Discussions on non-probabilistic convex modelling for uncertain problems

1 Discussions on non-probabilistic convex modelling for uncertain problems B.Y. Ni 1, C. Jiang 1,*, Z.L. Huang 2 1 State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, P. R. China 410082 2 School of Mechanical and Electrical Engineering, Hunan City University, YiYang City, P. R. China 413002 Abstract Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study. Keywords: uncertainty analysis; non-probabilistic convex modelling methods; correlation analysis; evaluation criterion. 1 Introduction The physical parameters used to describe a structure or system in practical engineering are often uncertain due to geometrical imperfections, model inaccuracies, external interferences, etc. As the principal way for quantification of these physical uncertainties, the probabilistic methods have been successfully applied to various engineering problems. However, in practical engineering, the credible probability models may not available if experimental data are insufficient [1]. To deal with the difficulty that sufficient information on the uncertain parameters are often unavailable due to limitations of test conditions or cost in practical engineering, the non-probabilistic convex modelling approach [2-6], has been developed since the early 1990s. The non-probabilistic convex modelling approach advocates representation of uncertain parameters by a convex set, which relies upon knowledge of the variation bounds of the parameters. Compared to the precise probability distribution functions, the variation bounds are much easier to obtain, since it generally needs only a small number of experimental data or just the experience of engineers. It should be pointed out that convex model approach does not represent only a single model;

2 instead it represents a model set. Presently, interval model and ellipsoid model are two kinds of the most commonly used models in this area. In interval model, the uncertainty of a single variable is described through its upper and lower bounds. The uncertainty domain is then depicted as a ‘multidimensional box’ for a multidimensional problem. For ellipsoid model, it is assumed that the parametric uncertainty lies within a ‘multidimensional ellipsoid’. The degree of uncertainty and the degree of correlation of the variables are described by the size and shape of the ellipsoid. In theory, interval model can deal only with problems involving independent uncertain variables, while the ellipsoid model can deal only with dependent variables. Based on interval model, the anti-optimization analysis [7-10] was performed to seek for the least favorable response of a structural system under imposed constraints of uncertain-but-bounded parameters. Under the assumption of small uncertainty, a first-order interval perturbation method was applied to determine the influence of interval parameters on eigenvalues [11] and static displacements [12] of structures. By taking into account the actual variation and dependency of uncertain parameters affecting the mass and stiffness matrices, the lower and upper bounds of the natural frequencies of a structure with uncertain-but-bounded parameters were evaluated [13]. To provide a measure for the individual influence of interval inputs on the range of the obtained interval outcome of structural systems, the interval sensitivity analysis was studied and applied to the envelope frequency response function analysis of uncertain mechanical structures [14]. To find the best ellipsoidal convex model fitting the given experimental data of uncertain parameters, a Gram-Schmidt orthogonalization procedure for rotation of the coordinate system was put forwarded [15]. To improve the efficiency in constructing the multidimensi

…(Full text truncated)…

This content is AI-processed based on ArXiv data.