Singular forces in the immersed interface method for rigid objects in 3D

In the immersed interface method, a boundary immersed in a fluid is represented as a singular force in the Navier-Stokes equations. An explicit approach was proposed recently for determining the singular force for the boundary of a rigid object with prescribed motion in 2D [Sheng Xu, The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow, J. Comput. Phys. 227 (2008) 5045-5071]. Necessary formulas for extending the approach to 3D are derived in this work. With the implementation of these formulas, the immersed interface method can accurately, stably, and efficiently simulate the prescribed motion of rigid objects in 3D.