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.

By a counter example, we show that there seem to be some problems in Ben-Haim's theory of robust reliability of dynamical systems based on convex models. We still point out that the property of the expansion of convex models is just the addition of a convex model and a real vector, and the property of the translation of convex models is just the scalar multiplication convex models. By means of the partial-order relation of the superscribed hyperrectangle or interval vectors of convex models, we present a correct criterion of reliability of the dynamical system with bounded uncertainty. Based on them, we propose the expansion function which is different from the one of Ben-Haim. Following Ben-Haim's thoughts, based on the new expansion function, we again define the input, failure, and overall reliability indices. By Ben-Haim's example, we obtain some results different from his. The conclusion and results may be thought of as to the further development of Ben-Haim's robust reliability.

In this study, a new, deterministic method is discussed for estimating the maximum, or least favorable frequency, and the minimum, or best favorable frequency, of structures with uncertain but non-random parameters. The favorable bound estimate is actually a set in eigenvalue space rather than a single vector. The obtained optimum estimate is the smallest calculable set which contains the uncertain system eigenvaluse. This kind of engenvalue problem involves uncertain but non-random interval stiffness and mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the generalized eigenvalue problem of the uncertain but non-random interval matrices. By applying the interval extension matrix perturbation formulation, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of uncertain but non-random interval stiffness and mass matrices. A perturbation method is developed which allows one to calculate eigenvalues of an uncertain but non-random interval matrix pair that always contains the system's true stiffness and mass matrices. Inextensive computational effort is a characteristic of the presented method. A numerical example illustrates the application of the proposed method. Copyright.

This paper is concerned with the problem of robust stability of linear dynamic systems with structured uncertainty by means of ellipsoidal set-theoretic approach. In this paper, the uncertainty in the physical parameters is expressed in terms of an ellipsoidal set in appropriate vector space. Two ellipsoidal set-theoretic approaches are presented for giving sufficient conditions for robust stability property of the systems with structured uncertainty. The bound produced by the ellipsoidal extension function theorem is shown to be less conservative than the one predicted by the Lagrange multiplier method. In order to introduce the ellipsoidal extension function theorem, in Appendix A of this paper, we try to present the theory of ellipsoidal algebra, following the thought of interval analysis. First of all, we give the concept of ellipsoidal numbers and define their arithmetic operations. Based on them, we finally introduce ellipsoidal vectors and ellipsoidal functions. In terms of the inclusion monotonic property of ellipsoidal functions, we present and prove the ellipsoidal extension function theorem.

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.

Anti-optimization technique, on the one hand, represents an alternative and complement to traditional probabilistic methods, and on the other hand, it is a generalization of the mathematical theory of interval analysis. In this study, in terms of interval analysis or interval mathematics, the arithmetic operations and the partial order relation of anti-optimization technique can be defined, and the convex model variables and the convex model extension function of convex models can also be introduced. The comparison of the Lagrange multiplier method with the convex model extension method for evaluation the region of static displacements of structures with uncertain-but-bounded parameters shows that the with of the upper the lower bounds on the static displacement of structures with uncertain-but-bounded parameters shows that the width of the upper and lower bounds on the static displacement yielded by the Lagrange multiplier method ofconvex models is tighter than those produced by the convex model externsion.

In this paper, we present a method for computing upper and lower bounds of natural frequencies of the structures with uncertain parameters. There parameters are unknown except for the fact that they belong to given bounded sets. The sel of possible system states can be described by interval matrices. By solving the interval matrix problem, we obtain the bounds on frequencies of the structure. The numerical results demonstrates the efficacy of the method.

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.