Most of the published research results about immersed finite element (IFE) methods deal with 2nd order elliptic interface problems. This project plans to extend the research on developing and analyzing IFE methods for interface problems to more sophisticated partial differential equations such as the Stokes system and the linear elasticity system which are of great importance in many applications of engineering and sciences. The proposed research consists of three key modules which complement each other. The first part is to develop new IFE functions and integrate them into modern finite element techniques for solving Stokes and linear elasticity interface problems. The intent is to find suitable IFE functions locally in interface elements that can handle interface jump conditions required by the interface problems and other conditions such as the inf-sup stability condition required by the finite element formulations to be used. The proposed research will result in efficient and robust IFE methods with an emphasis on DG formulations that simplify local h-, p-, and hp- refinements on Cartesian meshes. The second module is the theoretical analysis of IFEs, starting from their interpolation approximation capabilities and deriving error bounds for IFE solutions. The challenges are that the traditional analysis techniques have a limited use here. For example, using equivalent quotient norm in the scaling argument leads to an estimate for IFE interpolation useless for further deriving estimate of IFE solution unless it can be shown that the constants in the error bounds are independent of the interface. Also, 2D and 3D IFE methods are essentially non-confirming methods whose error estimations are often more complicated. The third module is about the applications of IFE methods. In addition to improving the IFE solver for the particle-in-cell simulator, this project will investigate the applications of IFE methods to be developed to multi-fluid Stokes flow problems and the multi-material shape/topology optimization problems involving the linear elasticity.
The IFE methods to be developed in this project can provide new and efficient simulation tools that can use structured/Cartesian meshes to solve challenging interface problems involving multi-scale and multi-physics with nontrivial interfaces in many areas of engineering and science, including flow problems, electromagnetic problems, shape/topology optimization problems, to name just a few. This research project will proceed in harmony with the development of IFE software packages to verify and support the theory, and address complex and realistic problems from a variety of disciplines. This strategy will enable theoretical innovation to become practice much more quickly than traditionally possible. The proposed research projects will have a great potential to impact on numerical simulations in the design/research of ion-propulsion engines for interplanetary deep space travel, optimal packaging of electronic devices, efficient and better image reconstruction in computer tomography, non- destructive/non-invasive detection of suspicious materials in security check, design of optimal shapes for lighter and stronger structures, and many other application areas of great federal interests.
