Variational methods for evaluating the design of mechanisms were first introduced by this group in the form of mathematical formulations generally applicable to open-and closed-loop mechanisms. This method is herein extended and demonstrated to the design of mechanical parts in the context of automatic parametrization of the geometry. The formulation is based on the development of constraint equations that govern the relation between geometry in a mechanical part as dictated by a designer. Instead of the tedious method of specifying mathematical relations between any two geometries of the part, it is proposed to use the notion of kinematic relations inherent in the formulation relating the connectivity between joints and links. Cut-joint constraints are introduced, kinematic joints in the formulation are combined, their variations evaluated, and a Jacobian is determined. Constraint violations are then compensated to compute an assembled mechanism, hence redesigning the part. It is shown that this kinematically-driven formulation is broadly applicable to 2D and 3D models. The method and algorithm are illustrated through a number of examples.

A numerical variational method is herein introduced for the automatic redesign of mechanical parts subject to a design change. This method is demonstrated to the design of mechanical parts in the context of concurrent engineering of 3D parts. Relations between dimensions in a mechanical part as dictated by a designer are modeled using mechanisms and their kinematic relations. Variational cut-joint constraints are introduced, kinematic joints in the formulation are combined, and a Jacobian is determined. Constraint violations are then compensated to compute an assembled mechanism, hence redesigning the part. It is shown that this formulation is broadly applicable to 2D and 3D models. The method and algorithm are illustrated through examples.