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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.

对从噪声、振动到电磁波的高频波场进行建模是一项具有挑战性的任务。对飞机、汽车或建筑物等大型复杂结构的波浪模拟主要基于一类称为有限元技术的方法，这种方法只有在结构的典型长度尺度与波长相当或小于波长的低频下才有效。例如，汽车工业中的噪声和振动建模只需使用高达500赫兹的有限元技术就可以可靠地执行。另一种技术，称为动态能量分析(DEA)，最近在诺丁汉开发，基于计算能量流方程。它已被改进为适用于实际规模的结构，如大型集装箱船或日本拖拉机制造商延马株式会社的拖拉机模型。这种方法现在工程界和工业界都在使用。数据包络分析具有丰富的基本数学结构，用一种被称为转移算子的算子来表示，这种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。这一建议的目的是利用功能分析的先进工具，使DEA建立在坚实的基础上，同时以系统的方式进一步提高该方法的效率。转移算符方法和数值分析的最新进展促进了这一点。前者在构造具有良好谱特性的新函数空间时具有更大的灵活性，后者是通过基于矩阵图算法的DEA矩阵的块压缩和重新排序技术来实现的，以提高求解器的效率并增强并行性。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操作员技术方面处于领先地位，诺丁汉大学提供了关于DEA当前实施的详细知识，诺丁汉特伦特大学拥有能量流动方程方面的数值分析技能。因此，该项目构成了一个最好的例子，在那里，纯数学为应用数学提供了信息，所产生的知识直接转化为工业应用。