Numerical Optimization of Planar Nozzle Shapes for Fused Deposition Modeling

Technical Summary: Numerical Optimization of Planar Nozzle Shapes for Fused Deposition Modeling

Problem Statement
In Fused Deposition Modeling (FDM), increasing printing speeds is essential for reducing manufacturing costs but is limited by pressure losses within the nozzle. High pressure drops necessitate greater feeding forces and can adversely affect the mechanical properties of printed parts. While standard nozzle designs are prevalent, the specific influence of nozzle geometry on pressure loss—particularly under viscoelastic flow conditions—requires further investigation. Existing literature often relies on generalized Newtonian fluid (GNF) models; however, viscoelastic models are better suited to capture elastic memory effects, stress relaxation, and the formation of recirculation zones (vortices) that can lead to material degradation and nozzle clogging. This work addresses the need to optimize nozzle geometry to minimize pressure loss while balancing computational cost and manufacturability.

Methodology
The study employs a numerical framework based on the steady, incompressible Navier–Stokes equations coupled with the Giesekus constitutive model to simulate viscoelastic polymer melt flow. The simulations utilize a logarithmic conformation formulation with an algebraic sub-grid scale method to ensure stability at high Weissenberg numbers.

The core of the work is a comparative shape optimization study using two distinct geometric parametrizations:

1. Angle-based Parametrization: A simple approach where the nozzle is defined by a single contraction angle ($\alpha$). This restricts the design space to low-dimensional, easily manufacturable geometries.
2. Spline-based Parametrization: A more flexible approach using a spline curve to define the nozzle wall. To prevent mesh entanglement and reduce computational complexity, the x-coordinates of the control points are coupled to the contraction angle, while only the y-coordinates vary individually. A monotonicity constraint is applied to ensure the contraction decreases smoothly, preserving manufacturability.

The optimization objective is to minimize the pressure drop ($\Delta p$) across the nozzle. The COBYQA algorithm, a derivative-free trust-region method, is used to solve the optimization problem. The study considers three common FDM polymers (PLA, PET-G, and PA6/66) and various extrusion velocities (50–140 mm/s). A mesh sensitivity study confirmed that the selected "fine" mesh provides sufficient accuracy with negligible numerical uncertainty compared to the expected optimization gains.

Key Findings

• Local Minima and Flow Regimes: The angle-based optimization revealed two distinct local minima for the pressure drop. One minimum corresponds to a smooth, vortex-free flow (lower optimal angles, approx. 50°–56°), while the other corresponds to a configuration with pronounced recirculation regions in the upper corner (higher optimal angles, approx. 75°–90°). Although the recirculation cases yielded slightly lower pressure drops, they were deemed undesirable due to the risk of material degradation and clogging caused by prolonged residence times in high-shear zones. Consequently, the study focuses on the vortex-free solutions.
• Material and Velocity Sensitivity: For PLA and PET-G, the optimal contraction angle remains relatively stable (approx. 48°–52°) across different extrusion velocities, with a consistent pressure loss reduction of roughly 4–5% compared to a standard 30° nozzle. In contrast, PA6/66 exhibits higher sensitivity, with the optimal angle increasing from 54.4° to 63.5° as extrusion velocity rises from 50 to 140 mm/s.
• Spline vs. Angle Optimization: The spline-based parametrization achieved only marginal additional reductions in pressure loss compared to the optimal angle-based designs. For PLA, the improvement increased from 4.48% (angle) to 4.80% (spline); for PET-G, from 4.43% to 4.88%; and for PA6/66, from 4.76% to 5.36%.
• Computational Cost: The spline-based approach required significantly more iterations (100–160) to converge compared to the angle-based approach (approx. 15 iterations), reflecting the increased dimensionality of the design space.

Significance and Claims
The paper claims to provide a systematic comparison between simple and advanced geometric parametrizations for FDM nozzle optimization under a consistent viscoelastic flow model. The primary contribution is the demonstration that most of the achievable pressure-loss reduction is captured by the simpler, more manufacture-ready angle optimization.

The authors conclude that while spline-based designs offer greater geometric flexibility, the resulting gains in pressure loss are marginal and may not justify the increased geometric complexity and associated manufacturing costs. Therefore, for the materials and conditions investigated, a simple, vortex-free contraction angle of roughly 50°–56° represents a robust and effective solution. The study emphasizes that the trade-off between performance gains and manufacturability favors the simpler approach in the context of the planar model used.

Limitations and Future Work
The authors acknowledge that the study is limited to a planar, two-dimensional geometry and assumes isothermal conditions with fully melted polymer. Future work is proposed to broaden the physical realism by incorporating axisymmetric geometries, thermal effects, and polymer melting dynamics, as well as validating the optimized shapes against experimental measurements of extrusion force and temperature.