Here 𝑖 = 1, … , 𝑁 where 𝑁 is the number of points in the deviation metric, and 𝑗 = 1, … , 𝑝 with 𝑝 representing the number of parameters. A parameter with very high variance compared to other parameters should therefore be discarded, as model outputs are not very sensitive to small changes to it.
5.3 Inverse uncertainty quantification
A deterministic calibration yields a point estimate in the parameter space, meaning that no statistical description about the estimator exist. Performing predictions with such a parameter setting is not best practice as no statistical confidence bounds in the results are obtained. Therefore, methods exist to account for the uncertainties in the observed data. Generally two viewpoints for the statistical inference problem exist, the frequentist and Bayesian viewpoints [74].
The frequentist view of probability is many times considered the traditional viewpoint. Here the probability for en experimental outcome is interpreted as the relative frequency of occurrence of that outcome in a long sequence of experiment repetitions [50, 74]. Thus, the estimator uncertainty is represented by the probability distribution obtained by applying an estimator to the experiment repetitions. Parameters of interest are viewed as deterministic but unknown. Structural dynamics methods taking the frequentist approach is common, see [61, 95]. A common estimator used is the maximum likelihood estimator (MLE) which gives a parameter setting that maximises the likelihood function [50]. From Equation (5.1), assuming that 𝐡𝑋 = 𝑣𝑒𝑐𝑡(𝐇𝑋)and 𝐡𝐹𝐸= 𝑣𝑒𝑐𝑡(𝐇𝐹𝐸), and with
the assumption that 𝐍𝐺 ∼ 𝒩(𝟎, 𝚺)the likelihood function can be formed as
𝑙(𝐡𝑥|𝜽) ∝ 𝑒𝑥𝑝 [− 1 2(𝐡𝑋− 𝐡𝐹𝐸) 𝑇 𝚺−1(𝐡 𝑋− 𝐡𝐹𝐸)] . (5.4)
Here ∝ denote proportionality between two expressions. The MLE estimator ̂𝜽𝑀𝐿𝐸is
then found as
̂
𝜽𝑀𝐿𝐸=arg max 𝜽
𝑙(𝐡𝑥|𝜽). (5.5)
Under certain conditions the MLE estimator ̂𝜽𝑀𝐿𝐸distribution can be shown to be [50]
̂
𝜽𝑀𝐿𝐸∼ 𝒩 (𝟎, ℱ𝑎 −1(𝜽)) . (5.6)
Here 𝑎
∼denote the asymptotically distributed symbol.
Another approach to find the sampling distribution of the MLE estimator is based on using bootstrapping [40, 52, 98, 95]. This approach is used in this thesis in Paper B and
Paper C. The procedure works by repeatedly drawing random datasets 𝐡𝑏
𝑋with replace-
ment from the original data set 𝐡𝑋. The calibration procedure is then performed towards
this new data set 𝐡𝑏
𝑋in Equation (5.2) so that a new calibration parameter setting ̂𝜽𝑏is
obtained. This procedure is repeated 𝑛𝑏times. Thus 𝑛𝑏vectors of calibrated parameters
̂
𝜽𝑏will be obtained, from which statistics can be computed [52, 98].
The Bayesian viewpoint, on the other hand, interprets probability as a degree of plausibility. Therefore, the parameters are considered random variables and a prior belief about their distribution can be used, if it exist. In structural dynamics much research is presently focusing on Bayesian techniques and especially their computational challenge using Markov chain Monte Carlo methods, see [9, 8, 14, 94]. The methods work by using Bayes’ theorem [12]
𝑝(𝜽|𝐡𝑋) =
𝑝(𝐡𝑋|𝜽)𝑝(𝜽)
𝑝(𝐡𝑋)
(5.7) where 𝑝(𝐡𝑋|𝜽)is the likelihood function, 𝑝(𝜽) the prior distribution and 𝑝(𝐡𝑋)a normal-
isation constant so that the posterior distribution 𝑝(𝜽|𝐡𝑋)becomes a valid probability
density function. The posterior distribution is therefore associated with the prior distri- bution, but is updated with respect to the data, and so new information is incorporated
in this framework. The posterior distribution can be used in an estimator. It is common to use the maximum a posteriori (MAP) estimator [50].
̂
𝜽𝑀𝐴𝑃=arg max 𝜽
𝑝(𝜽|𝐡𝑥) (5.8)
This estimator selects a parameter setting ̂𝜽𝑀𝐴𝑃that will maximise the posterior distri-
bution, i.e. find the posterior distributions mode. Another common estimator is the minimum mean squared error (MMSE) estimator [50].
̂
𝜽𝑀𝑀𝑆𝐸= ∫ 𝜽𝑝(𝜽|𝐡𝑥)𝑑𝜽 =E [𝜽|𝐡𝑥] (5.9)
Here the parameter setting ̂𝜽𝑀𝑀𝑆𝐸is chosen so that the estimator minimises the mean
square error (MSE). For other estimators see [50]. The computational burden associated with Bayesian techniques is very high for high dimensional parameter spaces (curse of dimensionality) which is one of the drawbacks of the method. The method has also been criticised as biased due to the prior distribution, which can sometimes be chosen arbitrarily. Furthermore, the implementation is fairly complex compared to the simpler frequentist methods.
Other approaches exist for the treatment of uncertainties and variability in model parameters for the inverse problem. Non-probabilistic methods have surfaced recently, such as the fuzzy set method [39, 80]. They have been introduced due to the inability of probabilistic methods to account for epistemic uncertainty [67], i.e. uncertainty that very little is known about or non-random uncertainty. In other words many authors propose that probabilistic methods should be avoided when the uncertainties are epistemic, i.e. lack-of-knowledge type uncertainties are present [10].
5.4 Forward uncertainty quantification
When uncertainties are associated with the simulation model one single deterministic simulation will generally not be sufficient for predictive purposes. That is because no confidence bounds for the results are present. In reality uncertainties about simulation models are always present. For instance when no experimental data about the model exist, or in cases when it is known that a particular parameter is representing a property created by a production process with some randomness with a known probability distribution. In such situations uncertainty propagation, or forward uncertainty quantification methods can be used. The most well known method is the Monte Carlo method, which is also often used as a reference for other methods. It can be seen as a brute force method in which the deterministic simulation is run very many times with different realisations of input parameters and controlled by the parameter distribution. Statistics about the output quantities are then obtained which in turn provide more confidence in the simulated results. This method is usually very computationally expensive as a single deterministic simulation can take many hours, and many simulations are needed for just one varying parameter. It is common to have hundreds of uncertain parameters, and hence a very
large parameter space. Therefore, other methods have been developed to circumvent this problem. Good overviews of methods are given in [91, 77, 78, 78, 89]. It should be noted here that the inverse quantification methods in the previous section generally solve the forward problems many times, and so the output uncertainties are implicitly obtained.
For probabilistic finite element analysis uncertainty propagation methods using pertur- bation techniques exist [54]. These are normally based on Taylor series expansion, which are generally well suited when the first and second moments, i.e. mean and variance, of the outputs are sought [91]. From reliability engineering the first and second order reliability methods exist which are well suited when the tail probabilities are sought [91], e.g. to assess a structures probability of failure. Further, spectral finite element methods exist [28] which strive to be suited for any type of problem. Although, these methods are not well suited for nonlinear problems and can be computationally intensive [89]. Sensitivity analysis is another important part of stochastic mechanics, which was briefly explained above where a local sensitivity analysis method was outlined in terms of the CRLB. It can be used to asses the importance of parameters and hence reduce the cost of simulations. Global sensitivity analysis on the other hand strives to quantify the output uncertainty due to uncertainty in all input parameters combined [91].
Most of the focus so far has been on parametric methods, in which specific model parameters are varied. Non-parametric approaches have surfaced recently based on random matrix theory [84, 87, 85, 86, 83]. The main idea is that the EOMs system matrices can be treated as random, i.e. the nominal model can be considered as the mean and very few parameters are used to represent the uncertainties in the matrices. The obvious disadvantage of the method is that no physical insight is usually retained, but on the other hand all types of uncertainties might be considered. This has resulted in an application for structural-acoustic modelling in automotive vehicles [17].
Possible ways of circumventing the computational burden of Monte Carlo methods is to use reduced order models, or surrogate models, e.g. through a multi-fidelity approach [35, 71, 69]. Another approach relies on substructuring, or component mode synthesis, where reduced order models are assembled to obtain system responses [41]. In comparison to the non-parametric methods, substructuring methods will generally allow for a greater physical insight as simpler components are coupled to form the complex system. Further, surrogate techniques exist, such as Gaussian processes [23]. Many of these techniques are computationally expensive in the creation of the surrogate models, but cheap in the evaluation of the surrogate models. Therefore, the sometimes millions of runs required by the Monte Carlo method can be performed cheaply. The drawback is that the surrogate models need to be recreated for other parameter configurations, and that surrogate models introduce errors in the system outputs.
Other approaches exist for the treatment of uncertainties and variability in model parameters, e.g. non-probabilistic methods. Common methods are the convex interval method [10] and the fuzzy set method [65]. These methods are especially useful in the treatment of epistemic uncertainty, i.e. lack-of-knowledge uncertainty rather than random uncertainty. The convex interval method is based on the assumption that many times the parameter range is constrained and as such it tends to form a convex set which simplifies
the analysis. The fuzzy set approach is based on fuzzy sets [99], which are sets where elements have a degree of membership associated with them, i.e. the degree of belonging to the set is defined by the membership function. The methods have surfaced due to the many times arbitrary assumption of the probability density function governing the uncertainties, e.g. when not enough data about the uncertain parameters exist which is many times the case. Experiments tend to be time consuming and expensive and only a limited number can be performed. These methods could thus be classified as a middle ground between deterministic simulations and stochastic methods.
6 Summary of appended papers
Paper A: Redundant information rejection in sensor localisation using system gramians
In this paper the method of effective independence (EfI) is shown to select redundant sensor positions in symmetric models and very detailed finite element models. An exten- sion of the EfI method is proposed where sensors are ranked by the EfI metric and sensor positions containing redundant information are identified and rejected from the final sensor set. The redundant sensor position criteria is based on system gramians and works for sensors measuring in any direction. The method is verified on a simple symmetric clamped plate.
Paper B: Calibration, validation and uncertainty quantification of nominally identical car
subframes
Vibration experiments of three nominal Volvo XC 90 (2015) front subframes are performed. Experimental models are acquired through system identification. Sensor placement is based on the method proposed in Paper A. The spread in dynamical and static properties is reported. It is found that within certain frequency regions the spread in dynamical prop- erties is considerable. The damping equalisation, frequency based, calibration method FEMcali is used to calibrate 9 physical parameters for the three subframes in independent calibrations up to 400 Hz towards the experimentally identified models. Considerable improvement in model predictability is achieved. A bootstrapping procedure, towards raw experimental data, is utilised in assessing the parameter uncertainty with respect to the measurement noise. It is found that one parameter is sensitive to the measurement noise for two components.
Paper C: Parameter estimation and uncertainty quantification of a subframe with mass
loaded bushings
One Volvo XC 90 (2015) rear subframe is considered. Vibration experiments, with sensor placement based on the method proposed in Paper A, are performed with and without
mass loaded bushings. Experimental models are identified using system identification. Model calibrations of configurations with and without mass loaded bushings are per- formed with the FEMcali method towards the experimentally identified models. High fidelity bushings models are created and the rubber stiffness updated towards the mass loaded experimental models. The updated subframe model is then further updated towards the configuration without mass loaded bushings for various physical parameters. Very good model predictability is achieved.
Paper D: A reduced interface component mode synthesis method using coarse meshes
A component mode synthesis (CMS) method with efficient interface reduction is proposed. CMS methods provide an efficient modelling methodology for large scale structural models. When a high density mesh is used, with detailed interfaces, the method’s efficiency is reduced. It is proposed that coarse mesh models are used in forming a reduction basis of the interface degrees of freedom. The method is verified on a simple plate model consisting of two substructures.
7 Conclusion and future work
This thesis deals with the development of methods for creation of predictive models under uncertainties for use in structural dynamics. The goal is twofold. Minimising mea- surement uncertainties and enabling computational models to account for uncertainties. The focus lies on complex industrial structures from the automotive industry. Therefore a component approach is considered so that physical insight is retained. The more complex system of interest can be obtained from an assembly of the individual components.In reducing measurement uncertainties an existing procedure for sensor position se- lection, the method of effective independence, is extended for rejection of sensor positions carrying similar information using system gramians. Further, a measurement software, implemented in MATLAB, for faster stepped sine vibration experiments is developed such that experimental data with low noise levels can be obtained relatively fast.
To quantify model parameter uncertainties a model updating method using frequency response functions and equalised damping with a bootstrapping uncertainty quantifi- cation procedure is used on two automotive structures. A front and a rear subframe are considered. Very good models are obtained, that can be used in a future synthesis. Physical parameters are updated, but it is noted that geometric parameters are highly desirable for a better parametrisation. It is found that the identification of an experimental system is very difficult, and tools for ease of use should be further developed. Spread in dynamical properties between nominally identical components of the two structures is also noted.
A component mode synthesis interface reduction method is also developed which enables fast computations of high fidelity finite element models. The method is based on the usage of coarse meshes to reduce the dimension of the linear system solved in a static condensation.
For future work the assembly of components will be considered, and how the vari- ability between nominally identical components propagate through such assemblies, in both experimental and computational models. Some possible methods to consider for propagation of uncertainties in built up structures are component mode based techniques [41, 48, 47].
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