自适应和迭代方法用于PNP-斯托克斯方程的纳米孔模拟
Adaptive and Iterative Methods for Simulations of Nanopores with the PNP-Stokes Equations
AI总结：
本文提出了一种基于PNP-斯托克斯方程的3D有限元求解器，结合自适应网格细化方法，用于模拟纳米孔传感器内大分子动力学。
摘要：
我们提出了一种用于电扩散的非线性泊松-能斯特-普朗克（PNP）方程的3D有限元求解器，并与流体动力学的斯托克斯系统耦合。该模型是模拟纳米孔传感器内大分子动力学的基础构建块。我们通过目标导向的自适应网格细化方法扩展了现有的数值方法。为了减少网格自适应的计算开销，我们的误差估计器使用更便宜的泊松-玻尔方程作为简化模型，这在启发式基础上得到了验证，但在实践中表现良好。为了解决完整PNP-斯托克斯系统中的非线性问题，提出了三种不同的线性化方案，并研究了两种分离迭代方法，这两种方法均优于简单应用牛顿法。在真实世界纳米孔传感器几何上进行了数值实验。我们还研究了两种不同的模型，用于通过PNP-斯托克斯方程研究目标分子与纳米孔传感器的相互作用。在一种模型中，分子具有有限尺寸，并明确构建到几何中；而在另一种模型中，分子位于单一点上，并仅在系统求解后进行隐式建模，这在计算上更具优势。我们比较了电场和速度场对分子作用的力谱，并得出点尺寸模型无法捕捉重要物理效应的结论，例如传感器电荷选择性对分子半径的依赖性。
英文摘要：
We present a 3D finite element solver for the nonlinear Poisson-Nernst-Planck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule dynamics inside nanopore sensors. We add to existing numerical approaches by deploying goal-oriented adaptive mesh refinement. To reduce the computation overhead of mesh adaptivity, our error estimator uses the much cheaper Poisson-Boltzmann equation as a simplified model, which is justified on heuristic grounds but shown to work well in practice. To address the nonlinearity in the full PNP-Stokes system, three different linearization schemes are proposed and investigated, with two segregated iterative approaches both outperforming a naive application of Newton's method. Numerical experiments are reported on a real-world nanopore sensor geometry. We also investigate two different models for the interaction of target molecules with the nanopore sensor through the PNP-Stokes equations. In one model, the molecule is of finite size and is explicitly built into the geometry; while in the other, the molecule is located at a single point and only modeled implicitly -- after solution of the system -- which is computationally favorable. We compare the resulting force profiles of the electric and velocity fields acting on the molecule, and conclude that the point-size model fails to capture important physical effects such as the dependence of charge selectivity of the sensor on the molecule radius.
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