Next generation finite element methods for wave problems

高效、准确地模拟波浪现象是跨科学和工程领域的关键技术。它的应用范围很广，包括声学和噪声控制、无损检测、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，甚至量子尺度模拟。但是，尽管潜在的偏微分方程通常是线性的，并且很容易理解，但每当波长与待模拟区域的直径相比很小时，波动现象就很复杂，很难模拟。模拟波动问题的一个主要和标准的计算工具是所谓的有限元法。该方法的思想是将计算域分解成小的元素，并以一种简单的方式逼近每个元素的解，例如作为线性变化。然而，只有当每个元素的直径与波长相比较小时，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，那么所需的元素数量以及相关的计算成本和存储是不可行的，因为它适用于非常复杂的波传播和散射问题，例如油气勘探中的地震波传播。最近，国际上对新颖的有限元公式产生了浓厚的兴趣，这些公式试图通过用本身就是波的函数来表示每个单元上的波场来解决这个问题。这允许更大的元素大小，因此大大减少了计算成本。然而，这些新颖的有限元方法仍处于起步阶段，人们对如何以最佳方式实现它们知之甚少。例如，一个关键的开放问题是使用哪种波函数的问题。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，其结果不会被计算机有限的精度所造成的影响所混淆。尽管大多数实际应用都是三维的，但这些问题和其他问题对于三维问题尤其不清楚。该奖学金针对这一广泛开放的研究领域。建立在关于如何以稳定的方式局部模拟波动现象的新思想之上，它结合了应用数学和计算数学的不同领域的基础研究，以便为波动问题开发下一代有限元方法。这些新方法有可能比目前的方法快几个数量级，从而实现目前无法实现的现象的数值模拟。在与科学和工业伙伴的密切合作下，新方法将应用于科学和工程领域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来实现的，斯伦贝谢剑桥研究中心将成为一个重要的项目合作伙伴。在整个奖学金期间，将在雷丁和斯伦贝谢组织关于波浪问题的下一代有限元方法的年度国际研讨会。这些会议将汇集学术界和工业界数值波模拟领域的领先研究人员，并将通过加强合作、开发和探索这些方法的应用领域，推动这一研究领域向前发展。波浪数值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动现象的能力。英国是波浪相关研究的领先国家之一。该奖学金将通过建立一个研究波浪问题的新型有限元方法的国际杰出研究小组来加强这一作用，这将在奖学金结束后很长一段时间内对波浪相关的研究和应用产生强大的影响。

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.