Fast algorithms for high fidelity simulation of viscous suspension flows

悬浮粒子在自然界中普遍存在，对悬浮粒子的研究涉及到基础科学和技术的许多领域。模拟它们有助于我们理解物质的性质，如干扰和相变，使我们能够研究生物结构的自发形成，如细胞膜和细胞骨架，它使我们能够设计大量的智能材料，从Kevlar和假体中的阻尼液到自组装纳米材料。然而，定义这些粒子系统的属性使它们的模拟非常具有挑战性，因为我们必须长时间准确地再现它们的行为。该研究项目的总体目标是为这些颗粒系统的数值模拟提供最先进的求解器的变革性改进。通过设计，本项目中的每一项贡献都对多物理场模拟和科学计算的数值方法产生了可预见的广泛影响，并最终通过使模拟方法弥合理论与实验之间的差距来提高我们的科学和技术能力。该项目与正在进行的教育活动相结合，包括开发新的应用数学课程，并为不同的学生群体提供研究机会。该项目将包括培养研究生。该研究项目涉及开发刚性颗粒悬浮液的快速模拟框架。这项工作的重点是用边界积分方法制定长程力，并采用基于优化的方法制定短程接触力，因为这是有效解决稠密粒子系统中多体相互作用的理想方法。该框架包括三个独立的贡献，旨在释放这一方法的全部潜力：（1）快速奇异和近奇异重新估值方案，用于解决远程粒子相互作用并分析球形、椭圆形和轴对称几何形状的积分算子，从而能够研究广泛的粒子系统和限制几何形状，（2）用于演化几何中的边界积分方程的结构化预条件子，其利用关于结构化矩阵和张量列分解的先前工作来加速所得线性方程组的解，以及（3）系统地，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Particle suspensions are ubiquitous in nature; their study is featured in many areas of fundamental science and technology. Simulating them helps us understand properties of matter like jamming and phase transition, enables us to study of spontaneous formation of biological structures like cell membranes and cytoskeletons, and it allows us to design a vast array of smart materials, from damping fluids in Kevlar and prosthesis to self-assembling nanomaterials. However, the very properties that define these particle systems make them very challenging to simulate, as we must accurately reproduce their behavior for long periods of time. The overarching goal of this research project is to provide transformative improvements to state-of-the-art solvers for the numerical simulation of these particulate systems. By design, each of the contributions in this project has, on its own, foreseeable broad impact in multiphysics simulation and in numerical methods for scientific computing, and ultimately, in advancing our scientific and technological capabilities by enabling simulation methods to bridge the gap between theory and experiment. This project is integrated with ongoing educational initiatives, including the development of novel applied mathematics curricula and providing research opportunities for a diverse group of students. The project will include training of graduate students.This research project involves the development of a fast simulation framework for rigid particulate suspensions. This work is centered around the formulation of long-range forces with Boundary Integral Methods and of short-ranged contact forces employing optimization-based methods, as this is ideal to effectively tackle the many-body interactions in dense particle systems. This framework features three separate contributions aimed at unlocking the full potential of this approach: (1) Fast singular and near-singular revaluation schemes to resolve long-range particle interactions and analyze integral operators for spheroidal, ellipsoidal and axis-symmetric geometries, enabling the study of a wide array of particle systems and confining geometries, (2) Structured preconditioners for boundary integral equations in evolving geometries leveraging prior work on structured matrices and tensor train decompositions to accelerate the solution of resulting linear systems of equations, and (3) Systematic, robust and adaptive acceleration method for optimization-based collision resolution using matrix-splitting schemes for viscous flow mobility matrices, which may be adapted to most formulations for particulate suspensions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.