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、工程或医学中的许多物理过程中。在大多数情况下，液体或气体在接触之前和之后位于结构之间，或者甚至在接触期间以非常薄的流体层的形式位于结构之间。该流体可能对接触动力学具有显著影响。因此，在许多情况下，一个完整的流固耦合（FSI）问题必须考虑。在这个项目中，我们开发了一个准确和强大的数值框架，模拟流固耦合接触。虽然存在大量的文献都流固耦合和接触力学单独，许多悬而未决的问题仍然存在的组合。许多常用于流固耦合的标准数值方法不能处理接触时流体区域的拓扑变化。能够处理接触的有前途的方法是全欧拉方法和基于通过Nitsche方法或使用拉格朗日乘子的耦合的拉格朗日-欧拉技术。然而，很少有研究存在的所有这些方法考虑的情况下，接触。在这个项目中，我们将比较现有的数值方法的情况下，薄壁结构的余维1和厚结构的等尺寸。对于离散化，我们计划使用一个几何不拟合的有限元方法，能够准确地处理移动接口，而无需重新meshing.To模型接触，新的方法的基础上，一个增广拉格朗日公式将进行调查，并与标准方法相比，使用拉格朗日乘子或惩罚力。该项目的最终目标是推导和分析厚结构和薄结构情况下接触的流体-结构相互作用的数值方法，并将其应用于实际应用中，如冠状动脉中二尖瓣的关闭。

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.