In industrial contexts, overconstrained assemblies are often used to ensure sufficient stiffness and accuracy of assembly. Such architectures are quite usual, however, analysis and synthesis of the tolerance are not easy to define and quantify. In these cases, the compliance of the assembly is not automatic and deformations may often occur, requiring a particular and difficult analysis of the assembly from scientific and computing points of view. The present work addresses such overconstrained mechanisms through a general and a sequential approach. Based on this, it is possible to determine the final assembly condition (with or without interference) as a function of part defects. First, the assembly procedure is performed based on polytope computations, assuming a rigid part behavior. From this, a stochastic simulation is performed and some non-compliant assemblies (assemblies with interferences) are identified. For these assemblies, the rigid behavior of parts is then overcome by means of finite element simulations and a typical procedure is set up to introduce part defects. We then deduce whether the assembly can be made from the load needed to assemble the parts. This procedure is applied to a flange composed of five pin / hole pairs, as this is a highly overconstrained mechanism. Nomenclature• t M−1,i/2,i : translation vector of surface i of part 1 in relation to surface i of part 2 at point M. If i = 0, we are considering the translation of the nominal geometry of part 1 in relation to the nominal geometry of part 2 at point M.• r 1,i/2,i : rotation vector of surface i of part 1 in relation to surface i of part 2. If i = 0, we are considering the rotation of the nominal geometry of part 1 in relation to the nominal geometry of part 2,• e M−1,i/1,0 : location defect vector of surface i of part 1 in relation to its nominal geometry at point M,• e M−2,i/2,0 : location defect vector of surface i of part 2 in relation to its nominal geometry at point M, 1• D n , D 1,i and D 2,i : the diameter of the nominal surface, the diameters of surface i of part 1 and surface i of part 2 respectively,• H − i, j : half-space of the contact constraint derived from the j th discretization point between surface i of part 1 and surface i of part 2,• P i : contact polytope of surface i of part 1 in relation to surface i of part 2,• P R : resulting contact polytope of part 1 in relation to part 2.

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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.