Chemotaxis Models in Biology and Texture Development in Materials: Numerical Methods, Analysis, and Modeling

拟议项目的重点是开发新的数值和分析技术来研究生物学和材料科学中的问题。这些问题给分析、建模和仿真带来了挑战性的问题。在这个项目中，直接从潜在的应用中选择了两个领域进行研究：生物学中的趋化和趋化模型以及材料微观结构的发展和演化。趋化性是指细胞对外界刺激(通常是化学刺激)做出反应时发生运动的机制。趋化作用在许多医学和生物学应用中是一个重要的过程，包括细菌/细胞聚集和模式形成机制，以及肿瘤生长。生物系统的数学模型是研究这些模式的重要工具。虽然关于这一主题的文献很多，但针对趋化模型提出的数值方法还很少。趋化系统通常用高度非线性、含时的偏微分方程组来描述。因此，发展准确、高效的数值方法对趋化性的建模和分析至关重要。此外，所有现有趋化系统的一个共同属性是它们能够对集中现象进行建模，这种集中现象在数学上导致解决方案，这些解决方案在聚集点/曲线的小社区快速增长。解决方案可能会爆炸，或者可能会表现出非常独特的尖峰行为。这种爆炸代表了对真实生物系统中发生的细胞集中现象的数学描述。在任何一种情况下，从数字上捕捉这样的解都是一个非常具有挑战性的问题。第一个项目的目标是设计、实施和分析新的准确和高效的数值方法，并开发新的分析方法来研究趋化模型以及与物理和生物学密切相关的问题。第二个项目的目标是研究材料中织构的发展和微观结构的演变。蜂窝网络在自然界中无处不在。它们在许多不同的长度和时间尺度上表现出行为，并且通常是亚稳定的。大多数技术上有用的材料是由无数称为颗粒的小晶体组成的多晶微结构，它们被称为晶界的界面隔开。晶界网络的能量学和连通性在决定材料在大范围尺度上的性质方面起着至关重要的作用。材料科学研究的一个中心目标是开发能够产生一种适合于所需材料特性的颗粒排列--一种纹理--的技术。第二个项目的主要目标是了解能量在材料纹理发展中的作用。为此，引入了一种新发现的晶界特征分布，并利用大规模的模拟和数学分析对其进行了研究。晶界特征分布是一种新的织构表征方法，它与界面能有很强的相关性。这项研究将带来新的分析/计算工具来研究颗粒网络，以及更好地了解晶界分布、晶界属性以及它们在材料加工过程中如何演变。通往新科学发现的道路取决于新的研究领域、跨学科合作和新的机会。拟议的项目将涉及跨学科研究，并将通过开发新的分析和计算工具来加强基础设施。拟议工作的第一部分将对开发新的数值和分析方法做出根本性贡献，这些方法将用于解决生物医学问题，例如，在更好地理解癌症以及启动学科之间的新合作方面。在提案的第二部分，项目的这一部分将产生新知识和新工具，用于提高所用材料的可靠性，例如用于飞机、微处理器和许多其他设备的材料。拟议项目的教育活动包括指导研究生和本科生以及开发新的建模课程。

The focus of the proposed project is to develop new numerical and analytical techniques to study problems in Biology and Materials Science. These problems give rise to challenging issues for analysis, modeling, and simulations. In this project two areas have been selected for the research directly from potential applications: chemotaxis and chemotaxis models in Biology and texture development and evolution of microstructure in Materials. Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Mathematical models of the biological systems are an important tool used in the study of these patterns. Although there is an extensive literature on this subject, only a few numerical methods have been proposed for chemotaxis models. Chemotaxis systems are usually described by highly nonlinear time-dependent partial differential equations. Therefore, development of accurate and efficient numerical methods is crucial for the modeling and analysis of the chemotaxis. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions, which rapidly grow in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. This blow-up represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems. In either case, capturing such solutions numerically is a very challenging problem. The goal of the first project is to design, implement and analyze novel accurate and efficient numerical methods, as well as develop new analytical methods to study chemotaxis models along with closely related problems in physics and biology. The goal of the second project is to study texture development and evolution of microstructure in Materials. Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small crystallites, called grains, separated by interfaces, called grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central goal of research in materials science is to develop technologies capable of producing an arrangement of grains -a texture- appropriate for a desired set of material properties. The main objective of the second project is to understand the role of energy in material texture development. For this, a recently discovered Grain Boundary Character Distribution is introduced and investigated by the use of a large scale simulation and mathematical analysis. Grain Boundary Character Distribution is a new characterization of the texture which is found to be strongly correlated to the interfacial energy. This research will lead to new analytical/computational tools to study grain networks, along with a better understanding of grain boundary distributions, grain boundary properties, and how they evolve during materials processing.The path to new scientific discoveries lies through new areas of research, interdisciplinary collaboration, and new opportunities. The proposed projects will involve interdisciplinary research and will enhance infrastructure through the development of new analytical and computational tools. The first part of the proposed work, will make fundamental contributions to the development of new numerical and analytical methods which will be used in solving biomedical problems, for example in developing a better understanding of cancer, as well as initiating new collaboration among disciplines. In the second part of the proposal, both new knowledge and new tools will emerge from this part of the project, which will be used to increase the reliability of materials used, for example in aircraft, microprocessors, and many other devices. Educational activities for the proposed projects include the mentorship of graduate and undergraduate students and the development of new modeling course.