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.

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.

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。

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.

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.