Many problems in biofluid dynamics involve the interaction between a
non stationary, incompressible viscous fluid and a visco-elastic biological
tissue, which may have
time dependent configuration, time-dependent elastic properties, or
both (e.g., the interaction between blood, heart muscles and heart valve
leaflets). Although these problems can be handled in a robust manner by
the Immersed Boundary Method and qualitatively good results be obtained,
this method suffers from a certain "lack of resolution" which is related
to limitations of computers such as speed and storage. Finer immersed boundary
geometric details and flow features can be adequately resolved only if
the computational mesh is dense. 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. A computational setting
for the Immersed Boundary Method which employs the Adaptive Mesh
Refinement Technique will be presented. This approach is capable
of removing the original uniform mesh restriction, enhancing the accuracy
of the method by covering locally an immersed boundary vicinity with a
sequence of nested, progressively finer rectangular grid patches which
dynamically follow the immersed boundary motion. Results obtained for a
two-dimensional model problem show that, with this approach, the accuracy
attained by refining only an immersed boundary vicinity is the same
as if the whole computation had been performed on an uniform mesh with
the resolution of the finest level used in the locally refined mesh.
Implementation issues, applications in sight and the work in progress will
be discussed.
Roma, A.M.: An adaptive Immersed Boundary Method: implementation issues, applications and work in progress, Eletronical Proceedings of the Symposium on Adaptive Methods for Partial Differential Equations, University of Utah, Salt Lake City, Utah, 06/22-24, 1998.