Transfer operator methods for modelling high-frequency wave fields - advancements through modern functional and numerical analysis

从噪声和振动到电磁波的高频波场建模是一项具有挑战性的任务。大规模复杂结构（如飞机、汽车或建筑物）的波浪模拟主要基于一类称为有限元技术的方法，这些方法仅在低频有效，结构的典型长度尺度与波长相当或小于波长。例如，汽车工业中的噪声和振动建模可以通过有限元技术可靠地进行，最高频率仅为500 Hz。另一种技术，称为动态能量分析（DEA），最近在诺丁汉开发，是基于计算能量流方程。它已被改进，以适用于真实的规模的结构，如大型集装箱船或拖拉机模型从洋马有限公司，一个拖拉机制造商从日本。该方法现在被工程界和工业界使用。DEA表现出丰富的数学结构，制定了一个运营商，被称为转移运营商，最初出现在混沌动力系统的理论。为了进一步提高该方法的适用性，需要进行深入的数学分析。该建议的目的是利用先进的工具，从功能分析，把DEA的坚实的基础上，同时，进一步提高效率的方法，在一个系统的方式。这是促进了最近的进展，转移算子方法和数值分析。前者可以增加构建新函数空间的灵活性，使算子具有良好的谱特性，后者是基于矩阵图算法对DEA矩阵使用块压缩和重新排序技术来实现的，以提高求解器效率并增强并行性。该项目的成员有专业知识，使这些不同的领域一起与玛丽皇后大学的伦敦领先的传输算子技术，诺丁汉大学带来了详细的知识，目前的实施DEA和诺丁汉特伦特大学具有数值分析技能的能量流方程的背景下。因此，该项目构成了一个典型的例子，其中纯数学通知应用数学和所产生的知识直接进入工业应用。

Modelling high-frequency wave fields ranging from noise and vibration to electromagnetic waves is a challenging task. Wave simulations for large-scale, complex structures such as aeroplanes, cars or buildings are mainly based on a class of methods, known as finite element techniques, which are efficient only at low frequencies with typical length-scales of the structure being comparable to or smaller than the wavelength. Noise and vibration modelling in the automotive industry, for example, can be performed reliably with finite element techniques only up to 500Hz. An alternative technique, termed Dynamical Energy Analysis (DEA), has recently been developed in Nottingham and is based on computing energy flow equations. It has been refined to be applicable to real scale structures such as a large container ship or a tractor model from Yanmar Co, Ltd, a tractor manufacturer from Japan. The method is now used both in the engineering community and by industry. DEA exhibits a rich underlying mathematical structure, formulated in terms of an operator, known as transfer operator, originally arising in the theory of chaotic dynamical systems. In order to advance the applicability of the method further, a thorough mathematical analysis is needed. The aim of this proposal is to exploit advanced tools from functional analysis to put DEA on sound foundations and, at the same time, improve the efficiency of the method further in a systematic way. This is facilitated by recent progress in transfer operator methods and numerical analysis. The former allows for an increased flexibility in constructing new function spaces on which the operator has good spectral properties, the latter is achieved using block compression and reordering techniques for the DEA matrix based on matrix graph algorithms to improve solver efficiency and enhance parallelism. The project members have the expertise to bring these diverse fields together with Queen Mary University of London leading in transfer operator techniques, the University of Nottingham bringing in detailed knowledge on current implementations of DEA and Nottingham Trent University having the numerical analysis skills in the context of energy flow equations. The project thus constitutes a prime example where pure mathematics informs applied mathematics and the arising knowledge is channelled straight into industrial applications.