Collaborative Research:Computational and Data-Enabled Science and Engineering: Characterizing Dynamics of Particle-based Systems

应用科学和工程领域的研究主要集中在离散对象构建的系统上。这包括与材料科学相关的系统，如干湿颗粒系统，也包括许多其他软物质系统，如泡沫、胶体和液晶。还有一个日益相关和活跃的活性物质领域，在这个领域中，我们感兴趣的系统是由受某种内力支配的粒子组成的，比如细菌和类似的东西。回到与材料科学相关的系统，人们可以注意到重要的应用，仅在美国每年就涉及数万亿美元。尽管由颗粒组成的系统广泛出现，但我们预测它们行为的能力远远落后于预测牛顿流体等更传统的材料。类似的，甚至更有力的结论，可以得出其他基于颗粒的系统，刚刚开始考虑。许多列出的系统缺乏基于连续体的模型，需要进行离散元素模拟，重点是模拟粒子-粒子相互作用。由于计算能力的提高，目前的模拟能够提供实验系统的真实描述，并且通常可以与预测能力一起使用。然而，描述粒子及其相互作用的空间和时间尺度的分离，以及在考虑整个系统的特性时感兴趣的描述中观或宏观尺度的分离，导致越来越大且基本上无法管理的数据量。本提案的重点是开发一种基于计算同调的技术，通过从这些大数据集中提取所需的信息，使我们能够更深入地了解所考虑的复杂系统的动态特性。本研究以拓扑数据分析为基础，特别关注基于持久同调的技术开发、非线性动力学的代数拓扑技术、计算能够识别和表征复杂时空系统非线性动力学的同调不变量的算法和软件。这些技术将应用于将并行开发的基于微粒的系统的离散元素模拟的结果。这些模拟将考虑在二维和三维空间中，由引力和排斥力相互作用的大量粒子。我们将考虑圆形/球形粒子，以及多边形/多面体形状的粒子。作为拟议项目的结果，我们期望对所考虑的系统的动态特性有更好的理解，然后将其传递给从事其应用的科学家和工程师。这个项目的成功意义深远。开发新的计算高效的数学工具来理解和预测大规模数据集上复杂模式的动态，为分析涉及复杂非平衡系统的广泛问题提供了基础。在以颗粒为基础的系统中，这包括(i)由颗粒通过排斥力相互作用形成的干燥颗粒物质，例如来自大自然的例子-包括雪崩，泥石流和地震，技术-煤，矿石和药品的加工；（ii）由粒子通过排斥和吸引的组合相互作用构建的“湿”系统，特别是与多孔介质应用相关的；（iii）许多多相系统，包括悬浮液和活性物质。在所有这些系统中，需要理解和预测特殊结构的复杂行为。

A significant part of the research in applied fields of science and engineering focuses on the systems built up from discrete objects. This includes the systems relevant to materials science, such as dry and wet granular systems, but also many other soft-matter systems such as foams, colloids, and liquid crystals. There is also an increasingly relevant and active field of active matter where the systems of interest are built out of particles governed by some type of internal forces, such as bacteria and the similar. Going back to the systems relevant to materials science, one could note important applications, involving trillions of dollars per year in the US alone. Despite wide-ranging appearance of systems built out of granular particles, our ability to predict their behavior lags far behind that for more conventional materials such as Newtonian fluids. Similar, even stronger, conclusions could be reached for the other particulate-based systems that are just becoming to be considered. Lack of continuum-based models for many of the listed systems requires carrying out discrete element simulations that focus on modeling particle-particle interactions. Due to increased computational power, current simulations are able to provide realistic description of the experimental systems and can often be used with predictive power. However, the separation of spatial and temporal scales describing particles and their interactions, and of those describing meso- or macro-scales that are of interest when considering properties of a system as a whole, leads to increasingly large and essentially unmanageable amount of data. This proposal focuses on development of a technique, based on computational homology, which allows us to reach deeper understanding of the dynamical properties of the considered complex systems by extracting required information from these large data sets.The proposed work is based on topological data analysis, and in particular it focuses on development of techniques based on persistent homology, algebraic topological techniques from nonlinear dynamics, and algorithms and software to compute homological invariants that are capable of identifying and characterizing the nonlinear dynamics of complex spatio-temporal systems. These techniques will be applied to the results of discrete element simulations of particulate-based systems that will be developed in parallel. These simulations will consider large number of particles interacting by both attractive and repulsive forces, both in two and three spatial dimensions. We will consider circular/spherical particles, as well as the particles of polygonal/polyhedral shapes. As an outcome of the proposed project, we expect to develop much better understanding of the dynamical properties of the considered systems, which will be then passed to scientists and engineers working on their applications. The implications of success in this project are far reaching. Developing new computationally efficient mathematical tools for understanding and predicting the dynamics of complex patterns on large scale data sets provides the foundations for the analysis of a wide range of problems involving complex nonequilibrium systems. In the context of particulate-based systems, this includes (i) dry granular matter built out of particles interacting by repulsive force, with the examples coming from nature - including avalanches, debris flows, and earthquakes, technology - processing of coal, ores, and pharmaceuticals; (ii) `wet' systems built out of particles interacting by a combination of repulsion and attraction, in particular relevant to porous media applications, and (iii) a number of multiphase systems including suspensions and active matter. In all of these systems, understanding and prediction of complex behavior of special structures is desired.