Numerical Optimization Study of Fused Deposition Modeling Nozzle Geometry

Table of Contents

1. Introduction

Fused Deposition Modeling (FDM) is a mainstream additive manufacturing technology, favored for its cost-effectiveness and material diversity. However, achieving high printing speeds without sacrificing printing accuracy remains a significant challenge, largely limited by the pressure loss inside the extrusion nozzle. While process parameter optimization is common, the geometric design of the nozzle itself is often overlooked, with most systems relying on standard conical structures. This study addresses this gap by proposing a numerical framework for optimizing nozzle geometry to minimize pressure loss, thereby enabling higher feasible printing speeds. The research critically compares two fundamental constitutive models for polymer melt flow: a temperature-dependent, shear-thinning viscous model, and an isothermal viscoelastic model.

2. Methodology

2.1. Flow Modeling

The core of the analysis lies in simulating the non-Newtonian flow of polymer melts. Two models were employed:

• Viscous Model: A generalized Newtonian fluid model, where viscosity ($\eta$) is a function of shear rate ($\dot{\gamma}$) and temperature (T), typically following the Carreau model or the power-law model: $\eta(\dot{\gamma}, T) = \eta_0(T) [1 + (\lambda \dot{\gamma})^2]^{(n-1)/2}$. This model captures shear-thinning effects but neglects elastic effects.
• Modeli ya Viscoelastic: Modeli ya isothermal inayozingatia athari za kumbukumbu ya majimaji na mkazo wa elastic, kwa kawaida hutumia mlinganyo wa msingi wa tofauti, kama vile modeli ya Giesekus au Phan-Thien–Tanner. Hii ni muhimu kwa utabiri wa matukio kama vile kuvimba kwa kusukumwa nje.

Mlinganyo wa udhibiti (uhifadhi wa wingi na uhifadhi wa kasi) wa mifano hii unatatuliwa kwenye kikoa cha nozzle kwa kutumia Mbinu ya Vipengele Visivyo na Kikomo (FEM).

2.2. Shape Parameterization

The nozzle shape is defined through parameterization to support optimization:

• Simple Parameterization: The nozzle contour is defined by a straight converging section with a variable half-opening angle ($\alpha$).
• Advanced Parameterization: The contour is described by a B-spline curve, controlled by a set of control points. This allows the generation of complex, non-conical shapes that cannot be represented by simple angles.

2.3. Optimization Framework

A gradient-based optimization loop is established. The objective function is the total pressure drop ($\Delta P$) from the nozzle inlet to the outlet. The design variables are the angle ($\alpha$) or the coordinates of the B-spline control points. This framework iteratively adjusts the geometry, remeshes, re-simulates the flow, and computes the sensitivity of $\Delta P$ to the design variables until a minimum is found.

3. Results and Discussion

3.1. Viscous Model Results

For the viscous model, the optimal half-opening angle ($\alpha_{opt}$) showsStrong dependence on volumetric flow rate (feed rate)。

• High flow rate: Preference for smaller convergence angles, with $\alpha_{opt}$ close to 30°. At high flow rates, a steeper convergence can minimize viscous dissipation within the long, narrow region of high shear.
• Low flow rate: Allows for a larger optimal angle (e.g., 60°-70°). Flow is less dominated by shear, and a gentler taper can reduce entrance effects.

Chart description: Graphs of $\Delta P$ versus $\alpha$ under different flow rates will show distinct minimum points. As the flow rate increases, the minimum point shifts to the left (the angle decreases).

3.2. Viscoelastic Model Results

In contrast, the viscoelastic model predicts the dependence of $\alpha_{opt}$ on the feed rate.Dependency is significantly weaker.Under different flow conditions, the optimal angle remains within a relatively narrow range. This is attributed to the competing effects of viscous shear stress and elastic normal stress, which have different sensitivities to geometry. The elastic stress, which cannot be captured by viscous models, alters the optimal flow path.

3.3. Comparison and Core Insights

1. Model selection is crucial: Constitutive models fundamentally alter optimization outcomes. Designs optimized using simple viscous models may be suboptimal for real viscoelastic melts, especially when elastic die swell affects deposition accuracy.

2. Diminishing returns of complexity: A key finding is that advanced B-spline parameterization yields only marginal improvement in reducing pressure loss compared to simple angle optimization.Marginal improvement. This indicates that for the primary objective of minimizing $\Delta P$, a simple conical nozzle with an appropriate angle is nearly globally optimal. The value of complex shapes may lie in addressing secondary objectives (e.g., controlling die swell, reducing stagnation zones).

3. Flow-rate-dependent design: For viscosity-dominated flows (or certain materials), the results support the adoption ofAdaptive or application-specific nozzle design, rather than a "one-size-fits-all" approach, especially when targeting a wide range of printing speeds.

4. Technical Details

The governing equations for incompressible flow are:

Mass Conservation: $\nabla \cdot \mathbf{v} = 0$

Conservation of momentum: $\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau}$

Where $\mathbf{v}$ is velocity, $p$ is pressure, $\rho$ is density, and $\boldsymbol{\tau}$ is the deviatoric stress tensor.

For the viscous model: $\boldsymbol{\tau} = 2 \eta(\dot{\gamma}, T) \mathbf{D}$, where $\mathbf{D}$ is the rate-of-deformation tensor.

For the viscoelastic model (e.g., the Giesekus model):
$\boldsymbol{\tau} + \lambda \stackrel{\triangledown}{\boldsymbol{\tau}} + \frac{\alpha_G}{\eta} (\boldsymbol{\tau} \cdot \boldsymbol{\tau}) = 2 \eta \mathbf{D}$
where $\lambda$ is the relaxation time, $\alpha_G$ is the mobility parameter, and $\stackrel{\triangledown}{\boldsymbol{\tau}}$ is the upper-convected derivative.

5. Analytical Framework Example

Case Study: Optimization for High-Speed PLA Printing

Objective: Design a nozzle for printing PLA at a layer speed of 150 mm/s.

Steps:

1. Material Characterization: Obtain rheological data of PLA at printing temperature (e.g., 210°C) to fit the parameters of the Carreau-Yasuda (viscous) model and the Giesekus (viscoelastic) model.
2. Baseline simulation: Simulate a standard 30° conical nozzle. Use two models for simulation to establish baseline $\Delta P$ and flow field.
3. Angle sweep (viscous first): Run the viscous optimization loop, varying $\alpha$ from 15° to 75°. Determine $\alpha_{opt}^{visc}$ (approximately 30-35° for high-speed printing).
4. Viscoelastic verification: Simulate the geometry obtained in step 3 using a viscoelastic model. Compare $\Delta P$ and observe the prediction of die swell.
5. Trade-off analysis: If the ΔP of the viscoelastic model is acceptable and swelling is controlled, adopt a simple tapered design. Otherwise, initiate multi-objective optimization using a B-spline framework (minimizing ΔP and swelling).

This structured approach prioritizes simplicity and model-based decision-making.

6. Matumizi ya Baadaye na Mwelekeo

• Multiphysics and Multi-objective Optimization: Future work must integrate heat transfer to simulate non-isothermal flow and combine flow optimization with objectives such as minimizing thermal degradation or improving interlayer bonding strength.
• Machine Learning-Enhanced Design: Techniques utilizing neural networks and other surrogate models, akin to advancements in aerodynamic shape optimization (see Journal of Fluid Mechanics, Volume 948, 2022), can significantly reduce the computational cost of exploring the complex design spaces supported by B-splines.
• Active or Multi-Material Nozzles: Explore nozzle designs featuring internal flow-guiding structures or sections made from materials with differing thermal properties to actively manage shear and temperature distributions.
• Standardization of Benchmark Testing: This field would benefit from standardized benchmark cases for FDM nozzle flow, similar to the 4:1 planar contraction benchmark used for viscoelastic flows, to facilitate the comparison of different models and optimization methods.

7. References

1. Bird, R. B., Armstrong, R. C., & Hassager, O. (1987). Dynamics of Polymeric Liquids, Vol 1: Fluid Mechanics. Wiley.
2. Haleem, A., et al. (2017). Role of feed force in FDM: A review. Rapid Prototyping Journal.
3. Nzebuka, G. C., et al. (2022). CFD analysis of polymer flow in FDM nozzles. Physics of Fluids.
4. Schuller, M., et al. (2024). High-speed FDM: Challenges in feeding mechanics. Additive Manufacturing.
5. Zhu, J., et al. (2022). Deep learning for aerodynamic shape optimization. Journal of Fluid Mechanics, 948, A34. (Reference on machine learning in optimization)
6. Open-source CFD software:OpenFOAM 和 FEATool For multiphysics simulation.

8. Expert Analysis: A Critical Perspective