Machine learning methods for particle stress development in suspension Poiseuille flows

Abstract

Numerical simulations are used to study the dynamics of a developing suspension Poiseuille flow with monodispersed and bidispersed neutrally buoyant particles in a planar channel, and machine learning is applied to learn the evolving stresses of the developing suspension. The particle stresses and pressure develop on a slower time scale than the volume fraction, indicating that once the particles reach a steady volume fraction profile, they rearrange to minimize the contact pressure on each particle. We consider the timescale for stress development and how the stress development connects to particle migration. For developing monodisperse suspensions, we present a new physics-informed Galerkin neural network that allows for learning the particle stresses when direct measurements are not possible. We show that when a training set of stress measurements is available, the MOR-physics operator learning method can also capture the particle stresses accurately.

Appendix A

A.1 Additional bidisperse suspension plots

In this section, we provide figures analogous to Figs. 7–11 for bidisperse suspensions with $\phi _B = 0.4$, $\lambda = 0.6$, and $\beta = 0.5$ (Figs. 28–30) and $\beta = 0.25$ (Figs. 33–35).

Fig. 31

Stress profiles $\sigma _{11}$ (top), $\sigma _{22}$ (middle), and $\sigma _{33}$ (bottom) for bidisperse steady flows of particles for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.5$, and for $\gamma = 0-260$

Fig. 32

Normal stress difference profiles for monodisperse steady flows of particles for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.5$

Fig. 33

Volume fraction profiles at accumulated strain $\gamma = 0-260$ for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.25$. The volume fraction is averaged of the channel center line at $y/a = 20.0$. The $(\cdot )^*$ notation denotes the profile is averaged over four reseeded simulations, as discussed in “Reseeding procedure” section

Fig. 34

Particle contact pressure profiles as defined by Eq. 4 at accumulated strain $\gamma = 0-260$ for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.25$

Fig. 35

Shear stress profiles for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.25$

Fig. 36

Stress profiles $\sigma _{11}$ (top), $\sigma _{22}$ (middle), and $\sigma _{33}$ (bottom) for bidisperse steady flows of particles for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.25$, and for $\gamma = 0-260$

Fig. 37

Normal stress difference profiles for monodisperse steady flows of particles for $\phi _B = 0.4$, $\lambda =0.6$, and $\beta = 0.25$

A.2 Galerkin neural network implementation

In order to apply Galerkin Neural Networks to the SBM formulation in “Learning the stress in monodisperse suspensions” section, we must first formulate it as a variational problem whose operator is symmetric and positive-definite. To accomplish this, we use a simple least squares variational approach:

$$\begin{aligned} \begin{aligned}&\sigma _{22} \in L^{2}(\Omega _{\gamma }; H^{1}(\Omega _{y})) \;: \\&\int _{\Omega } \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \sigma _{22}}{\partial y} \cdot \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \tau }{\partial y}\;d\Omega \\&\; \; + C\int _{\Omega _{\gamma }} \sigma _{22}\cdot \tau \;d\Omega _{\gamma } \\&= \int _{\Omega } \textbf{j}_{\perp }\cdot \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \tau }{\partial y}\;d\Omega \;\;\;\;\;\forall \tau \in L^{2}(\Omega _{\gamma }; H^{1}(\Omega _{y})). \end{aligned} \end{aligned}$$

The space $L^{2}(\Omega _{\gamma }; H^{1}(\Omega _{y}))$ is the Sobolev space (Adams and Fournier 2003) of all functions u such that

$$\begin{aligned} \vert \vert u \vert \vert _{L^{2}(\Omega _{\gamma }; H^{1}(\Omega _{y}))} := \left( \int _{\Omega _{\gamma }} \vert \vert u(\gamma ,\cdot )\vert \vert _{H^{1}(\Omega _{y})}^{2}\;d\Omega _{\gamma } \right) ^{1/2} \end{aligned}$$

is finite, where

$$\begin{aligned} \vert \vert v \vert \vert _{H^{1}(\Omega _{y})} := (\vert \vert v \vert \vert _{L^{2}(\Omega _{y})}^{2} + \vert \vert \nabla _{y}v \vert \vert _{L^{2}(\Omega _{y})}^{2})^{1/2}. \end{aligned}$$

For ease of notation, we define

$$\begin{aligned} \begin{aligned} a(\sigma ,\tau )&:= \int _{\Omega } \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \sigma _{22}}{\partial y} \cdot \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \tau }{\partial y}\;d\Omega \\&\;\;\;\;\;\;\;\; + C\int _{\Omega _{\gamma }} \sigma _{22}\cdot \tau \;d\Omega _{\gamma }\\ L(\tau )&:= \int _{\Omega } \textbf{j}_{\perp }\cdot \frac{2a^{2}}{9\eta _{f}}f(\phi )\frac{\partial \tau }{\partial y}\;d\Omega . \end{aligned} \end{aligned}$$

Here, $C>0$ is a constant that determines how strongly the boundary condition is enforced. We take $C=0.1$ in all examples.

The basis function $\Sigma _{i}^{NN}$ is obtained by solving the following optimization problem:

$$\begin{aligned} \Sigma _{i}^{NN} = \underset{\tau \in \mathcal{N}\mathcal{N}}{\mathrm {arg\;max}} \frac{L(\tau ) - a(\sigma _{22,i-1}, \tau )}{a(\tau , \tau )^{1/2}}, \end{aligned}$$

(23)

where $\mathcal{N}\mathcal{N}$ denotes the set of all realizations of a feedforward neural network of fixed width and depth. We denote by $\sigma _{22,i}$ the ith approximation to $\sigma _{22}$ using the first i basis functions $\{\Sigma _{j}^{NN}\}_{j=1}^{i}$; $\sigma _{22,i}$ is obtained by solving the discrete variational problem

$$\begin{aligned} \sigma _{22,i} \in S_{i} \;:\;\;\; a(\sigma _{22,i}, \tau ) = L(\tau ) \;\;\;\forall \tau \in S_{i}, \end{aligned}$$

where $S_{i} = \text {span}\{\Sigma _{0}^{NN},..., \Sigma _{i}^{NN}\}$. We take $\sigma _{22,0}$ to be the initial approximation to the PDE and set $\Sigma _{0}^{NN}:= \sigma _{22,0}$. For all of the simulations in “Physics-informed machine learning with the suspension balance model” section, we take $\sigma _{22,0} = 0$. This choice of initial approximation means that we do not assume any prior knowledge of the solution. One can of course use a more informed initial approximation if it exists, e.g. coarse result from another numerical method.

To evaluate the loss functional in Eq. 23, we approximate the integrals using high-order Gaussian quadrature rules such as the ones described in Davis and Rabinowitz (2007). In particular, for a given function $g: \Omega \rightarrow \mathbb {R}$, we use the quadrature rule given by the weights $\{w_{i}\}_{i=1}^{m}$ and nodes $\{(y_{i}, \gamma _{i})\}_{i=1}^{m}$ as follows:

$$\begin{aligned} \int _{\Omega } g(y,\gamma )\;d\Omega \approx \sum _{i=1}^{m} w_{i}g(y_{i},\gamma _{i}). \end{aligned}$$

(24)

The key idea behind Galerkin Neural Networks is the approximation $\Sigma _{i}^{NN} \approx \sigma _{22} - \sigma _{22,i-1}$ – that is, the ith basis function is an estimate of the error in the approximate solution $\sigma _{22,i-1}$. As the error $\sigma _{22}-\sigma _{22,i-1}$ decreases, its structure grows increasingly complex and contains high frequency modes. Thus, the first basis functions may be viewed as approximating low frequency error components while later basis functions approximate high frequency error components. A discussion and demonstration of the hierarchical nature of Galerkin Neural Networks alongside theoretical properties may be found in Ainsworth and Dong (2021).

A.3 MOR-Physics parameters

In this section, we provide additional details for the MOR-Physics operator parameterization used in “Data based machine learning for particle stresses” section. To perform training, we first partition the FCM simulations into 60% training data, 20% validation data, and 20% test data. Over the course of training, we track the validation loss and select the operator with the lowest validation loss as our learned operator. Below we list the hyperparameters we used in the studies,

Hyperparameter	Value
DNN width	64
DNN depth	32
DNN activation function	ELU
	(Clevert et al. 2015)
optimizer	Adam
	(Kingma and Ba 2015)
learning rate	$10^{-3}$
training steps	10000