Non-probabilistic Convex Model Research Articles

Probability and convexity concepts are not antagonistic

This study is devoted to two objectives to illustrate that the probability and convexity concepts are not antagonistic and to introduce a new non-probabilistic convex model for structural reliability analysis. It is shown that the new measure of safety is easier to evaluate than the corresponding measure utilizing the interval analysis. Moreover, interrelation between the classical probabilistic method and convex modeling method is demonstrated. The purpose of this study is not to replace the probabilistic approach by the convex modeling method, but to illustrate that the probability and convexity concepts are compatible. Some numerical examples are presented to illustrate the feasibility of the proposed methodology.

Non-probabilistic convex models and interval analysis method for dynamic response of a beam with bounded uncertainty

This paper is concerned with the comparison of two non-probabilistic set-theoretical models for dynamic response measures of an infinitely long beam. The beam is on an uncertain foundation and subjected to a moving force with constant speed. The steady state vibration is analyzed with finite element method. The dynamic responses of the beam are approximated to the first-order respect of the uncertainty variables. As a rule, in convex models and interval analysis, the uncertainties are considered to be unknown, but they give out their allowable vector space. Comparing the convex models with interval analysis in mathematical proofs and numerical calculations, it’s shows that under the condition of transform an interval vector to an outer enclosed ellipsoid, the dynamic response of the infinitely long beam predicted by interval analysis is smaller than that by convex models; under the condition of transform a hyperellipsoid to an outer enclosed interval vector, the dynamic response of the infinitely long beam calculated by convex models is smaller than that by interval analysis method.

Design optimization for dynamic response of vibration mechanical system with uncertain parameters using convex model

The concept of uncertainty plays an important role in the design of practical mechanical system. The most common methods for solving uncertainty problems are to model the parameters as a random vector. A natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution. However, the drawback of this approach is that the probability distribution is difficult to obtain. In this paper, we use the non-probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions. Using the convex model theory, a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived. Because the uncertain parameters can be selected as the optimization parameters, the present method can provide more information about the optimization results than those obtained by the deterministic optimization. The present method is implemented for a torsional vibration system. The numerical results show that the method is effective.

Comparison of static response of structures using convex models and interval analysis method

AbstractIn 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 © 2003 John Wiley & Sons, Ltd.

Stability of elastic bars on uncertain foundations using a convex model

A method is presented for evaluating the buckling load of weightless prismatic rigid bars resting on uncertain foundations which are modeled using linear extensional elastic springs. The uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model. An ellipsoidal bound is used which defines the uncertainty of the foundation in terms of a size parameter and the deviations of the elastic spring constants from their nominal values. The size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses. The semiaxes of the ellipsoid are the deviations of the foundation spring constants from their nominal values and they determine the shape of the ellipsoid. A first-order analysis shows that the reduction in the buckling load, when uncertainty in the foundation's spring stiffnesses is present, is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid. For the same uncertainty in the spring elements, different reductions in the buckling load result for beams with multimode buckling.

A non-probabilistic measure of reliability of linear systems based on expansion of convex models

In this paper we develop a rigorous quantitative alternative to the probabilistic theory of reliability. The intuitive concept of reliability which underlies our analysis is that a system is reliable if it is robust with respect to uncertainty. That is, a system is reliable if it can tolerate a large amount of uncertainty before failure can occur. Conversely, a system is unreliable if it is fragile with respect to uncertainty. We model uncertainty with non-probabilistic convex models, and measure the amount of uncertainty with the expansion parameters of the convex models. The measure of reliability developed here is the amount of uncertainty the system can tolerate before failure. We consider linear dynamic systems with uncertain inputs and uncertain output failure states. The reliability of these systems hinges on the disjointness of convex sets. Using the hyperplane separation theorem for convex sets, we introduce a concept of modal reliability.

Buckling of Elastic Columns Using Convex Model of Uncertain Springs

A procedure is described for evaluating the buckling load of weightless prismatic columns with uncertain rotational spring elements. The uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model. An ellipsoidal bound is used that defines the uncertainty in terms of a size parameter and the deviations of the spring constants from their nominal values. The size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses. The semiaxes of the ellipsoid are the deviations of the spring constants from their nominal values and they determine the shape of the ellipsoid. A first-order analysis shows that the reduction in the buckling load when uncertainty in the spring stiffnesses is present, is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid. For columns for which more than one buckling mode is possible, different reductions in the value of the buckling load result in each mode for the same uncertainty in the spring element stiffnesses.

A non-probabilistic concept of reliability

Uncertainty can be modelled either probabilistically or non-probabilistically. The former option leads to the concept of reliability as the probability of no-failure. In this paper non-probabilistic convex models of uncertainty are used to formulate reliability in terms of acceptable system performance given uncertain operating environment or uncertain geometrical imperfections. It is shown that probabilistic reliability can be very sensitive to small inaccuracy in the probabilistic model. Consequently, the non-probabilistic concept of reliability is useful when insufficient information is available for verifying a probabilistic model. In addition, a theorem is presented showing that analogous convex and probabilistic models of input uncertainty can lead to very different predictions of the range of output variation.