Composites Part B: Engineering · 2026

Derivable geodesic weaving: enabling tension-compression anisotropic topology optimization of densified continuous fiber paths with controlled-spacing

A dual-field topology optimization framework that converts differentiable geodesic fields into dense, continuous, fabrication-ready fiber paths while accounting for the bi-modulus behavior of printed composites.

Yifan Guo^1,3, Chang Su¹, Jikai Liu^1,*, Shuzhi Xu^2,*, Takayuki Yamada³

¹ School of Mechanical Engineering, Shandong University, China

² Graduate School of Engineering, The University of Osaka, Japan

³ Graduate School of Engineering, The University of Tokyo, Japan

^* Corresponding authors

Paper PDF DOI Citation

Overview of the dual-field geodesic weaving framework

The workflow couples derivable geodesic fields, a homogenized continuous-fiber model, tension-compression-aware analysis, and post-processing for executable fiber trajectories.

Abstract

Conventional homogenized optimization is computationally efficient, but its density and orientation fields do not directly provide printable fiber paths. This work closes that gap with two orthogonal geodesic fields whose gradients define globally consistent path directions. Structural topology, field phases, orientation, and fiber spacing are optimized together. A bi-modulus material model steers reinforcement toward tensile load paths, while a dedicated alignment-and-bridging procedure converts the optimized fields into continuous G-code-ready trajectories. Numerical studies and printed specimens show that the method improves path continuity, fiber utilization, stiffness, and peak load capacity.

Dual geodesic fields

Two interleaved path families avoid orientation ambiguity and local path conflicts.

Controlled spacing

A fiber-density variable controls local spacing and overall reinforcement usage.

Bi-modulus mechanics

Tension-compression asymmetry is embedded directly in analysis and optimization.

Fabrication ready

Field alignment, path extraction, and bridge selection produce continuous toolpaths.

Method

Five coupled design fields are optimized: material density, two selective variables, fiber orientation, and fiber spacing. The selective variables split the domain into two phases, each associated with a derivable geodesic field. Their orthogonal streamline families provide a continuous representation of fiber direction and allow the homogenized constitutive model to remain consistent with the final printable paths.

Construction of two derivable geodesic fields — Density and phase-selection fields are transformed into two complementary geodesic fields.

Continuous fiber path generation and homogenized model — Geodesic gradients define continuous path directions and the local homogenized fiber response.

Bi-modulus analysis coupled with fiber paths — Signed principal stress separates tensile and compressive response along the fiber direction.

Continuous fiber path post-processing pipeline — Field alignment, iso-contour extraction, endpoint matching, bridge screening, and smoothing produce continuous paths for G-code conversion.

Numerical Results

L-bracket, double-clamped beam, and bridge-shaped benchmarks evaluate the influence of dual fields, fiber-density limits, and tension-compression asymmetry. Across the examples, the optimized paths remain continuous and concentrate reinforcement along the mechanically relevant trajectories.

6.036 Lowest L-bracket compliance with the dual-field formulation

0.2194 Dual-field fiber fraction under the same density cap

4.40 Compliance at maximum fiber density d₀ = 0.5

0.3617 Global fiber fraction at d₀ = 0.5

Single-field and dual-field L-bracket comparison — The dual-field design is less sensitive to initialization and achieves a denser, more uniform fiber layout with lower compliance.

Fiber density study and convergence history — Increasing the allowable fiber density first stabilizes the topology and then densifies the principal load paths.

Optimized designs for different bi-modulus ratios — Lower compressive modulus shifts the optimized architecture toward tension-dominant members.

Bridge topology and fiber-path results — Bridge-shaped designs show the same coupled influence of fiber density and bi-modulus behavior.

Fabrication and Experiments

Material tests identify the anisotropic, tension-compression-dependent constitutive parameters of the printed composite. The double-clamped beam is then re-optimized with these measured properties. Optimized boundaries are exported as STL geometry, while geodesic iso-contours are converted to fiber toolpaths and G-code for dual-nozzle printing.

+21.3% Increase in measured structural stiffness

+39.5% Increase in average peak load

0.365 Stiffness with bi-modulus design (kN/mm)

2.550 Average peak load with bi-modulus design (kN)

The compared specimens have similar average mass, enabling a fair mechanical comparison.
The bi-modulus-aware design suppresses premature compression-driven buckling.
Continuous fibers are preferentially aligned with tensile load-transfer paths.
The resulting load-displacement response remains more stable before final failure.

Printing workflow and fabricated specimens — From optimized geometry and geodesic iso-contours to slicing, G-code, and printed specimens.

Mechanical testing and quantitative experimental results — Compression tests confirm higher stiffness and peak load for the design that includes bi-modulus behavior.

Failure sequence without bi-modulus consideration — Without bi-modulus modeling, local compression buckling triggers progressive damage and multiple load drops.

Failure sequence with bi-modulus consideration — The bi-modulus-aware design delays local instability and sustains a larger load before failure.

Citation

@article{guo2026derivable,
  title   = {Derivable geodesic weaving: enabling tension-compression
             anisotropic topology optimization of densified continuous
             fiber paths with controlled-spacing},
  author  = {Guo, Yifan and Su, Chang and Liu, Jikai and Xu, Shuzhi
             and Yamada, Takayuki},
  journal = {Composites Part B: Engineering},
  volume  = {317},
  pages   = {113604},
  year    = {2026},
  doi     = {10.1016/j.compositesb.2026.113604}
}

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 52475290. The authors also acknowledge Professor Bin Zou for providing the continuous-fiber 3D-printing equipment used in the experiments.