Development of Rheological Constitutive Modeling Method Using a Sparse Identification Algorithm: A Case Study for Extensional Flows

This study validates the applicability of the Rheo-SINDy framework to extensional flows by demonstrating its ability to accurately recover the Giesekus model and derive a predictive approximate constitutive model for FENE dumbbell data through a manually designed library.

The Gist

Imagine you are trying to teach a robot how to predict how a sticky, stretchy fluid (like melted cheese or a polymer solution) behaves when you pull it apart. This is called "extensional flow."

Usually, scientists have to write very complex math equations by hand to describe this behavior. But in this paper, the authors tried a different approach: they let a computer "learn" the rules directly from data, using a method called Rheo-SINDy.

Here is a simple breakdown of what they did and what they found, using everyday analogies:

1. The Goal: Teaching a Robot the "Rules of the Road"

Think of the fluid as a car and the flow as the road. Scientists want to know the exact laws of physics (the constitutive model) that tell the car how to move when the road stretches.

• The Old Way: Experts write the rulebook based on theory.
• The New Way (This Paper): The computer looks at a massive amount of driving data and tries to figure out the rulebook itself by finding the simplest pattern that fits.

2. The Tool: A "Sparse" Detective

The method they used is called Sparse Identification. Imagine you are a detective trying to solve a crime. You have a giant list of 1,000 possible suspects (variables).

• Most detectives might accuse everyone.
• This "Sparse" detective is very picky. They know that usually, only two or three people are actually involved. They use a special algorithm to ignore the 997 innocent suspects and find the tiny handful of real culprits that explain the crime.
• In this study, the "crime" is the fluid's movement, and the "suspects" are mathematical terms (like stress, speed, and their combinations).

3. The Test Drive: Two Scenarios

To see if their detective method works, they ran two tests using computer-generated data (simulations):

Test A: The "Perfect" Puzzle (The Giesekus Model)

• The Setup: They created data using a known, perfect mathematical rule (the Giesekus model).
• The Challenge: Could the computer look at the data and rediscover the exact rulebook that created it?
• The Result: Yes! The computer successfully found the exact equation, proving the method works when the answer is already known. It's like giving a student a math problem with the answer key, and watching them perfectly reverse-engineer the steps to get that answer.

Test B: The "Mystery" Puzzle (The FENE Dumbbell Model)

• The Setup: They used a more complex model (FENE dumbbell) that describes how tiny polymer chains stretch. This model is so complicated that scientists cannot write down a simple, exact rulebook for it.
• The Challenge: Could the computer look at the messy data and create a good approximation (a "cheat sheet") that acts like the real thing?
• The Result: Yes, mostly. The computer didn't find the "perfect" equation (because one doesn't exist in simple form), but it found a simple, short equation that predicted the fluid's behavior very well.
  • It worked so well that it could predict what would happen in situations it had never seen before (like pulling the fluid much faster than in the training data). This is like a student who learns the concept of "gravity" and can then correctly predict how a ball falls on the Moon, even though they only practiced on Earth.

4. Why This Matters

The authors found that their "detective" method is powerful because:

1. It's accurate: It can find the exact laws when they exist.
2. It's efficient: The equations it finds are short and simple, making them easy for computers to use in real-world simulations.
3. It's robust: It can handle complex, messy data and still find a usable rule.

The Bottom Line

This paper is a proof-of-concept. It shows that you can use a smart, data-hunting algorithm to discover the mathematical laws of how stretchy fluids behave when pulled, without needing a human to guess the formula first. They successfully tested this on both simple and complex "stretchy" fluids, showing that the method is ready to be used for more difficult problems in the future.

Technical Summary

Technical Summary: Development of Rheological Constitutive Modeling Method Using a Sparse Identification Algorithm: A Case Study for Extensional Flows

Problem Statement
Deriving constitutive models (CMs) from numerical data has emerged as a systematic approach for rheological modeling. While the Rheo-SINDy framework (Rheological Sparse Identification of Nonlinear Dynamics) has previously demonstrated success in identifying approximate CMs from shear flow data, its versatility across other flow regimes remains uninvestigated. Extensional flow is a fundamental mode in rheology, critical for understanding complex fluids like polymer and surfactant solutions. However, experimental measurement of extensional properties is challenging, and existing phenomenological models often lack structural details, while mesoscopic models (e.g., FENE dumbbells) lack analytical CMs. This study addresses the applicability of the Rheo-SINDy framework to extensional flows, specifically testing its ability to recover known models and derive approximate models for systems without analytical solutions.

Methodology
The study employs a data-driven strategy extending the SINDy algorithm to rheology. The core assumption is that the time evolution of the stress tensor $\tau$ can be expressed via the upper convected derivative $\overset{\triangledown}{\tau}$ as a sparse linear combination of functions from a library matrix $\Theta(\mathbf{T}, \mathbf{K})$, where $\mathbf{T}$ and $\mathbf{K}$ represent time-series data for stress and velocity gradients, respectively:
$$\overset{\triangledown}{\mathbf{T}} = \Theta(\mathbf{T}, \mathbf{K})\Xi$$
where $\Xi$ is a sparse coefficient matrix determined via regression.

The study utilized two primary test cases:

1. Giesekus Model (Known CM): To validate the framework, training data were generated by numerically integrating the Giesekus model under oscillatory uniaxial and biaxial extensional flows. The library matrix included polynomials up to the third degree constructed from stress and velocity gradient components.
2. FENE Dumbbell Model (Unknown CM): To test the derivation of approximate models, training data were generated via Brownian dynamics (BD) simulations of a Finite Extensible Nonlinear Elastic (FENE) dumbbell model. Since this model lacks an analytical CM, the library matrix was designed using rheological knowledge derived from the analytically tractable FENE-P (Pre-averaged) dumbbell model.

Regression and Optimization
Two sparse regression algorithms were tested: Sequentially Thresholded Ridge (STRidge) and adaptive Lasso (a-Lasso). The selection of the optimal hyperparameter $\alpha$ (controlling sparsity) was based on a normalized total mean squared error (MSE) defined as:
$$\text{MSE} = w \cdot \text{MSE}_{\text{training}} + (1-w) \cdot \text{MSE}_{\text{solved}}$$
where $\text{MSE}_{\text{training}}$ measures the fit to the derivative data, and $\text{MSE}_{\text{solved}}$ measures the error in stress predictions obtained by integrating the identified model. A weight factor of $w=0.5$ was used to ensure numerical stability and prevent the selection of unphysical, diverging models.

Key Results

• Recovery of the Giesekus Model: The Rheo-SINDy framework successfully identified the exact expression of the Giesekus model from extensional flow data. Both STRidge and a-Lasso recovered the correct terms, though a-Lasso yielded sparser solutions. The identified equations matched the theoretical form, confirming the framework's ability to handle extensional kinematics.
• Approximate Modeling of the FENE Dumbbell: For the FENE dumbbell model, the framework identified a relatively simple approximate CM (14 terms) using the library designed from FENE-P theory.
  • The identified model captured the essential dynamics, including the role of trace terms ($\text{tr}(\tau)\tau$) which are critical for FENE behavior.
  • Predictive Performance: The identified CM reasonably predicted extensional rheological properties for the FENE dumbbell model. It successfully reproduced BD simulation results for oscillatory flows and demonstrated extrapolation capabilities to steady extensional flows with strain rates higher than those in the training data (up to $\dot{\epsilon}=30$).
  • Accuracy: While the model showed fair agreement (14% deviation) at lower strain rates and good agreement (1% deviation) at higher strain rates for major stress components, minor stress components exhibited deviations. However, the model remained bounded and physically plausible.
• Computational Efficiency: The identified CM offers a significant computational advantage over BD simulations. While BD requires small time steps and large ensembles of dumbbells, the identified CM solves only three deterministic differential equations.

Significance and Claims
The paper claims that these findings demonstrate the "fundamental validity" of using Rheo-SINDy under extensional flow conditions. The study establishes that:

1. The framework can accurately recover known phenomenological models from extensional data.
2. By incorporating rheological knowledge into the library design, the framework can derive useful approximate CMs for mesoscopic models that lack analytical solutions.
3. The resulting models can extrapolate to flow histories and strain rates outside the training set, suggesting potential utility in predicting transport phenomena governed by extensional flows (e.g., capillary thinning, melt spinning).

The authors maintain a modest tone, acknowledging that the current approach relies on specific training datasets and that future work is needed to refine the method for more complex models, handle noise, and address multiple relaxation modes. The study positions Rheo-SINDy as a promising tool for rheological modeling but emphasizes the need for further refinement before it becomes a standard tool for complex systems.