The application of the Immersed Boundary Method (IBM) coupled with Adaptive Mesh Re nement (AMR) is considered to be one of the powerful tools for solving complex viscous incompressible ow problems. In this paper, the Kajishima-cut-cell IBM is combined with AMR to solve the ow of complex incompressible, viscous uid, and rigid body problems. In the IBM, the solid and uid motions at the interface are controlled by a body force that can be calculated with a fraction of solid volume. The objective is to develop an automatic adaptive mesh re nement strategy to enhance the solution in proximity to the uid-structure interface. This is necessary as the ow eld might be signi cantly a ected by the structure boundaries; therefore, it is essential to capture the boundary layers precisely. The capability of this method can be demonstrated through computational results to improve the ow resolution near the uid structure. The proposed approach is validated using a 2D laminar ow numerical example. The approach is validated in terms of accuracy and performance. The combined IBM-Adaptive mesh re nement approach showed a promising outcome for the investigated problem. The result of the implemented method achieved an acceptable error performance within a reasonably low computation time.