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

物理学的许多方面都可以用数学方程来描述，例如，海水的运动，金属在加热和冷却时的膨胀和收缩，或者无线电天线中的电流。科学家们用数学符号来写这些方程。然而，当我们希望用计算机模拟一个特定的问题时，例如一个特定发动机部件的热膨胀，我们需要做几件实际的事情。首先，我们需要能够准确地描述发动机部件的形状。通常，形状可以通过简单的网格来描述（例如，任何二维形状都可以划分为三角形，三维形状可以划分为三棱锥，称为四面体）。通过将复杂的形状分割成更小、更简单的形状，方程可以更容易地求解。其次，也是最重要的，描述物理的数学方程需要转换为计算机可以理解的形式。在过去，这对程序员来说是一项繁琐的任务，但在FeNiCS科学软件库（fenicsproject.org）中，通过将类似于方程的数学语言直接转换为计算机代码，这变得容易得多。沿着一些关于网格和外部条件的附加信息（例如，在上面的例子中，热源），可以计算完整的模型，并可以绘制和分析结果。并非所有的模拟都如此简单。例如，气泡在水中运动的方程将包含表面张力项。表面张力对网格曲面的曲率敏感，但由三角形组成的网格始终具有平坦的多面曲面。在模拟中，当气泡移动时，网格也需要变形以考虑气泡的新位置。在另一种情况下，对于喷气发动机发出的噪声，方程可能需要复数，或者是高度非线性的。在这两种情况下，都需要编写计算机代码来处理特定问题。科学家们希望进行计算机模拟，而不必担心编程这些细节的复杂性。在这个项目中，我将解决五个主要问题：复数，曲面，困难的非线性问题，网格变形和网格格式。FeNiCS软件框架将被扩展，允许用户访问这些功能，而无需自己直接编程。它将开辟新的研究领域，例如表面张力和液-气相互作用、多相流、声学、量子力学和稳定性分析，并使软件更容易在HPC平台上使用。软件开发本身不足以推动科学和工程的进步。该项目的另一个重要方面是与应用科学家合作，帮助他们充分利用软件库，了解他们的科学问题，并帮助他们将其转化为计算机代码。我将与EPSRC博士培训中心合作，提供大规模使用新软件开发的培训，包括在超级计算机上使用。软件维护是任何科学项目的另一个重要方面，我将鼓励下一代科学家使用版本控制和自动测试的现代技术，这将提高他们的软件项目的质量，并延长他们的寿命。

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.