## Why Cartesian Grids Are Good

The fine folks at Mentor Graphics publish a lot of good white papers. One of their recent ones is is How to Choose an Effective Grid System for CFD Meshing.

I think it’s mis-titled. The paper should be called Why Cartesian Grids are Good. They actually undersell the benefits of Cartesian grids. They can be extremely useful in a variety of situations. (Keep in mind that Pointwise doesn’t generate Cartesian grids so I don’t have any stake in this game.)

## The Case I’d Make for Cartesian Grids

1. Compared to body-fitted grids, generating a Cartesian grid is trivial. And that’s a person’s time you’re saving.
2. The price you pay for that triviality is complexity in how the boundaries are treated (i.e. cut cells). But that complexity is handled by the solver. Understand its limitations and you’re golden.
3. Solutions on Cartesian grids tend to converge better than those of body-fitted methods.
4. Other practitioners have been very successful with Cartesian methods.

## What the White Paper Actually Says

A Cartesian grid is the most common such system in practical use…” I take this to mean that the majority of all CFD analyses performed in the world today use Cartesian grids. This doesn’t jive with my personal experience but because I don’t claim to be omnipresent I’d like to see the data.

With respect to the issue of grid quality, “The velocity components solved for are almost always those aligned in the Cartesian coordinates directions.” In other words, you’re solving for Vx, Vy, and Vz so why not use an x-y-z aligned grid. Other than seeming like a bit of circular reasoning, I think that in reality what we want is grid alignment with the local flow direction, not the coordinate system in which the Navier-Stokes equations are derived.

The grid quality issue that’s undersold is how Cartesian systems don’t require additional numerics to account for a cell’s non-orthogonality. Boundary-fitted systems need additional terms to correct the diffusion flux and these additional terms have a tendency to make the system of equations “stiffer” and impede convergence of the solution. In other words, solutions on Cartesian grids converge better.

Representing complex geometry boils down to the matter of cut cells where the geometry passes through the Cartesian grid. The advantage cited is “Good grid quality is ensured, with the benefits explained earlier.” Nothing is farther from the truth. Cut cells are to Cartesian grids what hole cutting is to overset meshing – a gray area of black magic. That doesn’t make it a drawback, just something that should be addressed explicitly.

Certainly you can point to the work of Patankar, Spalding, Aftosmis, and Dawes as examples of good CFD with Cartesian grids but you have to make sure you’re comparing apples to apples. Aftosmis’ Cart3D is inviscid so you can’t make too general a conclusion about “the Cartesian-grid results are of comparable quality with results from body-fitted and more-complex grid systems.” Certainly that’s true in many cases but perhaps not all. (Aftosmis’ Cart3D is a superb example of the great things that can be accomplished with Cartesian mesh-based CFD – and I’m not just saying that because he’s a fellow Syracuse alum.)

# Choosing Your Grid System

You really ought to checkout Mentor Graphic’s white paper and if it the benefits of Cartesian grid-based CFD resonate with you give them a call to find out more about their CFD software.

On the other hand, if your CFD solver of choice doesn’t support Cartesian grids perhaps you should take a look at Pointwise for your multi-block, structured hex, unstructured tet, and hybrid meshing.

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### 9 Responses to Why Cartesian Grids Are Good

1. I didn’t read Mentor’s white paper since the web page required me to sign up for it. But, I thought one of the weak points with Cartesian grids was high Reynolds flow? I’ve heard of hybrid systems (Cartesian + Body fitted grid) for high Re flow. But, the body fitted grid and Cartesian grid need to be connected together, on way or another, and therefore a problematic area. In addition to generating the body fitted grid. Has the state of the art advanced?

2. Cartesian grids methods based on ghost cells (as opposed for cut-cells) work great (especially for compressible flows). Also, for higher Reynolds number, a wall function can be used to simulate turbulent flows (of course this is subjected to all the restrictions of wall functions).

If we look at the world of CFD as an asymptotic solution with big CFD as the outer solution and reduced order methods as the inner solution. I think that methods based on Cartesian grids would be the matching solution as they present the best tradeoffs between flow complexity and ease of complex geometry manipulation.

3. John Chawner says:

Martin: The system Mentor Graphics describes in the white paper includes boundary layer resolution (via an extrusion technique if I remember correctly) that couples to the Cartesian mesh. That approach is not unique having been used by others in the past.

Patrick: I too am intrigued by Cartesian methods which is why I read Mentor Graphics’ white paper in the first place and then blogged about how, in my opinion, they didn’t sell it strongly enough and instead muddied the waters with debatable commentary.

4. In regards to high Re flow and immersed Cartesian grids, NASA has a code called LAVA (Launch Ascent and Vehicle Aerodynamics) which “consists of two solvers: an off-body immersed-boundary Cartesian solver with block-structured adaptive mesh refinement and a near-body unstructured body-fitted solver which includes conjugate heat transfer. Two-way coupling is achieved through overset connectivity between the off-body and near-body grids.” (quoted from the following paper)

http://www.nas.nasa.gov/assets/pdf/papers/ICCFD7-3102_paper.pdf

I think a distinction needs to be made between Cartesian grids and immersed boundary conditions. I assume any grid, structured or unstructured, can have an immersed boundary condition. And, Cartesian grids can be used with either structured (overset) or unstructured solvers. It’s just a matter of how one couples (ties) the Cartesian grid to the other grids. For example, IMO, when one combines, in user defined surface regions, a Cartesian grid with an extruded near-body unstructured body-fitted grid using overset connectivity, that’s an overset (Chimera) approach in those regions. On the other hand, for codes other than LAVA, if the body-fitted grid is coupled to the off-body grid using unstructured cells, then it’s an unstructured approach. In neither case, IMO, is it an immersed B.C. The B.C.s used are probably simply no slip with extrapolated pressure and density.

Cartesian grids, independent of the B.C.s used, have multiple advantages which decrease the time to solution, some of them are: 1) reduced memory requirements (speeds up memory access), 2) simplified math (as John pointed out), and 3) a flavor of ADI can be used (assuming the CFD code is implicit). Also, as far as I know, it’s easier to apply higher order spatial discretization methods to Cart grids. I’m sure there are other pluses… On the down side, a Cart grid’s error and/or generated noise for a converged (either steady state or dual time stepping approach) may be higher than a well done polyhedral grid.

Unfortunately, at least for an overset method which lags fringe points, a body fitted grid which is coupled to a Cartesian grid will bring down the time step or CFL number. Thus the convergence time slows down. One of the key tidbits of knowledge then is how the body fitted grid is coupled to the off body grid. For example, with OVERFLOW the fringe points are lagged. I’m not sure what LAVA does.

Following the vehicle assent theme, the nice part about using immersed B.C.s for initial vehicle assent is that the tower and surrounding structures are geometrically complex and are in low Re flow. On the other hand complex geometries in high Re flow (such as the launch abort tower on the Saturn V, the milk stools and open regions in between some rocket stages, or protuberances) should probably not be solved with immersed B.C.s.

But, some of these are only my opinions and I wish I had data to back them up.

5. John Chawner says:

Martin, you make a good point about the use of terminology. Most every overset grid has a Cartesian background grid (at least the ones I’ve seen) yet no one is calling them Cartesian. And your point about the manner in which the Cartesian grid and the near-body grid (whether that’s the surface grid of the body itself or some off-body grid for boundary layer resolution) is well taken.

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