Immersed Boundary Ghost Cell Method
I am currently writing a student research paper on the use of the IBM to model the human heart valves (a not quite new topic, I'm afraid, but in my department this hasn't been done yet). Currently I'm doing literature research and finding out about the various methods of the IB concept. At the moment, I am having trouble understanding the Ghost Cell Method (GCM).
In particular, I do not exactly understand the interpolation. But first let me tell you what I understand so far:
- A forcing term is added to the discrete system of governing equations
- This forcing term is derived from the boundary conditions (e.g. no slip: u=0)
- In general, the derivation of the forcing term is done by estimating the velocity field and and correcting it at the boundary to fit the boundary conditions.
- Ghost cells are defined inside the immersed body, so that each GC has at least one neighbor in the fluid
- If the artificial value for the flow parameters were known in these cells, the computation could stop right there w/o having to solve in the fluid
- However, there is no information in the GC themselves, but nearby there is the boundary condition (as before, a no slip condition would be an example)
- The value of the GC flow parameters can then be extrapolated (using a bilinear or trilinear scheme, depending on the case) from the boundary conditions and the fluid cells close by.
My question is: what is the point of calculating the value of the GC flow parameters if the boundary condition is already known?
Isn't it just more work to do so, because the flow parameters inside the body are purely artificial?
I think this has something to do with the fact that the IB does not necessarily coincide with the mesh points and therefore one cannot impose boundary conditions on a discrete equation if these conditions do not lie on the mesh points. Therefore, another cell has to be added with a value that approximates the boundary conditions between the already existing cells and the new GC. Am I completely off-track with this line of thought?
I am working on the same method and implemented it for flow over a cylinder problem. I think you are not off the track. In ghost cell method the boundary doesn't coincide with the Cartesian mesh how to satisfy the boundary condition. Here we take help of ghost points. We interpolate the value at ghost point from neighboring fluid points such that boundary condition is satisfied. We dont solve for the solid cell which are completely inside the solid and doesn't have any fluid point as a neighbor.
I hope some doubts are cleared.
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