# CFX Comparison - Numerical Methods

## Numerical method in Palabos

To simulate the flow inside the aneurysm, some of the new features of Palabos that can be found in version 1.0 were used. Specifically, a Guo off-lattice boundary condition was utilized. The aneurysm surface geometry was read from an STL file and processed appropriately. The new parallel voxelizer was used to voxelize the computational domain and distinguish between lattice nodes that belong to the interior of the artery from the ones that lay outside of it. Lastly, the aneurysm application code incorporates the concept of smooth grid refinement until convergence achieved. The word "smooth" indicates that the whole grid was doubled in each direction, and successive solutions were obtained until a grid independent solution was finally achieved.

## Numerical method in CFX

AnSys(R) CFX is a commercial software package using traditional highly efficient first- and second-order finite volume methods on a variety of different grid types. Only pure tetrahedral meshes were considered in the present benchmark. These meshes were refined close to the arterial wall and were prefered from other more complicated hybrid-type grids (like the ones that consist of prism layers near the walls) because the results they produced in this specific test were considered to be of a higher quality. All computational meshes were constructed by the AnSys(R) Icemcfd code. Concerning CFX, a fully second-order accurate finite volume solver was used for all simulations.

## Convergence Study: benchmark resolution

At which resolution must a program run to yield sufficient accuracy? And which criteria must be met so one can argue that CFX and Palabos have reached the same level of accuracy? There is no single answer to these questions, especially as the two numerical tools have a different convergence behavior, depending on the monitored quantity. In Palabos for example, the average energy appeared to converge much faster with respect to the grid resolution, while CFX showed a distinctly higher speed of convergence for the pressure. In the end, we selected the RMS vorticity as our main criteria of convergence, as it seemed to best reflect the global convergence behavior of both tools.

Performing a grid convergence study is particularly easy in Palabos, because the grid is generated automatically. Thus, a single batch job can be started to obtain the data for an array of grid sizes. For the CFX benchmark data, a fixed mesh with approximately 1 million grid points was generated, with a relative error of 3e-2 with respect to its own high-resolution grid (see next section). In Palabos, a grid convergence study (see picture on the right), showed that 700'000 grid points are required to obtain the same relative error. Again, the error is measured relative to Palabos' own high-resolution grid.

## As large as possible: reference resolution

To get reference results, the simulation was executed on the largest possible grid size. In the case of our CFX run, the maximal grid size was limited by the abilities of our (non-parallel) grid generation tool. In the case of Palabos, the size was limited by our wish to get the results within a few days on a medium-sized parallel machine.

In the following, all numerical results are displayed at both the benchmark and the reference resolution. The following table summarizes the grid points used by both methods at both these resolutions