The phrase "topology-optimized" carries a certain authority in structural engineering. A topology-optimized bracket, a topology-optimized heat exchanger, a topology-optimized airframe rib: these are objects that have been computationally wrung to their mathematical ideal, achieving maximum structural performance per unit of material. They are, however, frequently unfabricable—or rather, they are fabricable only with significant geometric compromise. The optimization assumes unlimited geometric freedom; the manufacturing process does not.
The standard response to this is to constrain the optimizer: add overhang penalties, enforce minimum feature sizes, bias the design toward self-supporting forms. The result is a part that can be printed without supports—but a part that is no longer the topologically optimal design. A recent paper from the University of Manchester, titled "Can Any Model Be Fabricated? Inverse Operation Based Planning for Hybrid Additive–Subtractive Manufacturing," proposes a different resolution: rather than constraining the design to fit the manufacturing process, redesign the manufacturing process to accommodate the design.
Planning in Reverse
The core insight of the work, led by Yongxue Chen and Charlie C.L. Wang, is that the combinatorial difficulty of hybrid manufacturing process planning is much more tractable when approached backward. Instead of asking "what do I print next?"—a forward search that frequently walks into dead ends when an overhanging section requires support structure that the current geometry no longer permits inserting—the algorithm asks: "how do I disassemble this completed object?"
In this inverse logic, the finished target model is progressively *nullified*: additive operations are represented as erosion (removing material from the top), and subtractive operations are represented as accretion (adding material back where a tool can reach). Every step in the nullification sequence is checked for kinematic feasibility. The reversed sequence then becomes a valid, collision-free manufacturing plan.
"By planning backward from the completed part, the algorithm guarantees that every step of the manufacturing sequence is feasible—eliminating the dead ends that make forward planning for complex overhangs so intractable."
Stability Without Global Analysis
A persistent challenge in hybrid manufacturing planning is ensuring part stability through intermediate stages. A structure that is strong in its final form may become disconnected or mechanically precarious partway through assembly. The Manchester team addresses this with a localized stability check: rather than computing global structural integrity at every step (an operation that scales prohibitively with voxel count), the algorithm verifies local connectivity within a $\Delta$-neighborhood of the current operation. This approximation is physically justified—a component is stable as long as its material is locally connected in the vicinity of the active operation—and computationally efficient, enabling the processing of models approaching one million voxels in minutes.
Across test cases spanning triply periodic minimal surfaces, topology-optimized brackets, and complex internal lattice structures, the algorithm consistently finds valid manufacturing sequences. The physical validation was conducted on a custom five-axis hybrid platform, alternating between FDM extrusion and CNC milling. The fabricated parts include topology-optimized designs produced without any geometric constraints on the optimizer—designs that conventional additive-only slicers cannot handle.
The 30.5 Percent Argument
The performance comparison is striking. A topology-optimized bracket produced without manufacturing constraints, using the hybrid planner, demonstrates a 30.5 percent improvement in stiffness compared to the same bracket designed with additive-only constraints imposed. The difference is not marginal. It is the cost of forcing an optimizer to think about its tools.
What the Manchester team has produced is, ultimately, a theoretical guarantee: any voxelized model is, in principle, manufacturable through the hybrid additive-subtractive sequence this algorithm generates. Whether every such model is *economically* manufacturable—whether the number of required process transitions is acceptable, whether the surface quality of the subtractive passes meets specification—remains a function of the specific application. But the guarantee that fabricability is not a matter of luck or designer experience, but a computationally solvable property of any geometry, represents a meaningful shift in what we can ask of the design-to-manufacturing pipeline.