Keywords: robustness, fuzzy arithmetic, uncertainty analysis, sensitivity analysis, vibrations, crash.

A problem often encountered in the numerical simulation of real-world systems consists of the fact that exact values for the parameters of the model are used, even though, in reality, the parameter values exhibit a rather high degree of uncertainty. Those uncertainties arise due to imprecision or lack of information, due to variability or scatter, or as a consequence of model simplification. In any case, the uncertainties are finally reflected in uncertain model parameters and sometimes also in uncertain initial or boundary conditions. Consequently, the results that are obtained for simulations which only use one specific set of values as the most likely ones for the model parameters cannot be considered as representative for the whole spectrum of possible model configurations. To incorporate the uncertainties in the simulation procedure of the models, fuzzy arithmetic proves to be a promising approach [1]. In this novel concept of robustness analysis, the uncertain parameters are described by fuzzy numbers [2] in contrast to random numbers, which are used in stochastic approaches. A fuzzy number is a special case of a fuzzy set, as introduced by Zadeh in 1965 [3], and complies perfectly with the human perception of quantifying uncertainty.

To be able to work with fuzzy numbers as a numerical representation of uncertain model parameters, the transformation method [1] can be used. This technique allows a complete inclusion of the uncertainty information in the model, and it shows how the uncertainties are propagated through the calculation procedure. The transformation method avoids the possibly serious drawbacks of conventional fuzzy arithmetic, such as the overestimation effect [1]. Moreover, it can be used to determine the degrees of influence of each fuzzy parameter, i.e. the proportion to which the uncertainty of each model parameter contributes to the overall uncertainty of the model output. Furthermore, due to the specific properties of the transformation method, the method can be coupled to any existing software environment for system simulation and, in particular, to any commercial finite element code.

As applications, the finite element simulations of two mechanical structures with uncertainties in the geometry and material parameters are presented.

In the first example, the transformation method is applied to simulate and to analyze the vibration behavior of a circuit board as part of a state-of-the-art automotive control unit. In reality, the circuit board is fixed in a slot in the engine compartment of a car with a thin rubber layer between the unit and the slot. In the model, the circuit board is regarded as a thin undamped plate which is clamped on two sides. To model the various types and different sizes of electronic components located on the circuit board, two planar masses are added, the location of which are considered as uncertain to simulate the variations in different samples of the circuit board. Additionally, taking into account the real support conditions of the plate, a 'nearly clamped' plate is modeled by introducing uncertain boundary conditions, which complies much better with the real assembly situation than the idealized assumption of a `perfectly clamped' plate. Thus, the effect of loosening fixity points on the frequency response function of the circuit board can be investigated. Finally, the effects of these uncertainties on the frequency response function of the board are predicted and discussed.

In the second example, the front bumper of a vehicle that crashes against an inclined rigid wall, starting from a specific initial velocity, is investigated. The model is characterized by a number of geometry, material and interface parameters, three of which will be assumed to be uncertain in the following: the interface friction factor quantifying the friction between the bumper and the wall, the impact angle, i.e. the angle of inclination of the rigid wall with respect to the vertical (yaw) axis of the vehicle, as well as the sheet thickness of the bumper. Turning out to be a rather revealing result, an interesting conclusion can be drawn that whenever the crash is expected to be not completely straight on, but even at a slightly oblique angle, the sheet thickness of the bumper is of significantly less influence, although the actual setting of this quantity has often been defined on the basis of a preliminary structural optimization procedure. Hence, keeping in mind that perfectly straight frontal crashes represent a strong idealization of reality, the usefulness of sophisticated preliminary optimization procedures without the inclusion of uncertainties turns out to be rather questionable.

Summarizing, against the background that in mechanical structures, uncertainties usually exhibit rather drastic effects on the predicted behavior of the structure, it is strongly recommended that in the future the increasing computer power should be used for the inclusion of uncertainties, rather than to progress with a further refinement of the finite element model, which provides a fake exactness that is not reasonable in most cases.

1

M. Hanss, "Applied Fuzzy Arithmetic - An Introduction with Engineering Applications", Springer, Berlin, 2005.

2

A. Kaufmann and M. M. Gupta, "Introduction to Fuzzy Arithmetic", Van Nostrand Reinhold, New York, 1991.

3

L. A. Zadeh, "Fuzzy sets", Information and Control, 8:338-353, 1965. doi:10.1016/S0019-9958(65)90241-X

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