Determination of intervals in systems of parametric interval linear equilibrium equations in applied mechanics with the method of quantifier elimination

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2020
• Θέματα: Intervals; Interval analysis; Interval arithmetic; Interval variables; Uncertain variables; Uncertainty; Systems of linear equations; Parametric interval systems; Equilibrium equations; Bars; Trusses; Loads; Reactions; Quantifiers; Universal quantifier; Existential quantifier; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Solution sets; Symbolic computations; Computer algebra systems; Mathematica; Διαστήματα; Ανάλυση διαστημάτων; Αριθμητική διαστημάτων; Μεταβλητές διαστήματος; Αβέβαιες μεταβλητές; Αβεβαιότητα; Συστήματα γραμμικών εξισώσεων; Παραμετρικά συστήματα με διαστήματα; Εξισώσεις ισορροπίας; Ράβδοι; Δικτυώματα; Φορτία; Αντιδράσεις; Ποσοδείκτες; Καθολικός ποσοδείκτης; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Σύνολα λύσεων; Συμβολικοί υπολογισμοί; Συστήματα υπολογιστικής άλγεβρας

The method of quantifier elimination is an interesting computational tool in computer algebra with many practical applications including problems of applied mechanics. Recently, this method was used in applied mechanics problems with uncertain parameters varying in known intervals (interval parameters) including systems of parametric interval linear equilibrium equations, direct and inverse problems and the computation of resultants of interval forces. Here the case of systems of parametric interval linear equilibrium equations is further considered by using related quantified formulae including not only the existential quantifier (as is the case with the united solution set of such a system), but also both the universal and the existential quantifiers in the quantified formula (a more general case) with respect to the parameters of the problem including the external loads applied to the mechanical system. Two problems of applied mechanics related to systems of parametric interval linear equilibrium equations are studied in detail: (i) the problem of a simply-supported truss with two external loads recently studied under uncertainty (interval) conditions by E. D. Popova and (ii) the problem of a clamped bar with a gap subjected to a concentrated load recently studied again under uncertainty (interval) conditions by E. D. Popova and I. Elishakoff. Here, in both these problems, by using the method of quantifier elimination both (i) complete solution sets for the unknown quantities (here mainly reactions) and (ii) separate intervals for each unknown quantity are computed on the basis of related quantified formulae. The present results are compared to the results obtained by E. D. Popova and I. Elishakoff on the basis of both the classical interval model and the new algebraic interval model, the latter recently proposed by E. D. Popova.