3D printable biomimetic rod with superior buckling resistance designed by machine learning

The study addresses the challenge of improving buckling resistance in slender columns or rods subjected to axial compression, a common failure mode that limits load-carrying capacity. Inspired by biological structures—such as plant stems and roots, animal quills, seashells, and honeycombs—whose external shapes and internal porous architectures enhance buckling resistance, the authors seek to design manmade rods with superior performance. The context includes extensive prior biomimicry in engineering and design, with examples ranging from lotus-inspired self-cleaning surfaces to pummelo-inspired damping structures and bamboo-inspired composites. The purpose of the study is to combine biomimetic external cross-sectional shapes (e.g., circular, square, triangular, star/flower-like) with internal porous architectures (e.g., continuous hollow cylinders, honeycomb, scattered pores) to develop rods with higher buckling loads per mass than classical solid or hollow cylinders. Given the complexity of these geometries, Euler buckling equations are not directly applicable; therefore, finite element analysis (FEA) is used to generate a large training dataset, validated experimentally, and machine learning is employed to identify and optimize designs that exceed the performance of natural counterparts.

The paper situates its contribution within a broad body of biomimicry and machine learning in materials/structural design. Biomimetic inspirations include lotus leaf superhydrophobicity for self-cleaning surfaces; plant fluid transport inspiring microfluidics and moisture-management textiles; velcro inspired by plant burrs; pummelo rind for low-weight, high-damping structures; bamboo fiber helicity for composite reinforcement; tendril-inspired polymeric artificial muscles; and microscale studies via SEM enabling understanding of hydrophobic plant surfaces. In structural optimization, prior work shows non-uniform/drum-shaped columns can increase buckling load; hollow/porous members often outperform solids for buckling given material distribution. Machine learning has been used for property prediction in polymers (KRR, GPR), polymer thermal conductivity discovery (neural networks), cement/concrete property prediction (SVM), and as surrogates for FEA (deep learning, regression trees, ensemble methods) in biomechanics and composites. These studies motivate using ML to explore vast design spaces of biomimetic rods, which is infeasible manually.

Selection and design of biomimetic rods: The authors idealized biological cross-sections by combining external stem shapes (rice, bamboo, cactus, square/mint/cup plant, bulrush, papyrus, she-oak) with internal porous architectures (roots, hedgehog quills, seashells, honeycomb). They created 21 basic biomimetic rods and expanded them to a dataset of 1,500 unique rods by varying pore shapes, sizes, distributions, and locations. Seven groups were organized by external shape (Table S1). All rods had the same height (10 cm) and the same overall volume as a reference solid circular rod of 1 cm diameter (volume 7.85 cm³). Control rods included solid and hollow cylinders (outer diameters 1.0, 1.5, and 2.0 cm; hollow with 1.0 cm OD and 0.5 cm ID). FEA setup: ANSYS Workbench (Release 18.1) was used for static and Euler buckling analyses. Material was 3D-printable PLA (Hatchbox). Compression properties were characterized per ASTM D695-15 using a Q-TEST 150 machine (2 mm/min); stress–strain data at room temperature were imported into ANSYS as the constitutive law (Table S2, Fig. S3). Boundary conditions: one end fixed and the other pin-supported. A constant axial compression load of 1000 N was applied for eigenvalue buckling analysis; ANSYS returned a buckling factor multiplied by 1000 N to give the actual buckling load. Meshing used hexahedral elements with an element size of 0.1 mm after convergence study. For each rod, buckling load, stress, displacement, mass, and volume were recorded. Experimental validation: Representative rods (solid, bamboo-like, cactus-like, square-like) were modeled in SolidWorks and 3D printed using an extrusion-based Creality CR-10 S printer with PLA filament (extruder 210 °C; filament/nozzle thickness reported as 1.75 mm). Coarse layer resolution was used for all prints; STL files were sliced via Ultimaker Cura (v4.7.1) to generate g-code. Minimal post-processing was required. Buckling tests were performed on a Q-TEST 150 machine at 2 mm/min; mass was measured prior to testing; the onset buckling load was recorded. Machine learning pipeline: Feature identification (fingerprinting) converted each geometry into a machine-readable vector capturing external shape ID, internal features (e.g., large inner circle, small cylindrical pores), and their coordinates relative to an origin (outer circle centered at origin). For example, a bamboo-inspired rod might be fingerprinted as: outer circle ID and (0,0), inner circle ID and (0,0), followed by multiple small circle IDs with (x,y) bottom-center coordinates; some designs contained up to ~400 pores. Supervised learning used inputs: mass, volume, and all fingerprint features; output: buckling strength. Data were split 90/10 train/test with fivefold cross-validation in MATLAB (v9.3). Multiple algorithms were evaluated (SVM, GPR, neural networks, decision trees, KNN, gradient boosting). Based on RMSE, an Ensemble Bagged Trees model (leaf size = 8) was selected as best suited for the high-dimensional fingerprint vectors. The ensemble divides data into subsets, trains individual trees, and averages predictions. Reported error was <10% for most test data points; the authors note future work should compare RMSE between training and testing to assess potential overfitting. Forward design and optimization: A MATLAB code generated over a million new fingerprint combinations inspired by the biomimetic patterns. Initial manual heuristics on porosity bounds and pattern counts reduced the set. Further filtering used Excel (IF, >/<, INDEX) and MATLAB logical indexing to retain designs with predicted buckling loads exceeding those in the semi-optimal 1,500-rod dataset. This yielded 160 optimized fingerprints. These were converted to CAD in ANSYS and analyzed under the same uniaxial compression and buckling conditions for verification (Fig. 6).

- Biomimetic rods substantially outperform classical solid and hollow cylinders at the same mass: normalized buckling capacities are more than twice those of solid and hollow rods (Fig. 2A). - Compressive stress: For rods of equal mass, biomimetic rods exhibit peak compressive stresses similar to solid rods; even the lightest designs have stresses far below the PLA compressive strength (Table S2), indicating compression failure did not precede buckling (Fig. 2B). - Axial displacement under load is similar for biomimetic vs. solid/hollow controls at equal mass (Fig. 2C). - Geometric influences: External shapes matter—cactus- and square-shaped stems slightly outperform bamboo-like circular stems (Fig. 2A). Internal architecture matters—continuous hollow-cylinder-like porosity (e.g., bamboo xylem) outperforms scattered pores (e.g., hedgehog quill), and stem-like hollow cylinders outperform root/quill/seashell-inspired scattered/porous cross-sections in buckling capacity and compression strength (Fig. S1, Table S4). - Experimental validation: 3D-printed rods (solid, bamboo-like, cactus-like, square-like) showed normalized buckling loads consistent with ANSYS simulations; small discrepancies are attributed to low 3D printing resolution and adjusted boundary conditions (Fig. 3, Table S3). - Machine learning performance: Among tested algorithms (ensemble trees, SVM, GPR, neural nets), the bagged ensemble tree model provided the best agreement between predictions and FEA observations for the high-dimensional fingerprint data (Fig. 4), with most test predictions within 10% error. - Optimization outcome: 160 new optimized rods were generated that outperform all 1,500 rods in the training set (Table S5). ANSYS verification showed these optimized rods have buckling strengths nearly double those of the initial biomimetic rods (Fig. 6). - Overall, ML-designed rods achieved up to approximately 150% improvement in buckling resistance over the training database and several-fold improvement over classical solid/hollow cylinders at comparable mass.

The study demonstrates that combining biomimetic external cross-sections with tailored internal porosity distributions can markedly enhance buckling resistance while maintaining comparable axial stiffness and stress levels relative to mass-matched solid/hollow controls. The FEA–experiment agreement validates the modeling approach and confirms that the designs fail by buckling rather than compressive material failure under the examined loads. Machine learning serves as an effective surrogate to traverse a vast design space that is impractical to explore manually, identifying non-intuitive configurations that surpass nature-inspired baselines. The finding that continuous, strategically placed hollow regions (akin to plant xylem) outperform scattered porosity supports the structural mechanics principle of relocating material away from the neutral axis to maximize bending stiffness. The framework enables rapid forward prediction of buckling strength for new fingerprints, facilitating iterative optimization and enabling translation to lattice structures and lightweight truss elements. The improved buckling capacity at a given mass suggests potential applications in bridges, buildings, and truss members, where weight savings and stability are critical. Future improvements in printing resolution and comprehensive ML validation (e.g., training/testing RMSE comparison) can further bolster confidence and extend applicability across materials and scales.

Biomimetic rods, inspired by plant stems/roots and other natural porous structures, were designed, modeled, validated, and optimized using a combined FEA and machine learning approach. From 1,500 initial biomimetic designs, a bagged ensemble tree model captured the relationship between geometric fingerprints and buckling load. Using ML-guided forward design and filtering, 160 new rods were discovered that outperform all rods in the training dataset, with nearly double the buckling strength relative to initial biomimetic rods and several times higher than classical solid or hollow cylinders at comparable mass. This work establishes a pathway to create lighter, more stable columns/rods with superior buckling resistance and suggests integration into advanced lattice and structural systems. Future work may address broader material systems, refine printing/measurement fidelity, and deepen ML validation to guard against overfitting while expanding the design space.

- Experimental-printing limitations: Slight discrepancies between simulations and experiments are attributed to low 3D printing resolution and the coarse layer thickness used for all prints. - Boundary condition mismatch: Simulation boundary conditions were adjusted to match experimental setups, which may differ from idealized conditions. - Machine learning validation: While most test predictions had <10% error, the authors note that comparing RMSE between training and testing data would better assess potential overfitting of ensemble trees; this is deferred to future work. - Material and geometry scope: Validation was conducted with PLA and specific rod dimensions (10 cm height) and constant overall volume, which may limit generalizability across materials/scales without further study. - Design space coverage: Despite generating over a million candidate fingerprints and filtering to 160 optimized designs, the full combinatorial design space remains only partially explored; early manual truncation used heuristic porosity bounds.