The usual type of problem that is solved when using Immersed Boundary (IB) methods consists of a solid (usually in Lagrangian form) fully immersed in a fluid domain (usually in Eulerian form). The IB method is used to interpolate velocities from the fluid to the solid nodes and to spread forces from the solid to the fluid nodes. Nonetheless, this method deals with the fluid-structure interaction at the interface between the fluid and the immersed solid. On the other hand, at the physical boundaries of the fluid domain, the easiest approach is to impose periodic boundary conditions. If no-slip boundary conditions are required, Griffiths (2009) proposes to add a virtual immersed boundary parallel to the physical boundaries where the no-slip boundary condition is imposed using a penalty-type approach. Problems arise when the solid approaches the physical domain and the smooth Dirac delta interpolator no longer selects a constant number of points to interpolate velocities (or spread forces).
Hence, how can boundary conditions be imposed at fluid domain's physical boundaries to avoid the issue of nearby moving solids or for solids that share Dirichlet nodes with the fluid's physical domain?
The immersed boundary method has received an astonishing highlight during the last years, despite of the fact that is a quite old method.
The key is to understand that two general approaches can be used, that one due originally to Peskin, dated 50 yrs ago and a more recent one. The former set the BCs. by means of a locally distributed forcing function (linked to the discrete Dirac function) and is very suitable for elastic membrane interacting with the fluid. However, if the membrane is rigid that leads to a stiff problem.
On the other hand, a second approach is substantially based on an interpolation approach in which the Dirichlet condition is introduced in the interpolation at the location of the fictious boundary.
A review of the methods is illustrated here Article Immersed boundary method
Some more recent literature is present but generally falls in one of the two methodologies.
Be aware, that mass conservation can be one of the main problem in setting the approximation in the BC.
I think that Paolo Lampitella has worked a lot on this problem.
As mentioned by Filippo Maria Denaro , there are several IB approaches. I am no expert in the original, Peskin like, method you seem to be using but the issue, I think, can be related to other implementations as well, that is: do you include other boundary points in the interpolation/delta stencil or do you modify the stencil when other boundaries (immersed or not, I guess it has little relevance) are met (i.e., one of its points would be included in the stencil)?
I don't know much for the Peskin approach, but in the discrete forcing one some people add also other boundary points (leading to an implicit system to be solved) while others don't.
In my case, the deciding points were a) need to apply wall functions as well (if they apply then, it is wrong to interpolate to the boundary with the usual method, so the boundary point has to be excluded) and b) a unique framework/approach which is consistent for whatever bc. As a consequence, my interpolation stencil never includes boundary points and has to adapt near the boundaries (but this is automatic for a correct implementation on unstructured grids).
However, in your case it might indeed be different
Thank you very much Filippo Maria Denaro Paolo Lampitella and Ijaz Durrani
for your helpful answers. I am indeed using the Peskin-like IB approach. For interpolation stencils which have a part that falls out of the fluid physical domain, I am modifying the weights of the stencil points that fall at the fluid physical domain boundary so that the normalization of the stencil is maintained. I doing this modifications in such a way that it's equivalent to expanding the domain in a couple of extra node layers which have the same Dirichlet BCs as the real fluid physical domain boundary. For the force spreading operation this is not a problem since Dirichlet BCs are set at the fluid physical domain boundary and thus forces have no effect. It will be an issue when I try to set Neumann BCs. I'll look into it.
David Oks
I am not sure about what you are doing, do you want to use the Peskin's approach of elastic forces on a rigid wall?
I'm using Peskin's approach to solve an elastic body immersed in a fluid. When one of the nodes of the elastic body approaches the fluid domain boundary, part of the interpolation stencil is left out of this domain. So I just renormalize the nodes that fall at the boundary so the normalization of the stencil is not lost.
David Oks
I worked many year ago using the Peskin's approach (see Article 2-D Transmitral flows simulation by means of the immersed bo...
)and close to a wall the velocity is much more small than the interior fluid, I found useful to define a local distribution of the spreading law, the radius being directly linked to the local measure of the computational cell. This way I had not problem in having part of the interpolation stencil out of the domain
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