Analysis of Mechanical Systems Using Interval Computations Applied to Finite Element Methods

Dessombz, Olivier; Thouverez, Fabrice; Lainé, Jean-Pierre; Jézéquel, L.

doi:10.1006/jsvi.2000.3191

Cited by 91 publications

(43 citation statements)

References 16 publications

(13 reference statements)

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“…Nevertheless, uncertainties are sometimes epistemic due to imprecise or incomplete information, and that is why non-probabilistic approaches were developed [9]. Among the first developments was the interval theory, usually applied when the focus lies in identifying the bounds of the interval in which the output of interest varies, such as to problems in mechanics [10,11] and robotics [12]. The fuzzy set theory is another nonprobabilistic approach that uses the fuzzy logic which is based on human reasoning rather than rigid calculations [13,14].…”

Section: Introductionmentioning

confidence: 99%

Uncertainties in structural dynamics: overview and comparative analysis of methods

Daouk

Louf

Dorival

et al. 2015

Mechanics & Industry

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-When studying mechanical systems, engineers usually consider that mathematical models and structural parameters are deterministic. However, experimental results show that these elements are uncertain in most cases, due to natural variability or lack of knowledge. Therefore, engineers are becoming more and more interested in uncertainty quantification. In order to improve the predictability and robustness of numerical models, a variety of methods and techniques have been developed. In this work we propose to review the main probabilistic approaches used to model and propagate uncertainties in structural mechanics. Then we present the Lack-Of-Knowledge theory that was recently developed to take into account all sources of uncertainties. Finally, a comparative analysis of different parametric probabilistic methods and the Lack-Of-Knowledge theory in terms of accuracy and computation time provides useful information on modeling and propagating uncertainties in structural dynamics.

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Section: Introductionmentioning

confidence: 99%

Uncertainties in structural dynamics: overview and comparative analysis of methods

Daouk

Louf

Dorival

et al. 2015

Mechanics & Industry

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“…Solving parametric linear systems of equations is an interesting computational issue encountered, for example, when computing the characteristic solutions for differential equations (see [5]), when dealing with coloured Petri nets (see [13]) or in linear prediction problems and in different applications of operation research and engineering (see, for example, [6], [8], [12], [15] or [19] where the elements of the matrix have a linear affine dependence on a set of parameters). More general situations where the elements are polynomials in a given set of parameters are encountered in computer algebra problems (see [1] and [18]).…”

Section: Introductionmentioning

confidence: 99%

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Diaz–Toca¹,

González-Vega²,

Lombardi³

2005

SIAM J. Matrix Anal. & Appl.

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“…In most engineering design problems, models in operational research, linear prediction problems, etc. [2], [4], [5], [6] usually there are complicated dependencies between coefficients. The main reason for this dependency is that the errors in several different coefficients may be caused by the same factor.…”

Section: Introductionmentioning

confidence: 99%

“…Except for [8], by now there is no other work that would apply and study the results of this method. Remarkably, the parametric Rump's method was recently reinvented in slightly different notations [2]. A significant effort has been devoted to eliminate the sources of overestimation by several construction methods that reduce the number of parameters in the system to be solved [4], [5], [6].…”

Section: Introductionmentioning

confidence: 99%

Parametric Interval Linear Solver

Popova

2004

Numerical Algorithms

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Abstract.IntervalComputations'LinearSystems' is a Mathematica package supporting tools for solving parametric and non-parametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results in validated interval arithmetic. The package is designed to be easy to use, versatile, to provide a necessary background for further exploration, comparisons and prototyping, and to provide some indispensable tools for solving parametric interval linear systems.This paper presents the functionality, provided by the current version of the package, and briefly discusses the underlying methodology. A new hybrid approach for sharp parametric enclosures, that combines parametric residual iteration, exact bounds, based on monotonicity properties, and refinement by interval subdivision, is outlined.

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Analysis of Mechanical Systems Using Interval Computations Applied to Finite Element Methods

Cited by 91 publications

References 16 publications

Uncertainties in structural dynamics: overview and comparative analysis of methods

Uncertainties in structural dynamics: overview and comparative analysis of methods

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Parametric Interval Linear Solver

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