Aaron Baier-Reinio

Aaron Baier-Reinio, Patrick E. Farrell, Charles W. Monroe

We present a broad family of high-order finite element algorithms for simulating the flow of electroneutral electrolytes. The governing partial differential equations that we solve are the electroneutral Navier--Stokes--Onsager--Stefan--Maxwell (NSOSM) equations, which model momentum transport, multicomponent diffusion and electrical effects within the electrolyte. Our algorithms can be applied in the steady and transient settings, in two and three spatial dimensions, and under a variety of boundary conditions. Moreover, we allow for the material parameters (e.g. viscosity, diffusivities, thermodynamic factors and density) to be dependent on the local state of the mixture and thermodynamically non-ideal. The flexibility of our approach requires us to address subtleties that arise in the governing equations due to the interplay between boundary conditions and the equation of state. We demonstrate the algorithms in various physical configurations, including (i) electrolyte flow around a microfluidic rotating disk electrode and (ii) the flow in a Hull cell of a cosolvent electrolyte mixture used in lithium-ion batteries.

Kars Knook, Aaron Baier-Reinio, Patrick E. Farrell

The Onsager-Stefan-Maxwell (OSM) equations are an important model of mass transport in multicomponent flows with multiple chemical species. They describe the coupling of diffusive fluxes between species, accounting for their interactions through frictional and thermodynamic driving forces. In this work we propose an augmented Lagrangian preconditioner and prove its discretization-robustness for a Picard linearization of the stationary OSM equations in the isobaric, isothermal, ideal gaseous setting. For the Newton linearization we employ the augmented Lagrangian preconditioner as a block diagonal smoother inside a monolithic geometric multigrid iteration and combine with vertex star Schwarz methods. This strategy is shown to be applicable in a wide variety of settings which incorporate cross-diffusion, nonideal mixing, thermal, pressure, convective, and electrochemical effects. We demonstrate robustness or mild dependence with respect to mesh refinement and polynomial degree of the proposed monolithic preconditioning strategy for different types of multicomponent flows in several applications: cross-diffusion in the human airways, separation of gases under a temperature gradient, nonideal mixing of benzene and cyclohexane, and electrolytic transport in a Hull cell undergoing electroplating.

Aaron Baier-Reinio, Hans De Sterck

We use neural ordinary differential equations to formulate a variant of the Transformer that is depth-adaptive in the sense that an input-dependent number of time steps is taken by the ordinary differential equation solver. Our goal in proposing the N-ODE Transformer is to investigate whether its depth-adaptivity may aid in overcoming some specific known theoretical limitations of the Transformer in handling nonlocal effects. Specifically, we consider the simple problem of determining the parity of a binary sequence, for which the standard Transformer has known limitations that can only be overcome by using a sufficiently large number of layers or attention heads. We find, however, that the depth-adaptivity of the N-ODE Transformer does not provide a remedy for the inherently nonlocal nature of the parity problem, and provide explanations for why this is so. Next, we pursue regularization of the N-ODE Transformer by penalizing the arclength of the ODE trajectories, but find that this fails to improve the accuracy or efficiency of the N-ODE Transformer on the challenging parity problem. We suggest future avenues of research for modifications and extensions of the N-ODE Transformer that may lead to improved accuracy and efficiency for sequence modelling tasks such as neural machine translation.