Physics-Informed Dynamical Modeling of Extrusion-Based 3D Printing Processes

Amrita Basak; Mandana Mohammadi Looey; Marissa Loraine Scalise; Satadru Dey

arxiv: 2512.11048 · v2 · submitted 2025-12-11 · ⚛️ physics.flu-dyn · cs.SY· eess.SY

Physics-Informed Dynamical Modeling of Extrusion-Based 3D Printing Processes

Mandana Mohammadi Looey , Marissa Loraine Scalise , Amrita Basak , Satadru Dey This is my paper

Pith reviewed 2026-05-16 22:40 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.SYeess.SY

keywords 3D printingreduced-order modelingextrusionNavier-Stokesdynamical systemsreal-time controlCFD validation

0 comments

The pith

A reduced-order model from spatially averaged Navier-Stokes equations captures transient extrusion flows in 3D printing.

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

The paper develops a simplified dynamical model for flow inside the nozzle, across the gap, and in the deposited layer during extrusion-based 3D printing. High-fidelity CFD simulations provide detailed data but run too slowly for online control, so the authors average the governing equations over space and introduce parameters that depend on process inputs. They fit the resulting low-order model to CFD runs across varied printing conditions using nonlinear least squares and then test it on held-out cases. The model reproduces the dominant transients in all three regions while remaining simple enough for real-time use.

Core claim

The reduced-order dynamical flow model, derived from the Navier-Stokes equations through spatial averaging and input-dependent parameterization, is identified from CFD data and validated to match the transient behavior within the nozzle, nozzle-substrate gap, and deposited layer across multiple combinations of printing conditions.

What carries the argument

Spatially averaged, input-parameterized reduced-order dynamical model obtained from the Navier-Stokes equations.

If this is right

Real-time control and optimization algorithms can now use a physics-based flow model instead of full CFD.
The same identification procedure can be repeated for new materials or nozzle geometries without rebuilding the full simulation.
Model predictions remain accurate in the nozzle, gap, and layer regions simultaneously under the conditions examined.

Where Pith is reading between the lines

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

The approach could be combined with feedback controllers that adjust extrusion rate on the fly to correct layer height errors.
Extending the parameterization to include temperature or material viscosity as explicit inputs would widen the operating envelope.
The same averaging technique might apply directly to other nozzle-based deposition processes such as direct ink writing.

Load-bearing premise

Spatial averaging together with input-dependent parameterization preserves the essential transient flow physics across the tested printing conditions.

What would settle it

Large, systematic deviations between the reduced-order predictions and new CFD simulations for printing parameters outside the training set would show that the averaging step has lost critical dynamics.

Figures

Figures reproduced from arXiv: 2512.11048 by Amrita Basak, Mandana Mohammadi Looey, Marissa Loraine Scalise, Satadru Dey.

**Figure 2.** Figure 2: Block diagram representation of the input-output dynamics of the 3D printing process. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic of sub-system 1. In sub-system 1, the material is propelled by a constant force along the y-direction, producing a quasi-steady inlet mass flow rate at the nozzle. The nozzle is assumed rigid, while the build plate translates at a constant velocity Us, imposing kinematic boundary conditions on the extrudate. As depicted in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Schematic of sub-systems 2 and 3. Now, we consider our last sub-system, namely sub-system 3 (refer to [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: (a) High-fidelity computational domain. All dimensions are in mm. (b) A representative DIW strand. The [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Average velocity profiles for a) sub-system 1, b) sub-system 2, c) sub-system 3. d) Average pressure profile [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: Performance of the dynamical model trained with simulation cases 1, 2, 3, 7, 8, and 9. The model is evaluated [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: RMSE distribution obtained from the dynamical model across all sub-systems for training and testing [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: Probability distribution of model errors for the interpolative mass flow rate training scenario. The dynamical [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: RMSE distribution of the dynamical model across all the datasets and sub-systems for the extrapolative [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 11.** Figure 11: RMSE distribution of the dynamical model across all the datasets and sub-systems for different interpolative [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: RMSE distribution of the dynamical model across all the datasets and sub-systems for random (a) 67-33, [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗

read the original abstract

The trade-off between model fidelity and computational cost remains a central challenge in the computational modeling of extrusion-based 3D printing, particularly for real time optimization and control. Although high fidelity simulations have advanced considerably for offline analysis, dynamical modeling tailored for online, control-oriented applications is still significantly underdeveloped. In this study, we propose a reduced order dynamical flow model that captures the transient behavior of extrusion-based 3D printing. The model is grounded in physics-based principles derived from the Navier Stokes equations and further simplified through spatial averaging and input dependent parameterization. To assess its performance, the model is identified via a nonlinear least squares approach using Computational Fluid Dynamics (CFD) simulation data spanning a range of printing conditions and subsequently validated across multiple combinations of training and testing scenarios. The results demonstrate strong agreement with the CFD data within the nozzle, the nozzle substrate gap, and the deposited layer regions. Overall, the proposed reduced order model successfully captures the dominant flow dynamics of the process while maintaining a level of simplicity compatible with real time control and 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.

Desk Editor's Note private letter to a colleague

The paper gives a CFD-matched reduced-order model for extrusion printing flow via spatial averaging and input-dependent parameters, but the validation stays entirely inside simulations.

read the letter

The main point is a reduced-order dynamical model for extrusion-based 3D printing that starts from the Navier-Stokes equations, applies spatial averaging to cut the dimension, and adds parameters that vary with inputs such as speed or pressure. Those parameters are identified by nonlinear least squares on CFD data, and the model is then checked against the same simulations in the nozzle, the gap, and the deposited layer. Agreement looks solid across the tested conditions, and the resulting equations stay simple enough for real-time control loops. That combination of physics structure plus input dependence is the concrete step forward here; it is not a new averaging technique but a tailored version for this process. The work is useful for anyone building model-based controllers or optimizers in additive manufacturing who needs something faster than full CFD yet still tied to the governing equations. The soft spot is the complete reliance on simulation data for both fitting and testing. No physical printer measurements appear, so effects like real rheology drift, surface tension changes, or substrate wetting that CFD often idealizes are not checked. Because the quantitative values come from fitting the same CFD runs, the model risks being more of an interpolator than a predictor outside the simulated range. Spatial averaging itself can also suppress transients that matter in practice. This is worth sending to peer review. The method is clear, the engineering goal is stated plainly, and the results are presented without overclaim, but referees will need to see how the authors plan to address experimental validation before the model can be treated as ready for control use.

Referee Report

2 major / 1 minor

Summary. The paper proposes a reduced-order dynamical model for the transient flow in extrusion-based 3D printing. The model is derived from the Navier-Stokes equations via spatial averaging and input-dependent parameterization, then identified by nonlinear least squares on CFD data spanning multiple printing conditions and validated against the same CFD data in the nozzle, nozzle-substrate gap, and deposited-layer regions. The central claim is that the resulting low-order model captures the dominant dynamics while remaining simple enough for real-time control and optimization.

Significance. If the spatially averaged model generalizes beyond the CFD data used for identification, the work would supply a computationally tractable, physics-informed dynamical description suitable for online process optimization in additive manufacturing. The derivation from first principles and the multi-condition validation protocol are positive features; however, the absence of any experimental comparison leaves open whether the averaging step preserves the transients that matter under real rheology, surface tension, and substrate conditions.

major comments (2)

[Validation section] Validation section (and abstract): all quantitative agreement is reported exclusively against the CFD simulations from which the input-dependent parameters were fitted via nonlinear least squares. No experimental measurements, error-bar analysis, or hold-out physical data are presented; this directly undermines the claim that the model is suitable for real-time control, because non-ideal effects omitted from the CFD may violate the spatial-averaging closure.
[Model derivation and identification] Model derivation and identification: the input-dependent parameterization is obtained by fitting to CFD; the manuscript does not provide a sensitivity study or a priori bounds showing that the identified parameters remain valid when the printing conditions deviate from the training set, which is load-bearing for the real-time-control assertion.

minor comments (1)

[Abstract] The abstract states 'strong agreement' without supplying any scalar error metric (e.g., L2 norm, maximum relative error) or table of quantitative results; adding such numbers would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed review. We appreciate the positive assessment of the physics-based derivation from the Navier-Stokes equations and the multi-condition validation protocol. We address each major comment below and describe the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Validation section] Validation section (and abstract): all quantitative agreement is reported exclusively against the CFD simulations from which the input-dependent parameters were fitted via nonlinear least squares. No experimental measurements, error-bar analysis, or hold-out physical data are presented; this directly undermines the claim that the model is suitable for real-time control, because non-ideal effects omitted from the CFD may violate the spatial-averaging closure.

Authors: We acknowledge that the quantitative comparisons are performed against CFD data (including hold-out testing scenarios across printing conditions, as stated in the manuscript). This demonstrates that the spatially averaged model captures the dominant transients under the CFD assumptions. We agree that the absence of experimental data leaves open questions about real rheology, surface tension, and substrate effects. As this is a computational study focused on deriving and validating a reduced-order model in silico, we do not have experimental measurements available. We will revise the abstract, validation section, and conclusions to explicitly qualify the scope as CFD-validated, discuss potential closure violations from omitted physics, and outline the need for future experimental benchmarking to support real-time control claims. revision: partial
Referee: [Model derivation and identification] Model derivation and identification: the input-dependent parameterization is obtained by fitting to CFD; the manuscript does not provide a sensitivity study or a priori bounds showing that the identified parameters remain valid when the printing conditions deviate from the training set, which is load-bearing for the real-time-control assertion.

Authors: The model structure is obtained by spatial averaging of the Navier-Stokes equations, with input-dependent parameters identified to account for condition-specific effects. Validation was already performed on separate testing scenarios not used in fitting. To directly address the concern, we will add a new sensitivity analysis subsection that evaluates the model on input values (e.g., extrusion flow rate and print speed) outside the original training range and reports the resulting prediction errors. This will supply empirical bounds on parameter validity and clarify the operating envelope for which the real-time control suitability holds. revision: yes

standing simulated objections not resolved

Absence of experimental measurements for validation against physical data, which cannot be addressed without conducting new experiments outside the scope of the current computational study.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The reduced-order model structure is obtained from the Navier-Stokes equations via spatial averaging and input-dependent parameterization, which are independent of the subsequent fitting step. Parameters are identified by nonlinear least squares on CFD data and validated on held-out CFD scenarios across printing conditions. This is standard physics-informed system identification against an external benchmark (CFD simulations), not a reduction of the central claim to a self-definition, fitted input renamed as prediction, or self-citation chain. The derivation remains self-contained with independent content from the NS equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the Navier-Stokes equations as the starting point, the validity of spatial averaging for transient behavior, and the assumption that CFD data adequately represent real printing physics.

free parameters (1)

input-dependent model parameters
Parameters are adjusted according to printing inputs and fitted via nonlinear least squares to CFD data.

axioms (1)

standard math Navier-Stokes equations govern the incompressible flow inside the nozzle and gap
The reduced model is explicitly derived from the Navier-Stokes equations.

pith-pipeline@v0.9.0 · 5500 in / 1231 out tokens · 44696 ms · 2026-05-16T22:40:42.517989+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

spatial averaging technique on the spatio-temporal variables in momentum equations to get an averaged linear ordinary differential equation (ODE)-like mathematical structure... parameterized reduced order model is given by: d v̄1 / dt = β1 p_d1 + β2 v̄1 + β3 ṁ
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The model is grounded in physics-based principles derived from the Navier–Stokes equations and further simplified through spatial averaging and input-dependent parameterization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Extrusion-based additive manufacturing with cement-based materials–production steps, processes, and their underlying physics: a review,

V . Mechtcherine, F. P. Bos, A. Perrot, W. L. Da Silva, V . Nerella, S. Fataei, R. J. Wolfs, M. Sonebi, and N. Rous- sel, “Extrusion-based additive manufacturing with cement-based materials–production steps, processes, and their underlying physics: a review,”Cement and Concrete Research, vol. 132, p. 106037, 2020

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A general simulation- based study on printability of inks in direct ink writing,

M. A. Sourov, S. I. Emu, M. S. H. Thakur, M. M. Ikram, A. Morshed, and M. M. Rahman, “A general simulation- based study on printability of inks in direct ink writing,”Scientific Reports, vol. 15, no. 1, pp. 1–16, 2025

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[4] [4]

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[5] [5]

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G. Wagner, J. M. Silva, J. Ribeiro, B. Figueiredo, P. J. Cruz, and J. M. N´obrega, “A novel and flexible approach to modeling the additive manufacturing extrusion of cementitious materials,”Journal of Manufacturing Processes, vol. 151, pp. 1081–1094, 2025

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Experimental validation of a numeri- cal model for the strand shape in material extrusion additive manufacturing,

M. P. Serdeczny, R. Comminal, D. B. Pedersen, and J. Spangenberg, “Experimental validation of a numeri- cal model for the strand shape in material extrusion additive manufacturing,”Additive Manufacturing, vol. 24, pp. 145–153, 2018

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A control-oriented model for melt-pool volume in laser powder bed fusion additive manufacturing,

Q. Wang, “A control-oriented model for melt-pool volume in laser powder bed fusion additive manufacturing,” inDynamic Systems and Control Conference, vol. 59148, p. V001T10A002, American Society of Mechanical Engineers, 2019

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Reduced-order multivariable model- ing and nonlinear control of melt-pool geometry and temperature in directed energy deposition,

Q. Wang, J. Li, M. Gouge, A. R. Nassar, P. Michaleris, and E. W. Reutzel, “Reduced-order multivariable model- ing and nonlinear control of melt-pool geometry and temperature in directed energy deposition,” in2016 Ameri- can Control Conference (ACC), pp. 845–851, IEEE, 2016. 18 APREPRINT- DECEMBER24, 2025

work page 2016

[9] [9]

A reduced-order model of thermo-viscoelastic filaments in a material extrusion process,

J. Meng, J. Liu, H. Xia, X. Ao, and W. Zhang, “A reduced-order model of thermo-viscoelastic filaments in a material extrusion process,”Additive Manufacturing, vol. 91, p. 104343, 2024

work page 2024

[10] [10]

Control-oriented modeling and layer-to-layer stability for fused deposition modeling: A kernel basis approach,

E. C. Balta, D. M. Tilbury, and K. Barton, “Control-oriented modeling and layer-to-layer stability for fused deposition modeling: A kernel basis approach,” in2019 American Control Conference (ACC), pp. 4727–4733, IEEE, 2019

work page 2019

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Analytical model to predict the extrusion force as a function of the layer height, in extrusion based 3d printing, addit. manuf. 38 (2021) 101791,

G. Percoco, L. Arleo, G. Stano, and F. Bottiglione, “Analytical model to predict the extrusion force as a function of the layer height, in extrusion based 3d printing, addit. manuf. 38 (2021) 101791,” 2020

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Combined parabolic and elliptic velocity profile-based low-dimensional model in falling film,

A. Samanta, “Combined parabolic and elliptic velocity profile-based low-dimensional model in falling film,” Physical Review Fluids, vol. 9, no. 12, p. 124002, 2024

work page 2024

[13] [13]

Modelling and parameter optimization for filament deformation in 3d cementitious material printing using support vector machine,

Z. Liu, M. Li, Y . Weng, Y . Qian, T. N. Wong, and M. J. Tan, “Modelling and parameter optimization for filament deformation in 3d cementitious material printing using support vector machine,”Composites Part B: Engineer- ing, vol. 193, p. 108018, 2020

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Nonlinear least squares,

J. J. Dennis, “Nonlinear least squares,”State of the art in numerical analysis, pp. 269–312, 1977. 19

work page 1977