Generalized interval-based polynomial approximations to functions in applied mechanics by using the method of quantifier elimination

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Γλώσσα: English
• Έκδοση: Αυτο-έκδοση 2020
• Θέματα: Intervals; Interval parameters; Uncertainty; Ranges; Polynomial approximations; Interval-based approximations; Generalized approximations; Parameterized approximations; Interval enclosures; Beams; Winkler foundation; Oscillators; Critical damping; Displacements; Trusses; Perturbations; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Quantifiers; Symbolic computations; Computer algebra systems; Διαστήματα; Παράμετροι διαστήματος; Αβεβαιότητα; Πεδία τιμών; Πολυωνυμικές προσεγγίσεις; Προσεγγίσεις βασιζόμενες σε διαστήματα; Γενικευμένες προσεγγίσεις; Παραμετροποιημένες προσεγγίσεις; Περιβλήματα με διαστήματα; Δοκοί; Βάση Winkler; Ταλαντωτές; Κρίσιμη απόσβεση; Μετατοπίσεις; Δικτυώματα; Διαταραχές; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Ποσοδείκτες; Συμβολικοί υπολογισμοί; Συστήματα υπολογιστικής άλγεβρας

The method of quantifier elimination constitutes an interesting computational approach in computer algebra already implemented in few computer algebra systems. In applied mechanics, this method was already used for the determination of ranges of functions. Here the application of the same method, quantifier elimination, is generalized to the determination of generalized interval-based polynomial approximations to functions again in applied mechanics. The main idea behind the present application is the use of linear interval enclosures for the approximation to functions and, more generally, the use of parameterized solutions to parametric interval systems of linear algebraic equations. This idea is mainly due to Lubomir V. Kolev. Here the present method is at first applied to two simple examples concerning (i) a rational function and (ii) the exponential function with their variables lying in intervals. Next, the same method is also applied to functions in applied-mechanics problems with variables also lying in intervals: (i) the problem of a beam on a Winkler elastic foundation with related function the dimensionless deflection of the beam, (ii) the problem of free vibrations of an oscillator with critical damping with related function the dimensionless displacement of the oscillator and (iii) the problem of a seven-member truss with related functions the nodal displacements. In this application, the stiffness of a bar is an uncertain, interval variable and, moreover, the classical perturbation method is also used. From the present results it is concluded that the method of quantifier elimination constitutes a useful tool for the derivation of simple parameterized interval-based polynomial approximations to functions in applied mechanics.