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arxiv: 2511.21449 · v2 · pith:K5MDZNR2new · submitted 2025-11-26 · 💻 cs.CE

Numerical Optimization of Planar Nozzle Shapes for Fused Deposition Modeling

Steffen Tillmann , Felipe A. Gonz\'alez , Stefanie Elgeti This is my paper

Pith reviewed 2026-05-17 04:48 UTC · model grok-4.3

classification 💻 cs.CE
keywords Fused deposition modelingNozzle shape optimizationPressure loss minimizationGiesekus viscoelastic modelAngle-based parametrizationSpline-based parametrizationRecirculation in nozzles
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The pith

Angle-based optimization of FDM nozzle shapes captures most pressure-loss reductions, while spline parametrization adds only marginal gains at the expense of manufacturability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates how nozzle geometry influences pressure loss in fused deposition modeling to support higher printing speeds. It sets up a numerical optimization framework that compares a basic approach of varying the nozzle convergence angle against a more flexible spline-based description of the inner contour. The flow of the polymer melt is modeled with the Giesekus constitutive equation to capture viscoelastic effects. Results show two local minima for the pressure-loss objective under angle optimization: one with smooth flow and one with recirculation zones that lower the drop further but are viewed as problematic. Across materials and speeds, the spline method delivers only small extra reductions while making the nozzle harder to produce.

Core claim

For planar nozzles minimizing pressure loss under Giesekus viscoelastic flow, angle-based parametrization already secures nearly all achievable reduction; spline-based parametrization yields only marginal further improvement while lowering manufacturability. A secondary minimum featuring internal recirculation produces lower pressure drop but is set aside because of longer residence times and higher risk of degradation or clogging.

What carries the argument

Angle-versus-spline geometric parametrization inside a numerical optimization loop that evaluates pressure loss for Giesekus-modeled polymer melt flow.

If this is right

  • Angle optimization already recovers the bulk of the pressure-loss benefit for high-speed FDM.
  • Spline parametrization supplies only small extra reductions across different materials and velocities.
  • Nozzles with recirculation achieve lower pressure drop yet are rejected on manufacturability and reliability grounds.
  • The simpler angle-based designs remain preferable when production constraints are considered.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical nozzle design for FDM may safely stop at angle optimization without needing complex contours.
  • Adding a manufacturability penalty directly to the objective could steer future optimizations toward even more usable shapes.
  • The same comparison of parametrizations could be tested on three-dimensional axisymmetric nozzles or different viscoelastic models.

Load-bearing premise

Recirculation regions inside the nozzle are undesirable because they raise residence times and raise the chance of material degradation or clogging, so pressure-loss minimization can be treated as the main objective.

What would settle it

High-speed printing trials that measure actual clogging frequency or material degradation in nozzles designed at the recirculation minimum versus smooth-flow minima at matched pressure drop.

read the original abstract

Purpose: In fused deposition modeling (FDM), the nozzle plays a critical role in enabling high printing speeds while maintaining precision. Despite its importance, most applications still rely on standard nozzle designs. This work investigates the influence of nozzle geometry on pressure loss inside the nozzle, a key factor in high-speed printing performance. Design/methodology/approach: We focus on optimizing the nozzle shape to minimize the pressure loss and establish a framework that allows both simple angle-based optimization and more advanced spline-based parametrization. To model the polymer melt flow, we use a Giesekus model to account for viscoelastic effects. Findings: For angle-based optimization, the pressure-loss objective exhibits two local minima: one associated with smooth flow and another with pronounced recirculation regions inside the nozzle. While the latter yields a lower pressure drop, such flow patterns are generally undesirable due to increased residence times and the associated risk of material degradation and nozzle clogging. The splinebased parametrization results in only marginal additional reductions in pressure loss compared to angle optimization, while decreasing the manufacturability of the nozzle considerably. Originality/value: This paper presents a comparative study of FDM nozzle shape optimization using a Giesekus model. We introduce a flexible optimization framework that accommodates both simple and advanced geometric parametrizations. The main contribution is the systematic comparison between angle- and spline-based parametrizations across materials and extrusion velocities, showing that most of the achievable pressure-loss reduction is already captured by the simpler and more manufacture-ready angle optimization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This manuscript investigates the optimization of planar nozzle shapes for fused deposition modeling (FDM) to minimize pressure loss in polymer melt flow. Using the Giesekus viscoelastic model, it compares a simple angle-based parametrization (revealing two local minima, one with recirculation) against a more flexible spline-based parametrization. The central finding is that spline optimization yields only marginal additional pressure-loss reductions while considerably decreasing manufacturability; the work introduces a flexible framework for both parametrizations and compares results across materials and extrusion velocities.

Significance. If the numerical results hold after verification, the paper provides a practical optimization framework for FDM nozzle design, showing that angle-based methods capture most pressure-loss benefits with superior manufacturability. This could guide high-speed printing nozzle development and reduce reliance on complex geometries. The systematic comparison adds value, though the absence of self-referential predictions limits broader impact.

major comments (2)
  1. [§4 (Numerical Results)] §4 (Numerical Results) and abstract findings: The central claim that spline-based parametrization yields only 'marginal additional reductions' in pressure loss compared to angle optimization lacks reported mesh-convergence studies or grid-independence checks for the optimal shapes. Without these, the small difference cannot be confirmed as physical rather than numerical artifact, directly affecting the comparison's robustness.
  2. [Abstract findings and §5 (Discussion)] Abstract findings and §5 (Discussion): The assertion that spline parametrization decreases manufacturability 'considerably' is not supported by any explicit quantitative metric (e.g., maximum curvature, minimum wall thickness, or production tolerance) or tabulated comparison. This makes the qualifier un-auditable and weakens the recommendation favoring angle optimization.
minor comments (2)
  1. [Methods] The Giesekus model parameters (e.g., relaxation time, mobility factor) and their values for different materials should be tabulated in the methods section for reproducibility.
  2. [Figures] Figure captions for optimal nozzle shapes could include the corresponding pressure-loss values and recirculation indicators to aid direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review. We agree that the robustness of the comparison between parametrizations requires explicit verification of the numerical results, and that the manufacturability claim would benefit from quantitative support. We will revise the manuscript accordingly to address both points.

read point-by-point responses

  1. Referee: [§4 (Numerical Results)] §4 (Numerical Results) and abstract findings: The central claim that spline-based parametrization yields only 'marginal additional reductions' in pressure loss compared to angle optimization lacks reported mesh-convergence studies or grid-independence checks for the optimal shapes. Without these, the small difference cannot be confirmed as physical rather than numerical artifact, directly affecting the comparison's robustness.

    Authors: We acknowledge the validity of this concern. Mesh-convergence studies were performed during the optimization process for both parametrizations to ensure reliable pressure-loss values, but these checks were not explicitly documented for the final optimal shapes. In the revised manuscript we will add a new subsection (or appendix) presenting grid-independence results for the reported optima, including the number of elements, refinement levels, and the observed variation in pressure loss (typically <1% between successive meshes). This will confirm that the marginal differences are physical rather than numerical artifacts. revision: yes

  2. Referee: [Abstract findings and §5 (Discussion)] Abstract findings and §5 (Discussion): The assertion that spline parametrization decreases manufacturability 'considerably' is not supported by any explicit quantitative metric (e.g., maximum curvature, minimum wall thickness, or production tolerance) or tabulated comparison. This makes the qualifier un-auditable and weakens the recommendation favoring angle optimization.

    Authors: We agree that the current qualitative statement is insufficiently supported. In the revision we will introduce two explicit manufacturability metrics: (1) maximum wall curvature (in 1/mm) and (2) minimum local wall thickness relative to the nozzle diameter. These will be computed for all reported optima and presented in a new table in §5, allowing direct, auditable comparison between angle-based and spline-based designs. The discussion will then reference these numbers when qualifying the manufacturability trade-off. revision: yes

Circularity Check

0 steps flagged

No circularity: forward numerical optimization against external Giesekus model

full rationale

The paper performs direct numerical shape optimization of nozzle geometry to minimize pressure loss computed from the Giesekus viscoelastic flow model. Angle-based and spline-based parametrizations are compared via repeated forward solves; the reported marginal additional reduction from splines and the qualitative manufacturability note are outputs of those solves rather than quantities defined by or fitted to the same objective inside the paper. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the derivation chain. The central findings remain independent of any internal redefinition or renaming of results.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Giesekus constitutive model for viscoelastic flow, the assumption that pressure loss is the primary performance metric, and the numerical discretization of the planar nozzle domain. No new physical entities are introduced.

free parameters (2)
  • nozzle convergence angle
    Varied as the primary design variable in the angle-based optimization; its optimal values are found by minimizing the pressure-loss objective.
  • spline control-point positions
    Free geometric parameters in the advanced parametrization; their number and placement determine the additional pressure-loss reduction.
axioms (2)
  • domain assumption Giesekus model accurately captures the viscoelastic behavior of the polymer melt inside the nozzle
    Invoked to model the flow; the abstract states the model is used to account for viscoelastic effects.
  • domain assumption Recirculation regions increase residence time and therefore raise degradation and clogging risk
    Used to dismiss the lower-pressure-drop local minimum; appears in the findings paragraph.

pith-pipeline@v0.9.0 · 5576 in / 1586 out tokens · 37711 ms · 2026-05-17T04:48:39.186985+00:00 · methodology

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Reference graph

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