Adaptive Numerical Methods for Optoelectronic Devices

液晶显示器已经成为现代生活的一个组成部分。从洗衣机上简单的黑白白色显示器到巨大的彩色电视机，液晶显示器被用来向用户提供各种信息。无论是告诉你要等多久才能洗完衣服，还是告诉你《老大哥》中的参赛者是如何应对的，人们对越来越复杂的显示器的需求越来越大。液晶显示器的销售数字年复一年地增加，并且随着许多主要电子公司关闭CRT电视工厂以集中于液晶显示器生产，液晶显示器的这种扩散在近期或中期的将来不太可能减缓。虽然一些显示器的性能足以满足许多应用，但仍有一些关键的市场领域需要改进。产生高清晰度图像的能力对于现代大屏幕电视以及较小的便携式显示器至关重要，在较小的便携式显示器中，降低所使用的功率也很重要。随着电视变得越来越大（最大的液晶电视现在是70）和便携式显示器变得越来越复杂，这些问题正在推动对新液晶显示技术的研究。特别地，利用液晶材料内的缺陷的能力已经导致了许多新颖的多稳态显示器，使它们能够在不从电池或电源获取电力的情况下显示图像。这将使显示器制造商能够生产更节能、更高清晰度的显示器。但是有一个问题。为了充分开发这种显示器，以最大限度地减少能源需求并最大限度地提高显示器的速度和分辨率，工业公司通常考虑其显示器的数学模型，以便可以对许多虚拟实验进行数值模拟。这些新型显示器的问题在于，虽然模型已经根据微分方程组来制定，但是没有能够近似方程的精确且鲁棒的数值算法。的微分方程，模拟复杂的三维区域中的流体动力学，以及通过显示器的离子和电压的运动，是非线性的，并表现出各种局部化的现象，包括奇点，严重降低标准的数值方法的性能。除了这些问题之外，为了将模型与实验进行比较，必须考虑光通过显示器时的行为，并且由于在这些设备中发现的复杂结构，还必须使用特殊的数值技术。然而，在使用自适应，高阶有限元方法的最新进展表明，这种方法是准确的和强大的意义上说，性能不退化时，缺陷或突变的结构存在。因此，本项目的主要目的是发展必要的理论，实现强大的和有效的高阶数值方法的三维数值模拟的液晶显示设备。这项工作将与位于布里斯托的惠普实验室合作进行，他们需要这些可靠的数值方法来将他们的新显示技术推向下一个发展阶段。特别令人兴奋的是，我们将能够立即将我们的新数值方法应用于最先进的工业应用。这些高阶有限元方法是当前数值分析研究中的一个热门话题，这一领域的进展将对求解偏微分方程的许多其他科学和工程领域产生重要影响。

Liquid crystal displays have become an integral part of modern life. From the simple black and white display on a washing machine to enormous colour televisions, liquid crystal displays are used to provide a variety of information to the user. Whether that display tells you how long you have to wait until the washing is done or how the contestants in Big Brother are coping, there is an increasing demand for more and more sophisticated displays. Year after year the sales figures for liquid crystal displays increase and with many major electronics companies shutting down their CRT television factories to concentrate on liquid crystal display production, this proliferation of liquid crystal displays is not likely to slow in the near or mid-term future. While the performance of some displays is sufficient for a number of applications there are key market areas where advancements are needed. The ability to produce a high definition image is crucial for modern large screen televisions as well as in smaller portable displays where reducing the power used is also important. As televisions become larger (the largest liquid crystal television is now 70 ) and portable displays become more sophisticated, these issues are fuelling research into new liquid crystal display technologies. In particular, the ability to harness defects within the liquid crystal material has led to a number of novel displays which are multistable, giving them the ability to display an image with no power taken from the battery or power source. This will enable display manufacturers to produce more energy efficient, higher definition displays. There is however a problem. In order to fully develop such displays, to minimise the energy requirements and maximise the speed and resolution of the display, industrial companies often consider mathematical models of their displays so that many virtual experiments can be simulated numerically. The problem with these new types of displays is that, whilst the models have been formulated in terms of sets of differential equations, there are no accurate and robust numerical algorithms capable of approximating the equations. The differential equations, which model the fluid dynamics in complicated three dimensional regions as well as the motion of ions and the voltage through the display, are nonlinear and exhibit various localised phenomena including singularities that severely degrade the performance of standard numerical methods. In addition to these problems, to compare the model to experiments, the behaviour of light as it passes through the display must be considered and because of the complex structures found in these devices special numerical techniques must also be used. However, recent advances in the use of adaptive, high-order finite element methods have shown that such methods are accurate and robust in the sense that the performance does not degenerate when defects or abrupt changes in the structures exist.The main aim of this project is therefore to develop the theory necessary to realise robust and efficient high-order numerical methods for three-dimensional numerical simulation of liquid crystal display devices. The work will be undertaken in collaboration with Hewlett-Packard Laboratories in Bristol, who need these reliable numerical methods to take their new display technologies to the next level of development. It is particularly exciting that we will be able to immediately apply our new numerical approach to state-of-the-art industrial applications. These high-order finite element methods are a hot topic in current numerical analysis research and advances in this area would have important implications for solving partial differential equations in many other areas of science and engineering.