Automatic derivation of positivity conditions inside boundary elements with the help of the REDLOG computer logic package
The very well-known shape functions are used in classical boundary element analysis for the construction of the polynomial interpolation function p for the approximation to the unknown field quantity u. In this note, we show the usefulness of the recent REDLOG Reduce computer logic package of Dolzmann and Sturm for the construction of positivity and analogous parametric feasibility conditions, based on the nodal values of the polynomial p, inside the whole boundary element. A simple one-dimensional case, based on the classical quadratic element, and a more difficult two-dimensional case, based on a quadratic triangular element, are used for the illustration of the approach, whereas several related conclusions are also drawn. The present results are applicable to cases where the positivity or the negativity or just the boundedness of the unknown quantity is required (such as the case of the pressure distribution in contact problems and the opening displacement in crack problems) and significantly extend the already available recent related computational quantifier elimination research results.
boundary elements computer logic interpolation functions feasibility conditions quadratic elements triangular elements quantifier elimination computer algebra inequality constraints positivity/negativity conditions