The immersed boundary methods are well known as an efficient flow solver for engineering problems involving fluid structure interactions. However, in order to obtain better results, higher resolutions near the immersed boundary points are desired. Non-uniform Cartesian mesh can easily fulfill this task without introducing a dramatic increase on the cost of computation and coding. In the current paper, an immersed boundary method with non-uniform Cartesian mesh is demonstrated. The Poisson problem is solved with assistance of a scientific parallel computational library PETSc. The code is validated with a three-dimensional flow over a stationary sphere. Then, a fluid-structure interaction model is coupled and validated with two-dimensional vortex induced vibration problems. Comparisons with previous studies are presented. The ultimate goal is to couple the fluid-structure interaction model with the three-dimensional immersed boundary method.

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