One method for modeling geometric variations in hyperstatic (i.e. overconstrained) systems is to use sets of constraints. Different models have been developed in this way, e.g. domains, T-maps, and polytopes. In general, if the intersection of the contact constraints between two parts potentially in contact is nonempty, the parts can be assembled without interference, and their relative positions determined. In this study, the polytope method is used with a statistical approach to define the behavior of an assembly. In the first part, geometric variations including form deviations of individual parts are defined. The relations between these variations resulting from the architecture of a mechanism are then defined. In the second part, contact constraints are introduced and the general method to conform the constraints into double description polytopes is presented. The general process to simulate the compliance of the mechanism with respect to functional conditions is described. A failure rate is obtained for a simulated population of manufactured parts using the Monte Carlo method. In the third part, an application to a flange is described, an example from an industrial case study. We show how to take advantage of double description of polytopes when simulating the assembly and the misalignment of the two parts that make up the flange. Finally, we present our conclusions and prospects for future studies.Read less <