Fast algorithms for high fidelity simulation of viscous suspension flows

粒子悬浮物在自然界中无处不在；他们的研究在基础科学和技术的许多领域都有特色。模拟它们有助于我们理解物质的特性，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，从凯夫拉尔和假体中的阻尼流体到自组装纳米材料。然而，定义这些粒子系统的特性使得模拟它们非常具有挑战性，因为我们必须在很长一段时间内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解决方案的变革性改进。通过设计，本项目中的每一项贡献都将对多物理场模拟和科学计算的数值方法产生可预见的广泛影响，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，包括开发新颖的应用数学课程，并为不同群体的学生提供研究机会。该项目将包括研究生的培训。该研究项目涉及刚性颗粒悬浮液的快速模拟框架的开发。这项工作主要围绕着边界积分法的远程力和基于优化方法的短程接触力的公式展开，因为这是有效解决密集粒子系统中多体相互作用的理想方法。该框架有三个独立的贡献，旨在释放这种方法的全部潜力：(1)快速奇异和近奇异重估方案，用于解决远距离粒子相互作用和分析球面、椭球面和轴对称几何的积分算子，使研究广泛的粒子系统和限制几何成为可能；(2)利用结构化矩阵和张量列分解的先前工作来加速求解得到的线性方程组的演化几何边界积分方程的结构化预条件；(3)使用粘性流动迁移矩阵的矩阵分裂方案进行基于优化的碰撞解决的系统、鲁棒和自适应加速方法，该方法可适用于大多数颗粒悬浮液的公式。该奖项反映了美国国家科学基金会的法定使命，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.