In this article, we introduce a new method for discretizing micro-macro models of dilute polymeric fluids based on deterministic particles. Our approach integrates a finite element discretization for the macroscopic fluid dynamic equation with a deterministic variational particle scheme for the microscopic Fokker-Planck equation. To address challenges arising from micro-macro coupling, we employ a discrete energetic variational approach to derive a coarse-grained micro-macro model with a particle approximation first and then develop a particle-FEM discretization for the coarse-grained model. The accuracy of our method is evaluated for a Hookean dumbbell model in a Couette flow by comparing the computed velocity field with existing analytical solutions. We also use our method to study nonlinear FENE dumbbell models in different scenarios, such as extension flow, pure shear flow, and lid-driven cavity flow. Numerical examples demonstrate that the proposed deterministic particle approach can accurately capture the various key rheological phenomena in the original FENE model, including hysteresis and $\delta$-function-like spike behavior in extension flows, velocity overshoot phenomenon in pure shear flow, symmetries breaking, vortex center shifting and vortices weakening in the lid-driven cavity flow, with a small number of particles.

翻译：本文提出了一种基于确定粒子离散稀聚合物流体微观-宏观模型的新方法。该方法将宏观流体动力学方程的有限元离散与微观福克-普朗克方程的确定性变分粒子格式相结合。为应对微观-宏观耦合带来的挑战，我们首先采用离散能量变分方法推导出带有粒子近似的粗粒化微观-宏观模型，进而针对该粗粒化模型开发了粒子-有限元离散格式。通过将库埃特流中胡克哑铃模型的计算速度场与现有解析解对比，评估了该方法的精度。我们还将该方法应用于不同场景下非线性FENE哑铃模型的研究，包括拉伸流、纯剪切流和顶盖驱动方腔流。数值算例表明，所提出的确定性粒子方法能以少量粒子精确捕捉原始FENE模型中的多种关键流变现象，包括拉伸流中的迟滞效应和δ函数型尖峰行为、纯剪切流中的速度过冲现象，以及顶盖驱动方腔流中的对称性破缺、涡心偏移和旋涡减弱现象。