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.