Interval computations in various direct and inverse applied mechanics problems related to quantifiers by using 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 variables/parameters; Range; Uncertainty; Uncertain variables/parameters; Beams; Beams on elastic foundation; Winkler foundation; Deflection; Vibration problems; Damped harmonic oscillator; Critical damping; Quantifiers; Universal quantifier; Existential quantifier; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Systems; Input; Output; Control problem; Tolerance problem; Direct problems; Inverse problems; Symbolic computations; Mathematica; Διαστήματα; Ανάλυση διαστημάτων; Μεταβλητές/παράμετροι διαστήματος; Πεδίο τιμών; Αβεβαιότητα; Αβέβαιες μεταβλητές/παράμετροι; Δοκοί; Δοκοί σε ελαστική βάση; Θεμελίωση Winkler; Βέλος κάμψεως; Προβλήματα ταλαντώσεων; Αρµονικός ταλαντωτής µε απόσβεση; Κρίσιμη απόσβεση; Ποσοδείκτες; Καθολικός ποσοδείκτης; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Συστήματα; Είσοδος; Έξοδος; Πρόβλημα ελέγχου; Πρόβλημα ανοχής; Ευθέα προβλήματα; Αντίστροφα προβλήματα; Συμβολικοί υπολογισμοί

Quantifier elimination offers an interesting computational tool in many research areas including applied mechanics long ago. For example, quantifier elimination was recently applied to the computation of ranges of functions in problems of applied mechanics. Here we modify this approach by using the existential quantifier instead of the universal quantifier in the quantified formulae. This approach permits the reduction (by two) of the number of free variables. Yet, what is more important is that here we also extend this method based on quantifier elimination from the purely existential case to the mixed universal–existential case. The latter case is related to the classical interval tolerance and control problems so popular in interval analysis. Among the few implementations of quantifier elimination (in classical real analysis) in computer algebra systems again we selected the computer algebra system Mathematica for use in the present computations because it seems to offer the most efficient and user-friendly related implementation. Three applied mechanics problems are studied in detail: (i) a classical beam problem (beam fixed–simply-supported at its ends) under a uniform loading, (ii) a problem of a beam on a Winkler elastic foundation and (iii) the problem of free vibrations of the classical damped harmonic oscillator under critical damping. In these three problems, several quantified formulae were considered (of course, under appropriate assumptions) and the related QFFs (quantifier-free formulae) were easily derived. Moreover, the cases of (i) three interval variables and no parameter in the QFF, (ii) two interval variables and one parameter in the QFF and (iii) one interval variable and two parameters in the QFF were studied.