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。大型复杂结构（如飞机、汽车或建筑物）的波动模拟主要基于一类被称为有限元技术的方法，这种方法仅在低频时有效，并且结构的典型长度尺度与波长相当或小于波长。例如，汽车行业的噪声和振动建模可以通过有限元技术可靠地执行，最高可达500Hz。另一种技术，被称为动态能量分析（DEA），最近在诺丁汉发展起来，它是基于计算能量流方程。经过改进，适用于大型集装箱船或日本拖拉机制造商洋马公司的拖拉机模型等实际规模结构。该方法目前在工程界和工业界均有应用。DEA展示了一个丰富的潜在数学结构，用一个算子来表述，称为传递算子，最初出现在混沌动力系统理论中。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。本提案的目的是利用功能分析中的先进工具，为DEA奠定良好的基础，同时进一步系统地提高DEA的效率。最近在传递算子方法和数值分析方面的进展促进了这一点。前者允许在构造算子具有良好谱性质的新函数空间时增加灵活性，后者使用基于矩阵图算法的DEA矩阵块压缩和重排序技术来实现，以提高求解器效率并增强并行性。项目成员拥有将这些不同领域结合在一起的专业知识，伦敦玛丽女王大学在转移算子技术方面领先，诺丁汉大学在DEA的当前实施方面提供详细的知识，诺丁汉特伦特大学在能量流方程的背景下拥有数值分析技能。因此，该项目构成了一个主要的例子，其中纯数学为应用数学提供信息，并将产生的知识直接引入工业应用。

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.