Communication

When two blocks of different levels connect, a proper communication strategy must be developed. In the adopted formulation, we performed the communication such that it assures that the macroscopics (\(\rho, u_{\alpha\beta}, \sigma_{\alpha\beta}\)) are continuous thorough the simulation domain. The coupling between fine and coarse grid is illustrated below:

It’s important to notice that the top nodes of the fine blocks doesn’t coincide with the coarse level (only when the fine level ends). So to facilitate interpolation, this node is not used in the communication, serving as a ghost node, only to keep the recursiveness of the refinement.

Note

In order for not having invalid macroscopics, the values of these ghost nodes are copied from the neighbour nodes that coincide with the coarse level.

The communication of the coupling scheme is shown below:

Straight

The coarse-to-fine transference of information occurs in the blue nodes, whilst the fine-to-coarse in the orange. For concave and convex node, the communication is as follows:

Concave

Convex

It’s possible to notice that the ghost nodes of fine level does not participate in the communication.

Top domain refinement

One consequence of the ghost node in the fine block is an edge case for boundary conditions on top of the domain.

Example of two levels refinement on top of the domain.

When refining one or more levels in the top, it’s possible to notice that the top nodes of a level doesn’t coincide with the top one of the coarse leve. So if a boundary condition is applied in the top node of the coarse level, in the fine level it must be applied in the same position.

Fine to Coarse overlap

From the communication images, it’s possible to notice that there is an offset between the coarse to fine (C2F) and fine to coarse (F2C) communication borders. These are the F2C overlap nodes, and are measured using the coarse level node distance as reference.

For simulations of great turbulence, increasing this overlap from 1 to 2 or 3 gives the simulation more stability, as stated by Lagrava et al.[1]. In the macroscopics field it’s possible to see a smoother transition between levels and the decrease of checkerboard phenomenas (specially in pressure).

Example of F2C overlap equals to 2. Blue is C2F and yellow is F2C.

Macroscopics reconstruction

The populations of the fine grid for the nodes that coincide with the blue coarse nodes are initially unknown, and are reconstructed from the macroscopic values of the coarse grid in each time step of the fine grid simulation. For the fine nodes that are between the blue nodes not coinciding with the coarse grid, the macroscopic values of the fine grid are estimated through interpolation of the coarse grid macroscopics (\(\rho, u_{\alpha\beta}, \sigma_{\alpha\beta}\)). The adopted scheme can be found in Lagrava et al.[1]. Using these values, the populations are reconstructed. The fine grid solution must update the coarse grid boundary. Through the overlapping of grids, the macroscopics at orange coarse grid nodes are updated using the values from fine grid simulation.

The estimative of the macroscopics between coarse grid nodes uses a cubic four-point interpolation performed on the coarse-to-fine grid interface plane through the following expression:

(1)\[g\left(x\right) = \frac{9}{16}\left[g\left(x+h\right)+g\left(x-h\right)\right]-\frac{1}{16}\left[g\left(x+3h\right)+g\left(x-3h\right)\right]\]

where \(g(x)\) is the macroscopic considered. An illustration is presented below:

For a node that’s located near the fine grid-block border a cubic three-point interpolation is adopted with the following equation:

(2)\[g\left(x\right) = \frac{3}{8}g\left(x-h\right) + \frac{3}{4}g\left(x+h\right) - \frac{1}{8}g\left(x+3h\right)\]

A temporal interpolation of the populations at the boundaries is also performed since time-scale of fine grid is also different from coarse. We adopt a three-point Lagrangian scheme for that matter:

(3)\[f_{i}\left(t\right) = \sum_{k=-1}^{1} f_{i}\left(t_{k}\right)\left(\prod_{j=-1;k\neq j}^{1}\frac{t-t_{j}}{t_{k}-t_{j}}\right)\]

or:

(4)\[f_{i}\left(t\right) = 0.5\frac{1}{2}\left(\frac{1}{2}-1\right)f_{i}\left(t_{-1}\right) - \left(\frac{1}{2}-1\right)\left(\frac{1}{2}+1\right)f_{i}\left(t_{0}\right) + 0.5\frac{1}{2}\left(\frac{1}{2}+1\right)f_{i}\left(t_{1}\right)\]