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Tutorial 1: First Mesh and Run

Goal: Generate a mesh on a simple geometry and run the solver to see how AortaCFD controls mesh resolution and numerical schemes.

U-bend mesh overview U-bend pipe mesh generated by snappyHexMesh. This tutorial teaches meshing and solver basics on a simple curved pipe before moving to patient-like geometries.

Hands-On Exercises

Mesh generation: exercise_01_mesh.tar.gz (52 KB) | ~10 sec | OpenFOAM 12

tar -xzf exercise_01_mesh.tar.gz && cd exercise_01_mesh
source /opt/openfoam12/etc/bashrc && ./run_mesh.sh

Solver comparison: exercise_05_solver.tar.gz (1.5 MB) | ~5 sec | OpenFOAM 12

tar -xzf exercise_05_solver.tar.gz && cd exercise_05_solver
source /opt/openfoam12/etc/bashrc
cd case_robust && foamRun && cd ../case_standard && foamRun

Introduction

Before working with patient-specific aortic geometries, it is instructive to understand how AortaCFD generates meshes and drives the solver on a simple test case. This tutorial uses a U-bend pipe -- a curved tube with a single inlet and single outlet. The geometry is small enough that meshing completes in seconds and the solver finishes in under ten seconds, yet it exhibits curved-pipe flow physics (Dean vortices, asymmetric velocity profiles) that are directly relevant to aortic haemodynamics.

By the end of this tutorial you will be able to:

  1. Generate a volume mesh from an STL surface using snappyHexMesh.
  2. Read and interpret checkMesh quality output.
  3. Run the OpenFOAM solver with two different numerics profiles.
  4. Compare first-order (robust) and second-order (standard) solver behaviour.
  5. Identify healthy solver output in the log files.

Part 1: Mesh Generation

Extracting the Exercise

Download and extract the meshing exercise:

tar -xzf exercise_01_mesh.tar.gz
cd exercise_01_mesh
ls

You will see the following contents:

File Purpose
ubend.stl Surface geometry of the U-bend pipe (in millimetres)
run_mesh.sh Shell script that executes the meshing pipeline
system/blockMeshDict Background mesh definition
system/snappyHexMeshDict Surface-conforming mesh parameters
system/surfaceFeaturesDict Feature edge extraction settings

Running the Meshing Pipeline

Source the OpenFOAM environment and run the mesh script:

source /opt/openfoam12/etc/bashrc
./run_mesh.sh

The script executes three utilities in sequence:

blockMesh --> surfaceFeatures --> snappyHexMesh
  1. blockMesh creates a uniform hexahedral background mesh that encloses the STL geometry. This is the Cartesian starting grid.
  2. surfaceFeatures extracts sharp edges from the STL surface so that snappyHexMesh can preserve geometric features during snapping.
  3. snappyHexMesh carves the background mesh to conform to the STL surface in three stages: castellate (remove cells outside the geometry), snap (project cell vertices onto the surface), and optionally add boundary layers.

The entire process takes approximately 10 seconds on the U-bend geometry.

Understanding cells_per_diameter

The primary resolution control in AortaCFD is cells_per_diameter (cpd). This parameter determines how many cells span the local vessel diameter. Drag the slider to explore:

cpd Approximate cell count (U-bend) Use case
8 ~6,000 Quick debugging
10 ~10,000 Fast test
12 ~15,000 Tutorial default
15 ~30,000 Standard production
20 ~70,000 Fine resolution

For this exercise, cpd = 12 is used, producing approximately 15,000 cells. This is intentionally coarse -- sufficient for learning but not for publication-quality results.

How cpd Translates to Cell Size

AortaCFD reads the inlet diameter from the STL geometry and computes:

\[\text{cell size} = \frac{D_{\text{inlet}}}{\text{cpd}}\]

This cell size is then used to define the blockMeshDict grid spacing. Increasing cpd by a factor of 2 increases cell count by approximately a factor of 8 (cubic scaling in three dimensions).

Reading checkMesh Output

After snappyHexMesh completes, run checkMesh to assess mesh quality:

checkMesh

The output reports several quality metrics. The most important are:

Mesh non-orthogonality Max: 55.3 average: 8.7
Max skewness = 1.8 OK.
Max aspect ratio = 12.4 OK.
Mesh OK.
Metric Acceptable Warning Problematic
Max non-orthogonality < 60 deg 60--70 deg > 70 deg
Max skewness < 2 2--4 > 8
Max aspect ratio < 50 50--100 > 1000

Look for 'Mesh OK'

If checkMesh reports Mesh OK, the mesh passes all default quality thresholds. If it reports warnings about non-orthogonality exceeding 70 degrees, the mesh may require refinement or the robust numerics profile should be used.

Non-Orthogonality Above 70 Degrees

Meshes with non-orthogonality above 70 degrees can cause divergence with second-order numerical schemes. If your mesh reports values in this range, either increase cells_per_diameter to improve quality or use the robust numerics profile, which tolerates poor mesh quality at the cost of accuracy.

Part 2: Running the Solver

Extracting the Solver Exercise

Download and extract the solver exercise, which contains two pre-meshed U-bend cases:

tar -xzf exercise_05_solver.tar.gz
cd exercise_05_solver
ls
Directory Numerics Profile Schemes
case_robust/ robust Euler (time), upwind (convection)
case_standard/ standard backward (time), limitedLinearV (convection)

Both cases use the same mesh, the same boundary conditions, and the same physical properties. The only difference is the numerical discretisation.

Running Both Cases

source /opt/openfoam12/etc/bashrc

cd case_robust
foamRun
cd ..

cd case_standard
foamRun
cd ..

Each case completes in approximately 2--3 seconds.

foamRun vs Legacy Solvers

OpenFOAM 12 (Foundation version) uses the unified foamRun command with the incompressibleFluid solver module, replacing the legacy pimpleFoam solver. AortaCFD always targets foamRun.

Part 3: Comparing Profiles

Robust Profile (First Order)

The robust profile uses:

  • Euler temporal discretisation -- first-order implicit, unconditionally stable, introduces temporal diffusion.
  • upwind convection scheme -- first-order, maximum stability, smears velocity gradients.

This combination is extremely stable. It will run on virtually any mesh, including meshes with poor quality metrics. The trade-off is substantial numerical diffusion: velocity gradients near walls are artificially smoothed, causing wall shear stress to be systematically under-predicted.

Standard Profile (Second Order)

The standard profile uses:

  • backward temporal discretisation -- second-order implicit, unconditionally stable, accurate phase resolution.
  • limitedLinearV 1 convection scheme -- second-order TVD scheme, bounded, preserves sharp gradients while preventing non-physical overshoots.

This profile provides an appropriate balance between accuracy and stability for production cardiovascular CFD simulations. It requires reasonable mesh quality (non-orthogonality below approximately 65 degrees).

What to Look for in the Log

Examine the solver log for each case:

tail -30 case_robust/log.foamRun
tail -30 case_standard/log.foamRun

Key indicators of healthy solver behaviour:

Indicator Healthy Unhealthy
Courant number max < 1.0 > 2.0
PIMPLE convergence Converged in 2--4 iterations Not converged in 10 iterations
Pressure initial residual Decreasing trend Increasing or stagnant
Time step (deltaT) Steady or slowly varying Collapsing (< 1e-8)
Continuity error < 1e-6 > 1e-3 
Time = 0.02s
Courant Number mean: 0.04 max: 0.87
PIMPLE: Iteration 1
  smoothSolver: Solving for Ux, Initial residual = 0.0012, Final residual = 8.3e-08
  GAMG:  Solving for p, Initial residual = 0.31, Final residual = 2.1e-06
PIMPLE: Converged in 3 iterations

Comparing Residuals Between Profiles

The robust profile typically shows lower residuals because the first-order schemes introduce more numerical diffusion, making the system easier to solve. Lower residuals do not mean higher accuracy -- they mean the numerical system is less demanding, which is a consequence of the additional artificial diffusion.

Physical Differences

If you visualise both results in ParaView, you will observe:

  • Robust: smoother velocity field, rounded velocity profile at the bend, weaker secondary flow patterns.
  • Standard: sharper velocity gradients, more pronounced Dean vortices in the bend, steeper near-wall velocity profiles.

The standard profile resolves flow features that the robust profile smears out. For clinical applications where wall shear stress matters, the standard profile is the minimum acceptable choice.

Summary

Concept Key Takeaway
cells_per_diameter Primary mesh resolution control; cpd = 12 gives ~15K cells on U-bend
checkMesh Always inspect non-orthogonality, skewness, and aspect ratio
Robust profile 1st order (Euler + upwind); maximum stability, high diffusion
Standard profile 2nd order (backward + limitedLinearV); accurate, needs good mesh
Solver log Monitor Courant number, PIMPLE convergence, and residual trends
Mesh quality vs profile Higher-order schemes demand better mesh quality

What's Next

In Tutorial 2: Inlet and Outlet Conditions, you will learn how AortaCFD handles physiological boundary conditions -- pulsatile inlet flow waveforms and Windkessel outlet models that represent the downstream vasculature.

References

Full bibliography on the References page.

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