Elementary quantifier-free formulae in boundary elements
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Ioakimidis, Nikolaos
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Abstract
The use of elementary algebraic quantifier elimination techniques is suggested during the numerical solution of elasticity problems by boundary element methods in problems where the derived solutions need to be verified as far as the related physical constraints (in inequality forms) are concerned. The cases of crack problems, where the crack opening displacement must be non-negative, and of contact problems, where the pressure distribution between the bodies in contact must also be non-negative, constitute two such classical elementary examples where the derived ordinary numerical solutions need to be verified with respect to the aforementioned elementary constraints or simply rejected. As a simple such application, elementary quantifier-free formulae are derived (by using Sturm's theorem) for simple univariate polynomials of the first and of the second degree which should remain positive along a finite interval. These formulae are directly applicable to the aforementioned two elasticity problems.
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Keywords
Quantifier elimination, Quantifier-free formulae, Sturm sequences, Sturm's theorem, Inequality constraints, Boundary elements, Boundary element method, BEM, Elasticity, Crack problems, Contact problems, Maple