Misalignment topology optimization with manufacturing constraints

Topology optimization design problems aims at the minimization of an objective function while satisfying various constraints. Since Bendsøe and Kikuchi (1988) topology optimization has mostly been based on “compliance formulation” as it provides solutions where the displacements are globally controlled. However, this formulation doesn’t take into account special design requirements over local displacements or even relative displacements such as the misalignment between two gear axes. This point is of paramount importance to achieve the best efficiency in many mechanical transmission devices. Although critical in practical engineering designs, this question is especially challenging as very few contributions exist on the subject. Coupling topology optimization with the misalignment minimization can provide promising results once right formulation can be identified.
At first, the misalignment can be expressed in various ways. A few formulations have been tested on a simple case study composed of two gear axes to be align. This allowed us to choose a promising expression for the misalignment and furthermore to investigate its efficiency on 2D test cases consisting of a simplified one-stage-reduction box and a simplified differential. The objective is to minimize the misalignment between two beams representing the gears engaged with each other. The formulations have been implemented in our in-house MATLAB code. Different issues have been highlighted and solved. The first basic implementation leads to unclear optimized material distributions as well as non-converged solutions. Optimization results have been investigated and new design formulations have been elaborated to tackle the various issues. We have showed that imposing a constraint on the measure of non-discreteness is able to enforce black-and-white solution with actual engineering meaning. The second issue is a possible disconnection of the structure coming from an ill-posed nature of the optimization problem as only local constraints are taken into account and no global performance of the problem is required. This issue is partly tackled by imposing a constraint on the measure of discreteness, but another natural way is to introduce an additional constraint on the global compliance of the component.
According to our numerical experiments, the proposed formulation is able to yield optimized solutions that make sense from an engineering point of view in 2D but also in 3D applications. Presently we are also introducing manufacturing constraints such as minimum/maximum size and minimum gap to further improve the manufacturability of the optimized solutions. The 3D academic torsion problem (see below) illustrates the effectiveness of the proposed formulation including misalignment in a 3D case. The presentation will be illustrated with several industrial applications.