BEM++ Stage 2

现实世界中许多同质介质中的问题，如飞机的电磁散射或汽车的噪声辐射，都可以用域边界上的积分方程来表示。这种公式的优点是只需要对边界进行离散，而不需要对（可能无限的）介质进行离散。这种边界积分方程的解的计算是边界元法（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，漫射光学和声学方面的广泛问题。对电磁和弹性问题的支持即将到来。该库既可以从c++中使用，也可以直接从Python中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，并已成功地用于光学层析成像应用中的光扩散计算。在这个第二阶段的建议中，我们以成功的核心库为基础，并对其进行扩展，以解决现代异构计算架构上具有挑战性的应用问题。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集群上实现分层矩阵和快速多极方法。这些方法使具有数百万个未知数的边界积分方程问题的快速求解成为可能，对于非常大的应用问题是必不可少的。我们将bem++与Dune连接，Dune是一个著名的高性能库，用于基于大规模网格的应用程序，如有限元和有限体积方法。边界元公式适用于均匀介质问题，但不适用于经常使用有限元方法的非均匀介质。对于耦合问题，其中部分介质是均匀的，而其他介质是非均匀的，通常有利于有限元法和边界元法的耦合。Dune的接口将允许我们做到这一点，并解决大型并行计算架构上复杂的耦合问题。这些扩展将允许我们解决大规模的实际应用问题。我们将研究的第一个医学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瞄准肿瘤，通过局部热膨胀摧毁癌组织。然而，为了正确规划HIFU治疗，需要仔细的数值模拟，我们将建立一个基于bem++和Dune的HIFU工具箱来进行治疗计划模拟。这项工作是与国家物理实验室合作进行的，对于开发新的癌症疗法具有重大的工业意义。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质映射，我们使用库来模拟现实头部几何形状中光的漫射散射。这些技术对于绘制患者的大脑活动图谱非常有用，并且在治疗闭锁综合征患者方面具有潜在的应用前景。本提案研究的另一个大型应用领域是与英国气象局合作的大气科学。这里的兴趣是准确地模拟大气粒子（如冰晶）的电磁特性。云特性的遥感在很大程度上依赖于电磁散射模型，有关标准方法准确性的问题对我们理解水循环和云如何影响地球的辐射平衡具有潜在的深远影响。bem++对这些大气粒子散射的计算将与以前使用的渐近方法的结果进行比较，并研究其对遥感和大气辐射传输的影响。

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.