Interval Uncertainty Quantification in Numerical Models using Dynamic Fuzzy Finite Element Analysis

Uncertainty and variability are always to some extent present in numerical simulations. Especially in the model definition phase, lack of information, scatter in material properties or environmental conditions imply that a designer should incorporate the possible effects of this non-determinism into the design procedure. This can be achieved by incorporation of the non-determinism in the numerical model. Over the past decade, different methodologies have been applied to achieve this goal. Still, as for deterministic models, also for non-deterministic models thorough validation of the used models is required in order for the model to be of value in a design assessment procedure. This work shows how model uncertainty defined as intervals on model properties can be identified based on a limited number of measurements. The procedure is based on numerical fuzzy analysis. By defining fuzzy membership functions for the uncertain model properties, a large-scale sensitivity analysis on the range of the outcome of the analysis with respect to the interval on the uncertain inputs can be performed. This output range is than compared to the actual outcome of the physical testing. From this comparison, interval bounds on the numerical model properties are derived. The methodology is capable of handling partial initial information on properties by assigning membership functions that are compatible with the available information to the uncertain properties. The presented procedure focuses on dynamic frequency response function analysis. As the procedure requires an efficient implementation of the fuzzy analysis algorithm, the first part of the paper deals with a response surface based interval FRF analysis technique. The procedure for uncertainty identification is demonstrated on a spacecraft component. 1.0 INTRODUCTION In current mechanical design engineering, numerical analysis tools play an important and often decisive role in the design process. A profound numerical analysis of a new design can reduce the need for prototype testing substantially, resulting in a proportional reduction in associated costs. In order to extend the applicability of numerical modelling tools in the design process, thorough validation of the used models is required. Today's structural models are validated based on deterministic approaches that do not take into account the natural dispersion or scatter inherent to all physical mechanical assemblies. This results in limited confidence in the model, which can only be partially compensated by the use of safety factors. It is therefore necessary to consider the scatter as an integral part of the model and to establish correlation and validation techniques that take this scatter into account. A reliable and robust numerical model should exhibit a high correlation with measurement data. This work contributes to the development of such high fidelity numerical models in a non-deterministic context. An important prerequisite is the availability of non-deterministic numerical modelling tools, i.e. numerical modelling techniques that incorporate the model non-determinism, and are able to process the uncertain RTO-MP-AVT-147 15 1 Moens, D.; Vandepitte, D. (2007) Interval Uncertainty Quantification in Numerical Models using Dynamic Fuzzy Finite Element Analysis. In Computational Uncertainty in Military Vehicle Design (pp. 15-1 – 15-16). Meeting Proceedings RTO-MP-AVT-147, Paper 15. Neuillysur-Seine, France: RTO. Available from: http://www.rto.nato.int. UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED information to non-deterministic analysis results. Over the past decades, several methodologies are established, among which the probabilistic approach is by far the most popular. Recent developments however have clearly indicated that also the possibilistic concept can be very valuable for nondeterministic numerical analysis, as it requires less information and often is computationally more efficient than the probabilistic approaches. Two techniques are currently gaining momentum is the context of possibilistic finite element modelling: • The Interval FE (IFE) analysis is based on the interval concept for the description of nondeterministic model properties. The aim of an interval analysis is to calculate the range of possible outcomes of a numerical analysis, given that some of the model properties are contained within uncertainty intervals (see e.g.[1],[2],[3]). • The Fuzzy FE (FFE) analysis is basically an extension of the IFE analysis, and has been studied in a number of specific research domains, as e.g. static structural analysis (see [4], [5],[6]) and dynamic analysis (see [7], [8]). See [9] for a more general overview of non-probabilistic uncertainty treatment in finite element analysis. Recently, an interval finite element methodology to calculate envelope frequency response functions (FRF) of uncertain structures has been developed by the authors [10]. This procedure forms the basis for the implementation of the fuzzy finite element method. The goal of the interval analysis is to calculate the envelope of the FRF taking into account that the input uncertainties can vary within the bounded space defined by their combined intervals. For this purpose, a hybrid procedure involving both a global optimisation step and an interval arithmetic step has been developed. The resulting envelope response function gives a clear view on the possible variation of the response in the frequency domain. This paper increases the state-of-use of this approach by further development of the fuzzy finite element method envisaging its application in non-deterministic model validation procedures. In section 2.0, the main principles of fuzzy finite element analysis are briefly discussed, focusing on both the philosophy as well as the methodology for fuzzy FRF analysis. Next, section 3.0 concentrates on an efficient implementation strategy for performing the dynamic finite element analysis in a fuzzy context. The paper then discusses the possible use of the fuzzy concept for interval uncertainty identification in section 0. Finally, section 5.0 illustrates the methodology is illustrated using a numerical example from space industry. 2.0 FUZZY FINITE ELEMENT METHOD FOR DYNAMIC RESPONSE ANALYSIS 2.1 The fuzzy finite element method Fuzzy sets were introduced by Zadeh in 1965 [11]. They are capable of describing linguistic and other incomplete information in a non-probabilistic way. Where classical sets clearly distinguish between members and non-members, fuzzy sets introduce a degree of membership, represented by a membership function. The membership function describes the degree of membership of each element in the domain to the fuzzy set. A fuzzy set with its corresponding membership function is denoted as: (1) If for a certain element in the domain the membership value equals 1, this element is definitely a member of the fuzzy set. If the membership value equals 0, it is definitely not a member of the fuzzy set. In between, the membership is uncertain. The most used membership function shape is the triangular shape. Interval Uncertainty Quantification in Numerical Models using Dynamic Fuzzy Finite Element Analysis 15 2 RTO-MP-AVT-147 UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED Such a fuzzy number with support [a, b] the interval for which the membership value is positive and core c the point for which the membership value equals one is denoted (a/c/b). The objective of the fuzzy finite element method is to introduce uncertainty as fuzzy numbers into the model definition, and to propagate this uncertainty to a fuzzy number describing the corresponding uncertainty on the analysis result. It is clear that the application of the fuzzy concept in a numerical modelling procedure requires a procedure for calculating the result of numerical operations on fuzzy numbers. A possible implementation of fuzzy functions is the alpha-level strategy. The intersection of the membership function of each input parameter with a discrete number of alpha-levels results in an interval for each input parameter at each alpha-level. Using these input intervals, an interval analysis is performed at each alpha-level. The fuzzy solution is finally assembled from the output intervals obtained at each alpha-level. Figure 1 shows this procedure for a function of two triangular parameters. Figure 1: alpha-level strategy for a function of two triangular fuzzy parameters 2.2 Fuzzy FRF analysis Using the alpha-level procedure, it is clear that the fuzzy FE FRF analysis can be implemented as a sequence of interval FE FRF analyses. The goal of the interval FRF analysis is to calculate the bounds on the dynamic response of a structure in a specific frequency region given that a set of model parameters is uncertain but bounded. The intervals on these parameters are specified in an interval vector. The methodology for the envelope dynamic response analysis as developed by the authors is based on a hybrid interval solution strategy, consisting of a preliminary optimisation step, followed by an interval arithmetic step. In the first part of this procedure, the optimisation is used to translate the interval properties defined on the finite element model to the exact interval modal stiffness and mass parameters of the structure. The calculation of the envelope FRFs in the second part is done by applying the interval arithmetic equivalent of the modal superposition procedure on these interval modal parameters. The final envelope FRFs have been proved to contain only a very limited amount of conservatism. A brief overview of the basic principles of the method is given in this section. The complete mathematical description can be found in [12]. 2.2.1 The deterministic modal superposition principle For undamped structures, the deterministic modal superposition principle states that, considering the first nmodes modes, the frequency response function between degrees of freedom j and k equals: (2) Interval Uncertainty Quantification in Numerical Models using Dynamic Fuzzy Finite Element Analysis RTO-MP-AVT-147 15 3 UNCLASSIFIED/UNLIMITED UNCLASSIFIED/UNLIMITED with the i eigenvector of the system and K and M the system stiffness and mass matrices. Simplification of equation (2) yields: (3) with the modal parameters defined as: