Computational methods for multiphysics interface problems

科学技术中的许多问题都包含一个固定或移动的边界，在这个边界上，两个不同的物理系统相互耦合。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，血液的流体动力学耦合到动脉壁的固体动力学，在河流和河口中，自由流动耦合到渗透河床中的多孔介质流动。对这些系统的演化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更是令人生畏。实际上，没有已知的方法允许严格的数学分析，并且许多方法遭受取决于界面取向的稳定性或准确性的问题。数值计算最常在计算网格上执行，计算网格是将计算域分解为大量小构建块，即所谓的元素。我们提出的方法的一个重要特点是，接口可以通过计算网格的元素，或者换句话说，计算网格不需要适合接口。在多物理场问题的情况往往是复杂的事实，计算网格可能不适合接口，但两个系统的耦合必须发生独立的网格。这就是我们在本项目中要研究的情况。新的方法将被设计用于移动界面上的多物理场耦合。数学方法将被设计为具有鲁棒性和准确性，我们还将探索将两个系统解耦以实现有效时间推进的可能性。这可能会导致非常重要的节省计算时间，特别是对于非线性problems.Three重要的模型的情况下，将被认为是：耦合的两种流体，其中一个或两个可能是粘弹性的，耦合的自由流动和多孔介质流，最后耦合的流体和弹性结构。所有这些应用都在人体心血管系统建模中有重要的应用，但也在各种各样的其他应用中，如喷墨打印机，环境科学，化学工业等。

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.