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

复杂流体是两相共存的混合物。一些例子包括剃须膏、血液和显示器(LCD显示器)中使用的液晶，比如你现在可能正在阅读这篇摘要的那个。在微观尺度上，复杂流体的分子具有特殊的结构，在宏观尺度上影响应力应变的力学响应。例如，液晶分子在微观尺度上与电场反应，而电场在宏观尺度上改变通过材料的光的偏振。监视器利用这一特性，允许一定数量的红、绿或蓝光穿过每个像素。我们仅仅触及了复杂流体实现的可能性的皮毛。医学研究人员希望利用磁流体的微观特性来进行磁性药物靶向，以精确地控制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流体来组装纳米结构，如CPU中的硅电路。可以使用数学模型和计算机模拟来描述这些流体的动力学。这个研究项目的目标是设计和分析模拟液晶和磁流体行为的新的计算算法。这些算法将被用于模拟，这可能会补充并最终取代昂贵的物理实验。这一研究活动也有助于我们对复杂材料中图案形成的一般理解。铁磁流体和液晶的数学模型由偏微分方程组组成。由于所考虑的流体固有的细小尺度结构，这些偏微分方程组是高度非线性和耦合的。保持这些非线性偏微分方程解的能量平衡、长度和其他约束的离散形式，对于获得捕捉其动力学的真实场景的快速而稳定的数值格式是至关重要的。这项研究的目的是发展高效和收敛的有限体积和间断Galerkin方法，用于铁水动力学的Rosensweig模型、铁磁流体的多相流模型和液晶流的模型，在离散水平上模拟基本偏微分方程组的内在结构。所产生的算法将被实施并用于广泛的模拟，以与物理观测进行比较。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.