Computational methods for multiphysics interface problems

科学和技术中的许多问题都包括一个固定的或移动的边界，在这个边界上两个不同的物理系统耦合在一起。这种情况在医学和生物学系统中特别常见，例如：在人体动脉中，血液的流体动力学与动脉壁的固体动力学耦合，在河流和河口中，自由流动与渗透河床中的多孔介质流动耦合。对这些系统的演变进行精确的计算预测仍然是工程师面临的一个重要挑战，而对相关方法进行精确的数学分析则更加艰巨。事实上，没有已知的方法允许严格的数学分析，并且许多方法都存在依赖于界面方向的稳定性或准确性问题。数值计算通常在计算网格上进行，即将计算域分解为大量的小构建块，即所谓的元素。我们提出的方法的一个重要特征是接口可以切割计算网格的元素，或者换句话说，计算网格不需要适合接口。在多物理场问题中，计算网格可能无法适应接口，但两个系统的耦合必须独立于网格进行，这往往使情况变得复杂。这就是我们在这个项目中要研究的情况。将为移动接口上的多物理场耦合设计新的方法。数学方法将被设计为鲁棒和准确，我们也将探索解耦两个系统的可能性，以实现有效的时间推进。这可能会大大节省计算时间，特别是对于非线性问题。将考虑三种重要的模型情况：两种流体的耦合，其中一种或两种流体都可能是粘弹性的，自由流动和多孔介质流动的耦合，最后是流体和弹性结构的耦合。所有这些应用在人体心血管系统的建模中都有重要的应用，而且在喷墨打印机、环境科学、化学工业等各种其他应用中也有广泛的应用。

Many problems in science and technology includes a fixed or moving boundary over whichtwo different physical systems are coupled. This situation is particularly common in systems inmedicine and biology, for instance: in the human arteries the fluid dynamics of the blood couples to the solid dynamics of the arterial wall, in rivers and estuaries the free flow couples to the porous media flow in the infiltrated river bed. Making accurate computational predictions of the evolution of such systems remains an important challenge for engineers and the accurate mathematical analysis of the associated methods is even more daunting. Indeed no known methods allow for rigorous mathematical analysis and many suffer from problems of stability or accuracy depending on the orientation of the interface. Numerical computations are most often performed on a computational mesh, that is a decomposition of the computational domain in a large number of small building blocks, so called elements. An important feature of the methods that we propose is that the interface may cut through the elements of the computational mesh, or in other words, the computational mesh does not need to fit the interface.In multiphysics problems the situation is often complicated by the fact that the computational mesh may not be adapted to fit the interface, but the coupling of the two systems must take place independent of the mesh. This is the type of situation that we aim to study in the present project. New approaches will be designed for multiphysics couplings over moving interfaces. The mathematical methods will be designed so as to be robust and accurate and we will also explore the possibility to decouple the two systems for efficient time advancement. This may lead to very important savings in computational time, in particular for nonlinear problems.Three important model cases will be considered: the coupling of two fluids of which one or both may be viscoelastic, the coupling of free flow and porous media flow and finally the coupling of a fluid and an elastic structure. All of these applications have important applications in the modeling of the human cardiovascular system, but also in a wide variety of other applications such as ink-jet printers, environmental science, chemical industry and so on.