Design and Analysis of Structure Preserving Discretizations to Simulate Pattern Formation in Liquid Crystals and Ferrofluids

模拟液晶和铁磁流体中图案形成的结构保持离散化的设计和分析

• 批准号: 2409989
• 负责人: Franziska Weber
• 金额: $ 19.99万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-01-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2409989&HistoricalAwards=false
• 关键词: Design; Analysis; Structure; Preserving; Discretizations

Complex fluids are mixtures that have a coexistence between two phases. Some examples include shaving cream, blood, and the liquid crystals used in displays (LCD displays) like the one you are probably using right now to read this abstract. On a microscopic scale, the molecules of complex fluids have a special structure, which at a macroscopic scale affects the mechanical response to stress and strain. For instance, the molecules of liquid crystals react to electric fields on a microscopic scale, which on a macroscopic scale changes the polarization of the light passing through the material. Monitors take advantage of this property to allow a certain amount of red, green, or blue light through each pixel. We have barely scratched the surface of what is possible to achieve with complex fluids. Medical researchers hope to exploit the microscopic properties of ferrofluids for magnetic drug targeting, to control with precision the parts of the human body the drug is able to interact with. Materials engineers hope to use complex fluids to assemble nano-structures such as the silicon circuits in CPUs. Mathematical models and computer simulations can be used to describe the dynamics of these fluids. The goal of this research project is to design and analyze new computational algorithms that simulate the behavior of liquid crystals and ferrofluids. The algorithms will be used in simulations which may complement and ultimately replace expensive physical experiments. This research activity may also contribute to our general understanding of pattern formation in complex materials.Mathematical models for ferrofluids and liquid crystals consist of systems of partial differential equations. Due to the inherent fine scale structure of the fluids under consideration, these partial differential equations are highly nonlinear and coupled. Preserving discrete versions of energy balances, length and other constraints of the solutions of these nonlinear partial differential equations is crucial for obtaining fast and stable numerical schemes that capture realistic scenarios of their dynamics. The aim of this research project is to develop efficient and convergent finite volume and discontinuous Galerkin methods for the Rosensweig model of ferrohydrodynamics, multi-phase flow models of ferrofluids, and models of liquid crystal flows, that mimic the intrinsic structure of the underlying partial differential equations at the discrete level. The resulting algorithms will be implemented and used for extensive simulations to compare to physical observations.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.

复杂的流体是在两个阶段之间共存的混合物。一些例子包括剃须膏，血液和显示器中使用的液晶（LCD显示器），例如您现在可能正在使用的摘要来读取此摘要。在微观尺度上，复杂流体的分子具有特殊的结构，在宏观尺度上会影响对应激和应变的机械响应。例如，液晶的分子以微观尺度对电场反应，这在宏观尺度上改变了穿过材料的光的极化。监视器利用此属性，通过每个像素允许一定量的红色，绿色或蓝色光。我们几乎没有刮擦使用复杂的流体来实现的表面。医学研究人员希望利用铁体流体的显微镜特性来靶向磁性药物，以精确控制该药物能够与人体相互作用的部分。材料工程师希望使用复杂的液体组装纳米结构，例如CPU中的硅电路。数学模型和计算机模拟可用于描述这些流体的动力学。该研究项目的目的是设计和分析新的计算算法，这些算法模拟液晶和铁氟烷的行为。该算法将用于模拟中，这些算法可能会补充并最终取代昂贵的物理实验。这项研究活动也可能有助于我们对复杂材料中模式形成的一般理解。铁氟烷和液晶的数学模型由部分微分方程的系统组成。由于所考虑的流体的固有尺寸结构固有，这些部分微分方程是高度非线性和耦合的。保留这些非线性偏微分方程解决方案的能量平衡，长度和其他约束的离散版本对于获得快速，稳定的数值方案至关重要，以捕获其动力学的真实场景。该研究项目的目的是开发有限和收敛的有限体积和不连续的盖素方法，用于针对铁水力动力学的Rosensweig模型，铁氟烷的多相流模型以及液晶流的模型，这些模型模仿了离散级别的基础偏微分方程的固有结构。由此产生的算法将被实施并用于广泛的模拟，以与物理观察相比。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来获得支持的。