Fast numerical methods for large-scale multiphysics problems

在这个项目中要考虑的第一类问题是线性和非线性流体-结构相互作用问题。它们出现在从飞机设计到人体动脉、心脏和气道肺泡流动模拟的整个应用范围内。其目的是探索一种新的方法来解决这些问题，据我们所知，迄今为止还没有使用。这种方法的关键思想是重新制定，流体和结构问题的一个新的变量，应力张量。此外，通过使用适当的分裂方法，3D时间离散方程组可以简化为一组更简单，更容易解决的低维问题。 因此，由此产生的数值算法将显着更快，更容易在并行计算机上实现。这种新的流体-结构相互作用问题的设置的另一个非常令人愉快的结果是，由于在这两个域中，它只使用应力作为变量，它不需要显式跟踪的结构域的形状，通常是相当有问题的原始变量中的现有方法。 本项目的第二个主要问题是关于构造一种快速数值方法来求解描述球壳中大气和海洋动力学的方程。最明显的应用是研究地球大气和海洋中的流动。由于问题的规模是巨大的，算法设计必须是可用于非常大的并行计算机集群。我计划使用一个最近在我的小组中设计的高阶不可压缩方法，并修改它以放松高层大气中的不可压缩约束。它已被证明是非常理想的，特别是在大型并行计算机上。因此，要开发的算法将非常快，并将允许具有非常高的分辨率的模拟。其设计的最大挑战是双重的。一方面，它将需要使用具有局部细化的网格，这不容易与我们也打算采用的方向分裂算法联合收割机结合。最近，在我的小组中，我们开发了一种使用悬挂节点设计此类算法的方法，并证明了它在抛物问题中是无条件稳定的。这种方法将进一步扩展到包括一个有效的并行化策略（用于并行集群）。 另一个挑战来自域的几何形状，一个非常薄的外壳，这对并行化造成了特别的困难。 这将需要使用图分区方法。从该项目的这一部分产生的方法和软件将在质量上上级大多数目前可用的流体静力学数值模型，即它们以非常简化的方式纳入垂直方向的流动动力学。 此外，它们将允许更精细的模拟。

The first type of problems to be considered in this project are linear and nonlinear fluid-structure interaction problems. They appear in a whole range of applications starting from aircraft design, to simulation of flows in human arteries, heart, and the alveoli of the airways. The intention is to explore a new approach to such problems that, to our knowledge, has not been used so far. The key idea of this approach is to reformulate both, the fluid and the structure problems in terms of a new variable, the stress tensor. Further, by using a proper splitting approach, the 3D time-discrete system of equations can be reduced to a set of much simpler, lower-dimensional problems that can be resolved much easier. So, the resulting numerical algorithm will be significantly faster and easier to implement on parallel computers. Another very pleasant outcome of this new setting of fluid-structure interaction problems is that since in both domains it uses only the stress as a variable, it does not require explicit tracking of the shape of the structure domain that is usually quite problematic in the existing methods in primitive variables. The second major problem in this project is about the construction of a fast numerical method for solving the equations describing the dynamics of the atmosphere and the oceans in a spherical shell. The obvious application is to study the flow in the Earth's atmosphere and oceans in a coupled fashion. Since the size of the problem is enormous, the algorithm to be designed must be usable on very large parallel clusters of computers. I plan to use a high order incompressible method devised recently in my group and modify it to relax the incompressibility constraint in the upper atmosphere. It has been verified to be very optimal, particularly on large parallel computers. So, the algorithm to be developed will be very fast and will allow for simulations with a very high resolution. The biggest challenges of its design are twofold. On one hand, it will require to use meshes with local refinement which is not easy to combine with direction splitting algorithms that we also intend to employ. Recently, in my group we developed an approach for the design of such algorithms using hanging nodes and proved that it is unconditionally stable in case of parabolic problems. This approach will be further extended to include an efficient parallelization strategy (for the use on parallel clusters). The other challenge comes from the geometry of the domain, a very thin shell, that poses particular difficulties to parallelization. This will require the use of graph-partitioning methods. The resulting methods and software from this part of the project will be qualitatively superior to most of the currently available numerical models that are hydrostatic i.e. they incorporate the dynamics of the flow in the vertical direction in a very simplified manner. Besides, they will allow for much more refined simulations.