Inequality constraints in rectangular finite/boundary elements

Abstract

During the practical application of the finite element and boundary element methods to applied mechanics problems, we frequently encounter situations where a quantity of interest in such an element should satisfy an inequality inside the whole element. Here we consider the case of a simple two-dimensional rectangular element with eight nodes and we derive a quantifier-free formula (independent both of the universal quantifier and of the Cartesian coordinates) for the positivity of the fundamental quantity in this element on the basis of recent quantifier elimination results by Collins for the cubic univariate polynomial. The derived formula shows exactly for which numerical values of the fundamental quantity at the nodes of the element we must be sure about the satisfaction of the inequality involved (positivity) inside the whole element. Several applied mechanics problems, such as contact and crack problems in the boundary element method and problems concerning maximum permissible values for the displacement, stress and strain components in the finite element method, can be studied by the present approach. Several possible future generalizations are also suggested in brief and an alternative possibility, based on Sturm-Habicht sequences for polynomials, is studied in some detail both for the quadratic and for the cubic polynomial.