Abstract: The objective of this paper is to determine both the fuzzy eigenvalues and eigenvectors of a finite element model defined with fuzzy parameters. The proposed method introduces the concepts of mode shape pairing and the functional dependence of eigensolutions with respect to design parameters. High-order approximations are then introduced to limit the computational cost associated with variability management. Numerical test cases are used to highlight the abilities of this method to predict behaviour modifications due to variations in the physical parameters.

Abstract: Finite element simulations are well established in industry and are an essential part of the design phase for mechanical structures. Although numerical models have become more and more complex and realistic, the results can still be relatively far from observed reality. Nowadays, use of deterministic analysis is limited due to the existence of several kinds of imperfections in the different steps of the structural design process. This paper presents a general non-probabilistic methodology that uses interval sets to propagate the imperfections. This methodology incorporates sensitivity analysis and reanalysis techniques. Numerical interval results for a test case were compared to experimental interval results to demonstrate the capabilities of the proposed methodology.

Abstract: This paper presents an efficient method which allows one to characterize possible variations of the behaviour of a finite element model including imprecise data. The proposed method is integrated in the design phase of engineering structures and improves the prediction of the numerical models in the case of a static analysis. A numerical application to an industrial case highlights the effectiveness of these predictions with attractive computing times.

Abstract: This paper presents an efficient methodology to calculate fuzzy eigenvalues and eigenvectors of finite element structures defined by imprecise parameters. The material and geometric parameters are then described by fuzzy numbers. The proposed methodology, based on a-cut discretization of fuzzy numbers and Taylor's expansion, determines the extreme eigensolutions for each a-cut. The study of a finite element model and the comparison of results with a combinatorial approach, based on Zadeh's extension principle, show the efficiency of this methodology.

Abstract: In order to identify modal parameters with uncertain experimental data, a nondeterministic identi"cation method based on fuzzy formalism is proposed. The aim is to provide a degree of con"dence in the modal parameters identi"ed.

Abstract: The problem of structures' sensitivities to uncertain boundary conditions is presented in this paper and the effect of uncertain prescribed displacements on the static response of structures is discussed. The imprecise or uncertain prescribed displacements are treated as fuzzy quantities with known membership functions. For numerical computations, a fuzzy quantity can be approximated by sets of closed intervals with respect to specified -cuts. This transforms the problem into a set of interval equations at each -level. A numerical method is developed to solve the system of fuzzy equations. Simple stress analysis is also used to establish and illustrate the method. A numerical example is presented to demonstrate the capability of the method.

Abstract: The localization and correction of errors in computational models are discussed. A parametric optimization technique, based on the sensitivity analysis of measured eigensolutions and with regard to stiffness and mass parameters of a ÂŽ nite element model, was used. Uncertain eigensolutions are considered in the localization process due to fuzziness present in the experimental data used. To manipulate the uncertainties in the localization process, a fuzzy model is introduced and discussed. A numerical example is considered to illustrate the computational approach. The impact of the uncertain eigensolutions on the localization process is assessed by comparing the present numerical results with those given by the sensitivity-based method.

Abstract: This paper extends the fuzzy set theory to a dynamic finite element analysis of engineering systems which have uncertainties in material properties. A general algorithm which resolves the uncertain eigenvalue problem by using a reanalysis approach is considered. This algorithm is applied to the study of the modal behaviour of structures presenting uncertain material properties. Some indexes which determine the more sensitive eigenvalue to several uncertainty sources are also put forward. Finally, a plate structure as numerical path-test is analysed. The results of such a calculation determine the sensitivity of the modal behaviour to multiple simultaneous material parameters.

Abstract: This paper presents a new framework to predict a structure’s effective properties and sensitivities to multiple simultaneous uncertain endogenous parameters. The methodology is based on the use of fuzzy sets and this paper extends the fuzzy set theory to a dynamic finite element analysis of engineering systems containing uncertainty on material properties. A general algorithm, which can resolve the uncertain eigenvalue problem by using a Neumann expansion, is studied. This algorithm is applied to the study of the modal behavior of structures presenting uncertain material properties. Finally, the entropy and the specificity of fuzzy responses lead to the identification of a plate structure’s most sensitive eigenvalue to uncertain sources.

Abstract: Considers the problem of structures’ sensitivities to prescribed displacements uncertainties through the use of fuzzy numbers. Important properties of fuzzy subsets have been studied and used to model uncertainties, and to solve the fuzzy linear system resulting from taking into account the uncertainties on prescribed displacements. Develops a computer program to allow comparisons between several structural designs checking common specifications. Demonstrates the effectiveness of the present method for a cantilever structure modelled in the case of a static finite element analysis.

Abstract: The paper lays out an exact method, using the receptance strategy, to calculate the frequency response of a modified structure. A direct inversion of the modified impedance matrix is proposed, which reduces the computation time for successive calculations of an evolving design of the structure.

Abstract: An extension of the field of applications of classical modal reanalysis procedures is described, for the case when the number of degrees of freedom of a modified structure is greater than that of the initial structure. This feature is not developed classically, and such a case is usually resolved by the use of modal synthesis methods. A pure modal reanalysis procedure, with all the theoretical developments, is presented here. It is shown that any classical reanalysis algorithm can be used as the basis of the method. A characteristic example is presented, and modal results are discussed and compared with those of a classical reanalysis procedure.