Parameter Uncertainty and Variability in the Structural Dynamics ...

Parameter Uncertainty and Variability in the Structural Dynamics Modeling Process

Stijn Donders1, Joost Van de Peer2, Steven Dom1, Herman Van der Auweraer1, Dirk Vandepitte3 1

LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium; Noesis Solutions, Interleuvenlaan 70, B-3001 Leuven, Belgium; 3 Katholieke Universiteit Leuven (KULeuven), Department of Mechanical Engineering, PMA Division, Kasteelpark Arenberg 41, B-3001 Leuven, Belgium. [email protected], [email protected] ABSTRACT 2

Finite Element (FE) analysis is widely employed in today’s Computer Aided Engineering (CAE) to model realworld structures in an early design stage. FE models are deterministic, implicitly assuming that all design parameters are precisely known and that the manufacturing process produces identical structures. This is typically not valid. Parameter variability and uncertainty limit the predictive capabilities of FE models in the mid-frequency range, as even small parameter changes may have substantial effects on the product performance. For a realistic prediction of structural and vibro-acoustic behavior it is therefore required that variability and uncertainty are incorporated in the modeling process. Probabilistic approaches are generally used to model parameter variabilities. Uncertainties are typically assessed with possibilistic approaches to identify worst-case scenarios, as lack of knowledge on the parameter distribution functions prohibits the use of stochastic methods. In this paper, both method classes are demonstrated on a realistic automotive example: an assembly of a car body, a front cradle and a rear cradle with non-uniquely defined mount stiffness. Uncertainty of the stiffness is assessed with the Transformation Method, an approach based on fuzzy arithmetic that allows visualizing uncertainty on dynamic responses. Variability of the stiffness is assessed with several reliability analysis methods, namely Monte Carlo simulations, FORM and Importance Sampling. 1. INTRODUCTION Nowadays, the product design departments of the automotive industry are intensively using FE models to analyze and solve a variety of engineering problems related to the vehicle performance in terms of noise and vibration. These large numerical models are deterministic, i.e. it is implicitly assumed that all parameters are precisely known and that the manufacturing process produces identical structures. This is typically not valid, as two classes of parameter deficiency can be distinguished [1]. Variability refers to the variation inherent to the physical system or the environment under consideration, while uncertainty is a potential deficiency in any phase or activity of the modeling process that is due to lack of knowledge. Variability typically exists on the level of physical properties (geometric, material characteristics) and manufacturing tolerances, while uncertainty exists on the level of model inaccuracy (e.g. joint models between subsystems) and physical properties in an early design stage, when design decisions must still be taken, so that dimensions and material properties are not yet fixed. Increasing the knowledge may reduce uncertainty, whereas variability is an irreducible scatter of the parameter value. As even small parameter changes may have substantial effects on response predictions, a reliable method to assess the effect of uncertainty and variability is very important [2,3]. Variability is typically modelled with a probability density function (PDF); its effect can be assessed with well-established stochastic procedures. Uncertainty should not be described in a probabilistic manner: there is not enough information available, so that assigning a PDF changes the problem definition in a subjective way. The results may be erroneous, and there is no way to verify this from probabilistic results. This paper therefore aims to demonstrate that parameter uncertainty and variability are two different design problems that can be solved with possibilistic and probabilistic analyses, respectively. Section 2 presents a demonstration example: an assembly model of a car body. The connection stiffnesses between the substructures are typically not known exactly. Depending on the design problem, the connection stiffness can be modelled as an uncertainty and as a variability. Possibilistic analysis gives insight in worstcase scenarios on the Frequency Response Function (FRF) characteristics, but cannot be used for reliability analysis, as no measure of probability is provided on the output side. When parameter distributions are

available, an additional probabilistic analysis can be used to assess design reliability issues. Possibilistic analysis is performed with a numerical implementation of fuzzy arithmetic, namely the Transformation Method [11]. These concepts are explained in Section 3. Probabilistic analysis will be done with well-established reliability analysis methods, as described in Section 4. Analysis results on the demonstration model are given in Section 5. Finally, a conclusion is given in Section 6. 2. DEMONSTRATION EXAMPLE: FBS ASSEMBLY FRF-based Substructuring (FBS) predicts the global frequency characteristics of a coupled structure from the predetermined response functions of the uncoupled components and the coupling mounts [4,5]. Using LMS Virtual.Lab [6], an FBS assembly model has been created of a car body-in-white, a front cradle and a rear cradle, as a demonstration case in this paper. The front cradle (1934 nodes, 2221 elements) is a stiff framework that supports the engine in the car. The rear cradle (2016 nodes, 2462 elements) is a structure that adds extra stiffness to the rear side of the car body (20429 nodes, 23644 elements), and provides attachment points for the rear suspension. The response point (on the car body) is the connection rail for the driver seat.

Figure 1 – The demonstration model: an elastic FBS assembly of car body, front cradle and rear cradle, with non-uniquely defined coupling stiffness values. The component modes have been computed with the direct solution of MSC/NASTRAN [7], up to 150 Hz for the car body and up to 500 Hz for the components, with residual vectors in the connection points. In LMS Virtual.Lab [6] an FRF synthesis for each component yielded the FRFs between engine mounts, connection points and/or response points. The assembly has then been constructed with spring-damper connections. As the connection stiffness is typically not exactly known, and surely not identical in a range of manufactured products, a parameter scatter on the connection stiffness will be introduced, first as uncertainty and then as variability. The implementation can be found in Section 5. 3. POSSIBILISTIC ANALYSIS THEORY: TRANSFORMATION METHOD 3.1 Fuzzy Numbers and Fuzzy Arithmetic In classical set theory, the elements x of a set A either belong to the set entirely, i.e. the membership level is µA=1, or do not belong to the set at all, so that µA=0. This principle is generalized in fuzzy set theory [8]: a membership level µA(x)∈ [0,1] is assigned to all elements x, so that the elements can belong to the set to a certain degree. A fuzzy number [8] is a fuzzy set with some special properties: the set is convex, the membership function is piecewise continuous and the core (i.e. the subset for which µA=1) only contains a single element x. The shape of the membership function can be derived from expert knowledge or practical measurements. Two well established types, Gaussian and triangular, are shown in Figure 2. These membership functions are possibilistic distribution functions that denote if an input is possible (µA=1), impossible (µA=0) or something in between. Once the input uncertainty has been modelled, the output uncertainty can be computed by using fuzzy arithmetic. Application to structural dynamics problems is not straightforward, as fuzzy solvers are either slow or non-existing; for example, inversion of a fuzzy matrix is not (yet) possible. Numerical approximations have therefore been developed, such as interval arithmetic [9]. Each fuzzy number is represented by a set of intervals at different levels of membership, as in the example in Figure 3. Interval arithmetic can then be applied to the input interval extrema to find an interval representation of the output. Unfortunately, this often leads to overestimation of the output uncertainty [10].

Figure 2 – Examples of fuzzy numbers with a Gaussian and a triangular membership function

Figure 3 – a Gaussian number, subdivided into 4 intervals

3.2. The Transformation Method A recent evolution in numerical fuzzy arithmetic is the Transformation Method (TM), presented by M. Hanss [11] as a systematic approach to replace interval arithmetic with a set of deterministic computations. Hanss presented two variants of the Transformation Method, a Reduced form and a General form. Only the Reduced Transformation Method has been considered; this is the compact form, therefore computationally more attractive. It has been demonstrated in [12] that this variant is suitable to assess FRF uncertainty for structural design problems. From here on, the term „Transformation Method“ therefore denotes the Reduced form. Overestimation of the output uncertainty, which is the main disadvantage of interval arithmetic, does not occur with the Transformation Method, as the method only evaluates parameter combinations that actually occur. Consider an arithmetic function f(·) that depends on n uncertain input parameters (x1, x2, .. xn). Each parameter is modelled as a fuzzy number with a membership function µA(x) of arbitrary shape, so that the output q = f(x1, x2, … xn) is a fuzzy number as well. The input parameters are subdivided into m intervals at equidistant levels of membership. The Transformation Method then creates a systematic set of parameter combinations (x1, x2, ... xn) that combine the extreme values of each interval in every possible way. This yields a set of 1 + m ·2n experiments. In a Design-of-Experiments (DOE) [13] terminology, a Full-Factorial DOE is created at each interval level. Figure 4 shows a geometric interpretation of the Transformation Method; In essence, the Transformation Method is a systematic sampling plan along the 2n-1 diagonals in the parameter space.

Figure 4 – Geometrical interpretation of the Transformation Method for a problem with 3 input parameters x1, x2, x3, modelled as triangular-shaped fuzzy numbers that are subdivided into 5 intervals. The TM evaluates a set of parameter combinations (shown as black dots) along the diagonals of the parameter space (x1, x2, x3).

The designed set of experiments is evaluated via conventional arithmetic, i.e. with deterministic, non-fuzzy computations. The 1 + m ·2n deterministic outputs are stored in arrays Z(j) that contain (for j = 0,1,..m-1)

k (j)

z , i.e. the k = 1..2n deterministic outputs at the j-th membership level

(for j = m)

z(m), the single deterministic output at the highest level

(eq. 1)

From these deterministic outputs, the fuzzy output q is then approximated as the interval representation [a(j), b(j)] of the fuzzy output, which is obtained with the recursive procedure (for j = 0,1,..m-1)

a(j) = min ( a(j+1), kz(j) )

(eq. 2)

k

b(j) = max ( b(j+1), kz(j) ) k

a(m) = z(m) = b(m)

(for j = m)

At the highest level of membership µ=1, the fuzzy output is determined by the single deterministic output at the core level. At lower interval levels µ = α, the lower boundary a(j) of the interval representation of the fuzzy output is the minimal value attained by kz(j) at a membership level µj > α. Equivalently, the upper boundary is the maximum value of kz(j) in the parameter range for which µj > α. The Transformation Method can easily be applied to structural dynamics problems in the frequency domain, as has been demonstrated in [12]. Again, the n uncertain inputs are modelled as fuzzy numbers that are subdivided into m intervals. This yields a set of 1 + m ·2n parameter combinations that are evaluated with deterministic computations. In the frequency domain, the deterministic computations yield FRF vectors kz(ƒ)(j) instead of scalar outputs kz(j) as in (eq. 1). In analogy with (eq. 1), the deterministic FRFs are stored in arrays Z(ƒ) (j) that contain (for j = 0,1,..m-1)

k

z(ƒ)(j), the 2n deterministic FRF outputs at the j-th membership level

(for j = m)

z(ƒ)(m), the single deterministic FRF output at the highest level

In essence, application to the frequency domain requires that a frequency dimension is added to the problem. With a frequency increment ∆ƒ, the frequency range is given by [ƒmin, ƒmin+ ∆ƒ, ... ƒmax], consisting of Nƒ samples (abbreviated as [ƒmin, ƒmax] from here on). The recursive algorithm (eq. 2) must be applied at each sampled frequency, yielding the interval representation of the fuzzy FRF amplitude at that frequency, given by ( j)

A ( j ) ( f ) = [ A ( f ), A ( j ) ( f )] , for j = 0..m ( j)

and

f ∈ [ f min , f max ]

(eq. 3)

( j)

where A ( f ) and A ( f ) denote the lower and upper boundary, respectively, of the fuzzy FRF amplitude at the frequency ƒ. Effectively, the amplitude at each sampled frequency is computed as an independent fuzzy number. These fuzzy amplitudes A(j)(ƒ) approximate the interval representation of the fuzzy FRF characteristics at each input level of membership. In Section 5, this fuzzy FRF will be visualized with a grayscale plot, where a black color denotes the highest membership level, µ=1. Light gray denotes the lowest membership level at µ=0, and intermediate shades of gray denote the intermediate membership levels. The TM envelopes are validated with a sufficiently large number NMC of Monte Carlo (MC) simulations, uniformly distributed over the input parameter range (as all possible outputs should be assessed). Two validation criteria have been used for this purpose [12], which are graphically represented in Figure 5. The Overall Inclusion Percentage (OIP) is defined as the percentage of Monte Carlo FRF data samples that lies between the TM envelopes at the lowest membership level µA=0. The OIP has a maximal value of 100%, if all Monte Carlo data is contained by the TM envelopes.

Figure 5 – Validation of the TM envelopes is done with the OIP criterion (left) and the VAF criterion (right).

The Variance-Accounted-For (VAF) is a shape conformity factor [14] of a reference vector y and its estimate ŷ, defined as

 var( y − yˆ )   ⋅ 100 % VAF ( y, yˆ ) = 1 − var( y )  

(eq. 4)

This requires that the upper and lower envelope of all Monte Carlo data is computed as reference. The shape of the global Transformation Method envelopes can then be validated on these global reference envelopes, using (eq. 4). A VAF of 100% denotes that both envelopes are identical. 4. PROBABILISTIC ANALYSIS THEORY: RELIABILITY ANALYSIS A probabilistic analysis of the effect of variability in the connection stiffness will be done with well-established structural reliability methods that are briefly described in this section, namely FORM, Monte Carlo and Importance Sampling. The goal of a reliability analysis is to estimate the probability that a structure will fail to meet a pre-defined criterion [15]. Figure 6 visualizes a typical reliability problem, with two stochastic parameters X = [x1, x2] with a Gaussian distribution. The ovals of increasing size in Figure 6 represent lines of equal variance of the parameter steps with respect to the mean value [µ1,µ2]. Using the terminology of [15], the monitored output parameter gS(X) is called the load effect. The limit value gR, which should not be exceeded by the load effect, is known as the resistance effect. The intersection of load and resistance is the Limit State Function (LSF), the subset where g(X) = gR - gS(X) = 0. When projected on X-space, the LSF forms the boundary between the safe area and the failure area in the parameter space X. The nearest failure point in the parameter space is commonly known as the design point X*, lying at a distance β from the mean value. This distance is called the Reliability Index, as it is a direct measure for the reliability: it denotes how many variance steps lie between the parameter mean and the design point X*.

Figure 6 – A typical reliability problem. The load effect gS(X) depends on the inputs X. When the resistance gR is exceeded, a failure occurs. The intersection of load and resistance is the Limit State Function that bounds the safe and failure domain in the parameter space. A reliability problem is generally not evaluated in X-space, but first transformed to the standard normal space U, where the parameters are uncorrelated and have a zero-mean, unit variance normal distribution. This transformation prevents numerical problems and simplifies the algebra [15]. In the remainder of this section, all reliability analysis methods are visualized in U-space so that they can easily be compared.

4.1. Monte Carlo Simulation The most straightforward numerical approach to solve a reliability problem is by performing a number of simulations with randomly selected parameter combinations for the given distribution, and to verify for each combination if this results in a failure. One then divides the number of failures by the number of simulations, which yields an approximation of the actual failure probability. This is the standard Monte Carlo (MC) simulation approach [15], visualized in Figure 7. This method is widely used for benchmarking and validation purposes, as it is almost always applicable. A large disadvantage of the method is that a lot of simulations are typically required to accurately estimate the failure probability, especially when the actual failure probability is low. The reason is the low failure density, see Figure 7. For example, if the failure probability is Figure 7 – Monte Carlo Simulation 0.003, only three of 1000 samples are expected to produce a failure. But as the simulations are random, there might as well be 2 or 4 failures, thus changing the estimate with as much as 33%. This means that a fixed number of failures is required to obtain a required accuracy, to reduce the influence of a single failure case on the total estimate. This influence can be quantified as the coefficient of variation (c.o.v.), the ratio of the standard deviation and mean value of the estimated failure probability [15]. 4.2. First Order Reliability Method The First Order Reliability Method (FORM) is a gradientbased search algorithm to locate the design point. In (zero-mean, unit variance) U-space, the design point U* is located at a distance β from the origin, and the lines of equal variance are represented by circles, as can be seen in Figure 8. The algorithm creates a linear approximation of the Limit State Function, which is used as an approximate boundary between the safe and failure domain [15]. Using this boundary, an estimate of the failure probability can be computed. For sufficiently large β and/or linear LSFs, this approximation yields accurate results. Several implementations of the FORM algorithm have been reported in literature, see e.g. [1519]. In this paper, the HL-RF method [18, 19] has been applied. This is a widely used algorithm that is generally fast and reliable. Each iteration step consists of a numerical gradient evaluation to find the direction of steepest descent towards the design point, followed by an optimization loop to set an optimal step in that direction. The algorithm has two user-defined termination criteria that monitor if the LSF has been approached closely enough, and if the vector β is perpendicular to the LSF with a certain degree of accuracy; the search algorithm is terminated if both criteria are met. These criteria limits should be chosen with care, as a higher accuracy is generally obtained at the cost of higher CPU time.

Figure 8 – FORM algorithm

4.3. Importance Sampling The general idea behind importance sampling is to put more samples in the failure domain in the parameter space, in order to increase the failure density of the sampling session. The most straightforward approach for this purpose is the Importance Sampling Monte Carlo

Figure 9 – ISMC approach

(ISMC) approach, where the sampling center of a standard Monte Carlo sample set is moved from the origin in U-space to the FORM design point U* over a distance β [15], as can be seen in Figure 9. The ISMC failure density now depends on the shape of the LSF in the vicinity of the design point, not on the linear approximation of the LSF. Therefore, the ISMC approach can be used to improve or validate the FORM estimate of the failure probability. This still requires a reasonably high number of samples, typically much higher than the FORM method itself needs, as a sufficient number of failures is needed to reduce the coefficient of variation of the estimate to an acceptably low level. However, the ISMC method often requires much less samples than the standard Monte Carlo approach to yield an acceptable result; this advantage of ISMC increases if the actual failure probability decreases. 5. ANALYSIS RESULTS OF THE FBS ASSEMBLY MODEL The FBS assembly in Section 2 has been evaluated with the possiblistic and probabilistic methods described in Sections 3 and 4. To better compare the results of the uncertainty and variability analysis, the same distribution functions are assumed; i.e. the membership functions of the uncertain parameters have the same shape as the probability density functions of the variable parameters. The front cradle connections are the main parameters of interest, as they largely determine the propagation of the engine vibrations to the driver seat. The 4 connections between front cradle and car body are modelled as uncorrelated Gaussian parameters, with a mean value of 2.00·105 N/m and a standard deviation of 0.10·105 N/m in the Z-Z direction; see Figure 10. The stiffness in X-X and Y-Y direction is assumed to be a third of this value. The rear cradle is connected to the car body with the same mean stiffness values, but without scatter on this value. The fuzzy FRF is then computed, with the input at an engine mount (+Z direction) and the output at the driver seat connection rail (+Z dir.). The described algorithms have been implemented in MATLAB [20]. Process management is performed with LMS OPTIMUS [21], post processing with both MATLAB and OPTIMUS. 5.1. Transformation Method Results For the possibilistic analysis, the parameter scatter described above is modelled with the membership function in Figure 10. The distribution is set to zero outside the boundary [-6σ, +6σ] with respect to the nominal value; if this is omitted, the input ranges from [-∞,+∞], so that all input parameter values would be possible. As can be seen in Figure 10, the input range has been subdivided into 5 interval levels of membership. For this problem with 4 inputs and 5 intervals, the Transformation Method requires that 1 + 5 ·24= 81 parameter computations are computed to obtain the fuzzy FRF in Figure 11. The fuzzy FRF has been visualized with a gray scale plot, created with MATLAB [20], where a darker color denotes a higher level of membership, ranging from Figure 10 – Gaussian membership function of black (µA=1) down to light gray (µA=0). The range with the connection stiffness between front cradle a non-white color therefore represents the range of and car body, subdivided into 5 intervals. possible outputs. The validity of this range has been checked with 500 Monte Carlo simulations, uniformly spread in the same parameter range. Figure 12 compares the global TM envelopes (i.e. the boundary at µA=0 in Figure 11) with the global MC envelopes [20]. The V.A.F. criterion (eq. 4) is equal to 94.0 for the lower envelopes and 90.4 for the upper envelopes. The shape conformity is therefore high but not perfect, as can also be seen in Figure 12. The O.I.P. criterion (see Section 3.2) has a value of 99.7%. This means that almost all Monte Carlo data samples are bounded by the TM envelopes. It can be concluded that the fuzzy FRF in Figure 11 is a reliable visualization of all possible outputs for the given problem. The fuzzy FRF in Figure 11 allows the engineer to identify worst-case scenarios on the FRF characteristics. A sensitive range has been found around 20 Hz. Figure 13 shows a magnification of the FRF in Figure 11. For nominal parameter values, an anti-resonance is found at 22 Hz, and a resonance with an FRF amplitude of 10-2 kg-1 is found at 22.5 Hz. As a result of parameter uncertainty, the amplitude at 22.5 Hz can reach values above 10-1 kg-1. The fuzzy FRF only shows that this event is possible, but doesn’t estimate a probability of this event. To estimate this probability, the reliability analysis methods described in Section 4 have been applied in the next section.

Figure 11 – Fuzzy FRF between engine mount (+Z) and driver seat (+Z). The 4 connections between front cradle and car body are modelled as fuzzy parameters with Gaussian membership functions, as shown in Figure 10. A darker color represents a higher level of membership of the fuzzy FRF output.

Figure 12 – The global envelopes of the fuzzy FRF in Figure 11 (dashed) and the global envelopes of 500 Monte Carlo FRFs in the same parameter range (solid).

Figures 13 – Magnification of the fuzzy FRF in Figure 11 in the range around 22 Hz (with a brighter color scale). The solid dark curve represents the core FRF. The resonance peaks in the fuzzy FRF denote that this frequency range is quite sensitive to the parameter uncertainty. Subsequent to the fuzzy analysis, a reliability analysis has been performed, where an upper boundary has been set to the FRF amplitude at 22.5 Hz (see eq. 5). 5.2. Reliability Analysis results Following the terminology in Section 4, a stochastic parameter vector X = [x1, x2, x3, x4] is introduced, containing the 4 connection stiffness values that are modelled as uncorrelated Gaussian parameters with a mean of 2.00·105 N/m and a standard deviation of 0.10·105 N/m. The amplitude at 22.5 Hz is denoted as the stochastic load effect A22.5(X). The resistance effect is the maximal amplitude that is allowed (i.e. the „Limit Value“ in Figure 13), which is set to A22.5,max = 1.0·10-1 kg-1. The Limit State Function (LSF) is then equal to g(X) = A22.5,max – A22.5(X) = 1.0·10-1 – A22.5(X)

(eq. 5)

A failure is obtained if the amplitude A22.5(X) attains a value above 1.0·10-1 kg-1. For this problem, a reliability analysis has been done with the methods described in Section 4. The results are given in Table 1. Table 1 – Reliability Analysis results for the problem given in (eq. 5)

Method FORM Monte Carlo ISMC

Number of LSF evaluations

Number of Failures

failure probability (pf)

reliability index (β)

coefficient of variation (cv)

36 1000 400

not relevant 13 16

0.0823 0.0130 0.0156

1.390 2.226 2.153

not available 0.28 0.24

The FORM algorithm required 5 iterations (and 36 LSF evaluations) to locate the design point X* at [x1, x2, x3, x4] = [2.073, 2.074, 2.064, 2.067]·105 kg-1. This point lies quite near in the parameter space (with respect to the parameter mean value), at a distance β = 1.39. Table 1 shows that the simulation methods (Monte Carlo and ISMC) predict a much lower failure probability. As the results with Monte Carlo and ISMC are in good agreement (i.e. within the range set by the coefficients of variation), the FORM estimate seems to be too high. A possible explanation is that the FORM design point lies in a local failure region in the parameter space, and that the LSF does not consist of a smooth surface as shown in Figure 8. FORM makes a linear approximation of the LSF in the design point, and then assumes that all parameter combinations beyond this LSF approximation produce a failure. If the FORM design point lies in a local failure region, such an approach is not valid. In that case, there exist „safe“ parameter combinations beyond the LSF approximation in the FORM design point, so that FORM will overestimate the actual failure probability. As expected, the simulation methods (Monte Carlo and ISMC) require a much higher amount of simulations than FORM. The Monte Carlo approach uses 1000 LSF evaluations to obtain 13 failures; this allows to estimate the failure probability of 0.013 with a coefficient of variation of 0.28. The ISMC approach is much cheaper in terms of CPU usage: only 400 LSF evaluations produce 16 failures, which yields a failure

probability of 0.016 with a c.o.v. of 0.24. Clearly the failure density of the ISMC simulations (around the FORM design point) is higher than the failure density of the Monte Carlo simulations (around the origin in the parameter space). This indicates that the FORM design point is located in the correct region in the parameter space. However, as the FORM algorithm produces a high estimate of the failure probability, it can be assumed that not all parameter combinations beyond the linear LSF approximation in the FORM design point produce a failure. To verify this assumption, the simulation results are further studied with MATLAB [20] and OPTIMUS [21]. The failure surface cannot be plotted as in Figure 6, as this would require an impossible 5-dimensional drawing (with the 4 parameters and the g(X)-value on the 5 axes). However, lower dimensional plots already provide insight in the nature of the problem. Figure 14 plots x1 (the right front connection stiffness between Front Cradle and Car Body) against x2 (the left front connection stiffness) [20]. Failures seem to be concentrated in the parameter domain where x1 and x2 have values above the mean values. The FORM design point is also located in this region with higher failure density. When other parameter combinations are plotted in this manner, comparable results are obtained. With LMS OPTIMUS [21], the Monte Carlo results are again visualized in Figure 15. The values of x1 and x2 (the filled squares in Figure 14) are plotted against the amplitude A22.5(X) in (eq. 5); an amplitude larger than 0.10 results in a failure. The parameter combinations that produce a failure seem to pop up from the set of „safe“ combinations. This indicates that „safe“ combinations exist beyond the FORM linearization of the LSF, which explains the overestimation of the failure probability with the FORM algorithm.

Figure 14 – The results of Table 1, projected on the x1–x2 space: the (x1,x2) combinations for the mean (filled circle), the FORM design point (star) and the failures found with MC (filled squares) and ISMC (crosses).

Figure 15 – The (x1,x2) combinations of all 1000 MC experiments for Table 1 are shown in the horizontal plane, plotted against the amplitude A22.5(X) on the vertical axis. Amplitudes above 0.10 kg-1 produce a failure.

6. CONCLUSIONS AND DISCUSSION In this paper, possibilistic and probabilistic methods have been described as two different approaches that deal with parameter uncertainty and variability, respectively. Both classes of methods can be used to reveal different aspects of the same design problem. This has been demonstrated on an FBS assembly model taken from the automotive industry: a car body to which a front cradle has been attached with not uniquely defined connection stiffness values. A possibilistic analysis with the Transformation Method allows to visualize worstcase scenarios on the FRF characteristics of the assembly. The uncertain FRF showed that around 22 Hz, the amplitude is very sensitive to the uncertainty in the parameter values. The fuzzy FRF shows a high resonance peak here that has is missing from the FRF obtained at nominal parameter values. The amplitude A22.5(X) at a frequency of 22.5 Hz has then been used as the input to a probabilistic analysis problem. The stiffness parameters have been modelled as stochastic parameters, and an upper limit has been set to A22.5(X). Several reliability analysis methods, namely FORM, Monte Carlo and ISMC, have then been used to estimate the probability that the limit value is exceeded. The FORM algorithm provided a higher estimate of the failure probability than Monte Carlo and ISMC. The latter methods are in good agreement, which indicates that FORM overestimates the actual failure probability. Post processing of the simulation

results indicates that „safe“ parameter combinations do exist beyond the FORM linearization of the LSF, which explains the overestimation of the failure probability with the FORM algorithm. The FORM design point has been located at stiffness values above the mean stiffness values. Based on this analysis, a designer can decide to use a lower stiffness in order to reduce the FRF amplitude at 22.5 Hz. The design can also be improved by adding damping to the connection mounts. As the connection mounts are usually made of rubber, a material that introduces damping to the connection (which has not been taken into account in the analysis), one can expect that the investigated resonance is less critical in reality. Furthermore, note that an actual design problem usually deals with reliability criteria based on eigenfrequency values rather than on amplitude values. The problem addressed in this paper should therefore be seen as a demonstration case for methods to assess parameter uncertainty and variability, and not as an actual industrial problem. ACKNOWLEDGEMENT This work was made possible under the IST Marie Curie Industry Host Fellowship ‘VIPROM’. The aim of this research project is to develop Finite Element methods for the “medium frequency range” (e.g. 200-600 Hz for cars). The effects of parameter variability and uncertainty become increasingly important in this range, hence the interest in the topics addressed in this paper. The EC is greatly acknowledged for their support.

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