DGTO: Derivable geodesics-coupled topology optimization for multi-axis hybrid additive and subtractive manufacturing with curved layer generation

This paper presents a concurrent optimization model named DGTO (Derivable Geodesics-coupled Topology Optimization), aiming at simultaneously designing the structure, slices, and sequences for multi-axis hybrid additive and subtractive manufacturing. The proposed method involves two variable fields: the density field representing the structure, and the auxiliary variable field for generating geodesic distance-based curved slices for additive manufacturing (AM). A novel heat diffusion equation and Poisson equation-based approach is proposed to generate the curved layers while ensuring all slicing-related information derivable. Additionally, sequence division thresholds are optimized to determine the timings of alternating between AM and subtractive machining (SM) operations. An excellent feature of the proposed sequence division method is that, it eliminates the initial-guess dependency issue, i.e., the quantity of AM-SM alternations has a reducing trend during optimization and converges to the most compact solution. To realize the synchronous optimization and ensure the successful manufacturing, a coupled and differentiable optimization model is established, including a self-support constraint to eliminate the need for support structure, a direction constraint to avoid the collision between the laser head and printing platform, and a curvature constraint to guarantee the layer thickness uniformity. In addition, a collision-free constraint, coupled with the planned sequences, is introduced to prevent cutting tool collision during SM. To validate the proposed method, a number of 2D and 3D numerical examples are studied. The above figure demonstrates one of the results and the associated manufacturing process simulation.