Vehicle System Dynamics International Journal of Vehicle Mechanics and Mobility ISSN: 0042-3114 (Print) 1744-5159 (Online) Journal homepage: https://www.tandfonline.com/loi/nvsd20 Uncertainty quantification in vehicle dynamics Christine Funfschilling & Guillaume Perrin To cite this article: Christine Funfschilling & Guillaume Perrin (2019): Uncertainty quantification in vehicle dynamics, Vehicle System Dynamics, DOI: 10.1080/00423114.2019.1601745 To link to this article: https://doi.org/10.1080/00423114.2019.1601745 Published online: 08 Apr 2019. Submit your article to this journal View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=nvsd20 VEHICLE SYSTEM DYNAMICS https://doi.org/10.1080/00423114.2019.1601745 Uncertainty quantification in vehicle dynamics Christine Funfschillinga and Guillaume Perrinb a SNCF Innovation & Recherche, Saint-Denis, France; b CEA DAM DIF, Arpajon Cedex, France ABSTRACT ARTICLE HISTORY This paper presents a review of various works that highlight the importance of introducing the variability of the road-track/vehicle system into dynamic simulations as soon as this latter is meant to be predictive. The first section of the paper presents the Uncertainty Quantification, Verification and Validation method (UQ-VV). This latter proposes tools to model uncertainties, to associate a confidence to the prediction of quantities of interest and to estimate the probability of occurrence of different scenarios. The method is illustrated by various examples mainly from the rail domain but also from the road sector. The second section summarises application examples of predictive modelling, robust optimisation and calibration. Received 8 February 2019 Revised 21 March 2019 Accepted 25 March 2019 KEYWORDS Uncertainty quantification; vehicle dynamics; predictive modelling; robust optimisation; calibration 1. Introduction Because of its complexity, the railway dynamic system has only been studied experimentally for decades. The main track and rolling stock developments, the rules to specify, to maintain and to certify were built empirically based on expert judgment and experimental campaigns. In the road dynamic domain, simulation has been used earlier, but essentially at conception stage and for driving simulators. The need for shorter conception and development durations as well as the necessity of cost cutting, however lead to a constant increase of the use of simulation. Nevertheless, resorting to simulation for specification, conception, maintenance and certification, changed the expectation of the engineers toward it: the simulation is no longer just a tool to explain and to understand the physical phenomena, it must be predictive. Building a predictive dynamic model of the vehicle-road/track system is however difficult because it is complex and contains several sources of uncertainties that must be taken into account. The main uncertainty sources of the system are recalled here-under. Track - Road • The track superstructure is composed of random media (soils, ballast, concrete). • The roads and the railway tracks present, from their construction, geometrical irregularities. These stem from the loadings of the vehicles, from the weather solicitations and from defects of the different elements of the infrastructure. These irregularities evolve during the life cycle of the system. CONTACT C. Funfschilling christine.funfschilling@sncf.fr © 2019 Informa UK Limited, trading as Taylor & Francis Group 2 C. FUNFSCHILLING AND G. PERRIN Rolling-stock • The mechanical responses of new suspension elements submitted to the same loadings are variable for several reasons: at manufacturing stage, there is a dispersion between suspensions built by different manufacturers but also within a batch stemming from the same manufacturer. The mechanical behaviour moreover depends on the weather (temperature, humidity), on the age and the loading history of the suspension, on its health state. Elastomeric components are particularly affected by these phenomena. • The masses and inertia of the car bodies change with the load carried (number of passengers, goods . . . ). The masses and inertia can also be modified because of parts changes during maintenance operations. Vehicle-road / track interaction • The friction coefficient between the wheel and the road or the track depends on the weather (rain, snow, temperature) and changes along the track. It is also changing along the rail profile. • The wear of tyres alters the adhesion. Wear also modifies the wheel and rail profiles leading to time and space dependent conicities. Finally the variability can also stem from the nature of the observed phenomena. Indeed, different random physical phenomena occur in the road / track - vehicle system, for instance fatigue of different structures (carbodies, rail and railway wheels, . . . ), ruptures, yaw instabilities, electronic devices failures, turbulent flow etc. The Uncertainty Quantification, Validation, Verification methods (UQ-VV) provides tools to handle these uncertainties for simulation purpose. This paper rapidly describes the method and presents vehicle dynamic works introducing randomness in the simulation process. The third part of the paper is dedicated to application examples. These are sorted into four categories: predictive modelling, sensitivity analysis, robust optimisation and calibration. 2. Uncertainty quantification, verification and validation Simulation is more and more often used in decision-making processes. To play this role, simulation must not only be descriptive but also predictive. However the expected quality of the prediction, the robustness of the optimisation and the quality of the calibration depend on the simulation’s ability to integrate the various sources of uncertainty. But integrating these uncertainties changes the formulation of the problem and increases the complexity of the resolution. The framework classically adopted is described in Figure 1. X represents the vector of the uncertain input data and Y the vector of the random output responses. The numerical model is called G, (.)real denotes the real variables, (.)meas the measured variables and (.)sim the simulated variables. The Uncertainty Quantification, Verification and Validation method follows three main objectives: VEHICLE SYSTEM DYNAMICS 3 Figure 1. Uncertainty management methodology [1]. • Provide methods to integrate at best the available information (expert judgments, direct or indirect experimental measurements) and to build adapted models of uncertainties. This subject will be addressed in paragraph 2.1. • Propose a reformulation of traditional problems that takes into account uncertainties. This reformulation will be explained in the specific cases of robust optimisation and calibration in paragraphs 3.4 and 3.5. • Provide methods to solve efficiently these problems under uncertainty (required sample size, adapted designs of experiment, . . . ). This numerical aspect will be addressed several times in this paper but will not be covered in detail. 2.1. Uncertainty quantification One can distinguish two types of uncertainties for inputs X: • Epistemic uncertainties: some deterministic components of X are not perfectly known. These uncertainties are reducible using more precise characterisation methods or devices. • Random uncertainties: some components of X are random by nature (manufacturing dispersion, wind, natural radioactivity . . . ). These uncertainties are irreducible. In this case the output Y real is random. Measured and simulated outputs Y meas and Y sim are also subjected to uncertainties. On the one hand, controlling the simulation uncertainty ǫ sim := Y real − Y sim , is the role of UQVV approaches. One can distinguish three types of simulation uncertainties: 4 C. FUNFSCHILLING AND G. PERRIN • The model error: some of the physical phenomena are neglected, an expensive code has been replaced by an approximate code, . . . The control of this error is the role of validation. • The uncertain estimation of the (physico-numerical) parameters of the code. The reduction of the involved uncertainty is the role of calibration. • The uncertainty due to the numerical resolution of the system equations. The control of this uncertainty is the role of verification. On the other hand, controlling the measurement uncertainty, ǫ meas := Y meas − Y real is the objective of metrology. More details are given in the Guide to the expression of Uncertainty in Measurement (GUM) [2]. There are several methods to model the measurement and simulation uncertainties. The choice depends on the objective sought: • If the objective is to define a predictive model Y = G(X) in unmeasured values X, errors can compensate each other. • On the contrary, if the objective is to identify the input parameters X as close as possible to their physical values in order to study their evolution or reuse them in another code for example, the different errors have to be modelled carefully independently. In most industrial applications, these uncertainties are difficult to characterise. This is particularly true when the available information is limited, and when large dimensions inputs and outputs are considered. The aggregation of information from different experiences, different experts, different numerical codes constitute an other difficulty, especially when all this information seems inconsistent. In order to illustrate the uncertainty quantification, we rapidly describe in this section the modelling of two complex stochastic fields: the modelling of railway track geometry and the modelling of the geometry of ballasted grains. Illustration 1: track geometry modelling The use of simulation to certify the road-track vehicle system or to study different maintenance policies, leads to the need for representative road and track geometries. The shapes of the irregularities are however complex, they present strong spatial dependencies so that their stochastic modelling is complex. Perrin et al. [3] proposed to model the four track irregularity fields (lateral offset X1 (s), vertical offset X2 (s), cross-level X3 (s) and gauge X4 (s) represented in Figure 2) as follows: • X is written as a weighted sum of deterministic spatial functions thanks to a KarhunenLoève expansion. • The multidimensional distribution of these weights, which is non Gaussian, is modelled using a polynomial chaos expansion [4]. This representation allows the generation of realistic track geometries that are representative of the measured railway network. The stochastic modelling is validated from a statistical and frequency point of view comparing the mean number of up-crossings as well as the spectral content of a set of measured geometries and a set of generated geometries. Simulated dynamic responses (accelerations VEHICLE SYSTEM DYNAMICS 5 Figure 2. Parametrisation of the track irregularities (for each rail, the mean position is represented in black, whereas the real position is in grey). Figure 3. Statistical and spectral comparisons of dynamical quantities of interest obtained on measured ℓ and Q = Y r are the left and right transverse contact forces tracks and generated tracks. Q5 = YMC 6 MC ℓ and Q = Y ℓ are the left and right at the first wheelset of the first bogie of the motor car. Q7 = YPC 8 PC transverse contact forces at the second wheelset of the second bogie of the second passenger car [3, p.129]. and wheel-rail contact forces) of a high-speed train excited by measured or generated tracks are studied in the same manner. Figure 3 presents typical comparisons for transverse contact forces. The left hand side picture presents the comparison of the number of up-crossing for different levels. The right hand side presents the comparison of the power spectral densities. The second illustration concerns the modelling of the ballast grain shapes. Illustration 2: modelling of the ballast grain shapes The stochastic modelling of the shape of ballast grains is useful to study the links between geometry and mechanical properties of the granular material. Their characterisation thanks to specific spatial shapes could moreover be used to specify and to sort the particles to optimise the mechanical behaviour of the ballast layer. To facilitate modelling, [5] proposes the following pre-processing: first the grains are centred on their centre of mass and oriented according to their principal inertia directions. 6 C. FUNFSCHILLING AND G. PERRIN Figure 4. Representation of the three-dimensional surface of the first shape functions: mode 1 to mode 4 [5, p.5]. The grains are then represented by 800 radii from their centre of mass to the surface along isotropically distributed directions. These sets of radii are modelled by a random vector R. The modelling procedure is similar to the one developed for track geometry: • R is written as a weighted sum of deterministic spatial functions thanks to a KarhunenLoève expansion. These spatial functions, each of which represents a three-dimensional surface, form a basis of shape functions. To build a reduced order model, a truncation of the basis is then realised. The first four shape functions, i.e. the most important ones, are given in Figure 4. • The coordinates on this basis of the grains of the experimental learning set are finally modelled with a vector-valued non Gaussian field identified thanks to kernel smoothing [6]. To assess the validity of the model, statistical comparisons of geometrical characteristics between the learning set and a validation set are raised. To model the mechanical behaviour of these heterogeneous or granular media one can also adopt a stochastic continuous mechanical model as proposed for instance in [7,8]. 2.2. Verification and validation Once the uncertainties are quantified, the quality of the model has to be assessed. The UQ-VV method proposes to distinguish the verification process and the validation process [9]: • The verification consists in determining whether a numerical model conforms the mathematical model it is supposed to solve (in other words verify that the model is ‘solving VEHICLE SYSTEM DYNAMICS 7 the equation the right way’). This refers to numerical analysis classically on reference solutions and the study of algorithms. • The validation consists in determining whether a model is representative of reality, within its scope of use (in other words verify that the model is ‘solving the right equations’). This typically involves confrontation with measurements and the use of statistical tests (χ2 tests for example). It has to be noted that the model can be valid for one application and not for another one, for one quantity of interest and not another one. Moreover it is easier to determine that a model is not valid than the contrary. Application of the verification process to vehicle dynamics The Manchester Benchmarks [10,11] propose to compare different codes on simple typical vehicles/track systems. Both eigenvalues and transient simulations are considered. Application of the validation to vehicle dynamics In vehicle dynamics, the vector-valued responses can be compared in the time domain or in the frequency domain. The time domain comparison can be very poor in case of phase-shifted signals, even if the model is good. Because of the non-linearities of the system, the spectral responses are on the other hand very dependent on the excitations which have thus to be modelled carefully. Different misfit or cost-functions have moreover been proposed: least-square differences over the whole signal, distance between mean values, peaks or chosen quantiles, comparison of cumulative distributions [12], etc. In railway dynamics, the model validation has been first studied for certification purposes, notably in the european DynoTrain project [13]. The validation process developed is based on comparisons between on-track measurements and simulated dynamic response on 24 different track sections whose characteristics are defined by the EN14363. Mean values and extreme values (0.15 and 99.85 quantiles as stipulated in the leaflet) of different accelerations and wheel-rail loads are compared on each of the selected track sections in the time domain (see [14] for details). The evaluated differences being very sensitive to phase shifts, the synchronisation of the experimental and numerical signals is achieved. The mean and the standard deviations of the obtained differences over the different track sections are finally compared to thresholds (see Table 2 [14, p.137]). Figure 5 presents Figure 5. Validation criteria developed in the DynoTrain project [15, p.7]. 8 C. FUNFSCHILLING AND G. PERRIN an example of validation for the vertical acceleration. It has to be noted that the absolute nature of the criterion makes it necessary to define one threshold per quantity of interest. To circumvent this requirement, Kraft et al. [16] propose a two-step validation process with relative criterion: • The quasi-static behaviour of the model is first validated comparing measurements and simulations of bogie rotation and sway tests. • Second, the dynamic behaviour is validated computing on each track section (k) the following least-square misfit function in the time domain: (k) L = min  T (k) 0  (k) T 0 (Y meas (t) − Y sim (t))2 dt .  T (k) (Y meas )2 (t) dt, 0 (Y sim )2 (t) dt (1) • The cumulative distribution of these distances L(k) over the track sections is finally computed and a validation criterion is defined as a threshold for a given quantile. Figure 6 gives examples of cumulative distributions for the bogie and car-body accelerations of a high-speed train. It can be noticed that, as expected, the vertical behaviour is better reproduced than the lateral as well as the bogie movements which are better reproduced than car-body movements. Instead of comparing one measurement to one simulation in a deterministic way, and because of the presence of uncertainties in the system, a probabilistic framework can also be adopted. Lebel et al. [17] propose for instance, for calibration purpose, to introduce the modelling and measurement uncertainties as an additive stochastic Gaussian process B on the simulated spectral response. The quality of the simulation is then evaluated computing the log likelihood L of the available measurement Y mes (ω) and the stochastic simulated response Y sim (ω). Figure 6. Cumulative distribution of the least-square misfit functions L(k) computed on a set of track sections. Bogie acceleration on a high-speed train on the left, car-body acceleration on the right [16, p.1497]. VEHICLE SYSTEM DYNAMICS 9 3. Application examples As discussed in introduction, the applications of simulation in vehicle dynamics are numerous. We will summarise in this section a few works dealing with probabilistic simulations. The first subsection is dedicated to predictive modelling; the second presents risk assessment studies; the third deals with sensitivity analysis; the fourth presents robust optimisation and the last subsection focuses on model calibration. 3.1. Predictive modelling 3.1.1. Damage modelling Damage modelling is a typical example of probabilistic predictive modelling. As example we will consider road and track damage. This damage strongly depends on the vehicle loadings and thus on the speed of the vehicles, their masses and their mechanical characteristics, on the irregularities of the road or of the track surface [18]. Several physical modelling of the degradation can be found in the litterature. These are usually based on more or less complex laboratory tests or on track measurements (see for example [19,20]). These types of models present the advantages of being constructed from physical characteristics of the system and can thus give indications on the causes of the observed degradation. However, in order to be predictive, they often suffer from the need of a large number of experimental characterisations (local stiffness of the different track layers for example) that are hard to achieve. More recently, data-driven models have been developed to characterise the global track degradation. Different evolution laws have been proposed: linear with random parameters, exponential smoothing or Gamma processes [21,22]. These models allow the description of the mean degradation as well as the variability observed. Ramos Andrade and Fonseca Teixeira [23] moreover proposes to use Hierarchical Bayesian models to infer the degradation of a local track portion from a global model and a few measurements on the studied portion. These data-driven models do however not take profit of the mechanical properties of the vehicle / track interaction that partly explain the variability of the degradation behaviour. There are finally models mixing physical modelling and data analysis. To model the evolution of one particular defect, Lestoille et al. [24] developed for example an ARMA type model indexed on the simulated wheel / rail interaction force. The damage predictions seem comparable to the observations. 3.1.2. Fatigue modelling Fatigue damage is another classical application of predictive modelling under uncertainties since the crack initiation and propagation are random phenomena. Many infrastructure and vehicle elements that play a role in vehicle dynamics are subjected to fatigue [25]. One can cite bridges, catenaries [26], axels, wheels, rails, supension elements, carbodies, bogies. The physical phenomena involved are also various: bending and torsion fatigue [27], thermal fatigue [28], contact and rolling contact fatigue but also fretting fatigue [29]. For safety reasons and to limit the maintenance costs, it is interesting to be able to predict crack initiation and propagation, to propose optimised systems and monitoring devices. Ugras [30] proposes for example an original on board real time damage estimation of fatigue. 10 C. FUNFSCHILLING AND G. PERRIN As illustration example, we will concentrate on rail rolling contact fatigue that has been widely studied. The physical phenomena involved are complex: plastic flow, phase change, wear . . . and have origins at different scales: • The relative position of the rail and the wheel as well as the contact forces are partially governed by the vehicle / track interaction and can be estimated thanks to railway dynamics simulations (macro-scale). • At the meso-scale one can observe the development of abrasion, friction, elasto-plastic strains. The stresses can be evaluated thanks to finite element modelling. • The orientation and the stress state of the different grains promote the development of micro-cracks (micro-scale). Hence one can find both experimental and numerical works. To predict rail rolling contact fatigue, the most classical model is the ‘energetic criterion’ [31]. This model links rolling contact fatigue to the quantity τγ directly given by railway dynamic simulations:  (2) τγ = τ .γ dS S with γ the pseudo-slip and τ = σ .n − (σ .n, n)n the tangential surface stress in the contact patch and S the contact surface of normal n. This quantity, usually called energy, is a friction dissipated power but has the unit of a load. The material characteristics are only taken into account through their rheological models in the railway dynamic code. This global criterion is generally analysed thanks to a multi-linear curve identified on damage observations (see for instance Figure 7). Four functioning situations can be observed: • at very low τγ no damage occurs, • for higher τγ rolling contact fatigue is observed, Figure 7. Rail damage with respect to τγ . VEHICLE SYSTEM DYNAMICS 11 Figure 8. Mean value of the fatigue index in the rail section: empty slow vehicle, wide curve (left); full fast vehicle, tight curve (right). [35, p.1788]. • wear enters then in competition with fatigue, and this latter lowers linearly with the raise of the energy, • finally wear becomes prominent and rolling contact fatigue isn’t observed anymore. The points delimiting these four functioning modes are identified on on-track measurements. The obtained damage is used to understand the involved phenomena but also to predict the crack initiation with a reasonable accuracy [32] which is helpful for maintenance purposes. Another strategy consists in adopting a multi-scale approach. In [33,34] a four step model is developed. • A railway dynamic modelling of the track vehicle interaction is built. The stresses vector distribution σ .n in a specific zone is extracted and introduced as a loading in a finite element modelling of the rail. • The mechanical behaviour of the rail is modelled by an elasto-plastic law with cinematic hardening. The asymptotic mechanical state is directly computed thanks to a direct stationary algorithm in a finite element code. • The risk of crack initiation is computed thanks to the meso / macro Dang Van fatigue criterion. • The Miner’s additive rule is used to estimate the mechanical state of the rail. To limit the numerical cost of this modelling strategy and allow variance-based global sensitivity analysis, Panunzio et al. [35] propose to replace the finite element modelling and the fatigue post-processing with a meta modelling. The meta modelling consists in replacing the actual model by an approximation fast to evaluate. In the considered work, the input-output relationship is represented by a polynomial chaos. Examples of the mean value and the coefficient of variation of the fatigue index are given Figures 8, 9. 3.2. Risk assessment The assessment of the probability of accident and the associated risk is another classical application of the UQ-VV methods. 12 C. FUNFSCHILLING AND G. PERRIN Figure 9. Coefficient of variation of the fatigue index in the rail section: empty slow vehicle, wide curve (left); full fast vehicle, tight curve (right). [35, p.1789]. Two types of risks are usually studied :  • The collective risk: CR = i Wi Ai where Wi denotes the occurrence probability of a hazard in a certain amount of time or equivalent (for example, accidents per km for railway and road vehicles) and Ai its consequence (for example in terms of injuries, fatalities or costs due to damage).  • The individual risk: IR = i Vi Wi Ai with Vi the probability of presence of the individual when the hazard occurs. Different strategies can be adopted to define an acceptable threshold for these risks. One can cite: • The ‘As Low As Reasonably Practicable’ method (ALARP), originally developed in the United Kingdom, which proposes to reduce the residual risk as far as reasonably practicable [36,37]. The threshold can be estimated so that the cost involved to further reduce the collective risk is not disproportionate to the benefit. • The ‘Globalement Au Moins Aussi Bon’ (GAMAB) or ‘Globalement Au Moins Equivalent’ (GAME) principle, developed in France, requires that the total risk of any new system shall not be higher than those of existing systems with comparable performance characteristics and operating conditions [38,39]. • The ‘Minimum Endogenous Mortality’ (MEM) method, frequently applied in Germany, prescribes that the individual risk should be less than 10−5 per year. This number stems from the requirement that a new technique should not increase the minimum endogenous mortality, which is about 2.104 per year in industrial countries, by more than 5%. To handle the very low probabilities involved, adapted distributions models and propagation methods have to be considered. To model rare events, Generalised extreme value distributions [40] are usually adopted. The Gumbel and Weitbull laws, commonly used in mechanics, belong to this family [41]; the Gompertz model is more specifically used to model mortality rate [42]. VEHICLE SYSTEM DYNAMICS 13 To estimate the low probabilities of the output, different propagation methods have been developed. The First- and Second-Order Reliability Methods (FORM, SORM) propose to achieve a linear or quadratic analytical approximation of the failure domain around the most critical functioning point (see [43] for details). The subset sampling proposes to reduce the required number of evaluations expressing the small failure probability as the product of larger conditional probabilities. In other words, the rare event problem is divided into a series of more frequent problems easier to analyse (see [44]). To illustrate the risk assessment, a few analysis of the cross-wind effects on vehicle ride and some gauge studies will be described here-under. Illustration 1: vehicle overturning. A large number of analyses of the cross-wind effects on road and rail vehicles can be found (see [45]). They essentially study the risk of overturned vehicles as it is required for railway vehicles certification by the Technical Specification for Interoperability (TSI), but also course deviations, excitation of suspension modes or fatigue of the road drivers when driving in gusty winds. To avoid accidents the wind speed is estimated thanks to on-site measurements or thanks to weather forecasts. If this latter is too high, the running conditions or the route are adapted. In order to assess the overturning risk, one can find experimental and numerical studies. Different full-scale or reduced-scale tests have been achieved in wind tunnel or on-site. These tests naturally consider one part of the observed variability (the turbulence for example). One can cite the works achieved in the PoliMi wind tunnel [46] to analyse the effect of the environment on the overturning risk and the effect of the movement of the train (see Figure 10). For high-speed trains the overturning criterion usually adopted consists in verifying that the wheel unloading is lower than 0.9: Q Q − Q0 = < 0.9. Q0 Q0 (3) with Q the vertical load on the studied wheel and Q0 the nominal load on the wheel. Indeed the evaluation of the risk itself is not required by the safety authorities. The overturning risk can also be assessed thanks to vehicle dynamic simulations (see for example [47]). In this work, a stochastic framework is adopted, the uncertainties are propagated through the dynamic model thanks to Monte-Carlo sampling and stochastic Figure 10. Lorry model on flat ground – dynamic test in tunnel exit [46, p.63]. 14 C. FUNFSCHILLING AND G. PERRIN abacus are build. Moreover, the numerical approach allows the estimation of the margin between the unloading criterion and the overturning. The required aerodynamic resultant forces and moments can be estimated thanks to experimental tests or to Computational Fluid Dynamics Simulations even if the turbulence is very hard to model precisely (see Figure 11). Kwon et al. [48] proposes for example the estimation of the resultant loads at different speeds and different yaw angles with 5% reduced-scale tests for two trains: KTX high-speed train and AREX conventional train. Favre [49] studies the aerodynamic response of ground vehicles submitted to sudden strong wind disturbances with Detached Eddy Simulations. The accuracy and the sensitivity to the mesh are carefully assessed. Illustration 2: gauge assessment. Another classical application of risk assessment in vehicle dynamics concerns the gauging process; ensuring the gauge clearance is indeed a safety criterion. In [50], Evans and Berg emphasise the necessity of considering the effects of suspension tolerances but also of degraded suspensions, of covering the full range of possible vehicle speeds and of possible track geometries within the maintenance limits. The authors also depict the method developed in the United Kingdom based on a the statistical processing of hundreds of 20 km long simulations covering a wide range of speeds, curve radii and cant deficiencies as well as different vehicle conditions. The vehicle movements are post-processed with the ClearRouteTM software to assess the clearance at each structure, defined by the mean+2.12 standard (µ + 2.12σ ) deviation of the simulated displacements. The obtained simulation results could also be post-treated with the Wilk’s quantile which gives the minimal size of the required realisation set and allows the evaluation of the misleading rate [51]. Figure 11. CFD computations of wake vortex flows behind trains in a cross wind, ICE. The figure represents the main vortices that develop on this form of train, with detachments at the level of the nose (downwind), the roof etc. On the right hand side, the colour bar represents the pressure in the fluid. [45, p.996]. VEHICLE SYSTEM DYNAMICS 15 3.3. Sensitivity analysis Once the model is built, a sensitivity analysis is often realised to estimate the contributions of the different parameters to the variability of the studied model response. The objectives are multiple: • Estimate qualitatively the parameters that have only little influence on the model outputs variations to reduce the number of parameters to be considered. • Quantify the effect of the input parameters variability on the model outputs variations. This can be useful to define measurement campaigns or advanced modelling works to enhance the representation of these input parameters. It can also help to define the inputs elements that could be optimised and should be treated with care during construction or maintenance. Two types of sensitivity analysis can be achieved: local (around a functioning point) or global (over the whole input distributions). A large number of methods exist [52, p.101–122]; they can be sorted in two main families (see Figure 12): • Screening methods that only require few code evaluations to explore a large set. A classic example is the one-at-a-time Morris method [53]. • Quantitative methods based on variance decomposition. One can cite the well-known Sobol indices [54]. Figure 12. Classification of sensitivity analysis methods according to the complexity of the model and the number of required code evaluations. 16 C. FUNFSCHILLING AND G. PERRIN The choice of the sensitivity analysis depends on the studied quantity of interest (mean value, high or low quantile for example) but also on the number of parameters, their nature (continuous or categorical), their dimension (scalar, vectorial), their statistical dependencies, the complexity and regularity of the studied model, its computational cost and the affordable number of simulations. Often a screening method is first used to reduce the model complexity; a quantitative method is then used to classify the inputs. To illustrate the use of sensitivity analysis in vehicle dynamics, two papers are rapidly summarised. Illustration 1: sensitivity of the critical speed. The first one proposes to study the sensitivity of the critical speed with respect to the suspension parameters which is useful at conception stage [55]. The parameters (dimension 24) are modelled with Gaussian distributions. The distribution of the critical speed is obtained thanks to 200 simulations of the program DYnamics Train SImulation (DYTSI) sampled with the Latin Hypercube method. Total Sensitivity Indices which give the contribution of one parameter to the total variance, including its interactions with other parameters, are then computed. To limit the computational cost an ANOVA- highdimensional model representation has been used. Typical results of such studies are given in Figure 13. Illustration 2: sensitivity analysis of the subjective assessment of vehicle dynamics. The second paper [56] proposes an original sensitivity analysis of the subjective dynamic assessment of road vehicles to design tolerances. To objectify the driving experience and to make predictions of the subjective assessment of new vehicles, the authors propose to use two Artificial Neural Networks: • First, a self organising-map (unsupervised network) is built to classify measured vehicles from objective metrics (ex: lateral accelerations, roll rate gradient . . . ). It also allows the prediction of subjective characteristics (like steering feel for example) of a new vehicle looking at the proximity in the map of this vehicle with known vehicles. Figure 13. Distribution of the critical speed obtained using Latin Hypercube sampling and pie plot of the Total Sensitivity Indices on the reduced stochastic model giving the importance factors of the mechanical characteristics of different system elements. PSL stand for primary suspension of the leading bogie. The following L and T stand for leading and trailing wheel sets. Finally K2 and K3 stand for longitudinal and vertical springs. [55, p.101–122]. VEHICLE SYSTEM DYNAMICS 17 Figure 14. ‘Sensitivity analysis of the tolerances of the Objective Measurements-requirements for an SUV example, represented by dashed lines with triangles. (a) and (b) show two requirements (yaw and roll control straight) for which their tolerances produce large movements within the Self OrganizingMap. (c) gives an example of an Objective Measurements-requirement (roll control cornering) with low impact on the position in the Self Organizing-Map.’ [56, p.162]. • A General Regression Neural Network (supervised network) is superimposed to the first one to allow the quantification of subjective assessment. The paper finally proposes to use the developed simulation tool to achieve a sensitivity analysis of the steering feel to the building tolerances during the development process. As illustration, a one at a time method is used to analyse the effect of the tolerance of one Objective Measurement (OM) on the Subjective Assessment (SA) for a SUV (see Figure 14). The axes on the picture indicate the neuron index in the 100 ∗ 100 neuron grid, whereas the level lines represent the value of the Subjective Assessment. Finally, the rhombus represents the position of the studied SUV in the self organising-map and the dashed line the movement of this position when changing one objective measurement within its tolerance interval. 3.4. Robust optimisation Numerical optimisation tools are more and more used at conception stage. Calling X the conception parameters (geometry, materials, . . . ) and Y(X) the conception criteria (cost, performance, . . . ), the deterministic optimisation problem is written: X ∗ ≈ arg min C (Y(X)), X∈X (4) with C a cost function aggregating the different conception criteria and X the search domain. The design parameters have however to be robust, that is to say they must have an adapted behaviour even in non-nominal configurations. Most of the time, the integration of the robustness involves the introduction of the uncertainties in the optimisation process. The optimisation problem under uncertainties is reformulated: X ∗ ≈ arg min C (Y(X; Z)), X∈X (5) 18 C. FUNFSCHILLING AND G. PERRIN with Z a random quantity. In this case Y(X, Z) is a random variable so that an order relationship is necessary to determine if a design point X is better than an other one X ′ . Göhler et al. [57] lists 108 metrics associated with robustness. One can cite: • Worst case approach: X is better than X ′ if max[Y(X; Z)] ≤ max[Y(X ′ ; Z)]. Z Z (6) • Mean case approach: X is better than X ′ if E[Y(X; Z)] ≤ E[Y(X ′ ; Z)]. (7) • Multi-criteria approach based on a compromise between the minimisation of: E[Y(X; Z)] and Var[Y(X; Z)]. (8) Illustration: design of vehicle suspensions to optimise ride comfort Drehmer et al. [58] and Bathou et al. [59] propose a method to achieve a robust design of the mechanical characteristics of a road vehicle. After having introduced the uncertainties in the simplified multi-body vehicle model (see Figure 15), cost functions are built. [58] considers a multiobjective function combining ride comfort, road holding capability and suspension working space seat damping. Batou et al. [59] uses as cost function the mean value and the variance of the random comfort vector. Weighted spectral responses to account for the non-linear human sensitivity are chosen as comfort index in both cases. The minimisation is then achieved using the classical Monte Carlo simulation method as stochastic solver in the second study and the particle swarm optimisation in the first study. This method was originally developed to model the displacement of groups of birds and in socio-psychology. Both studies underline the importance of the mechanical characteristics of the seat suspensions. Figure 15. Vehicle multi-body model [59, p.93]. VEHICLE SYSTEM DYNAMICS 19 3.5. Calibration The calibration consists in estimating the physico-numerical parameters of a code. In a deterministic calibration, we are interested (most often) in a single value of the input parameter β. Classically, this value is searched as the minimum of: β ∗ ≈ arg min C (Y mes (X), Y sim (X; β)), β∈B (9) where C is a cost function (e.g. least squares), which can integrate measurement errors in the form of weighting, B is a search domain and X are the system parameters. When working on a complex system submitted to several sources of uncertainties, the Bayesian formalism can be adopted. In that case, unknown parameters β are assumed to be random. The ranges of likely values for β, knowing the observations Y meas (X), are then searched as follows (Bayesian inference): P[β|Y meas ] ∝ P[β]P[Y meas |β], (10) with P[Y meas |β] the probability to have Y meas knowing β and P[β] the prior knowledge on β [60]. Illustration 1: identification of the mechanical properties of damaged suspensions. The applications of calibration are numerous; we will focus here on the identification of abnormal mechanical behaviours of suspension elements. Several methods have been proposed in the last decades to monitor the health-state of vehicle suspension elements [61–63]. Their goal is either to detect fault, to identify the faulty element or to estimate the mechanical characteristics of the faulty suspension. In order to detect faulty suspension, data-driven methods have been developed: see for example the inverse regression in the frequency domain proposed in [64], the use of Random Decrement Technique and Nearest Neighbours categorisation on spectral accelerations in [65] or the use of functional ARX in [66]. One can also find numerous works resorting to model-based methods: Extended Kalman filters [67], parallel Kalman Filters [68,69], Interacting Kalman Filters [70,71], Kalman filters associated with Rao-Blackwellized particle filter [72], spectral analysis of the different components of the Kalman filter [73]. To estimate the mechanical characteristics of the faulty suspension elements despite the presence of various system uncertainties, the resolution of the inverse problem with a complete multi-body simulation seems promising. Lebel et al. [17] proposes to use Bayesian calibration to identify the mechanical properties of six mechanical parameters of air-springs, primary axial joints, vertical primary dampers, yaw dampers and upper inter-car-body dampers. Measured spectral dynamic responses are compared with simuR lated responses (on Vampire  ) thanks to a likelihood function. For numerical reasons, the required evaluations of the likelihood functions are achieved thanks to a surrogate model. The global method is described in Figure 16. The proposed method is validated on numerical experiments. Figure 17 presents one example showing a very good agreement between the identified parameters and their true value. The method is also applied to on-track measurements. Illustration 2: calibration of vehicle-soil interaction models. As second illustration, we describe in this paragraph the calibration of vehicle-soil interaction models. 20 C. FUNFSCHILLING AND G. PERRIN Figure 16. Diagram of the Bayesian calibration method [74, p.11]. Figure 17. Results of the numerical experiment. The input parameters values (black triangles) are compared to the mean of the posterior PDF marginals (blue dots). The blue lines represents the 98% confidence intervals around these calibrated values [17]. Lee proposes in [75] to calibrate and to validate a tyre-snow interaction model. There are many uncertainties in the interaction, among them: • The model uncertainties as the model fails to capture the discrete mechanical behaviour of snow, its sensitivity to temperature gradient, liquid water content, strain state . . . • The uncertainties on field tests: the terrain profile, the composition and the geometry of the tyres, the physical interaction between tyre and snow. Both the noisy test data and the physical models are thus replaced by metamodels (Gaussian processes), computationally very efficient. The calibration, which constitutes a multidimensional optimisation problem, is finally achieved thanks to a differential evolution genetic algorithm. The validation is then achieved comparing measured and simulated results with different metrics. Figure 18 presents a comparison between measured data and VEHICLE SYSTEM DYNAMICS 21 Figure 18. ‘Prediction of the mean (red) and 95% interval (blue) of traction for the left-front wheel [··· ] when compared with test data.’ [75, p.297]. Figure 19. ‘Statistical procedure.’ [76, p.11]. the metamodel predictions. The calibrated and validated models are finally used to achieve prediction. The global statistical procedure developed is described in Figure 19. One can also cite the work of Dettwiller et al. [77] who proposes a Bayesian calibration of Vehicle-Terrain Interface models, based on empirical equations, with laboratory and field 22 C. FUNFSCHILLING AND G. PERRIN testing data. The Metropolis algorithm (Markov Chain Monte Carlo) is used to estimate the expected posterior distributions. 4. Conclusions and perspectives Simulation is more and more used in vehicle dynamics. To be used in decision-making process it has however to be predictive. The introduction of uncertainties is thus necessary but it modifies the problem and the way it is solved. The Uncertainty Quantification, Validation and Verification method proposes tools to achieve the transformation of the problem. Various applications to risk analysis, damage prediction, calibration or robust optimisation can be found in the literature, showing an increasing interest for this topic. However, even if the contributions that can be expected are important for design and certification, robust optimisation and predictive maintenance, the introduction of these methods to vehicle dynamics is still in its infancy (particularly in industrial processes). 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