Quantifier-elimination-based interval computations in beam problems studied by using the approximate methods of finite differences and of finite elements

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Γλώσσα: English
• Έκδοση: Αυτο-έκδοση 2020
• Θέματα: Intervals; Interval arithmetic; Interval variables; Interval parameters; Uncertainty; Ranges; Finite differences; Finite elements; Systems of linear algebraic equations; Beams; Parametric loading; Deflection; Rotation; Bending moment; Quantifiers; Universal quantifier; Existential quantifier; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Symbolic computations; Computer algebra systems; REDLOG; Διαστήματα; Αριθμητική διαστημάτων; Μεταβλητές διαστήματος; Παράμετροι διαστήματος; Αβεβαιότητα; Πεδία τιμών; Πεπερασμένες διαφορές; Πεπερασμένα στοιχεία; Συστήματα γραμμικών αλγεβρικών εξισώσεων; Δοκοί; Παραμετρική φόρτιση; Βέλος κάμψεως; Στροφή; Ροπή κάμψεως; Ποσοδείκτες; Καθολικός ποσοδείκτης; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Συμβολικοί υπολογισμοί; Συστήματα υπολογιστικής άλγεβρας

The rather recent interesting computational method of quantifier elimination already implemented in four computer algebra systems has been already used in many problems of engineering interest including several problems of applied and computational mechanics. Among the previous applications of interest here is mainly the problem of a beam with parametric inequality constraints and under the presence of a loading parameter. This problem was solved by the popular methods (i) of finite differences and (ii) of finite elements in combination with the method of quantifier elimination. Here the same approach is generalized to the case where the loading parameter belongs to an interval. The methods (i) of finite differences and (ii) of finite elements are used again (leading to parametric systems of linear equations) with the computation of the approximate intervals concerning (i) the dimensionless deflection, (ii) the rotation and (iii) the dimensionless bending moment on the whole beam computed on the basis of their values at the nodes used on the beam. In the application of the finite difference method both (i) the purely existential case and (ii) a mixed universal–existential case are considered evidently with respect to the interval loading parameter. The REDLOG computer logic package of the REDUCE computer algebra system has been used again in the present interval computations and the excellent convergence of the obtained approximate intervals computed with the finite difference method is observed. In the purely existential case, up to 3072 intervals on the beam have been successfully used and this is an extremely satisfactory situation in quantifier elimination because it concerns a total number of 3076 variables.