Fast numerical methods for large-scale multiphysics problems

本项目首先要考虑的问题是线性和非线性流固耦合问题。它们出现在从飞机设计到模拟人体动脉、心脏和呼吸道肺泡的流动的整个范围内。其目的是探索一种新的方法来解决这些问题，据我们所知，这些问题迄今尚未使用过。这种方法的关键思想是用一个新的变量--应力张量来重新表述流体和结构问题。此外，通过使用适当的分裂方法，3D时间离散方程组可以被简化为一组更简单、更低维的问题，可以更容易地求解。因此，所产生的数值算法将显著地更快且更容易在并行计算机上实现。这种新的流固耦合问题的另一个令人满意的结果是，由于在这两个域中只使用应力作为变量，它不需要显式跟踪结构域的形状，这在现有的原始变量方法中通常是很有问题的。

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.