Algebraic Dimensional Reduction for Microfluidic Simulation

Abstract

Microfluidic devices exhibit a high-aspect ratio in that their channel-widths are much smaller than their overall lengths. High-aspect geometry leads to an unduly large finite element mesh, making the (otherwise popular) finite element method (FEM) a poor choice for modeling microfluidic devices. An alternate computational strategy is to exploit well-known analytical solutions for fluid flow over the narrow channels of a device, and then either (a) assume the same analytical solutions for the cross-flow regions, or (b) exploit these solutions to set-up artificial boundary conditions over the cross-flow regions. Such simplified models are computationally far superior to brute-force FEM, but do not support the generality or flexibility of FEM. In this paper, we propose a third strategy for exploiting the analytical solutions: (c) directly incorporate them into standard FE-based analysis via algebraic reduction techniques. The advantages of the proposed strategy are (1) designers can use standard computer-aided design/computer-aided engineering (CAD/CAE) environments to model, analyze, and postprocess microfluidic simulation; (2) well-established dual-weighted residuals can be used to estimate modeling errors; and (3), if desired, one can eliminate the dependency on analytical solutions over selected regions, and instead revert to brute-force FEM. The simplicity and generality of the proposed method is inherited from the model reduction process, so are its theoretical properties, while simultaneously its computational efficiency is inherited from the use of analytical solutions.