Finite element modeling of charged species transport has enabled analysis, design and optimization of a diverse array of electrochemical and electrokinetic devices. These systems are represented by the Poisson-Nernst-Plank equations coupled with the Navier-Stokes equation, with a key quantity of interest being the current at the system boundaries. Accurately computing the current flux is challenging due to the small critical dimension of the boundary layers (small Debye layer) that require fine mesh resolution at the boundaries. We resolve this challenge by using the Dirichlet-to-Neumann transformation to weakly impose the Dirichlet conditions for the Poisson-Nernst-Plank equations. The results obtained with weakly imposed Dirichlet boundary conditions showed excellent agreement with those obtained when conventional boundary conditions with highly resolved mesh were employed. Furthermore, the calculated current flux showed faster mesh convergence using weakly imposed conditions compared to the conventionally imposed Dirichlet boundary conditions. This approach substantially reduces the computational cost of modeling electrochemical systems.

中文翻译：

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弱强加狄利克雷边界条件的电化学系统建模

带电物质传输的有限元建模使各种电化学和电动装置的分析、设计和优化成为可能。这些系统由 Poisson-Nernst-Plank 方程与 Navier-Stokes 方程结合表示，其中一个关键的关注量是系统边界处的电流。由于边界层（小德拜层）的临界尺寸很小，需要边界处的精细网格分辨率，因此准确计算电流通量具有挑战性。我们通过使用 Dirichlet-to-Neumann 变换对 Poisson-Nernst-Plank 方程弱强加 Dirichlet 条件来解决这一挑战。使用弱强加 Dirichlet 边界条件获得的结果与使用具有高分辨率网格的常规边界条件时获得的结果非常吻合。此外，与传统施加的狄利克雷边界条件相比，使用弱施加条件计算出的电流通量显示出更快的网格收敛。这种方法大大降低了建模电化学系统的计算成本。