Application of the method of quantifier elimination to the determination of intervals when the uncertain parameters satisfy an ellipsoidal inequality constraint

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Γλώσσα: English
• Έκδοση: Κανένας 2021
• Θέματα: Convex models; Ellipsoidal model; Ellipsoidal constraints; Inequality constraints; Uncertainty; Uncertain parameters; Uncertainty regions; Intervals; Correlation propagation; Non-probabilistic methods; Cubic equation; Structural mechanics; Shear frame buildings; Trusses; Quantifiers; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Symbolic computations; Computer algebra systems; Mathematica; Κυρτά μοντέλα; Ελλειψοειδές μοντέλο; Ελλειψοειδείς περιορισμοί; Ανισοτικοί περιορισμοί; Αβεβαιότητα; Αβέβαιες παράμετροι; Περιοχές αβεβαιότητας; Διαστήματα; Διάδοση συσχετίσεως; Μη πιθανοτικές μέθοδοι; Κυβική (τριτοβάθμια) εξίσωση; Δομική μηχανική; Πλαισιωτά κτίρια διατμήσεως; Δικτυώματα; Ποσοδείκτες; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Συμβολικοί υπολογισμοί; Συστήματα υπολογιστικής άλγεβρας

Quite frequently, problems that appear in applied mechanics should be solved under uncertainty conditions. Among the related non-probabilistic methods that based on interval analysis constitutes a very popular model. Here we consider another popular model: that based on an ellipsoidal inequality constraint among the uncertain parameters. This is the so-called ellipsoidal convex model. Generalized ellipsoidal convex models are also frequently adopted. Here the aim is to use the interesting computational method of quantifier elimination for the solution of such an uncertainty problem generally for the determination of the intervals of the responses of the system under consideration of course under the restriction that the total number of variables and the degrees of the polynomials involved are small. The present approach is applied to the problems of (i) a three-parametric cubic equation with respect to its real root, (ii) a two-storey shear frame building with non-linear stiffness, (iii) a three-member truss (with the adoption of several uncertainty models), (iv) a simple structural mechanics problem with symbolic intervals, (v) the correlation propagation in a system involving three uncertain parameters and (vi) a problem with a complicated uncertainty region for the uncertain parameters. The alternative, but essentially not so different, approach based on minimization and maximization is also considered in brief. The present results show us that the method of quantifier elimination can be successfully applied to simple systems with uncertain parameters satisfying an inequality constraint (such as an ellipsoidal constraint) and provide us the exact intervals of the responses of the system or even the exact regions showing their correlations.