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

TL;DR

The paper tackles extensional rheology by applying Rheo-SINDy to data from extensional flows. It demonstrates exact CM recovery for the Giesekus model and obtains an effective approximate CM for the FENE dumbbell model using a sparsity-promoting regression framework with a purposefully designed rheology-informed library. The learned models accurately reproduce extensional rheological responses and even extrapolate to unseen strain rates at a fraction of the computational cost of detailed Brownian dynamics. These results validate a data-driven, sparse identification approach for constitutive modeling in extensional flows and outline paths for refinement, such as symmetric training data and multi-mode extensions.

Abstract

Deriving constitutive models (CMs) from numerical data has been an attractive approach as a systematic CM building method. One recent study is Rheo-SINDy, which extended the sparse identification of nonlinear dynamics (SINDy) method to rheology. Although the Rheo-SINDy framework discovered an approximate CM from numerical data under shear flow, its versatility has not been investigated. To clarify its applicability to other types of flows, this study applied Rheo-SINDy to numerically generated data under extensional flow conditions. As baseline tests for extensional flow, we considered two problems: (i) whether the Rheo-SINDy framework can reproduce the famous Giesekus model from data generated by that model, and (ii) whether it can derive an approximate CM from data generated by a dumbbell model with a finite extensible nonlinear elastic (FENE) spring. For problem (i), we confirmed that Rheo-SINDy can identify the exact expression of the Giesekus model under extensional flow. For problem (ii), the Rheo-SINDy framework discovered a relatively simple expression of the approximate CM by manually designing the library matrix based on rheological knowledge. The identified approximate CM can reasonably predict extensional rheological properties of the FENE dumbbell model, including an extrapolation region. These findings demonstrate the fundamental validity of using Rheo-SINDy under extensional flow.

Figures (8)

• Figure 1: Schematic representation of the Rheo-SINDy framework.
• Figure 2: Hyperparameter ($\alpha$) dependence of (a) the total number of identified terms of the equations for $\overset{\triangledown}{\tau}_{xx}$, $\overset{\triangledown}{\tau}_{yy}$, and $\overset{\triangledown}{\tau}_{zz}$, and (b) the normalized MSE, obtained for the Giesekus model. We obtained these results from the training data, which include the data generated under uniaxial and biaxial extensional flows. The circles and squares correspond to the results obtained by STRidge and a-Lasso, respectively. In (b), values of $\alpha$ with no plotted symbols indicate the cases where the identified CM diverged or where the MSE was larger than the sparsest case ($\overset{\triangledown}{\boldsymbol \tau} = {\boldsymbol 0}$). The downward arrows in (b) indicate the (nearly) optimal $\alpha$ for each regression method.
• Figure 3: Hyperparameter ($\alpha$) dependencies of (a) the total number of identified terms and (b) the normalized MSE, both obtained by a-Lasso for the FENE dumbbell model. The downward arrow in (b) indicates the (nearly) optimal $\alpha$ value that has a minimum MSE ($\alpha = 3 \times 10^{-2}$). The number of terms at $\alpha = 3 \times 10^{-2}$ is $14$.
• Figure 4: The identified coefficient values for the FENE dumbbell model obtained by a-Lasso with $\alpha = 3 \times 10^{-2}$, which is the (nearly) optimum $\alpha$ value shown by the black arrow in Fig. \ref{['Fig03']}(b). The black circles, blue squares, and red triangles show the coefficient values of the equations for ${\overset{\triangledown}\tau}_{xx}$, ${\overset{\triangledown}\tau}_{yy}$, and ${\overset{\triangledown}\tau}_{zz}$, respectively. The inset shows the enlargement of the coefficient values for higher-order terms. ${\rm tr}_{(1)}$ and ${\rm tr}_{(2)}$ indicate ${\rm tr}_{(1)} = {\rm tr} {\boldsymbol \tau}$ and ${\rm tr}_{(2)} = {\rm tr} ({\boldsymbol \tau} \cdot {\boldsymbol \kappa}^{\rm T} + {\boldsymbol \kappa} \cdot {\boldsymbol \tau})$, respectively.
• Figure 5: The test simulation results for the oscillatory (a) uniaxial, (b) planar, and (c) biaxial extensional flows with $\omega = 1$. Here, the strain amplitude was $\epsilon_0 = 10$ for the uniaxial and planar extensional flows, and $\epsilon_0 = 5$ for the biaxial extensional flow. The thin solid, bold solid, and bold dotted lines represent the results obtained using the identified CMs, the FENE dumbbell model, and the FENE-P dumbbell model, respectively. The black, red, and blue lines show $\tau_{xx}$, $\tau_{yy}$, and $\tau_{zz}$, respectively.