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Infill Optimization for Additive Manufacturing -- Approaching Bone-like Porous Structures
Jun Wu, Niels Aage, Ruediger Westermann, Ole Sigmund
TL;DR
The paper addresses how to create lightweight, mechanically strong porous infill beyond regular additive-manufacturing patterns. It extends voxel-wise topology optimization with local-volume regulation and efficient global aggregation, producing detailed bone-like structures whose robustness and manufacturability are evaluated.
Problem
Regular infill patterns require a trade-off between load resistance, material use, and print time, motivating optimized lightweight interior layouts.
Method
The method extends voxel-wise topology optimization with local volume constraints, aggregated through a global p-norm, while stiffness optimization guides load-aligned porous structures.
Results
The optimized structures visually resemble trabecular bone and are reported as stiffness-optimized, lightweight, and robust to material deficiency and force variations.
Takeaways & Limitations
The optimized interiors are presented as candidates for application-specific infill in additive manufacturing.
Takeaways & Limitations
Because local constraints are approximated by a global p-norm, extremely high-stress locations can exceed prescribed volume limits and still develop wall-like structures.
Abstract
from arXiv · showhide
Porous structures such as trabecular bone are widely seen in nature. These structures exhibit superior mechanical properties whilst being lightweight. In this paper, we present a method to generate bone-like porous structures as lightweight infill for additive manufacturing. Our method builds upon and extends voxel-wise topology optimization. In particular, for the purpose of generating sparse yet stable structures distributed in the interior of a given shape, we propose upper bounds on the localized material volume in the proximity of each voxel in the design domain. We then aggregate the local per-voxel constraints by their p-norm into an equivalent global constraint, in order to facilitate an efficient optimization process. Implemented on a high-resolution topology optimization framework, our results demonstrate mechanically optimized, detailed porous structures which mimic those found in nature. We further show variants of the optimized structures subject to different design specifications, and analyze the optimality and robustness of the obtained structures.
1 INTRODUCTION
The paper develops bone-inspired porous infill through voxel-wise topology optimization, regulating local material distribution while optimizing stiffness under mechanical loads. It contributes a structural formulation and a parameter study of the resulting structures.
- Bone-like infill is motivated by trabecular bone, whose microstructures align with principal stress directions during natural adaptation.
- The method extends voxel-wise topology optimization by constraining local volume fractions to distribute sparse, stable material inside a design domain.The optimization maximizes stiffness for a prescribed material amount and given external loads.
- Local material constraints regulate interior structure while the stiffness objective automatically aligns porous structures with mechanical loads.
- The paper presents a novel formulation for generating porous structures grounded in structural optimization.
- A detailed parameter study provides mechanical insights into the optimal structures.
2 RELATED WORK
Related work covers computational fabrication, structural optimization, topology optimization, and bone-inspired generation. The paper distinguishes its unified optimization of local details from approaches based on reconstruction, biological simulation, or repeated microstructures.
- Research on additive manufacturing includes geometric and physical modeling, including toolpath generation and structural optimization.
- Prior systems improve structural soundness through finite-element detection followed by hollowing, thickening, or strut insertion operations.
- Topology optimization parametrizes the design domain volumetrically, allowing structures to emerge and adapt during iterative optimization.The paper uses a density-based SIMP formulation and approximate projection filtering for local volume constraints.
- Bone-inspired alternatives reconstruct microstructures from images or simulate tissue adaptation using biological models.
- Two-scale approaches use predefined microstructures but can produce discontinuities between cells and regular repetitions of limited cell types.
3 INFILL OPTIMIZATION
The paper formulates porous infill generation as topology optimization with local volume limits, then relaxes and aggregates the resulting constraints for efficient numerical optimization. The formulation distributes material throughout the domain, produces stress-aligned substructures, and exposes porosity and spacing controls through α and R.
- 3.1 Discrete Formulation: The discrete formulation assigns binary solid-or-empty values to voxels and minimizes compliance subject to elasticity, binary-design, and local-volume constraints.Each local neighbourhood limits its solid-voxel percentage to α, while the optimizer determines which voxels remain solid.
- 3.1 Discrete Formulation: The local-volume constraint limits material accumulation without prescribing specific solid voxels, encouraging a more even distribution resembling trabecular bone.For α = 0.6, each neighbourhood may contain at most 60% solid voxels.
- 3.2 Relaxations: Because the discrete problem can contain millions of variables and constraints, continuous design variables, filtering, projection, and p-norm aggregation enable gradient-based optimization.Filtering removes checkerboard patterns, projection sharpens the material field toward binary values, and p-norm aggregation approximates the maximum local constraint.
- 3.2.2 Continuous Design Variable, Filtering, and Projection: Increasing β sharpens the projection toward a binary classification, and parameter continuation doubles β after iterations to improve convergence behaviour.The procedure starts at β = 1 rather than applying a large value immediately.
- 3.4 Example: The local volume limit α controls local porosity, whereas the influence radius R controls spacing between substructures and larger radii yield stiffer structures.When R exceeds the design-domain size, local constraints become equivalent to a total-volume constraint.
4 EXTENSIONS
The extensions control total material, directional distribution, and sub-structure morphology, producing infills that trade material usage and stiffness while adapting connectivity and feature size.
- Local Volume and Influence Radius: Increasing the local volume limit from 0.4 to 0.6 reduces porosity, whereas increasing the influence radius R relaxes locality and increases stiffness.The same boundary conditions are used across the examples.
- Total Volume Control: A prescribed total volume ratio αtotal directly controls material usage; reducing it removes material from low-stress regions and approaches classical topology-optimization layouts.With local α = 0.6, progressively lower global limits make the distribution shrink from low-stress regions.
- Anisotropic Filter: For the 2D comparison, isotropic filtering gives compliance 22.6 with 59.8% volume, while anisotropic filtering gives compliance 34.6 with 51.7% volume.The anisotropic result uses less volume but has higher compliance.
- Anisotropic Filter: For the 3D comparison, isotropic filtering gives compliance 79.4 with 27.9% volume, while anisotropic filtering gives compliance 125.6 with 23.8% volume.The anisotropic result uses less volume and produces more bridge-like connections.
- Anisotropic Filter: Anisotropic filters replace unidirectional bars with connected sub-structures distributed across multiple orientations.In 2D, horizontal bars break into short horizontal and vertical sub-structures; in 3D, anisotropic filtering generates more bridge-like connections between xz-parallel planes.
- Truss- vs. Wall-like Structures: Increasing minimum feature size replaces some walls with sparse trusses and can reduce stiffness by up to a factor of 20%.The local volume constraint permits both thin walls and trusses, while feature size shifts their balance.
- Truss- vs. Wall-like Structures: The feature-size analysis assumes strictly enforced local constraints, but the global p-norm approximation can exceed the prescribed local volume at extremely high-stress locations.Wall-like structures may therefore still emerge in those locations.
5 RESULTS AND ANALYSIS
The proposed local-volume formulation generates bone-like porous infills that resemble trabecular structures and remain comparatively robust under force variations and material damage. Experiments show favorable stiffness, stress, convergence, and large-scale fabrication behavior.
- Bone-like structure: A 2D femur experiment shows that local volume constraints produce porous infill aligned with principal stress directions.The result is contrasted with classical topology optimization using a total volume constraint.
- 3D realization: A 5.56-million-element femur model was optimized and fabricated as a 12.32 cm × 5.85 cm × 16.00 cm physical replica.The 3D model used resolution 280 × 185 × 364 and selective laser sintering in strong flexible plastic.
- Bone-like structure: The optimized infill and a human femur sample both contain sparse trusses and a few walls.The comparison uses a cubic sample extracted from the optimized infill and a CT-scan sample from a human femur.
- Robustness: After localized material removal, compliance increased from 132.6 to 187.3 for the local-volume infill, versus 101.4 to 1763.8 for the total-volume infill.These values indicate substantially smaller compliance growth for the local-volume structure under the tested damage.
- Robustness: Beyond a 20° force-direction change, the local-volume structure has lower compliance than the total-volume structure; both reach worst-case compliance at 90°.At angles up to about 20°, the total-volume structure remains somewhat stiffer.
- Convergence: During optimization, compliance gradually decreases while the constraint stays below 0.0, and the density field becomes more discrete.The convergence examples correspond to iterations 79, 159, and 279; sharpness approaches the binary-field regime as optimization progresses.
- Comparisons: The porous infill is 1.5 times stiffer than the honeycomb structure and has maximum von Mises stress equal to 72.5% of the honeycomb maximum.The comparison uses identical material properties and boundary conditions, with compliance normalized against a fully solid shape.
- Comparisons: The porous infill is 1.14 times stiffer than the optimized rhombic structure, with comparable maximum stresses.Both optimized infill versions outperform the uniform grid in the reported comparison.
6 CONCLUSION
The paper presents stiffness-optimized porous interiors that visually resemble trabecular bone and are lightweight and robust to material deficiency and force variations. The authors identify these structures as candidates for application-specific additive-manufacturing infill.
- Conclusion: The method produces stiffness-optimized porous structures that visually resemble lightweight trabecular bone.The conclusion also characterizes the structures as robust to material deficiency and force variations.
- Conclusion: FDM replicas were produced for some models generated by the formulation.The conclusion’s application scope is additive-manufacturing infill.
APPENDIX
The appendix describes gradient computation for the numerical optimization scheme, including chain-rule derivatives and adjoint analysis.
- Gradient computation: The gradient of the objective c and constraint g with respect to the design variable φ is computed using the chain rule.This gradient is required by the numerical optimization process.
- Gradient computation: The derivative ∂c/∂ρ_i is calculated using adjoint analysis.
- Gradient computation: The remaining derivative components are derived separately after the adjoint-analysis term.