Physiological fluid dynamics generally involves the interaction of a viscous incompressible fluid with a visco-elastic biological tissue. The Immersed Boundary Method provides a robust framework for the computer simulation of biofluid dynamic systems, even when the configuration of the biological tissue is complicated, dynamic, and not known in advance. Although the Immersed Boundary Method is able to furnish qualitatively good results, it suffers from a certain "lack of resolution'", not related to the method itself but to limitations of computers such as speed and storage. Usually thin boundary layers develop along the immersed biological tissue, requiring a dense computational mesh to resolve adequately the flow there. If a uniform mesh is used, this requirement is inevitably extended to the entire computational domain, and the resulting mesh may exceed the storage capacity of the computer. The aim of this work is to present a new computational setting for the Immersed Boundary Method, in which the resolution can be enhanced locally by using the Adaptive Mesh Refinement Technique. In this new setting a vicinity of the immersed boundary is covered by a sequence of nested, progressively finer rectangular grid patches. This strategy concentrates the computational effort for the regions of the flow where it is most needed.
Also, an implicit formulation of the method is introduced to free the scheme from the time step restriction coming from the stiffness of the immersed boundary. The numerical results show that with this new setting the method succeeds in achieving the same accuracy as if the whole computation had been performed on an uniform mesh with the resolution of the finest grid patches that are used to cover the immersed boundary. Potential applications of the methodology presented include the computer assisted design of artificial heart valves and other biological problems where high resolution is needed in a neighborhood of the immersed elastic boundary to capture the boundary layers which are formed there.
 

Roma, A.M.: A multilevel self adaptive version of the Immersed Boundary Method. Ph.D. Thesis, CIMS-NYU, 01/1996. University Microfilms #9621828.