CAREER: Fast Algorithms for Particulate Flows

这个教师早期职业发展（CAREER）项目的目标是建立稳定，高精度和最佳的算法，用于颗粒流的直接数值模拟。在自然界和工程系统中，可变形颗粒在粘性流体中的稠密悬浮是普遍存在的。实例包括液滴、气泡、囊泡、游泳者和血细胞悬浮液。与简单的牛顿流体不同，由于可变形的微观结构和宏观尺度流动之间的复杂相互作用，描述其流动行为的定律还没有很好地建立。例如，软颗粒之间的相互作用修改它们的轨迹并引起剪切诱导扩散，并且与限制壁的相互作用控制悬浮液的空间组织。除了实验之外，直接数值模拟通常是了解这种复杂流体的非平衡行为的唯一手段。因此，需要可扩展的鲁棒和最佳算法。与研究工作相结合，该项目将开展教育，指导和宣传活动，包括一个新的跨学科研究生课程复杂流体的计算方法和互动教育模块，说明在复杂的流体系统中观察到的非直观现象，高中学生可以访问。 该项目的具体目标包括（i）高精度算法来计算边界积分方法中出现的近似奇异积分，当粒子在流动中彼此非常接近时，（ii）快速，高阶和自适应算法来模拟通过任意周期几何形状的多相流，以及（iii）用于刚性、非线性、时间相关的微分和积分微分方程的耦合系统的稳定时间推进和重新参数化方案，所述微分和积分微分方程控制粒子的演化和粒子表面上的某些物质浓度（例如，表面活性剂或多相脂质）。拟议的计算基础设施将导致更好的预测能力，通过复杂的几何形状，血小板和靶向载体的边缘化，以及新的微流体设备的设计工具的血液流动。它们可以应用于一个大类的颗粒流问题。除了流体力学，所提出的数值方法和计算基础设施可以应用于解决偏微分方程，出现在各种其他学科。研究成果将对科学和工程领域的广泛学科产生影响。

The objective of this Faculty Early Career Development (CAREER) project is to build stable, high-accuracy, and optimal algorithms for direct numerical simulations of particulate flows. Dense suspensions of deformable particles in viscous fluids are ubiquitous in natural and engineering systems. Examples include drop, bubble, vesicle, swimmer, and blood cell suspensions. Unlike simple Newtonian fluids, the laws describing their flow behavior are not well established, owing to the complex interplay between the deformable micro-structure and the macro-scale flow. For instance, interactions between soft particles modify their trajectories and cause shear-induced diffusion, and interaction with confining walls controls the spatial organization of the suspensions. Besides experiments, direct numerical simulations are often the only means for gaining insights into the non-equilibrium behavior of such complex fluids. Hence, there is a need for robust and optimal algorithms that are scalable. Integrated with the research effort, this project will undertake educational, mentoring, and outreach activities including a new interdisciplinary graduate-level course on computational methods for complex fluids and an interactive education module illustrating the non-intuitive phenomena observed in complex fluid systems that is accessible to high school students. The specific aims of this project include (i) highly accurate algorithms to compute the nearly singular integrals that arise within the context of boundary integral methods when particles approach very close to each other when subjected to flow, (ii) fast, high-order, and adaptive algorithms to simulate multiphase flows through arbitrary periodic geometries, and (iii) stable time-marching and reparameterization schemes for the coupled systems of stiff, nonlinear, time-dependent differential and integro-differential equations governing the evolution of particles and some material concentration on their surfaces (e.g., surfactants or multi-phase lipids). The proposed computational infrastructure will lead to better predictive capabilities for blood flow through complex geometries, margination of platelets and targeted carriers, and new design tools for microfluidic devices. They can be applied to a large class of particulate flow problems. Besides fluid mechanics, the proposed numerical methods and the computational infrastructure can be applied to solve partial differential equations that arise in various other disciplines. The research outcomes will have an impact on a broad spectrum of disciplines in sciences and engineering.