Unfitted Numerical Schemes for Fluid-structure Interaction and Applications

Mathematical modeling of fluid-structure interaction (FSI) has extensive practical applications in engineering and bioengineering. Considerable effort has been devoted to the devising and analysis of efficient numerical schemes for FSI problems. Designing accurate and efficient numerical schemes for FSI is challenging owing to the stiff coupling between the two domains. Methods are also limited by stability or accuracy issues that require severe timestep restrictions or are burdened by the amount of added-mass in the system when both fluid and solid have similar densities. Unfitted methods are attractive options when solving problems in a multi-domain setting since one can alleviate the cumbersome meshing process caused by the presence of complicated geometries and curved interfaces. The decisive asset is that background grids of unfitted methods are generated independently of the geometry, and the challenge is then to adapt the discretization scheme to the presence of cut elements.In this project, we will devise and analyze cutting-edge numerical schemes for FSI problems on unfitted meshes. Concerning space discretization, this project focuses on novel discontinuous Galerkin methods on polygonal meshes. These methods combine robustness and efficiency and are innovative to simulate FSI problems. An issue accompanying cut elements is an ill-conditioning of the finite element discretization in the presence of small cuts. To overcome this problem, a novel approach will be developed to ensure the robustness of the scheme with respect to small cuts. Both monolithic scheme and loosely coupled scheme free of the added-mass effect will be developed, which will provide fresh perspectives for the devising of loosely coupled schemes on unfitted meshes. Furthermore, high-order accurate time-stepping schemes will be proposed and analyzed, aiming to increase the temporal accuracy of the solution at a modest computational cost. We will then extend the proposed schemes to solve nonlinear FSI involving moving domains and/or interfaces. The accuracy and capabilities of the scheme will be verified by rigorous mathematical analysis and extensive numerical simulations for benchmark problems.This project is in the frontier of numerical analysis and scientific computing, and involves interfaces with practical applications. Moreover, the subject of unfitted methods is internationally very active and concerns many practical applications. Thus, the developed methodologies and mathematical tools will have the potential for a sizeable impact.