BEM++ Stage 2

均匀介质中的许多实际问题，如飞机的电磁散射或汽车的噪声辐射，可以用积分方程表示。这种公式的优点是只需要离散边界而不需要离散（可能是无限的）介质。计算这种边界积分方程的解是边界元法（BEM）的任务。我们开发了BEM++（www.bempp.org），这是一个现代开源边界元软件库，可以解决静电学，漫射光学和声学中的广泛问题。对电磁和弹性问题的支持即将到来。该库可以从C++或直接从Python通过方便的脚本接口使用。当前版本的BEM++已经在现代多核工作站上并行化，并已成功部署用于光学层析成像应用中的光漫射计算。在第二阶段的建议，我们建立在成功的核心库，并扩展它来解决具有挑战性的应用问题，现代异构计算架构。为了实现这一目标，将开发以下软件库组件作为项目的一部分。我们将在分布式CPU/GPU集群上实现层次矩阵和快速多极子方法。这些方法使得具有数百万个未知量的边界积分方程问题的快速求解成为可能，并且对于非常大的应用问题是必不可少的。我们将接口BEM++与沙丘，一个众所周知的高性能库的大规模网格为基础的应用程序，如有限元和有限体积法。虽然边界元公式适用于均匀介质问题，但它们不适用于非均匀介质，其中经常使用有限元法（FEM）。对于耦合问题，其中部分介质是均匀的，而另一些是不均匀的，耦合有限元和边界元方法通常是有益的。Dune的接口将允许我们这样做，并解决大型并行计算架构上的复杂耦合问题。这些扩展将允许我们解决大规模真实的世界应用问题。我们将研究的第一个医疗应用是高强度聚焦超声（HIFU）治疗。这里的原理是使用聚焦超声靶向肿瘤，通过局部热膨胀破坏癌组织。然而，为了正确规划HIFU治疗，仔细的数值模拟是必要的，我们将建立一个基于BEM++和沙丘的HIFU工具箱，用于此治疗规划模拟。这是与国家物理实验室合作的工作，对于开发新型癌症疗法具有巨大的工业利益。第二个医学应用是功能性神经成像。一个例子是漫射光学皮层映射，我们使用该库来模拟现实头部几何形状中的光漫散射。这些技术对于绘制患者的大脑活动是有用的，并且可以具有潜在的应用，例如在闭锁综合征患者的治疗中。该提案中研究的另一个大型应用领域是与气象局合作的大气科学。这里的兴趣是准确地模拟冰晶等大气粒子的电磁特性。遥感云的属性在很大程度上依赖于电磁散射模型和标准方法的准确性问题有潜在的深远影响，我们的水循环和云如何影响地球的辐射平衡的理解。BEM++计算这些大气粒子的散射将与以前使用的渐近方法的结果进行比较，遥感和大气中的辐射传输的影响进行调查。

Many real-world problems in homogeneous media, such as electromagnetic scattering from an airplane or the noise radiation from a car, can be formulated as integral equations over the boundary of the domain. The advantage of such formulations is that only the boundary and not a (possibly infinite) medium needs to be discretised.Computing solutions of such a boundary integral equation is the task of a boundary element method (BEM). We have developed BEM++ (www.bempp.org), a modern open source boundary element software library, which can solve a wide range of problems in electrostatics, diffuse optics and acoustics. Support for electromagnetic and elasticity problems is coming soon. The library can be used either from C++ or via a convenient scripting interface directly from Python.The current version of BEM++ already parallelises on modern multi-core workstations and has been deployed successfully for the computation of light diffusion in optical tomography applications. In this Stage 2 proposal we build on the successful core library and extend it to solve challenging application problems on modern heterogeneous computing architectures. To achieve this goal the following software library components will be developed as part of the project.1.) We will implement hierarchical matrix and fast multipole methods on distributed CPU/GPU clusters. These methods make possible the fast solution of boundary integral equation problems with millions of unknowns and are essential for very large application problems.2.) We will interface BEM++ with Dune, a well known high-performance library for large scale grid based applications such as finite element and finite volume methods. While boundary element formulations are suitable for homogeneous media problems they do not work for heterogeneous media, where finite element methods (FEM) are frequently being used. For coupled problems, where parts of the medium are homogeneous and others are heterogeneous it is often beneficial to couple FEM and BEM methods. The interface to Dune will allow us to do this, and to solve complicated coupled problems on large parallel computing architectures.These extensions will allow us to solve large scale real world application problems. The first medical application that we will study is High Intensity focused ultrasound (HIFU) treatment. The principle here is to use focused ultrasound targeted at a tumor to destroy the cancerous tissue by localized thermal expansion. However, to properly plan HIFU treatment, careful numerical simulations are necessary and we will build a HIFU toolbox based on BEM++ and Dune for this treatment planning simulation. This is work in collaboration with the National Physics Lab and is of large industrial interest for the development of novel cancer therapies. The second medical application is in functional neuroimaging. An example is diffuse optical cortical mapping, where we use the library to simulate the diffuse scattering of light in realistic head geometries. These techniques are useful to map the brain activities of patients and can have potential applications for example in the treatment of sufferers of locked-in syndrome. The other large application area studied in this proposal is atmospheric sciences in collaboration with the Met Office. The interest here is to accurately simulate the electromagnetic properties of atmospheric particles such as ice crystals. Remote sensing of cloud properties relies heavily on electromagnetic scattering models and questions regarding the accuracy of standard approaches have potentially profound implications for our understanding of how the water cycle and clouds influence the radiation balance of the earth. BEM++ computations of scattering by these atmospheric particles will be compared with results from previously used asymptotic methods, and the implications for remote-sensing and radiative transfer in the atmosphere investigated.