Computation of ranges of functions in problems of applied mechanics with the computational method of quantifier elimination

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Ioakimidis, Nikolaos
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The method of quantifier elimination constitutes an interesting and relatively modern computational tool in computer algebra. An efficient implementation of quantifier elimination is included in the computer algebra system Mathematica since 2003. Here after an introduction to the approach of quantifier elimination in Mathematica for the determination of intervals in some simple problems in arithmetic and ranges of functions in elementary algebra, we continue with the determination of ranges of functions concerning problems of applied mechanics. Four such kinds of problems are studied here: (i) Two classical beam problems, (ii) A problem of a beam on a Winkler elastic foundation, (iii) The problem of buckling of the Euler classical column and (iv) A problem of free vibrations of an oscillator with critical damping. In all cases, the ranges of the functions of interest are determined. Yet, Taylor–Maclaurin or minimax or similar approximations are necessary in problems where transcendental functions are involved. Naturally, the method is applicable to a variety of additional problems of applied mechanics although, unfortunately, its power is limited to problems with few variables (both quantified variables and free variables) and not very high degree(s) in the polynomial(s) involved. Therefore, the efficiency of the method is particularly clear mainly in problems involving only a single polynomial of degree about up to twenty and with only one parameter. The case of an interval with a parameter as one end is also in principle acceptable.
Intervals, Arithmetic, Elementary algebra, Beams, Beams on elastic foundation, Winkler foundation, Deflection, Columns, Buckling, Vibration problems, Critical damping, Damped harmonic oscillator, Upper bounds, Lower bounds, Maximization, Minimization, Symbolic computations, Universal quantifier, Quantified formulae, Quantifier elimination, Quantifier-free formulae, Functions, Range, Image, Computer algebra systems, Mathematica