Multiscale Modeling and Computation of Multiphase Energetic Materials

多尺度系统在科学和工程中大量存在。它们的综合治疗需要在精细尺度上对过程进行准确的数学建模，并需要一个跨尺度的信息流动的数学框架。这项建议在非均质炸药的背景下解决了这一问题。这些颗粒具有复杂的微观结构，颗粒聚集体中存在含能材料的碎片、空洞和气孔。当受到足够强烈的冲击时，就会引爆。虽然结晶均质炸药具有较高的点火阈值，但相对较弱的刺激足以引发非均质聚集体。这是由离散地点或热点的出现引起的，在那里燃烧开始，然后蔓延到吞噬整个体积。系统的多尺度性质是任何尝试从头开始模拟爆炸现象的一个令人生畏的障碍，至少目前是这样。提出了两种主要的数学建模方法。这两种方法都在宏观尺度上建立了连续介质方程，其中一定程度的均匀化是隐式的，细尺度过程被包括为亚网格模型。第一种方法以点火-生长模型为代表，将炸药视为两种不同组分的均匀混合物，在压力和温度平衡下，未反应炸药和反应产物。给每个组分分配一个状态方程，并假定炸药转化为产物的反应速率定律是单一的。第二种方法明确地识别了炸药混合物的两相特性。由此得到的模型有每个相的质量、动量和能量的分离平衡定律，以及一个允许由两相之间的压力差驱动固相压实的规则。出现了表示质量、动量和能量的界面交换的术语，对应于反应、阻力和热传递的非平衡过程。这些模型是双曲型偏微分方程组，可视为气体动力学欧拉方程的推广。这一建议是为了研究现有的连续介质模型以及对细尺度现象的研究。主要目标是研究热点的强度和分布如何定量地取决于炸药的组成特性和点火刺激的大小，从而提供可用于尝试多尺度描述的重要信息。科学和工程中的许多领域，包括天气、燃烧、污染和生物系统等，都涉及由微观尺度上发生的事件决定的观察尺度上的行为。对这类系统的彻底处理一方面需要对微观尺度上的过程进行基本的了解和准确的数学建模，另一方面需要开发一个理论和计算框架，以促进信息在不同尺度上的流动，以便能够可靠和准确地预测观察尺度上的行为。在以科学为基础的爆炸装置管理的背景下，对这类系统的关注尤其及时和恰当，正是来自这一领域的问题构成了本提案的核心。

Multiscale systems abound in science and engineering. Their comprehensive treatment requiresaccurate mathematical modeling of processes at the fine scales and a mathematical framework forflow of information across scales. This proposal addresses this problem in the context of heterogeneousexplosives. These have a complex microstructure with fragments of the energetic material, voids andpores existing within the granular aggregate. When subjected to a sufficiently strong shock, a detonationis initiated. Although the crystalline homogeneous explosive has a high ignition threshold, relativelyweaker stimuli are sufficient to initiate the heterogeneous aggregate. This is caused by the appearanceof discrete sites, or hot spots, where burning commences and then spreads to consume the entire bulk.The multi-scale nature of the system is a daunting obstacle in the way of any attempt at ab-initio modelingof the detonation phenomena, at least at the present time. Two major approaches to mathematicalmodeling have been proposed. Both generate continuum equations at the macro scale, wherein a certaindegree of homogenization is implicit and fine-scale processes have been included as subgrid models.The first approach, typified by the ignition-and-growth model, treats the explosive as a homogeneousmixture of two distinct constituents, the unreacted explosive and the products of reaction, at pressure andtemperature equilibrium. To each constituent is assigned an equation of state, and a single reaction-ratelaw is postulated for the conversion of the explosive to products. The second approach explicitlyrecognizes the two-phase character of the explosive mixture. The resulting model has separatebalance laws of mass, momentum and energy for each phase, plus a rule that allows compaction ofthe solid phase driven by pressure difference between the phases. Terms representing interfacialexchange of mass, momentum and energy appear, corresponding to the nonequilibrium processes ofreaction, drag and heat transfer. These models are systems of hyperbolic partial differential equationsthat can be considered as generalizations of the Euler equations of gasdynamics. This proposal is aimedat studies of existing continuum models as well as the investigation of fine-scale phenomena. The primaryobjective will be to study how the strength and distribution of hot spots depend, in a quantitative way,upon the constitutive properties of the explosive and the size of the igniting stimulus, thus providingimportant information that can be used in attempts at multi-scale descriptions.Many areas in science and engineering, including weather, combustion, pollution, and biological systems,among others, involve behavior at the scale of observation that is determined by the events occurring atmicro scales. A thorough treatment of such systems requires, on the one hand, a fundamentalunderstanding and accurate mathematical modeling of processes at the micro scales, and on the other, development of a theoretical and computational framework that would facilitate flow of informationacross scales, so that behavior at the scale of observation can be predicted reliably and accurately.Attention to such systems is particularly timely and apt in the context of science-based stewardship ofexplosive devices, and it is problems from this arena that form the core of this proposal.