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

用于模拟高频波场的传递算子方法 - 现代函数和数值分析的进步

基本信息

批准号：
EP/R012008/1
负责人：
Oscar Bandtlow
金额：
$ 80.89万
依托单位：
Queen Mary University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR012008%2F1
关键词：
Transfer operator methods modelling frequency

项目摘要

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.

建模从噪声和振动到电磁波的高频波场是一项艰巨的任务。用于大规模的复杂结构（例如飞机，汽车或建筑物）的波浪模拟主要基于一类方法，即被称为有限元技术，它们仅在典型的结构的典型长度尺度上有效，其结构的典型长度与波长相当或更小。例如，汽车行业中的噪声和振动建模可以通过仅高达500Hz的有限元技术可靠地执行。最近在诺丁汉开发了一种称为动力学分析（DEA）的替代技术，它基于计算能量流程方程。它已被完善，适用于来自日本的拖拉机制造商Yanmar Co，Ltd的大型集装箱船或拖拉机型号等真实结构。该方法现在在工程界和行业中都使用。 DEA表现出一种丰富的基本数学结构，该结构是根据操作员（称为转移操作员）最初在混乱的动力学理论中产生的。为了进一步提高该方法的适用性，需要进行彻底的数学分析。该提案的目的是利用高级工具从功能分析中利用DEA将DEA置于声音基础上，同时以系统的方式进一步提高了该方法的效率。转移操作员方法和数值分析的最新进展促进了这一点。前者允许在构建操作员具有良好光谱属性的新功能空间方面提高灵活性，后者是使用基于矩阵图算法的DEA矩阵的块压缩和重新排序技术实现的，以提高求解器效率并增强并行性。项目成员具有将这些不同领域的专业知识与伦敦皇后大学的转会操作员技术领导的专业知识，诺丁汉大学在能量流程方程的背景下具有有关DEA当前实施的详细知识和诺丁汉·特伦特大学的数值分析技能。因此，该项目构成了一个典型的例子，纯数学通知应用数学，而引起的知识直接将其引入工业应用。