ReviewNon-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances
Introduction
Non-deterministic approaches are gaining momentum in the field of numerical modelling techniques. The ability to include non-deterministic properties is of great value for a design engineer. It enables realistic reliability assessment that incorporates the uncertain aspects of the design. Furthermore, the design can be optimised for robust behaviour under varying external influences. Recently, criticism has arisen regarding the general application of the probabilistic concept in this context. Especially when objective information on the uncertainties is limited, the subjective probabilistic analysis result proves to be of little value, and does not justify its high computational cost (see, e.g., [1], [2]). Consequently, alternative non-probabilistic concepts have been introduced for non-deterministic numerical modelling.
In this context, interval and fuzzy approaches are becoming increasingly popular for the analysis of numerical models that incorporate uncertainty in their description. In the interval approach, uncertainties are considered to be contained within a predefined range. For each uncertainty, the analyst has to provide the lower and upper bound. The fuzzy approach extends this methodology by introducing a level of membership that represents to what extent a certain value is member of the range of possible input values. This concept provides the analyst with a tool to express a degree of possibility for a certain value. Based on the technique, the fuzzy analysis requires the consecutive solution of a number of related interval problems (see also Section 2.3). For that reason, current research activities on this subject mainly concentrate on the actual solution and implementation of interval analysis.
In recent literature, the application of both the interval and the fuzzy concept for the representation of parametric uncertainty during a classical finite element analysis has been studied extensively. While the problem at the core of the analysis, i.e., the solution of a set of interval equations, is easily formulated, the actual solution of this problem was proven to be extremely problematic [2]. Nevertheless, some solution schemes of fundamentally different nature have been developed. The intention of this paper is to give an overview of the recent additions to the state-of-the-art of numerical implementations of the interval and fuzzy finite element method in applied mechanics.
The following section briefly summarises the principle ideas and initial developments in the framework of interval and fuzzy finite element analysis. Section 3 focuses on some recent advances in interval finite element implementation strategies. Section 4 then gives an overview of recent developments in the context of fuzzy finite element analysis. Finally, Section 5 gives an overview of recent developments specifically aimed at the analysis of large finite element problems.
Section snippets
Fundamental strategies for interval and fuzzy finite element modelling
Interval finite element analysis is based on the interval concept for the description of non-deterministic model properties. The goal of the interval finite element analysis is to obtain the range of specific output quantities that corresponds to a given interval description of the uncertainty on some input parameters of the problem. As an example, consider the simple case of a cantilever beam subject to loading as illustrated in Fig. 1. In an interval framework, the uncertainty on this model
Parametric functions
A parametric approach was recently developed for the solution of systems of linear fuzzy equations. The method can be applied to solve systems of equations the coefficients of which are defined as fuzzy numbers. In essence, this translates to solving a system of linear interval equations at each considered . The method was introduced by Vroman et al. in [35]. The procedure is based on an explicit expression of the solution of a system of deterministic linear equations using Cramer's
Recent advances in fuzzy finite element implementations
This part of the paper discusses some recent advances related to the implementation of fuzzy finite element analysis. It first focuses on the general generic implementation strategy based on the transformation method. Next, some specific aspects of other interval procedures extended to fuzzy analysis are highlighted.
Interval and fuzzy analysis of large FE models
This part of the paper specifically focuses on non-probabilistic finite element analysis of very large models. Two recent developments are discussed that strongly reduce the cost of the evaluation of the objective function when it is re-evaluated with small variations in uncertain parameters. Therefore, these techniques are of specific interest not only in interval finite element implementations, but also in a probabilistic framework, e.g., for large-scale sampling.
Conclusions
This paper presents a general overview of the state-of-the-art and recent advances in interval and fuzzy finite element analysis. The main principles, strengths and shortcomings of the two fundamental classes of interval approaches are discussed, and the current research activities in both domains are summarised.
The interval arithmetic approach at first sight seems the most straightforward strategy for the solution of systems of interval equations. In addition, its inherent high computational
Acknowledgement
This work was supported by the European Commission through the Marie Curie Research and Training Network MRTN-CT-2003-505164 MADUSE: Modelling Product Variability and Data Uncertainty in Structural Dynamics Engineering.
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