Neural-Network-based Viscosity Closure for Non-Newtonian Multiphase Flows

This paper presents a practical workflow that integrates a neural network trained on experimental rheometry data as a viscosity closure within a Cahn–Hilliard–Navier–Stokes finite element solver, successfully validating the approach by accurately simulating the rise dynamics and shapes of non-Newtonian silicone inks without requiring solver modifications.

The Gist

Imagine you are trying to simulate how a drop of thick, gooey ink moves through a liquid bath. In the real world, this ink doesn't behave like water; it's "non-Newtonian." This means its thickness (viscosity) changes depending on how fast you stir it or squeeze it. If you push it hard, it might get thinner (like ketchup) or thicker (like cornstarch and water).

Traditionally, computer scientists trying to simulate this have to guess a specific mathematical formula (like a "Carreau–Yasuda" equation) to describe how the ink behaves. But if you invent a new ink, you have to stop, derive a new formula, and rewrite the computer code every time. It's like trying to drive a car where you have to manually rebuild the engine every time you change the fuel type.

This paper presents a smarter, more flexible way to do it using Artificial Intelligence (AI).

The "Smart Substitute" (The Neural Network)

Instead of forcing the computer to use a rigid mathematical formula, the researchers trained a "neural network" (a type of AI brain) to act as a smart substitute for the ink's behavior.

1. Learning from Experience: They took real-world data from a machine that measured how their specific silicone inks reacted to being stirred at different speeds.
2. The Training: They taught the AI to look at the speed of the stir and predict exactly how thick the ink would be at that moment.
3. The "Smoothness" Rule: To make sure the AI didn't get confused or make wild, unrealistic guesses (like predicting the ink turns into solid rock instantly), they added a rule called "Lipschitz regularization." Think of this as a speed limit for the AI's learning. It forces the AI to make smooth, gradual predictions rather than jagged, erratic ones.

The "Universal Translator" (ONNX)

Usually, if you train an AI, you have to rewrite your physics simulation software to understand that specific AI. That's slow and error-prone.

The researchers used a format called ONNX (Open Neural Network Exchange). Imagine this as a universal translator or a standard USB drive. They saved their trained AI in this format. Now, the physics simulation software can just "plug in" the AI file and ask it, "Hey, what's the viscosity at this speed?" without needing to be rewritten. The AI does the heavy lifting, and the simulation software just listens.

The Test Drive: The Rising Bubble

To prove this system works, they ran two types of tests:

1. The "Textbook" Test: They simulated a bubble rising in a fluid where they already knew the exact mathematical formula. They compared their AI-driven simulation against the known math.

   • Result: The AI matched the math perfectly. It proved the "plug-and-play" system works.

2. The "Real World" Test: They created two actual silicone ink mixtures in a lab. They filmed these ink drops rising through a special liquid (perfluorodecalin) using a high-speed camera.

   • They fed the real lab data into their AI.
   • They let the computer simulate the drops rising.
   • Result: The computer simulation predicted the speed and the shape of the rising drops almost exactly as they appeared in the real-life video. The simulated drops looked like the real drops, and they rose at the same speed.

Why This Matters (According to the Paper)

The paper claims this is a practical path forward for additive manufacturing (like 3D printing). When printing with complex materials (like the inks used in Digital Light Processing or Direct Ink Writing), the material's behavior is hard to predict.

This new workflow allows engineers to:

• Take real lab data from a new material.
• Train a small AI model on it.
• Plug that model directly into a simulation to see how the material will flow during printing.

In short: They built a system where you don't need to be a mathematician to describe a fluid's behavior. You just measure it, train a small AI, and let the computer figure out the rest, all while keeping the simulation running smoothly and accurately.

Technical Summary

Technical Summary: Neural-Network-based Viscosity Closure for Non-Newtonian Multiphase Flows

Problem Statement
Materials used in polymer-based additive manufacturing, such as Digital Light Processing (DLP) and Direct Ink Writing (DIW), typically exhibit complex non-Newtonian rheology. While standard constitutive models like Carreau–Yasuda and power-law formulations effectively describe basic shear-thinning or shear-thickening behaviors, they possess significant limitations. Their fixed, low-dimensional functional forms require researchers to derive new equations and re-implement them within flow solvers for every new material system or rheological behavior (e.g., viscoelasticity, thixotropy). Furthermore, empirical fits do not always guarantee physical credibility, and data-driven surrogates often require careful regularization to avoid non-physical oscillations when integrated into high-fidelity solvers. There is a lack of established workflows that deploy constitutive surrogates trained on experimental rheometry directly inside multiphase flow solvers without solver modification.

Methodology
The authors present a deployment workflow integrating a neural network (NN) trained on experimental rheometry data as a viscosity closure within a Cahn–Hilliard–Navier–Stokes (CHNS) finite element solver. The methodology comprises three core components:

1. Neural Network Formulation and Training:

   • A fully connected feedforward neural network is trained to map local shear rate ($\dot{\gamma}$) to viscosity ($\eta$).
   • To ensure the learned function is suitable for use as a constitutive closure, Lipschitz regularization is applied during training. This adds a penalty on the input gradient to the loss function, bounding the derivative of the network output. This prevents the network from overfitting noise with highly oscillatory functions and ensures smooth, physically plausible predictions, particularly during extrapolation outside the training shear-rate range.
   • The network architecture consists of two hidden layers with 64 neurons each, using hyperbolic-tangent (Tanh) activations. Training is performed in log–log space to handle data spanning several decades.

2. Solver Integration via ONNX:

   • The trained network is exported to the Open Neural Network Exchange (ONNX) format.
   • The CHNS solver queries the network at runtime via the ONNX runtime at every iteration. The solver passes the local shear rate to the model and receives the corresponding viscosity.
   • This approach allows the integration of neural constitutive models into existing multiphase flow frameworks without modifying the solver code or re-implementing the network architecture within the C++/Fortran solver environment.

3. Numerical Framework:

   • The solver utilizes a block-iterative time-stepping strategy, solving the Cahn–Hilliard equations followed by the Navier–Stokes equations.
   • The framework employs a parallel, octree-based Adaptive Mesh Refinement (AMR) infrastructure to concentrate resolution at the fluid interface, efficiently resolving steep gradients in phase boundaries.

Key Contributions

• Deployment Workflow: The paper establishes a practical path for deploying experimentally trained rheological surrogates inside finite element solvers using ONNX, eliminating the need for solver modification.
• Lipschitz Regularization: The application of Lipschitz regularization during training is demonstrated as a critical mechanism for producing smooth viscosity predictions suitable for constitutive closures, addressing the risk of non-physical oscillations in data-driven models.
• Validation on Synthetic and Real Data: The workflow is validated against both synthetic benchmarks (analytical Carreau–Yasuda laws) and real-world experimental data (silicone ink formulations), demonstrating robustness across monotonic and non-monotonic constitutive behaviors.

Results
The study validates the framework through two primary sets of experiments:

1. Benchmark Validation (Synthetic Data):

   • The solver was tested against benchmark bubble-rise cases in shear-thinning fluids (Pang and Lu [26]) with varying power-law indices and Weber numbers.
   • Simulations using the neural network closure (trained on synthetic Carreau–Yasuda data) produced bubble shapes and rise velocities that were visually indistinguishable from those using the analytical Carreau–Yasuda closure.
   • The neural network results also matched the reference literature values, confirming the solver's consistency and the fidelity of the neural pipeline.
   • A synthetic non-monotonic constitutive law (shear-thinning followed by shear-thickening) was successfully modeled, with the neural network reproducing the analytical results without architectural changes.

2. Experimental Validation (Real Data):

   • Two silicone ink formulations (Material A and B with varying fumed silica content) were characterized using rheometry.
   • Droplet rise dynamics were recorded via high-speed video in a perfluorodecalin bath.
   • The neural networks were trained on the experimental rheometer data and used to simulate the rise of these droplets.
   • Shape Agreement: The simulated steady-state droplet shapes agreed with experimental observations. Hausdorff distances ranged from 0.044 to 0.079, and Chamfer distances from 0.040 to 0.062 (normalized by average droplet size), indicating deviations within approximately 4–8% of the droplet size.
   • Velocity Agreement: Simulated rise velocities fell within the experimentally measured spread for all four independent cases (two materials, two needle gauges).
   • Computational Cost: The inference cost of the neural network was negligible compared to the linear algebra costs of the Navier–Stokes and Cahn–Hilliard solves.

Significance
The paper claims to contribute to the growing literature on integrating neural constitutive closures into multiphysics simulations. It demonstrates a practical, solver-agnostic method for replacing traditional parametric constitutive laws with data-driven surrogates trained on experimental measurements. By successfully reproducing complex rise dynamics and droplet shapes for real materials without requiring new analytical derivations or solver rewrites, the work provides a viable path for simulating non-Newtonian multiphase flows in additive manufacturing and other applications where material rheology is complex or difficult to capture with standard functional forms. The authors emphasize that the Lipschitz regularization is essential for ensuring the stability and physical plausibility of these data-driven closures within the solver.