Model reduction in geometric tolerancing by polytopes

There are several models used in mechanical design to study the behaviour of mechanical systems involving geometric variations. By simulating defects with sets of constraints it is possible to study simultaneously all the configurations of mechanisms, whether over-constrained or not. Using this method, the accumulation of defects is calculated by summing sets of constraints derived from features (toleranced surfaces and joints) in the tolerance chain. These sets are usually unbounded objects (R6-polyhedra, 3 parameters for the small rotation, 3 for the small translation), due to the unbounded displacements associated with the degrees of freedom of features. For computational and algorithmic reasons, cap facets are introduced into the operand polyhedra to obtain bounded objects (R6-polytopes) and facilitate computations. However, the consequence is an increase in the complexity of the models due to the multiplication of caps during the computations. In response to this situation, we formalized and tested a method for controlling the effects of cap facets. Based on the combinatorial properties of polytopes, we propose to trace the operand faces during the different operations. An industrial case is solved and discussed in order to show the significant gain in computational time when applying the new method. This example has been chosen to be as general as possible to illustrate the genericity of the method.