Multidimensional parallelepiped model—a new type of non‐probabilistic convex model for structural uncertainty analysis

Abstract

SUMMARYNon-probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions; thus, such models can be applied to uncertainty analysis of complex structures when experimental information is lacking. The interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non-probabilistic convex modeling. However, the former can only deal with independent variables, while the latter can only deal with dependent variables. This paper presents a more general non-probabilistic convex model, the multidimensional parallelepiped model. This model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex 'multi-source uncertainty' problems in which dependent variables and independent variables coexist. For any two parameters, the concepts of the correlation angle and the correlation coefficient are defined. Through the marginal intervals of all the parameters and also their correlation coefficients, a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters. Through the introduction of affine coordinates, the parallelepiped model in the original parameter space is converted to an interval model in the affine space, thus greatly facilitating subsequent structural uncertainty analysis. The parallelepiped model is applied to structural uncertainty propagation analysis, and the response interval of the structure is obtained in the case of uncertain initial parameters. Finally, the method described in this paper was applied to several numerical examples.

Uncertain static plane stress analysis with interval fields

“…UNCERTAIN STATIC PLANE STRESS ANALYSIS WITH INTERVAL FIELDS 1273 remarkable applicability of the probabilistic approach, it is gradually realized that there are situations where probabilistic analysis becomes incompetent to produce credible results because of the insufficiency of information on the uncertain system parameters [21,22].In order to adequately accomplish uncertainty analyses for engineering applications with insufficiency of data, the non-probabilistic uncertainty analysis framework has been proposed and implemented. The conventional non-probabilistic analysis, such as fuzzy analysis [23][24][25], interval analysis [26][27][28][29][30][31] and uncertainty analysis based on convex models [32][33][34], has the benefits to perform valid uncertainty analysis for engineering applications without straining on the availability of statistical information. Such uncertainty modelling relaxation has offered the complementarity between probabilistic and non-probabilistic methods, which has certainly enriched the implementation of uncertainty analysis in real-life engineering practice.However, in recent years, some researchers have noticed a critical issue of uncertainty analysis, which is the incorporation of spatial variations of uncertain system parameters within the uncertainty analysis.…”

A novel tolerance analysis for mechanical assemblies based on Convex Method and non-probabilistic set theory

“…Convex Method has many advantages over traditional tolerance analysis methods: (1) Convex Method does not need the distributions of the parameters and greatly reduces the demand of original data, (2) Convex Method can get a relatively reliable variation interval of the results depend on a small amount of data, and (3) the result of W-C method is overly pessimistic, the distribution assumptions of statistical methods are excessively ideal, while the result of Convex Method is closer to the engineering practice [29]. In this method, it is assumed that uncertainty of the parameters belongs to a convex region; thus, the uncertainty boundary can be obtained based on a small number of samples instead of an exact probability distribution.…”

Discussions on non-probabilistic convex modelling for uncertain problems

“…Different from the ME model, the MP model utilizes a parallelogram to quantify the uncertainty of two interval variables and a multidimensional parallelepiped for multiple intervals, as shown in problems [22], in which an affine coordinate system was employed to describe the uncertainty domain. It was improved in Ref.…”

“…4 correspond to those when the CCC c r equals to +0.6, 0 and −0.6, respectively. The CSM H used to construct the MP-I model is just the correlation matrix R, namely  HR (22) Note that although the MP-I model also uses a rhomb for correlation quantification, it differs with the MP-II model in definition of the CCC, which can also be illustrated by comparing Fig. 4 with the cases of the MP-II model in Fig.…”

“…As two typical convex models for uncertainty quantification, the ellipsoid model and the interval model were compared for carrying out dynamic response analysis and buckling failure analysis of bars [19]. Recently, some variants of the traditional convex models were also developed, including the multi-ellipsoid model [20,21], the multidimensional parallelepiped model [22,23], the super-ellipsoidal model [24], the convex model process [25], etc. Also, the convex model approach presently has been applied in various areas in structural mechanics, such as finite element analysis [26,27], non-probabilistic reliability analysis [5,18,[28][29][30][31], uncertain optimization [32][33][34][35][36][37], hybrid reliability analysis [38][39][40][41], etc.…”