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Fast Jacobi Spectral Methods and Closure Approximations for the Homogeneous FENE Model of Complex Fluids

Runkai Feng, Jie Shen, Haijun Yu

TL;DR

The paper tackles the high-dimensional, boundary-singular FENE Fokker–Planck equation by introducing two fast Jacobi-Spherical Harmonic spectral-Galerkin solvers (JG1 and JGinf) to achieve spectral accuracy in the 0+3D homogeneous setting. It forms a weighted weak formulation using the transform $f=(1-|m{q}|^2)^s h$ and a mapped radial coordinate, ensuring energy-stable time stepping with a semi-implicit $2^{nd}$-order BDF scheme. Beyond direct solvers, it scrutinizes macroscopic closure models (FENE-P, FENE-QE) and proposes a neural-network-accelerated QE closure (FENE-QE-NN) to combine physical fidelity with computational efficiency, validated against extensional and mixed-flow benchmarks. The results show spectral solvers provide high-accuracy reference solutions, FENE-P can misrepresent non-Gaussian features like bimodal distributions in strong extensional flows, while QE-based closures (especially FENE-QE-NN) offer superior accuracy and robustness with substantial speedups. This work advances multi-scale modeling of complex fluids by delivering rigorous numerical tools and data-driven closures that bridge micro-scale kinetics and macro-scale rheology, with potential to extend to non-homogeneous flows and more elaborate constitutive models.

Abstract

The Finitely Extensible Nonlinear Elastic (FENE) dumbbell model is a widely used mathematical model for complex fluids. Direct simulation of the FENE Fokker--Planck equation is computationally challenging due to high dimensionality and singularity of its potential. In this paper, we develop two fast Jacobi-Spherical Harmonic spectral methods for the spatially homogeneous FENE Fokker--Planck equation. These methods effectively resolve the singularity near the boundary by combining properly designed Jacobi polynomials with a weighted variational formulation. A semi-implicit backward differentiation formula of second-order (BDF2) is employed for time marching, and its energy stability is rigorously proved. The resulting linear algebraic system possesses a sparse structure and can be efficiently solved. Numerical results verify the spectral convergence and efficiency of the direct spectral solvers, establishing them as a reliable tool for generating reference solutions for challenging benchmark problems. Furthermore, to achieve an optimal trade-off between accuracy and efficiency, we compare several closure approximation models, including the industry workhorse Peterlin approximation (FENE-P), the quasi-equilibrium approximation (FENE-QE), and a novel neural network implementation for FENE-QE proposed in this paper (FENE-QE-NN). Numerical experiments in extensional and shear flows demonstrate the superior accuracy and efficiency of the proposed methods compared to traditional approaches.

Fast Jacobi Spectral Methods and Closure Approximations for the Homogeneous FENE Model of Complex Fluids

TL;DR

The paper tackles the high-dimensional, boundary-singular FENE Fokker–Planck equation by introducing two fast Jacobi-Spherical Harmonic spectral-Galerkin solvers (JG1 and JGinf) to achieve spectral accuracy in the 0+3D homogeneous setting. It forms a weighted weak formulation using the transform and a mapped radial coordinate, ensuring energy-stable time stepping with a semi-implicit -order BDF scheme. Beyond direct solvers, it scrutinizes macroscopic closure models (FENE-P, FENE-QE) and proposes a neural-network-accelerated QE closure (FENE-QE-NN) to combine physical fidelity with computational efficiency, validated against extensional and mixed-flow benchmarks. The results show spectral solvers provide high-accuracy reference solutions, FENE-P can misrepresent non-Gaussian features like bimodal distributions in strong extensional flows, while QE-based closures (especially FENE-QE-NN) offer superior accuracy and robustness with substantial speedups. This work advances multi-scale modeling of complex fluids by delivering rigorous numerical tools and data-driven closures that bridge micro-scale kinetics and macro-scale rheology, with potential to extend to non-homogeneous flows and more elaborate constitutive models.

Abstract

The Finitely Extensible Nonlinear Elastic (FENE) dumbbell model is a widely used mathematical model for complex fluids. Direct simulation of the FENE Fokker--Planck equation is computationally challenging due to high dimensionality and singularity of its potential. In this paper, we develop two fast Jacobi-Spherical Harmonic spectral methods for the spatially homogeneous FENE Fokker--Planck equation. These methods effectively resolve the singularity near the boundary by combining properly designed Jacobi polynomials with a weighted variational formulation. A semi-implicit backward differentiation formula of second-order (BDF2) is employed for time marching, and its energy stability is rigorously proved. The resulting linear algebraic system possesses a sparse structure and can be efficiently solved. Numerical results verify the spectral convergence and efficiency of the direct spectral solvers, establishing them as a reliable tool for generating reference solutions for challenging benchmark problems. Furthermore, to achieve an optimal trade-off between accuracy and efficiency, we compare several closure approximation models, including the industry workhorse Peterlin approximation (FENE-P), the quasi-equilibrium approximation (FENE-QE), and a novel neural network implementation for FENE-QE proposed in this paper (FENE-QE-NN). Numerical experiments in extensional and shear flows demonstrate the superior accuracy and efficiency of the proposed methods compared to traditional approaches.
Paper Structure (32 sections, 3 theorems, 65 equations, 5 figures, 4 tables)

This paper contains 32 sections, 3 theorems, 65 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The FENE Fokker--Planck equation and its weighted weak form
  3. Time discretization and stability analysis
  4. Spectral discretization in configuration space
  5. Spherical harmonic expansion and symmetry
  6. Fast Jacobi Galerkin method with $C^1$ continuity (JG1)
  7. Fast Jacobi Galerkin method with $C^{\infty}$ continuity (JGinf)
  8. Moment closure approximation models
  9. Derivation of the moment equation
  10. The FENE-P model
  11. FENE-QE model and PLA algorithm
  12. Neural network-accelerated FENE-QE model (FENE-QE-NN)
  13. Dataset and Permutation Invariance
  14. Network Architecture
  15. Hybrid Optimization Strategy
  16. ...and 17 more sections

Key Result

Lemma 3.1

For any $h \in H_s^1(\Omega)$ and $s > 1$, the following identity holds: Furthermore, by analyzing the quadratic form associated with the weighted norms, there exists a threshold $\gamma(s)$ such that for any $\gamma > \gamma(s)$, the following inequality holds: where

Figures (5)

  • Figure 1: Spectral convergence of the relative $L_{\omega}^2$ error with respect to the polynomial degree $N$.
  • Figure 2: Training diagnostics of the FENE-QE-NN constitutive model. (a) Loss history (MSE) showing the transition from Adam to L-BFGS. (b) Parity plot of the predicted vs. true Lagrange multipliers $\lambda$ on the validation set.
  • Figure 3: CDF contours for extensional flow under various conditions.
  • Figure 4: 1D cross-sections of the CDF comparing the exact solution with closure models.
  • Figure 5: CDF contours and cross-sections for mixed shear and extensional flow.

Theorems & Definitions (7)

  • Lemma 3.1
  • Theorem 3.1: Existence and Uniqueness
  • proof
  • Theorem 3.2: Stability
  • proof
  • Remark 3.1: Asymptotic behavior and stability at low $\mathrm{De}$ number
  • Remark 6.1: Numerical Definiteness