Numerical Simulation of Contact Problems with Fluid-Structure Interaction

不同固体结构之间的接触问题出现在自然界、工程或医学的许多物理过程中。在大多数情况下，液体或气体在接触前后的结构之间，甚至在接触过程中以非常薄的流体层的形式存在。这种流体可能会对接触动力学产生重大影响。因此，在许多情况下，必须考虑完整的流固耦合问题。在这个项目中，我们开发了一个精确和鲁棒的数值框架来模拟有接触的FSI。虽然对于FSI和接触力学都有大量的文献存在，但对于它们的结合仍然存在许多悬而未决的问题。通常用于FSI的许多标准数值方法不能处理接触时流体域的拓扑变化。能够处理接触的有希望的方法是完全欧拉方法和拉格朗日-欧拉技术基于耦合通过Nitsche的方法或使用拉格朗日乘子。然而，考虑到接触的情况，很少有研究存在所有这些方法。在这个项目中，我们将比较共维1的薄壁结构和等维厚结构的现有数值方法。对于离散化，我们计划使用几何上不拟合的有限元方法，该方法能够准确地处理移动界面而无需重新网格划分。为了模拟接触，将研究基于增广拉格朗日公式的新方法，并将其与使用拉格朗日乘子或罚力的标准方法进行比较。该方法在FSI中的应用是新颖的。该项目的最终目标是推导和分析在厚结构和薄结构情况下与接触的流体-结构相互作用的数值方法，并将其应用于实际应用，例如冠状动脉二尖瓣的闭合。

Contact problems between different solid structures arise in many physical processes in nature, engineering or medicine. In most situations, a liquid or gas lies between the structures before and after or even in form of a very thin fluid layer during the contact. This fluid might have a significant effect on the contact dynamics. Therefore, in many cases a full fluid-structure interaction (FSI) problem has to be considered.In this project, we develop an accurate and robust numerical framework for the simulation of FSI with contact. While a vast amount of literature exists for both FSI and contact mechanics alone, many open questions persist for their combination. Many of the standard numerical methods commonly used for FSI are not able to deal with the topology changes of the fluid domain at the time of contact. Promising approaches that are able to deal with contact are a Fully Eulerian approach and Lagrangian-Eulerian techniques based on a coupling by means of Nitsche's method or using Lagrange multipliers. However, very few studies exist for all these approaches considering the case of contact. In this project, we will compare the available numerical methods for both the case of thin-walled structures of co-dimension one and thick structures of equal dimension. For discretisation, we plan to use a geometrically unfitted finite element approach that is able to deal with the moving interfaces accurately without re-meshing.To model contact, new approaches based on an augmented Lagrangian formulation will be investigated and compared against standard methods using Lagrange multipliers or a penalty force. The application of this method in the context of FSI is novel.The final objective of the project is to derive and analyse numerical methodologies for fluid-structure interactions with contact both in the case of thick and thin structures and to apply it to applications of practical interest, such as the closure of the mitral valves in the coronary artery.