Review
Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances

https://doi.org/10.1016/j.finel.2010.07.010Get rights and content

Abstract

The objective of this paper is to give a general overview of recent research activities on non-probabilistic finite element analysis and its application for the representation of parametric uncertainty in applied mechanics. The overview focuses on interval as well as fuzzy uncertainty treatment in finite element analysis. Since the interval finite element problem forms the core of a fuzzy analysis, the paper first discusses the problem of finding output ranges of classical deterministic finite element problems where uncertain physical parameters are described by interval quantities. Different finite element analysis types will be considered. The paper gives an overview of the current state-of-the-art of interval techniques available from literature, focussing on methodological as well as practical aspects of the presented methods when their application in an industrial context is envisaged. Their possible value in the framework of applied mechanics is discussed as well. The paper then gives an overview of recent developments in the extension of the interval methods towards fuzzy finite element analysis. Recent developments in the framework of the transformation method as well as optimisation-based procedures are discussed. Finally, the paper concentrates specifically on implementation strategies for the application of the interval and fuzzy finite element method to large FE problems.

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 α-sublevel 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 α-level. 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.

References (87)

  • W. Dong et al.

    Vertex method for computing functions of fuzzy variables

    Fuzzy Sets and Systems

    (1987)
  • L. Chen et al.

    Fuzzy finite-element approach for the vibration analysis of imprecisely-defined systems

    Finite Elements in Analysis and Design

    (1997)
  • K. Abdel-Tawab et al.

    Uncertainty analysis of welding residual stress fields

    Computer Methods in Applied Mechanics and Engineering

    (1999)
  • T. Wasfy et al.

    Multibody dynamic simulation of the next generation space telescope using finite elements and fuzzy sets

    Computer Methods in Applied Mechanics and Engineering

    (2000)
  • M. Hanss

    The transformation method for the simulation and analysis of systems with uncertain parameters

    Fuzzy Sets and Systems

    (2002)
  • S. Chen et al.

    Interval optimization of dynamic response for structures with interval parameters

    Computers & Structures

    (2004)
  • L. Zadeh

    Fuzzy sets

    Information and Control

    (1965)
  • A. Vroman et al.

    Solving systems of linear fuzzy equations by parametric functions: an improved algorithm

    Fuzzy Sets and Systems

    (2007)
  • I. Skalna et al.

    Systems of fuzzy equations in structural mechanics

    Journal of Computational and Applied Mathematics

    (2008)
  • F. Massa et al.

    A fuzzy procedure for the static design of imprecise structures

    Computer Methods in Applied Mechanics and Engineering

    (2006)
  • F. Massa et al.

    A complete method for efficient fuzzy modal analysis

    Journal of Sound and Vibration

    (2008)
  • J. Sim et al.

    Modal analysis of structures with uncertain-but-bounded parameters via interval analysis

    Journal of Sound and Vibration

    (2007)
  • N.V. Queipo et al.

    Surrogate-based analysis and optimization

    Progress in Aerospace Sciences

    (2005)
  • A. Klimke et al.

    Computing expensive multivariate functions of fuzzy numbers using sparse grids

    Fuzzy Sets and Systems

    (2005)
  • M. Hanss et al.

    On the reliability of the influence measure in the transformation method of fuzzy arithmetic

    Fuzzy Sets and Systems

    (2004)
  • U. Gauger et al.

    A new uncertainty analysis for the transformation method

    Fuzzy Sets and Systems

    (2008)
  • O. Giannini et al.

    An interdependency index for the outputs of uncertain systems

    Fuzzy Sets and Systems

    (2008)
  • S. Donders et al.

    Assessment of uncertainty on structural dynamic responses with the short transformation method

    Journal of Sound and Vibration

    (2005)
  • O. Giannini et al.

    The component mode transformation method: a fast implementation of fuzzy arithmetic for uncertainty management in structural dynamics

    Journal of Sound and Vibration

    (2008)
  • H. De Gersem et al.

    Interval and fuzzy dynamic analysis of finite element models with superelements

    Computers & Structures, Special Issue on Computational Stochastic Mechanics

    (2007)
  • I. Elishakoff

    Possible limitations of probabilistic methods in engineering

    ASME Applied Mechanics Reviews

    (2000)
  • D. Moens et al.

    An interval finite element approach for the calculation of envelope frequency response functions

    International Journal for Numerical Methods in Engineering

    (2004)
  • E. Hansen

    Bounding the solution of interval linear equations

    SIAM Journal on Numerical Analysis

    (1992)
  • S. Shary

    Algebraic approach in the ‘outer problem’ for interval linear equations

    Reliable Computing

    (1997)
  • D. Moens et al.

    Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis

    Archives of Computational Methods in Engineering

    (2006)
  • U. Köylüogˇlu et al.

    Interval algebra to deal with pattern loading and structural uncertainties

    Journal of Engineering Mechanics

    (1995)
  • R. Mullen et al.

    Bounds of structural response for all possible loading combinations

    Journal of Structural Engineering

    (1999)
  • A. Neumaier et al.

    Linear systems with large uncertainties with applications to truss structures

    Reliable Computing

    (2007)
  • S. Rao et al.

    Fuzzy finite element approach for the analysis of imprecisely defined systems

    AIAA Journal

    (1995)
  • S. Rao et al.

    Analysis of uncertain structural systems using interval analysis

    AIAA Journal

    (1997)
  • S. Rao et al.

    Numerical solution of fuzzy linear equations in engineering analysis

    International Journal for Numerical Methods in Engineering

    (1998)
  • U. Köylüogˇlu et al.

    A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties

    Computers & Structures

    (1998)
  • B. Möller et al.

    Fuzzy structural analysis using α-level optimization

    Computational Mechanics

    (2000)
  • Cited by (239)

    View all citing articles on Scopus
    View full text