Quantifier-Free Formulae for Inequality Constraints Inside Boundary Elements

Abstract

In this chapter we reconsider the problem of construction of quantifier-free formulae for the positivity of the unknown function in the BEM (boundary element method) along the whole boundary element. Such a situation appears in crack problems in fracture mechanics, where the crack opening displacement should be a positive function so that no contact between the crack edges be present. It also appears in contact problems where the pressure distribution should also be a positive quantity. Here our previous related results on the basis of classical Sturm sequences are improved: (i) by studying the unknown function on the interval [0,∞), (ii) by using Mathematica for all of the present computations and (iii) by directly using the nodal values of the originally unknown function also in the cubic polynomial. Both cases of the quadratic polynomial (three nodes on the boundary element) and the cubic polynomial (four nodes on the boundary element) are studied.