Matched alternating direction implicit (ADI) schemes for solving the nonlinear Poisson-Boltzmann equation with complex dielectric interfaces

用于求解具有复杂介电界面的非线性泊松-玻尔兹曼方程的匹配交替方向隐式 (ADI) 方案

• 批准号: 1318898
• 负责人: Shan Zhao
• 金额: $ 25万
• 依托单位: University of Alabama Tuscaloosa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318898&HistoricalAwards=false
• 关键词: Matched; alternating; direction; implicit; ADI

The goal of the proposed project is to develop second order interface methods embedded in the alternating direction implicit (ADI) framework for solving the 3D nonlinear Poisson-Boltzmann (PB) equation with complex dielectric interfaces. Efficiency and accuracy are known to be the two major difficulties for solving the nonlinear PB equation numerically. The efficiency concern stems from the needs for solving the PB equation in demanding applications, such as one-time solution to systems with large spatial degrees of freedom, and/or million-time solutions in dynamical simulations. The accuracy concern is due to various challenging features of the PB model, including piecewisely-defined dielectric constants, a strong nonlinearity, singular point charges, and complex dielectric interfaces. Without addressing these features, fine meshes have to be used for a reliable simulation, which in turn impairs efficiency. In this project, a new pseudo-transient continuation formulation will be constructed based on a suitable regularization setting so that the singular charges are represented analytically. The nonlinear term of the PB equation will be integrated exactly with time splitting techniques. To deal with piecewise dielectric constants, a tensor product decomposition of 3D interface conditions will be carried out to derive essentially 1D jump conditions so that the dielectric interface can be accommodated along each Cartesian direction in an alternating manner. Fast algebraic solvers will be developed for solving matrices of each Cartesian direction. Consequently, the proposed matched ADI approaches not only maintain both the simplicity of Cartesian grids and the efficiency of the Thomas algorithm, but also achieve spatially second order accuracy in resolving complex dielectric interfaces.The electrostatic interactions are vital not only for the study of biological and chemical systems and processes at the molecular level, but also for the design of semiconductor devices at the nanoscale. The PB model, in which the electrostatic interactions are computed implicitly via a mean force approach, can surprisingly well describe the electrostatics of a charged system. This model finds broad applications in science and engineering, such as modeling the charged polymers and surfactants in interface and colloid science, studying transistors on very large scale integration (VLSI) semiconductor devices in nanotechnology, and analyzing structure, function, and dynamics of solvated biomolecules including proteins and DNAs in molecular biology. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in interdisciplinary fields involving computational mathematics, chemistry, biology, and electrical engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the electrostatic analysis of various physical, chemical, and biological systems/devices. In addition, this project will provide interdisciplinary research and training opportunities for students pursing careers in science and engineering.

该项目的目标是开发嵌入交替方向隐式（ADI）框架的二阶界面方法，用于求解具有复杂介电界面的三维非线性Poisson-Boltzmann（PB）方程。求解非线性PB方程的两个主要困难是效率和精度。效率问题源于在要求苛刻的应用中求解PB方程的需要，例如具有大空间自由度的系统的一次性解决方案，和/或动态模拟中的百万次解决方案。准确性问题是由于PB模型的各种挑战性特征，包括分段定义的介电常数，强非线性，奇异点电荷和复杂的介电界面。如果不解决这些特征，就必须使用精细网格来进行可靠的模拟，这反过来又会降低效率。在这个项目中，一个新的伪瞬态延拓公式将构建一个合适的正则化设置的基础上，使奇异电荷表示解析。PB方程的非线性项将用时间分裂技术精确积分。为了处理分段介电常数，将进行3D界面条件的张量积分解以导出基本上1D跳跃条件，使得介电界面可以以交替的方式沿着每个笛卡尔方向被容纳。快速代数求解器将开发用于求解每个笛卡尔方向的矩阵。因此，所提出的匹配ADI方法不仅保持了笛卡尔网格的简单性和托马斯算法的高效性，而且在求解复杂介电界面时达到了空间二阶精度。静电相互作用不仅对分子水平上的生物和化学系统和过程的研究至关重要，而且对纳米尺度上的半导体器件的设计也至关重要。PB模型，其中的静电相互作用计算隐式通过平均力的方法，可以令人惊讶地很好地描述带电系统的静电。该模型在科学和工程中有着广泛的应用，例如在界面和胶体科学中建模带电聚合物和表面活性剂，在纳米技术中研究超大规模集成（VLSI）半导体器件上的晶体管，以及在分子生物学中分析溶剂化生物分子（包括蛋白质和DNA）的结构、功能和动力学。拟议的数学建模，算法开发和数值计算将解决跨学科领域的关键科学挑战，涉及计算数学，化学，生物学和电气工程。计划中的研究活动将为计算数学带来新的进展，并为各种物理，化学和生物系统/设备的静电分析提供可靠的模拟工具。此外，该项目将为从事科学和工程职业的学生提供跨学科的研究和培训机会。