将非概率凸模型理论与摄动理论相结合，通过有界不确定参数结构的特征值问题，对凸模型理论的一次近似算法作出一种改进。改进后的算法由于在计算中不用特征值导数，与Elishakoff的算法相比，不仅拓广了凸模型理论的应用范围，而且还可提高算法的计算效率。

将非概率凸模型理论与摄动理论相结合，通过有界不确定参数结构的特征值问题，对凸模型理论的一次近似算法作出一种改进。改进后的算法由于在计算中不用特征值导数，与Elishakoff的算法相比，不仅拓广了凸模型理论的应用范围，而且还可提高算法的计算效率。

This paper is concerned with the problem of comparison of two non-probabilistic set-theoretical models for dynamic response and buckling failure measures of bars with unknown-but-bounded initial imperfections. Two kinds of non-probabilistic set-theoretical models are convex models and interval analysis models. In convex models and interval analysis models, the uncertain quantities are considered to be unknown except that they belong to given sets in an appropriate vector space. In this case, all information about the dynamic response and buckling failure measures of bars is provided by the set of dynamic responses and buckling failure measures consistent with the constraints on the uncertain quantities. The dynamic response estimate is actually a set in appropriate response space rather than a single vector. The set estimate is the smallest calculable set which contains the uncertain dynamic response, but it is usually impractical to calculate this set. Two set estimate methods are developed which can calculate the time varying box or hyperrectangle, i.e. interval vector in the response space that contains the true dynamic response. Comparison between convex models and interval analysis models in mathematical proofs and numerical calculations shows that, under the condition of the outer enclosed ellipsoid from a hyperrectangle or an interval vector, the set dynamic response predicted by interval analysis models is smaller than that yielded by convex models; under the condition of the outer enclosed hyperrectangle or an interval vector from an ellipsoid, the dynamic response set calculated by convex models is smaller than that obtained by interval analysis models.

Now by combining the finite element analysis and interval mathematics, a new, non-probabilistic, set-theoretical models, that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches. Interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures, in which bounds on the magnitude of uncertain parameters are only required, not necessarily knowing the probabilistic distribution densities. Instead of conventional optimization studies, where the minimum possible response is sought, here an uncertainty modeling is developed as an anti-optimization problem of finding the least favorable response and the most favorable response under the constraints within the settheoretical description. In this study, we presented the non-probabilistic interval analysis method for the dynamical response of structures with uncertain-but-bounded parameters. Under the condition of the interval vector, which contains the uncertain-but-bounded parameters, determined from probabilistic statistical information or stochastic sample test, through comparing between the non-probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples, we can see that the region of the dynamical response of structures with uncertain-but-bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach. In other words, the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method. This kind of results is coincident with the meaning of the probabilistic theory and interval mathematics.

In this paper, by combining the finite element analysis and non-probabilistic convex models, we present the numerical algorithm of non-probabilistic convex models and interval analysis method for the static displacement of structures with uncertain-but-bounded parameters. Under the condition of the box or interval vector determined from the ellipsoid of the uncertain-but-bounded structural parameter vector, by comparing the numerical algorithm of non-probabilistic convex models and the interval analysis method in the mathematical proof and the numerical example, we can see that the width of the maximum or upper and minimum or lower bounds on the static displacement yielded by the numerical algorithm of non-probabilistic convex models is tighter than those produced by the interval analysis method. Copyright.