques have been implemented; the Separable Shadow Hybrid Monte Carlo method (S2HMC) and the Hybrid Monte Carlo (HMC) technique. In the simulation the S2HMC method gave better results than HMC for both implementations when the time step is equal to and. Moreover, the S2HMC method is more efficient than the HMC method where it provides samples with a large step time which is not the case with the HMC method. The sampling rate for the HMC method significantly decreases when the time step increases. Also, a large time step provides better results when the S2HMC method is used to update an FEM. 10. References [1] Onãte E.: Structural Analysis with the Finite Element Method. Linear Statics, Vol. 1: Basis and Solids, Springer [2] Rao S. S: The Finite Element Method in Engineering, 4th ed., Elsevier Butterworth Heinemann, Burlington, [3] Friswell, M. I. and Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics, Kluwer, Academic Publishers, [4] Marwala. T.: Finite Element Model Updating Using Computational Intelligence Techniques, Springer Verlag, London, UK, [5] Bishop C.M.: Pattern Recognition and Machine Learning. Springer-Verlag, New York, [6] Yuen K.V.: Bayesian Methods for Structural Dynamics and Civil Engineering, Wiley, New York, [7] Cheung, S.H. and Beck, J.L.: (2009). Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters, Journal of Engineering Mechanics, 135(4), , [8] Boulkaibet I., Marwala T., Mthembu L., Friswell M. I. and Adhikari S.: Sampling Techniques in Bayesian Finite Element Model Updating, Proceedings of the Society for Experimental Mechanics, Vol. 29 pp , [9] Izaguirre J. A., Hampton S. S. and Comput J.: Shadow hybrid Monte Carlo: an efficient propagator in phase space of macromolecules Journal of Computational Physics, Vol. 200, pp , [10] Skeel R.D. and Hardy D.J.: Practical construction of modified Hamiltonians, SIAM J. Sci. Comput. Vol. 23, No 4, (2001) [11] Engle R. D., Skeel R. D. and Drees M.: Monitoring energy drift with shadow Hamiltonians, Journal of Computational Physics, Vol. 206, pp , [12] Neal R.M.: Slice Sampling, Technical Report, No. 2005, Department of Statistics, University of Toronto, [13] Hanson K. M.: Markov Chain Monte Carlo posterior sampling with the Hamiltonian Method, Proc. SPIE 4322, pp , [14] Ewins. D. J.: Modal Testing: Theory and Practice. Research Studies Press, Letchworth, [15] Guyan R.J.: Reduction of Stiffness and Mass Matrices, Am. Inst. Aeronaut. Astronaut, Vol. 11(5), pp , [16] Kraaij C.S.: Model updating of a clamped -free beam system using FEMTOOLS, Technische Universiteit Eindhoven, January, [17] Marwala T. and Sibisi S.: Finite element model updating using Bayesian approach. In Proceedings of the International Modal Analysis Conference, Orlando, Florida, USA, ISBN: [18] Bishop C.M.: Neural Networks for Pattern Recognition, Oxford University Press, Walton Street, Oxford, [19] Vapnik V. N.: The Nature of Statistical Learning Theory, Springer Verlag, New York, [20] Ching J. and Leu S. S.: Bayesian Updating of Reliability of Civil Infrastructure Facilities Based on Condition-State Data and Fault-Tree Model, Reliability Engineering & System Safety, Vol. 94, Issue 12, pp , December [21] Creutz M.: Global Monte Carlo algorithms for many-fermion systems, Physics Review D 38, Vol. D 38, pp , [22] Kennedy A.D. and Pendleton B.: Acceptances and autocorrelations in hybrid Monte Carlo, Nuclear Physics B Proceedings Supplements, Vol. 20, pp , 1991.

10 [32] Boulkaibet I., Mthembu L., Marwala T., Friswell M. I. and Adhikari S.: Finite Element Model Updating Using the Shadow Hybrid Monte Carlo Technique, Proceedings of the Society for Experimental Mechanics Series 2013, Vol. 6 pp [32] Sweet C. R., Hampton S. S., Skeel R. D., and Izaguirre J. A.: A separable shadow Hamiltonian hybrid Monte Carlo method, the Journal of Chemical Physic, Vol 131, Issue 17, [32] Beskos A., Pillai N. S., Roberts G. O., Sanz-Serna J. M., Stuart A. M.: Optimal tuning of the Hybrid Monte-Carlo Algorithm, Bernoulli Journal, December [26] Carvallo J., Datta B.N., Gupta A., Lagadapati M.: A direct method for model updating with incomplete measured data and without spurious modes, Mechanical Systems and Signal Processing 21 (7) [27] Datta B.N.: Finite element model updating, eigenstructure assignment and eigenvalue embedding techniques for vibrating systems, Mechanical Systems and Signal Processing 16 (2002), [28] Guyon I., Elisseeff A.: An introduction to variable and feature selection, Journal of Machine Learning Research 3 (2003), [29] Link M., Friswell M.I.: Generation of validated structural dynamic models - Results of a benchmark study utilizing the GARTEUR SM-AG19 Testbed, Mechanical Systems and Signal Processing, COST Action Special Issue, 17(1) January 2003, [30] Mthembu L.: Finite element Model updating, PhD thesis, Faculty of Engineering and the Built Environment, University of the Witwatersrand, 2012.