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，就像你现在正在阅读这篇摘要时使用的显示器一样。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，这种结构影响着对应力和应变的力学响应。例如，液晶分子在微观尺度上对电场起反应，而在宏观尺度上，电场改变了穿过材料的光的偏振。监视器利用这一特性允许一定数量的红色、绿色或蓝色光通过每个像素。我们仅仅触及了复杂流体可能实现的表面。医学研究人员希望利用铁磁流体的微观特性进行磁性药物靶向，以精确控制药物能够与人体相互作用的部位。材料工程师希望使用复杂的流体来组装纳米结构，比如cpu中的硅电路。数学模型和计算机模拟可以用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新的计算算法来模拟液晶和铁磁流体的行为。该算法将用于模拟，可以补充并最终取代昂贵的物理实验。这项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尺度结构，这些偏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的能量平衡、长度和其他约束的离散版本对于获得捕获其动态的现实场景的快速和稳定的数值格式至关重要。本研究项目的目的是为铁磁流体动力学的Rosensweig模型、铁磁流体的多相流模型和液晶流动模型开发有效和收敛的有限体积和不连续Galerkin方法，以模拟离散水平下潜在偏微分方程的内在结构。所得到的算法将被实现并用于广泛的模拟，以与物理观测进行比较。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.