Immersed boundary modal analysis and forced vibration simulation using step boundary method

Analysis using Cartesian background mesh with the geometry embedded or immersed in the mesh is gaining popularity. The primary advantage of this approach is that a traditional mesh, which conforms to the geometry, is not needed. Instead a background mesh that is independent of the geometry and has regular shaped undistorted elements is used because it is easy to generate automatically. Many methods for imposing Dirichlet boundary conditions on immerse boundaries have been studied. In this work step boundary method, where the trial and test functions are weighted using approximate step functions, has been used for imposing Dirichlet boundary conditions on boundaries that do not have nodes on them. This method has been shown to be effective for static problems in the past but has not been studied for dynamics. Step boundary method is extended to modal analysis and modal superposition as well as problems involving base excitation where the Dirichlet boundary conditions are functions of time. Several test examples are used to verify and validate the method.