Quantifier elimination in applied mechanics problems with cylindrical algebraic decomposition

• Main Author: Ioakimidis, Nikolaos
• Language: English
• Published: Pergamon Press (Elsevier Science) 2023

The method of cylindrical algebraic decomposition (CAD) of the k-dimensional space constitutes a classical technique for the efficient solution of quantifier elimination (QE) problems in algorithmic, computer-aided algebra. Here we apply this method to some applied mechanics problems under appropriate constraints. At first, we study the problem of a straight elastic beam under a restriction on the maximum permissible deflection along this beam (which can easily be reduced to the construction of a one-dimensional CAD) as well as the problem of a circular isotropic elastic medium where a stress component should not exceed a critical value (which requires the construction of a three-dimensional CAD). In both these problems, we also derive the required quantifier-free formulae (QFFs) not including the fundamental variables, but only the parameters involved. Much more difficult CAD/QFF-derivation applications concerning an elliptical elastic medium again with an upper bound for a stress component, a special case of failure by yielding in fracture mechanics, related to Sih's strain-energy-density factor and a frictionless contact problem for an elastic half-plane are also considered and explicitly solved with the help of already available CAD-produced results although, evidently, CAD is not expected to produce QFFs in extremely difficult problems. Finally, additional possible applications of CAD/CQE to applied mechanics problems are also suggested.