45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Confer 19-22 April 2004, Palm Springs, California
AIAA 2004-1672
Quantification of Uncertainty Using Inverse Methods Michael I. Friswell* Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK José R. Fonseca† School of Engineering, University of Wales Swansea, Swansea SA2 8PP, UK John E. Mottershead‡ Department of Engineering, University of Liverpool, Liverpool L69 3GH, UK and Arthur W. Lees§ School of Engineering, University of Wales Swansea, Swansea SA2 8PP, UK Parameter uncertainty is extensively used to analyze the robustness and reliability of a structure. However this analysis requires the specification of the uncertainty, and for many parameters this is not an easy quantity to estimate. This paper uses a maximum likelihood approach combined with either a perturbation or Monte Carlo method for the forward propagation of uncertainty. The method is demonstrated on a cantilever beam with a moving mass.
Nomenclature L l N m x y θ µ, Σ
= = = = = = = =
likelihood function log likelihood number of measurement sets number of measurements within a sample set physical parameters response vector parameters to define PDF of physical parameters mean vector and covariance matrix
I.
Introduction
The propagation of uncertainty in material or geometrical parameters through a finite element model to determine the uncertainty in the response has become a valuable and popular technique. Examples of the use of this approach range from robust design to reliability analysis, and computer code is available for stochastic finite element analysis and uncertainty propagation (examples of the latter are NESSUS and ST-ORM). These techniques work extremely well and give great insight into the robustness of structures. However all of these approaches rely on knowledge of the parameter uncertainty. For many parameters, such as panel thicknesses, direct measurement may be used to get the description of the uncertainty in terms of a probably density function. Tolerances may be used to give some insight into the variance for probabilistic approaches or the bounds for possiblistic approaches. However, direct measurement of many parameters is not possible. For example, the model of many joints is uncertain, but the nature of the modeling errors means that equivalent models must be used, based on generic elements, geometric parameters, or other techniques1,2. The uncertainty in the parameters must be obtained from measurements, using inverse methods, and this is the subject of this paper. *
Sir George White Professor of Aerospace Engineering,
[email protected] . PhD student, School of Engineering,
[email protected]. ‡ Alexander Elder Professor of Applied Mechanics, Department of Engineering,
[email protected]. § Professor of Mechanical Engineering, School of Engineering,
[email protected]. †
1 American Institute of Aeronautics and Astronautics Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Finite element model updating has become a viable approach to increase the correlation between the dynamic response of a structure and the predictions from a model. In model updating parameters of the model are adjusted to reduce a penalty function based on residuals between a measurement set and the corresponding model predictions. Typical measurements include the modal model (natural frequencies and mode shapes) and the frequency response functions. The choice of penalty function, and also the optimization approach, has been the subject of much research3. The weighted least squares approaches commonly used may be derived in terms of maximum likelihood estimates from Bayesian methods. The critical issues in model updating are how to parameterize a finite element model and how to regularize the resulting estimation equations to obtain a well-conditioned solution. In terms of uncertainty quantification, this equates to determining the regions of the model and the parameters that give rise to the uncertainty. This requires a model that has physical meaning. A good example of this is a recent European benchmark exercise that updated a simplified aircraft model, and then used this updated model to predict the change in response due to mass modifications to the wing and tailplane4,5. Figure 1 shows the structure, Fig. 2 shows the tail modification and Fig. 3 shows the set of parameters used for updating5. Table 1 shows the resulting improvement in the model. The modifications were implemented by increasing the size of just one of the masses on the wing shown in Fig. 1, and also by adding a mass to the tail, as shown in Fig. 2. Table 1 shows a significant improvement in the accuracy of the predicted natural frequencies for the updated model, particularly for the
Mode 1 2 3 4 5 6 7 8 9 10
Original Expt. Initial (Hz) FE (%) 6.55 0.88 16.61 2.05 34.88 3.15 35.36 3.09 36.71 1.40 50.09 1.58 50.72 6.34 56.44 5.07 65.14 -1.01 69.64 -3.87
Updated FE (%) -0.07 0.97 0.20 -0.20 -0.97 -0.35 0.02 -0.02 0.03 -0.02
Figure 1: The GARTEUR test structure used for the COST Action benchmark.
Figure 2: The tail modification to the GARTEUR structure.
Tail Modification Expt. Initial (Hz) FE (%) 6.54 0.95 13.94 1.71 32.36 -1.71 35.09 -0.10 35.52 2.67 38.10 -3.80 48.68 4.41 50.17 1.90 56.46 4.87 58.16 2.54
Updated FE (%) 0.04 -0.48 -1.92 0.49 -0.14 -0.33 -1.01 -0.49 -0.12 -1.83
Wing Modification Expt. Initial Updated (Hz) FE (%) FE (%) 6.31 4.69 -0.81 16.43 -13.69 0.88 27.28 16.59 5.85 35.36 -0.86 -0.80 36.46 0.04 -0.83 48.92 -25.08 -0.01 50.01 1.63 -0.59 54.34 -5.91 0.09 64.54 -8.25 -0.20 69.55 -14.25 0.09
Table 1: Results from the COST Action Benchmark. 2 American Institute of Aeronautics and Astronautics
Figure 3: The parameters used to update the GARTEUR benchmark structure. wing modification. The updated model clearly predicts the result of the modifications more accurately than the initial model.
II.
Approaches to Uncertainty Quantification
One approach to uncertainty quantification is to measure the response (for example natural frequencies) of a number of nominally identical structures. For each data set the parameters are estimated using the approaches of model updating. This will give some idea of the variation of the parameters, although it is unlikely that there would ever be sufficient measured data to derive accurate probability density functions. However the estimation of mean and variance should be possible. This however relies on a model that has physical meaning otherwise the parameter variation will have no physical meaning but be an artifact of the ill-conditioning in the equations. For multiple data sets this is made more difficult because regularization methods, such as the ‘L’ curve, use the measured data to determine the optimum relative weighting between the measurement residual and the parameter change. Consider a simple two degree of freedom example, where multiple data sets have been generated based on two parameters with a given probability density function. The optimum regularization parameter is obtained from the corner of the ‘L’ curve, and figure 4 shows a selection of these ‘L’ curves. Clearly there is an issue about whether the regularization parameter should vary, since this means that the penalty function changes for each Figure 4: The ‘L’ curves for a two degree of measurement. Indeed it is not even clear that the same freedom example. set of parameters should be used for each set of measured data. The importance of physical meaning and the avoidance of ill-conditioning is the key to uncertainty quantification.
III.
Uncertainty Quantification using Maximum Likelihood
The alternative approach is to parameterize the probability density function of the parameters and to estimate these parameters from multiple measured data sets. For example, if the parameters were assumed to be taken from a Gaussian distribution then the mean vector and covariance matrix is all that is required. This is essentially a form of regularization, since a smaller number of parameters are estimated for a given measured data set. The important issues are the form of the penalty function, which will be based on a maximum likelihood criterion, and the evaluation of the penalty function through perturbation and Monte Carlo approaches. Note that this method is the only one possible to deal with random fields, where the uncertainty is spatially distributed and may be simply parameterized by a correlation length (in the simplest model) 3 American Institute of Aeronautics and Astronautics
A. Maximum Likelihood The key to the maximum likelihood approach is to parameterize the probability density functions (PDFs) of the parameters. The procedure is to determine the probability that the measurements occur given the PDF of the parameters. An optimization procedure, such as the simplex algorithm, may then be used to determine the optimum parameters describing the PDFs. Suppose that the physical parameters, x, follow a certain probability distribution belonging to a probability distribution family parameterized by θ (for example the mean, µ, and covariance matrix, Σ). For a given θ, the output PDF, f ( y | θ ) can be approximated using one of the uncertainty propagation methods. Let the measurements be y 1 , y 2 ,…, y N . The measurements are assumed to be independent, therefore the measurements likelihood is L ( θ ) = f ( y 1 , y 2 ,… , y N | θ ) =
N
∏ f ( y i | θ)
(1)
i =1
or taking logarithms, l ( θ ) = log L ( θ ) =
N
∑ log f ( y i | θ ) .
(2)
i =1
The maximum likelihood estimator, θˆ , is the value of θ for which l ( θ ) is a maximum. A non-gradient based optimization method, such as the simplex method, can be employed to allow the use of standard uncertainty propagation methods. The drawback of this approach is its iterative nature. The uncertainty propagation methods are by themselves computationally intensive and to execute one in an iterative optimization loop would be prohibitive for most interesting applications. Ways to efficiently integrate the maximum likelihood estimation with the two most common propagation methods are discussed below. Fonseca et al.6 has more detail. B. Perturbation Method In the perturbation approach, the response outputs, y, are expanded as a Taylor series in the physical parameters x, about the parameters x 0 . Thus if y = g ( x ) , then y = g ( x 0 ) + J 0 ( x − x 0 ) + higher order terms
(3)
where J 0 is the sensitivity or Jacobian matrix evaluated at x 0 . If the physical parameters are derived from a normal
distribution with mean µ and covariance Σ, denoted N ( µ, Σ ) , then y is
(
)
N µ y = g ( x 0 ) + J 0 ( µ - x 0 ) , Σ y = J T0 ΣJ 0 .
(4)
Using the standard PDF for the normal distribution produces the log likelihood function as N 1 l ( µ, Σ ) = − Nm log 2π + N log Σ y + yi −µ y 2 i =1
∑(
)
T
Σ −y1 y i − µ y ,
(
)
(5)
where m is the number of elements in the vector y, and this is a straight-forward optimization problem for the parameter mean µ and covariance Σ. Ideally x 0 would be the mean µ but the mean is not known beforehand, so a guess must be made for its initial
value. Depending how far that initial guess is from the estimated mean µˆ , it may be necessary to recompute g ( x 0 )
and J 0 to more accurately describe the response surface near µ. However, this computation does not have to be 4 American Institute of Aeronautics and Astronautics
performed for every evaluation of Eq. (5). For most applications an approximate knowledge of the mean value is available reducing the need for such calculations. C. Monte Carlo Method The Monte Carlo approach samples physical parameters based on the assumed PDF. These samples are used to calculate the response of the structure and the PDF of y may then be approximated using a kernel density estimator. Using this standard method is very time consuming because when the parameters describing the PDF of x (such as the mean and variance) change then a new set of samples must be calculated and the results propagated through the model. The efficiency of the method may be improved significantly by retaining the results for the samples we have already computed. When the parameter PDF changes the contribution of these samples to the response PDF are simply re-weighted6.
IV.
A Cantilever Beam with a Moving Mass
The example is a cantilever beam with a point mass placed at an uncertain position along the beam length, shown in Fig. 5. The beam has a length of 1m, a rectangular section of 0.100m x 0.010m and is made of steel (Young modulus, E = 210 x 109 Pa and density ρ = 7800kg/m3). The moving mass is m = 0.700kg and its position x follows a normal distribution N ( µ = 0.618, σ = 0.005 ) . This example was designed
Figure 5: The cantilever beam example with a with future experimental verification in mind, since lumped mass at an uncertain position. the actual parameter distribution is easily measured. Figure 6 shows the variation of the natural frequencies with mass position, and shows that for very small position variation, the natural frequencies vary almost linearly and the perturbation approach becomes attractive because of its computational efficiency. However for large position variations and for the higher modes, the linear approximation will break down. Figure 7 shows the log likelihood function for the given problem using the perturbation approach for the first natural frequency. The function is very steep for mean values away the real mean and low variance, but it is very flat for large variances. More importantly there is only one minimum, which is very close to the actual optimum point, and convergence is achieved from any initial guess. The log likelihood function for the Monte Carlo approach is almost identical for this example. As mentioned above, the optimum results for the perturbation approach are obtained when the linearization is made about the mean value of the parameters. However, this point is not known and therefore an estimate must be made. Figure 8 shows the effect that performing the linearization away from the real mean has on the estimation error. The mean position estimate is generally much more accurate than the standard deviation Figure 6: The variation of the natural frequencies as a function and over a much wider range of linearization of mass position. 5 American Institute of Aeronautics and Astronautics
points, which is expected. Notice that the minimum error is not necessarily obtained at the real mean point. In this example there is a single parameter, and therefore measuring a single natural frequency (e.g. the first) would suffice to estimate the parameter. Furthermore, in this example the higher natural frequencies vary more with mass position than the lower frequencies (as shown in Fig. 6). This highlights an undesirable property of the perturbation approach, since adding more information (in the sense of adding new natural frequencies rather than more samples) makes the estimates of the mean and variance worst. This is shown in Fig. 9. Also shown is the Monte Carlo approach, and in this case the extra measured data does improve the estimation accuracy.
V.
Conclusion
This paper has given some insight into the issues involved in the quantification of parameter uncertainty. Many of the issues are similar to those in finite element model updating, particularly the requirements for regularization and parameterization and the need for physically meaningful updated models. The most promising approach is to parameterize the probability density function of the parameters (for example using the mean and covariance) and to identify these parameters using the maximum likelihood approach. The perturbation approach has problems when the linear approximation to the response is not very accurate, and can lead to more information giving higher parameter estimation errors. The Monte Carlo approach works well, but the algorithm must be carefully designed to ensure that the computational effort is realistic. The next stage is to test the methods on experimental examples, and initially this will use experimental data from a cantilever beam with a moving mass.
Acknowledgments The authors acknowledge the support of the Engineering and Physical Sciences Research Council (United Kingdom) through grants GR/R34936 and GR/R26818. Friswell acknowledges the support of a Royal SocietyWolfson Research Merit Award. Foncesa acknowledges the support of the Portuguese Foundation for Science and Technology through the scholarship SFRH/BD/7065/2001.
References 1
Friswell, M. I., Mottershead, J. E. and Ahmadian, H., “Finite Element Model Updating using Experimental Test Data: Parameterisation and Regularisation,” Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, special issue on Experimental Modal Analysis, Vol. 359, 2001, pp. 169-186. 2 Mottershead, J. E., Friswell, M. I., Ng, G. H. T. and Brandon, J. A., “Geometric Parameters for Finite Element Model Updating of Joints and Constraints,” Mechanical Systems and Signal Processing, Vol. 10, No. 2, 1996, pp. 171-182. 3 Friswell, M. I. and Mottershead, J. E., Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, 1995. 4 Link, M. and Friswell, M. I., “Working Group 1. Generation of Validated Structural Dynamic Models - Results of a Benchmark Study Utilising the GARTEUR SM-AG19 Testbed,” Mechanical Systems and Signal Processing, COST Action Special Issue, Vol. 17, No. 1, January 2003, pp. 9-20. 5 Mares, C., Mottershead, J. E. and Friswell, M. I., “Results Obtained by Minimising Natural-Frequency Errors and using Physical Reasoning,” Mechanical Systems and Signal Processing, COST Action Special Issue, Vol. 17, No. 1, 2003, pp. 39-46. 6 Fonseca, J. R., Friswell, M. I., Mottershead, J. E. and Lees, A. W., “Uncertainty Identification by the Maximum Likelihood Method,” Journal of Sound and Vibration (submitted).
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Figure 7: Log likelihood for the perturbation approach.
Figure 8: The influence of the linearization point on the estimation error.
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Figure 9: The effect of using higher natural frequencies in the estimation.
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