Expressive Finite Element Modelling for HPC: enabling advanced techniques for scientists

Many aspects of physics can be described by mathematical equations, for example, the motion of water in the sea, the expansion and contraction of metal as it is heated and cooled, or the flow of electricity in a radio antenna.

Scientists write these equations using mathematical symbols. However, when we wish to model a specific problem with a computer, for example the thermal expansion of a particular engine part, there are several practical things which we need to do.
Firstly, we need to be able to describe the shape of the engine part accurately. Usually, the shape can be described by a simple mesh (for example, any two-dimensional shape can be divided into triangles, and three-dimensional shapes can be divided into triangular pyramids, known as tetrahedra). By dividing the complex shape into smaller, simpler shapes the equations can be solved more easily.
Secondly, and most importantly, the mathematical equations describing the physics need to be converted to a form the computer can understand. In the past, this has been a tedious task for programmers, but in the FEniCS scientific software library (fenicsproject.org), this has been made much easier by translating from a mathematical language, similar to the equations, directly into computer code. Along with some additional information about the mesh, and the external conditions (e.g. in the example above, the source of heat), a full model can be computed and the results can be plotted and analysed.

Not all simulations are so straightforward. For example, the equations for air bubbles moving through water would contain terms for surface tension. Surface tension is sensitive to the curvature of the mesh surface, but a mesh made of triangles always has a flat, faceted surface. In a simulation, as the bubbles move, the mesh would also need to be distorted to take account of the new position of the bubbles. In another context, for noise emitted from jet engines, the equations may require complex numbers, or be highly non-linear. In both of these cases, the computer code needs to be written to take care of the specific issue. Scientists would like to make computer simulations without having to worry about the complexities of programming these details.

In this project, I am addressing five main issues: complex numbers, curved surfaces, difficult non-linear problems, mesh deformation, and mesh formats. The FEniCS software framework will be extended to allow users to have access to these features without having to program them directly themselves. It will open up new areas of research, for example in surface tension and liquid-gas interaction, multiphase flow, acoustics, quantum mechanics, and stability analysis, as well as making the software more accessible on HPC platforms.

Software developments alone are not enough to drive forward scientific and engineering advances. Another important aspect of this project is to engage with application scientists and help them to get the most out of software libraries, understand their scientific problem, and help them translate it into computer code. I will provide training, in association with EPSRC doctoral training centres, to use the new software developments at large scale, including on supercomputers. Software maintenance is another important aspect of any scientific project, and I will encourage the next generation of scientists to use modern techniques of version control and automatic testing, which will enhance the quality of their software projects, and improve their longevity.

The developments of this project have the potential to impact in many areas, including industry, healthcare and society both in the UK and internationally.

The software development proposed will enhance the existing highly popular FEniCS finite element package.
FEniCS has been used for the modelling of: patient-specific physiology, tidal turbine array optimisation, permafrost, glacial flows, the float glass process, protein interaction in solvents, turbulent flow, micro-magnetics, damage and fracture mechanics, strength-weight optimisation, elastic deformation, magma flow, oil reservoirs, carbon capture and storage (CCS) and many other problems. The new developments will enable advances in acoustics, surface tension, moving interfaces, and complex non-linear problems.

Some of these applications have direct industrial impact, as they can be used to model real industrial processes, for example the float glass process, acoustic noise from a jet engine, surface tension in paint droplets, elastic deformation of a rotor blade, or flow in an oil reservoir. By better understanding the way these systems behave, companies can improve their efficiency, and make financial or environmental savings. Direct applications in healthcare include the modelling of patient-specific heart valves, aneurysms or spinal columns, which can help clinicians decide on the best course of treatment for a patient. Finally, some applications have implications for policy, such as the feasibility of CCS storage schemes.

In addition to the software library developments, this project extends into providing a service to scientists and engineers, helping them to produce more robust software, more quickly, and scale up to supercomputers with ease. For industry, this reduces the time from design to obtaining results, and gives the benefit of a more responsive design cycle. Scientists who are supported to use the latest computational tools will also be able to solve problems which they previously thought too large or difficult. This can have a huge impact. Many scientific models are in 2D, simply because the scientists cannot scale up their computations to 3D. With the right support, more accurate modelling in 3D is possible. Larger, more accurate full body medical models for patients may follow, and industrial designs with billions of degrees of freedom which can be solved in minutes.