BEM++ Stage 2

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.

High-quality open source software forms the fundamental component of modern scientific computing. Modern high-performance finite element libraries such as deal.II or DUNE allow the solution of complex simulation problems on thousands of cores. But while there exist a number of high-performance libraries for finite element discretizations on massively parallel systems, only little is available for the solution of boundary integral equations. Yet, boundary integral equations are important tools in a wide range of applications, such as medical imaging, atmospheric modelling, and electro-magnetic or acoustic shielding in the aeronautical and automotive industries.

BEM++ is closing this gap by delivering a free, highly performant modern boundary element library that can be easily adopted for boundary integral equation problems in all areas of science and engineering. In Stage 1 of the project we have developed the core library foundations. In Stage 2 we present a wide program to i) extend the library for very large problems on heterogeneous computing clusters, ii) interface it with the well established DUNE library to solve complex coupled finite element and boundary element problems (FEM/BEM coupling), and iii) apply the library to complex applications in medical imaging and atmospheric sciences. This work will bring significant impact in several ways.

As a general tool for solving complex coupled boundary integral equation/partial differential equation problems BEM++, in combination with DUNE, will give researchers across science and engineering a highly performant software toolbox to solve large-scale problems in homogeneous or coupled homogeneous and heterogeneous media, enabling them to perform simulations that otherwise would not have been possible for them. Similarly, industry will profit from the availability of a highly performant library for coupled FEM/BEM simulations, which can be used for in-house simulations or integrated into their products.

Significant impact will also be generated from the applications worked on during the course of this proposal. A large application area is for medical imaging problems. Here, we will work on two areas, namely High-intensity Focused Ultrasound (HIFU), and functional neuroimaging. In HIFU focused ultrasound is used to destroy cancerous tissue. In order to generate a treatment plan for patients, careful numerical simulations need to be performed to target the ultrasound correctly. Based on BEM++ and DUNE, and with collaboration with the National Physics Lab, we will build a HIFU software toolbox, which will make possible fast and accurate simulations for treatment planning. This application is of direct industrial relevance and may lead to further commercialization. Another application is functional neuroimaging. In Stage 1 we have already started work on Diffuse Optical Cortex Mapping and will continue this work and include other functional imaging modalities. Functional neuroimaging is one of the most fruitful and challenging areas of applied research. Mapping neural activity has potential in a range of applications, from helping sufferers of "locked-in" syndrome to the development of brain-interface devices. Efficient tools to simulate functional neuroimaging modalities are essential to drive this area forward.

Another application area targeted in Stage 2 is the large-scale simulation of electromagnetic scattering by atmospheric particles in collaboration with the Met Office. This work will help the verification of currently used models for atmospheric particles, and provide a pathway into novel applications of boundary elements for the remote sensing of the atmosphere.

Many other applications of the library are likely arise over time, thanks to its generic and polymorphic nature, and will lead to impact in areas of strategic importance to EPSRC such as "Manufacturing the Future".