Abstract: Interpolation in the overset domain using connectivity information for a cell-centered CFD flow solver will typically use a least squares procedure to determine the interpolation weights. The weights produced using the least squares procedure are not bounded between zero and one. Thus, the interpolation can be non-monotonic and will introduce new extrema in the solution, which can cause difficulties with the CFD solution. Interpolation using a dual grid, which connects primal cell centers to form dual grid cells, can be used with trilinear interpolation to produce weights bounded between zero and one. A global, dual-grid approach where a single grid connects all cell centers can be expensive to store. While using a set of local dual grids, where each primal grid element has an associated local dual grid independent of neighboring local duals, can reduce memory requirements by loading only the set of local dual grids required for interpolation. In this article, compressible CFD solutions using the least square interpolation weights are compared with solutions using the global dual grid interpolation weights. These results show that the non-monotonic interpolation using the least square interpolation weights can cause solution instabilities. The CFD solutions are observed to be more stable when using the dual grid interpolation weights.
The overset or chimera grid methodology utilizes a set of overlapping grids to discretize the solution domain. The component grids may be fitted without regard to other portions of the geometry and can simplify the grid generation process to a great extent. The result is a flexible computational simulation framework that can be an enabling force in many situations. It has been widely used to simplify the structured grid generation requirements for complex geometries. Using an overset grid system is also a promising solution for the simulation of bodies in relative motion, such as a fuel tank being dropped from an aircraft, and rotorcraft.
The stencil of points and the method of defining the interpolation weights are critical factors while determining the flow solution on an overset composite grid system. The accuracy and smoothness of this interpolation will affect the flow solution accuracy and stability. The location of the flow-dependent variables in the grid affects the interpolation stencil and the method used to determine the interpolation weights.
A primal grid consists of grid points, where the cell connectivity between those grid points are as produced by the grid generation software. The set of cell types will typically be tetrahedra, hexahedra, prisms, and pyramids.
A dual grid consists of cell centers of the primal grid and connectivity of those dual-grid points to form dual cells. The dual-grid cell connectivity must be defined, as we are provided with only the primal grid cell connectivity. The type of dual-grid cells constructed will significantly impact the scheme used for interpolation. A disadvantage of the weights computed using the least squares procedure is that the weights are not bounded between zero and one. A dual grid for unstructured grids will provide the same benefits as those for a structured grid but at the cost of more memory and time.
A. Structured Dual Grid
For a structured grid, the connectivity between the cell-centered locations is implicit in the same way as for the points/nodes. As the dual grid connects cell centers, they will not cover the same volume as the nodal or primal grid. The dual-grid cells do not cover the space between the boundary cell centers and the grid boundary. Hence, a donor in the dual grid must use extrapolation for any fringe located in the void between the boundary cell centers and the block boundary face. The dual grid can be extended across point-match block-to-block interfaces if the connection is regular and topologically consistent.
B. Unstructured Dual Grid
Figure 1 shows an unstructured primal grid composed of triangles (black lines) and the cell centers shown as blue dots. One possible dual grid with triangles is shown in light blue, which connects the cell centers. The unstructured simplex dual-grid cell in three dimensions will be tetrahedra. Clearly, multiple dual-grid triangles are required to cover the primal triangles. Thus, a donor search in the primal grid must be augmented by a donor search in the dual grid to find the correct dual donor cell.
Figure 1. Cell-centered unstructured primal and dual grids.
C. Structured Dual-Grid Donor Hexahedron
The donor search within the Suggar++ software will initially operate on the primal grid even if a dual-grid donor is requested. The primal grid donor cell is used as one corner of the dual-grid donor hexahedron, and the procedures try to use appropriate neighbors to form the remaining corners of the hexahedron. As the dual grid connects the cell centers, note that if a fringe falls in the volume between the cell center and a boundary face, the fringe location will be outside the dual-grid donor, and extrapolation will occur.
D. Reduction of Dual-Grid Donor
As discussed earlier, the dual grid will not cover the same volume as the nodal or primal grid and leaves a void near the boundaries. A fringe point located in this void will find a primal-grid donor, but the dual-grid donor will require extrapolation, possibly resulting in non-monotonic interpolation with the interpolated value not bounded by the donor member values.
Monotonic interpolation can be retained if the user accepts a reduction in the accuracy of the interpolation in the boundary normal direction. The dual-grid donor members can be reduced to a quadrilateral, but when there is no single normal direction to the boundary, the reduction procedure will reduce the donor cells to a single donor member: the primal cell. A similar reduction process is used for the unstructured dual grid, where the donor will be automatically degraded to a triangle, line, and point when the fringe point lies near a boundary face and is outside the dual tetrahedral grid.
E. Benefits of Using Dual Grid for Cell-Centered Donor
A cell-centered dual-grid donor can provide significant benefits relative to the default cell-centered donor. As mentioned, the donor members for the default cell-centered donor are the donor cell and its neighbors, with the donor weights computed using a least-squares procedure. These least-squares weights may not be bounded by zero and one and may result in non-monotonic interpolation. In addition, the interpolated value may not be continuous as the fringe location moves from one donor cell into a neighboring cell, as the donor members will change. In contrast, the dual-grid donor uses an interpolation cell connecting the cell centers in the region of the primal cell.
For a structured grid, the dual-grid donor will be a hexahedron connecting the donor cell center with adjacent cell centers. For an unstructured grid, a simplex dual-grid donor will be a tetrahedron connecting cell centers in the region of the primal-dual cell. The benefit of the dual-grid tetrahedral donor interpolation must be weighed against the cost of storage for the unstructured dual-grid and the second donor search within the dual grid. The grid points in the dual grid will be the cell centers of the primal grid. An estimate of the number of tetrahedra in the dual grid will be 6 – 7 times the number of cell centers in the primal grid. Therefore, the dual-grid storage can be quite large.
Figure 2. Cell-centered dual-grid donor with fringe near a boundary.
Evaluation Cases for Dual Grid Capability
Two geometries were chosen for simulation in RavenCFD to assess the dual-grid capability in Suggar++. The first is the hypersonic HIFiRE-1 cone/cylinder/flare geometry, and the second is the NASA Common Research Missile. These geometries were chosen because of the existence of experimental data for comparison and to assess a range of flow regimes.
The HIFiRE-1 is a hypersonic cone/cylinder/flare geometry assessed by the hypersonic international flight research and experimentation (HIFiRE-1) flight-test program. In the current work, the Mach 7.16 case with a wall temperature of 300 K was chosen for RavenCFD simulations, which were conducted in three forms to evaluate the dual-grid donor approach.
- Least squares donor weighting without interpolation function clipping
- Least squares donor weighting with interpolation function clipping
- Dual-grid donor weighting without interpolation function clipping
In these cases, the interpolation region was pushed very close to the HIFiRE-1 body to ensure that the interpolation occurred in regions of high gradients. Figure 3 is a slice along the y-axis, which shows the behavior of wall heat transfer on the HIFiRE-1 geometry surface and the bulk flow behavior in the region of the flare for simulation using least squares weighting with interpolation function clipping.
The dual grid requires significantly more memory, but the overall donor search wall clock times show a minimal difference in this case. As mentioned before, the least-squares procedure has a serious disadvantage in that the weights may not be bounded between zero and one. Thus, the interpolation can be non-monotonic and introduce new extrema into the solution causing difficulties for the flow solver. This behavior was exemplified in simulating the HIFiRE-1 geometry in RavenCFD.
When using the least-squares donor weighting without clipping, the interpolation function resulted in numerical difficulties, and the simulation crashed. While the simulation using the least-squares donor weighting with interpolation function with clipping proceeded to convergence. As expected, the simulation using the dual-grid donor without the interpolation function clipping was able to proceed to convergence. Therefore, the dual grid effectively bounded the interpolation function to eliminate the introduction of erroneous extrema into the flow solver solution.
Figure 3. RavenCFD simulation results show surface heat flux on the HIFiRE-1 flare region and Mach number in the bulk flow.
B. NASA Common Research Missile (CRM)
For the CRM geometry, three simulations were conducted:
- Least squares donor weighting with interpolation function clipping
- Dual-grid donor weighting without interpolation function clipping
- Conformal grid
In this case as well, the overset grid boundary was forced very close to the CRM, so overset interpolation took place in regions of high gradients for this work. Figure 4 shows the CRM grid (black) cut close to the body on multiple levels of increasingly coarse Cartesian background grids as the distance from the missile body increases.
As expected, memory usage is greater for the dual grid, as is the donor search time. The increased difference in donor search time between the least squares and dual-grid donor search algorithms compared to the HIFiRE-1 case can be attributed to the increased number of component grids for the CRM case. The multiple levels of background Cartesian refinement require Suggar++ to search multiple overlapping grids for suitable donors.
Figure 4. Component grids within the CRM simulation.
However, using the dual-grid donor search algorithm again effectively bounds the interpolation function between zero and one, and thus, allows the CRM simulation using the dual-grid to be performed without clipping the interpolation function. The results for each of the CFD simulations are very similar, with the overset simulations exhibiting nearly identical behavior to the conformal simulation, albeit with some slight oscillations that can most likely be attributed to the interpolation in the high gradient regions. Overall, the RavenCFD simulations shown in this work have demonstrated that the bounding of the interpolation function when using the dual-grid approach can eliminate spurious extrema introduction into the flow solver, which results in numerical difficulty.
An overset grid system used by a flow solver that stores its dependent variables at the cell centers will typically use an interpolation donor. Second-order interpolation requires that a least-squares approach can be used to find the interpolation weights. These interpolation weights are typically not bounded between [0−1], and new extrema can be introduced into the solution via the interpolation. A simple approach to remove the extrema is to clip the interpolated values to the minimum/maximum value of any of the donor members, which reduces the interpolation to the first order.
The effectiveness of using unstructured dual grid interpolation was investigated for compressible flow CFD solutions using the RavenCFD code for a cone/cylinder/flare at hypersonic conditions and a complex missile configuration at low supersonic conditions. This effort compared the results using the standard least squares donor and weights with and without interpolation function clipping and global dual grid interpolation. For these compressible CFD solutions, the least square interpolation with clipping is a more cost-effective approach for moving body problems, although the interpolation function, when using the dual-grid approach, can eliminate spurious extrema introduction into the flow solver. The use of dual grid interpolation with incompressible solutions should reduce noise in the solution arising from the non-monotonic interpolation and increase solution robustness, convergence, and accuracy.
- Noack, Ralph W., Wyman, Nicholas J., McGowan, G., and Brown, C., “Dual-Grid Interpolation for Cell-Centered Overset Grid Systems,” AIAA paper no. 2020-1407, January 2020.
Request a Demo
If you’d like to try Fidelity Pointwise on your CFD applications to see the benefit of Dual-Grid Interpolation for Cell-Centered Overset Grid Systems, request a demo today.