Next generation finite element methods for wave problems

波浪问题的下一代有限元方法

• 批准号: EP/H004009/2
• 负责人: Timo Betcke
• 金额: $ 77.49万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH004009%2F2
• 关键词: Next; generation; finite; element; methods

Efficient and accurate simulation of wave phenomena is a key enabling technology across science and engineering. Applications span diverse areas, and include the whole of acoustics and noise control, non-destructive testing and ultrasonic and microwave technologies for medical imaging, problems of seismic and radar propagation and imaging, and even quantum scale simulations. But even though the underlying partial differential equations are usually linear and well understood, wave phenomena are complex and hard to simulate whenever the wavelength is small compared to the diameter of the region to be simulated.A main and standard computational tool for simulations of wave problems is the so-called finite element method. The idea of the method is to break up the computational domain into small elements and to approximate the solution on each of them in a simple way, e.g. as a linear variation. However, this gives accurate solutions only if the diameter of each element is small compared to the wavelength. Thus the number of elements needed and the associated computational cost and storage is infeasible if the diameter of the region to be simulated is very large compared to the wavelength, as it is for very many complex problems of wave propagation and scattering, e.g. seismic wave propagation for hydrocarbon exploration.Recently, there has been strong international interest in novel finite element formulations that try to solve this problem by representing the wave field on each element by functions that are themselves waves. This allows much bigger element sizes and so a significant reduction of the computational cost. However, these novel finite element methods are still in their infancy and it is poorly understood how to implement them in an optimal way. For example, one key open problem is the question of which wave functions to use. Another open question is how to achieve numerical stability, i.e. an algorithm whose results are not garbled by effects resulting from the limited accuracy that computers have. These and other questions are particularly unclear for three dimensional problems, although most practical applications are three dimensional.The fellowship addresses this wide open research area. Building upon novel ideas about how to locally model wave phenomena in a stable way it combines fundamental research in diverse areas of applied and computational mathematics in order to develop the next generation of finite element methods for wave problems. These new methods have the potential to be orders of magnitude faster than current methods allowing for numerical simulations of phenomena that are currently out of reach. In close collaboration with partners in science and industry the new methods will be applied to exciting research problems in science and engineering. In particular, a major part of the hydrocarbon exploration business is enabled through the modelling and inversion of large scale 3D seismic and electromagnetic data sets, and Schlumberger Cambridge Research will be a key project partner. Throughout the fellowship annual international workshops on next generation finite element methods for wave problems will be organised, at Reading and Schlumberger. These will bring together leading researchers in the area of numerical wave simulations from academia and industry and will drive this research area forward by intensifying collaborations and developing and exploring application areas for these methods.Numerical wave simulations are an essential technology in science and engineering. Innovations in many areas depend upon the ability to simulate complex wave phenomena. The UK is one of the leading countries for wave-related research. This fellowship will enhance this role by building up an internationally outstanding research group on novel finite element methods for wave problems that will have a strong impact on wave-related research and applications long after the duration of the fellowship.

高效、准确地模拟波浪现象是跨越科学和工程的关键使能技术。应用跨越多个领域，包括整个声学和噪声控制、无损检测和用于医学成像的超声波和微波技术、地震和雷达传播和成像问题，甚至量子规模的模拟。但是，即使基本的偏微分方程组通常是线性的并且被很好地理解，但当波长与要模拟的区域的直径相比很小时，波动现象是复杂的，并且很难模拟。该方法的思想是将计算域分解成小元素，并以简单的方式近似每个元素的解，例如作为线性变分。然而，只有当每个元素的直径与波长相比很小时，这才能给出准确的解。因此，如果要模拟的区域的直径与波长相比非常大，则所需的单元数量以及相关的计算成本和存储是不可行的，因为对于非常复杂的波传播和散射问题，例如用于油气勘探的地震波传播。最近，国际上对试图通过用本身是波的函数来表示每个单元上的波场来解决该问题的新型有限元公式有着强烈的兴趣。这允许更大的元素大小，因此显著降低了计算成本。然而，这些新的有限元方法仍处于初级阶段，人们对如何以最佳方式实现它们知之甚少。例如，一个关键的悬而未决的问题是使用哪些波函数的问题。另一个悬而未决的问题是如何实现数值稳定性，即一种算法，其结果不会受到计算机精度有限造成的影响。这些问题和其他问题对于三维问题尤其不清楚，尽管大多数实际应用是三维的。研究员解决了这一广泛的开放研究领域。它建立在如何以稳定的方式局部模拟波浪现象的新想法的基础上，结合了应用数学和计算数学不同领域的基础研究，以开发用于波浪问题的下一代有限元方法。这些新方法有可能比目前的方法快几个数量级，从而允许对目前无法实现的现象进行数值模拟。在与科学和工业伙伴的密切合作下，新方法将被应用于科学和工程中令人兴奋的研究问题。特别是，油气勘探业务的主要部分是通过大规模三维地震和电磁数据集的建模和反演实现的，斯伦贝谢剑桥研究公司将成为主要的项目合作伙伴。在整个联谊会期间，将在雷丁和斯伦贝谢组织关于波浪问题的新一代有限元方法的年度国际研讨会。这些将汇聚学术界和工业界在波浪数值模拟领域的领先研究人员，并将通过加强合作和开发和探索这些方法的应用领域来推动这一研究领域的发展。数值波浪模拟是科学和工程中的一项重要技术。许多领域的创新依赖于模拟复杂波浪现象的能力。英国是波浪相关研究的领先国家之一。这项奖学金将加强这一作用，建立一个国际上杰出的研究小组，研究波浪问题的新型有限元方法，这将在奖学金期限结束后很长一段时间内对与波浪有关的研究和应用产生重大影响。

项目成果

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

矩阵多项式不变对的扰动、提取和细化

• DOI: 10.1016/j.laa.2010.06.029
• 发表时间: 2011
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Betcke T

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

声学中组合势积分算子的条件数估计及其边界元离散化

• DOI: 10.1002/num.20643
• 发表时间: 2010
• 期刊: Numerical Methods for Partial Differential Equations
• 影响因子: 3.9
• 作者: Betcke T

Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering

声散射中边界积分算子的谱分解和非正态性

• DOI: 10.1093/imanum/drt002
• 发表时间: 2013
• 期刊: IMA Journal of Numerical Analysis
• 影响因子: 2.1
• 作者: Betcke T

The factorization method for three dimensional electrical impedance tomography

• DOI: 10.1088/0266-5611/30/4/045005
• 发表时间: 2013-12
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: N. Chaulet;S. Arridge;Timo Betcke;David Holder

Solving Boundary Integral Problems with BEM++

• DOI: 10.1145/2590830
• 发表时间: 2015-02
• 期刊: ACM Transactions on Mathematical Software (TOMS)
• 作者: W. Smigaj;T. Betcke;S. Arridge;J. Phillips;M. Schweiger