[aerosayan]

Quadtree grids can be body fitted, instead of cutting the cells and creating triangles or polygons.
I didn't see anyone else use this method. I guess I might have made an original discovery.

METHOD:
1. Create quadtree grid to a sufficient refinement.
2. Apply laplacian/poisson smoothing like in hyperbolic grid generators to move the points of the cells intersecting the boundary, until the cell edge/s fits the body.
3. Apply refinement if necessary.
4. Apply laplacian smoothing to some extent away from the body, to ensure the grid is not skewed.
5. Keep rest of the untouched domain as the original quadtree.

See the images I took from the net, you'll get some idea of what I mean.
I don't want to give away every detail, if you get it, you get it.


Using this method we will have pure quad cells throughout the domain, and no garbage triangular cells.
Cells near the boundary will fit the body and form prism like layers, and cells away from the body remain like original quadtree grid.

On solver side, we'll get a performance boost since the original quadtree cells don't need to store surface normals and surface areas. They can be calculated on the fly. The body fitted prism like layers will give better performance in high reynold's number flows. Of course, writing a solver for this grid will a little bit complex, and any naive implementation will only mess things up. I don't want to share the optimal implementation details of the solver.

[LuckyTran]

Pardon me for not seeing the novelty but there's volumes and volumes of real math that deals with creating these types of grids.

I get that you avoiding the so-called trimmer meshes but you can always take a pure quad mesh and stretch it to fill a domain. Question is, what happens to the skewed cells and these will always be a problem.

And have you seen meshers like GridPro?

[praveen]

You will have the problem of very small cells which will impact time step in explicit schemes.

There are some works that deal with small cut cells and resulting time step issue, e.g.,

https://doi.org/10.1016/j.jcp.2010.07.014

Also see the latest ICOSAHOM2020 which had many talks on cut cell issue.

Personally I prefer to use body-fitted structured grids close to boundaries and then you are free to fill the rest with quad-tree type grids. You may get the best of both this way.

[ssh123]

Originally Posted by praveen

You will have the problem of very small cells which will impact time step in explicit schemes.

There are some works that deal with small cut cells and resulting time step issue, e.g.,

https://doi.org/10.1016/j.jcp.2010.07.014

Also see the latest ICOSAHOM2020 which had many talks on cut cell issue.

Personally I prefer to use body-fitted structured grids close to boundaries and then you are free to fill the rest with quad-tree type grids. You may get the best of both this way.

He does not use Cartesian cut cell. It's body fitted quadtree grid.

[aerosayan]

Originally Posted by LuckyTran

Pardon me for not seeing the novelty but there's volumes and volumes of real math that deals with creating these types of grids.

I get that you avoiding the so-called trimmer meshes but you can always take a pure quad mesh and stretch it to fill a domain. Question is, what happens to the skewed cells and these will always be a problem.

And have you seen meshers like GridPro?

Cells won't be skewed, as cell quality checks will be implemented and enforced. Costly 3rd party meshers can't be used for automatic refinement by an adaptive solver. Even if they can be used, the cost would be high. Automating quad mesh creation, then stretching it to fill a domain would be challenging. I would be interested in reading any paper that creates body fitted quadtree grid automatically, with prism layers and 100% quad cells (ignoring that interface cells between two levels of a quadtree aren't exactly quad cells, but they can thought of as quad cells). If such a paper exists, it would save me a lot of time and effort from reinventing the same thing again.

[LuckyTran]

Originally Posted by aerosayan

Automating quad mesh creation, then stretching it to fill a domain would be challenging.

It is straightforward. All you do is move the grid points by solving a constrained optimization problem (the poisson problem with additional constraints). The domain body you are trying to fit is the constraint. If you are familiar with the method of lagrange multipliers, you know how straightforward it is to calculate the solution to this problem.

I know firsthand I can do this in GridPro. I've done it. Give me any CAD and I can fit a cube to it. You can fit prism layers after the fact, and GridPro does this too.

But as mentioned already, you will end up with skewed cells. You have to come up with a metric that balances local versus global skewness and this part is probably not so straightforward. Less naive implementations will produce higher quality grids but there are singularities one way or another, it's a topology problem. You can show this with just pen and paper if you try to fit a 2x2 quad mesh to a triangle. There is skewness, no matter what. Any shape that is not a quadrilateral (and even some quadrilaterals) will produce skewness. You can repeat this for other size grids, i.e. a 3x3 or 3000x3000 but the skewness does not go away. The question is how many cells are skewed, and the degree of skewness in those cells. You can concentrate the skewness into 1 cell or distribute it over the entire grid. But it's there. Mathematicians deal with this on a theoretical level all the time in differential geometry.

I'm not at all involved in grid generation. But the proceedings of the international society of grid generation comes to mind. Also, there's a fair number of references in the GridPro. I'm not trying to advertise for GridPro or anything, but I can confirm that at least one tool exists and it has existed for decades.

Again, if there is some novelty greater than fitting a quad mesh to an arbitrary surface, then I have missed it.