Worst-case and statistical tolerance analysis based on quantified constraint satisfaction problems and Monte Carlo simulation

Abstract

This paper deals with the mathematical formulation of tolerance analysis. The mathematical formulation presented in this paper simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion (existential quantifier: “there exists”; universal quantifier: “for all”). It takes into account not only the influence of geometrical deviations but also the influence of the types of contacts on the geometrical behavior; these physical phenomena are modeled by convex hulls (compatibility hull, interface hull and functional hull) which are defined in parametric space. With this description by convex hulls, a mathematical expression of the admissible deviations of parts integrates the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier: “there exists”), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier: “for all”). To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.