A New Type Of Non-probabilistic Convex Model And The Corresponding Structural Uncertainty Analysis Techniques

Uncertainty widely exists in practical problems, and in most cases the values ofthe uncertainties are very small, but it may lead to non-negligible deviation of theresponds of the system, when the uncertainties of several parameters are coupled toeach other. Thus, it is necessary to consider the uncertainties of parameters in thestructural optimization process. The probabilistic method is the primary method todeal with uncertainty, but it requires a lot of uncertain information to construct theexact probability distribution to describe the uncertainty of parameters, which is verydifficult in many practical engineering problems. Non-probabilistic convex modelsonly need boundaries of parameters rather than their exact probability distributions;thus, such models can be applied to uncertainty analysis of complex structures whenexperimental information is lacking. The interval model and the ellipsoidal model arethe two most common methods in the field of non-probabilistic convex modeling.However, the former can only deal with independent variables, while the latter canonly deal with relevant variables. Therefore, it is very important for the uncertaintyanalysis and the reliability design of the complex structure to develop a more generalnon-probabilistic convex model that simultaneously takes into account theindependence and correlation of the variables.This article has conducted systematic research for non-probabilistic convexmodel and the corresponding uncertainty analysis techniques, and the main work is asfollows:(1) This paper presents a more general non-probabilistic convex model—Multidimensional Parallelepiped Model. This model can include the independent andrelevant uncertain variables in a unified framework and can effectively deal withcomplex “multi-source uncertainty” problems in which relevant variables andindependent variables coexist. In addition, for any two parameters, the concepts of therelevant angle and the correlation coefficient are defined. Through the marginalintervals of all the parameters and also their correlation coefficients, amultidimensional parallelepiped can easily be built as the uncertainty domain forparameters. Finally, the parallelepiped model in the original parameter space isconverted to an interval model in the affine space through the introduction of affinecoordinates, thus greatly facilitating subsequent structural uncertainty analysis. (2) The parallelepiped model is applied to structural uncertainty propagationanalysis, and the response interval of the structure is obtained in the case of uncertaininitial parameters. Because the multidimensional parallelepiped model can include theindependent and relevant uncertain variables in a unified framework, the structuraluncertainty propagation problem can be handled conveniently, where exists“multi-source uncertainty”. Because the level of uncertainty of parameters isgenerally small, the response function can be expanded to the first-order Taylor series,and then the lower and upper bounds of the response function can be obtainedaccording to the interval extension.(3) Based on the multidimensional parallelepiped convex model, we propose anew method for non-probabilistic structural reliability analysis in which marginalintervals are used to express scattering levels for the parameters, and relevant anglesare used to express the correlations between uncertain variables. Using an affinecoordinate transformation, the multidimensional parallelepiped uncertainty domainand the limit-state function are transformed to a standard parameter space, and areliability index based on multidimensional parallelepiped model is defined tomeasure the structural reliability.(4) Multidimensional parallelepiped model is introduced into the field ofreliability-based design optimization (RBDO) and corresponding RBDO problembased on multidimensional parallelepiped model (MP-RBDO) is built. Due to thiskind of problem, a reliability-based design optimization method based onmultidimensional parallelepiped model (MP-RBDO) has been proposed. The methodavoids a two-layer nesting optimization in RBDO problems, and can handle theRBDO problems efficiently.