Immersed Boundary Methods for Fluid-Structure Interaction and Shape Optimization within an FEM-Based PDE Toolbox

One of the main challenges in a classical mesh-based FEM-approach is the representation of complex geometries. This challenge is often tackled by a computationally costly mesh generation process, where the resulting mesh’s facets represent the boundary. An alternative approach, that we employ here, is the immersed boundary (IB) approach. This uses instead a computationally cheaper structured adaptive Cartesian mesh and an explicit boundary representation, where the challenge mainly lies in the boundary condition (BC) imposition on the mesh cells intersected by the geometry’s boundary. One IB method is Nitsche’s method that we employ here for fluid-structure interaction (FSI) and shape optimization problems. The simulation of such complex physical systems modeled by PDEs requires a combination of sophisticated numerical methods. Implementing a FEM-based simulation software that computes a particular PDE’s solution often requires the reusage of existing methods. In order to make our approach public and also to prove the modularity of it, we integrated our IB methods in an existing FEM-based PDE toolbox of the Trilinos project, called Sundance.