Regularization in Space-Time Topology Optimization for Multi-Axis Additive Manufacturing

TL;DR

This work addresses manufacturability in space-time topology optimization for multi-axis additive manufacturing by enforcing a monotonic fabrication sequence through a heat-equation–based regularization. A virtual temperature field is produced by solving a steady-state heat equation with a spatially varying diffusivity $\kappa=\rho\mu$, and the time field is obtained as $\bm{t}=\bm{1}-\bm{\tau}$, ensuring monotonic progression across layers. The method is integrated into a full optimization framework with density and diffusivity design variables, and sensitivity analysis via adjoints is developed to enable gradient-based optimization. Through 2D and 3D experiments under gravity and thermomechanical loads, the approach consistently avoids local minima, reduces process-induced distortion, and supports concurrent optimization of structure and fabrication sequence, demonstrating its practical potential for robust multi-axis additive manufacturing.

Abstract

In additive manufacturing, the fabrication sequence has a large influence on the quality of manufactured components. While planning of the fabrication sequence is typically performed after the component has been designed, recent developments have demonstrated the possibility and benefits of simultaneous optimization of both the structural layout and the corresponding fabrication sequence. The simultaneous optimization approach, called space-time topology optimization, introduces a pseudo-time field to encode the manufacturing process order, alongside a pseudo-density field representing the structural layout. To comply with manufacturing principles, the pseudo-time field needs to be monotonic, i.e., free of local minima. However, explicitly formulated constraints are not always effective, particularly for complex structural layouts. In this paper, we introduce a novel method to regularize the pseudo-time field in space-time topology optimization. We conceptualize the monotonic additive manufacturing process as a virtual heat conduction process starting from the surface upon which a component is constructed layer by layer. The virtual temperature field, which shall not be confused with the actual temperature field during manufacturing, serves as an analogy for encoding the fabrication sequence. In this new formulation, we use local virtual heat conductivity coefficients as optimization variables to steer the temperature field and, consequently, the fabrication sequence. The virtual temperature field is inherently free of local minima due to the physics it resembles. We numerically validate the effectiveness of this regularization in space-time topology optimization under process-dependent loads, including gravity and thermomechanical loads.

Figures (14)

• Figure 1: (a) Illustration of a component and a pseudo-time field, with a smaller value indicating that the corresponding point is materialized earlier in fabrication. (b) The isolines of the pseudo-time field segment the part into a set of consecutive layers. (c) A series of intermediate structural layouts during the fabrication process. Note that, in stages 1 to 3, a patch of material, corresponding to a local minimum in the pseudo-time field, is in isolation from the rest of the component. This renders the fabrication sequence infeasible.
• Figure 2: Schematic of curved layers for fabricating an $L$-shaped component. Two successive intermediate structures during fabrication are illustrated on the left and right.
• Figure 3: Comparison of the Euclidean and the geodesic distance field as the initialization of the time field for fabrication sequence optimization. (a) A 2D component. The Euclidean distance and the geodesic distance from location $p$ to the build plate at the bottom are shown by the green line and the black polyline, respectively. (b) The Euclidean distance field. (c) The geodesic distance field. (d) The optimized time field using (b) as initialization. (e) The optimized time field using (c) as initialization. The sequences corresponding to the time fields in (d) and (e) are shown in (f) and (g), respectively.
• Figure 4: The workflow of deriving a series of intermediate structures $\bm{\rho}^{\{j\}}$ from the thermal diffusivity field $\bm{\mu}$ and density field $\bm{\rho}$.
• Figure 5: Heat conduction on a given component (a). The start points are the nodes on the bottom boundary. The generated virtual temperature field by solving Equation (\ref{['eq:heatFEM']}) is shown in (c). The light yellow indicates the largest temperature, and the dark blue indicates the smallest virtual temperature. The corresponding time field is shown in (c). The sequence generated with the time field shown in (d) which is composed of 10 layers.