3-D Dynamic Cantilever

Tutorial Aims

This tutorial demonstrates how to:

• simulate the large-deformation vibration of a three-dimensional cantilever
• create a beam mesh and rotate it into its initial vertical configuration
• apply a time-dependent end force and zero end moment
• use the implicit Newmark time-integration scheme
• compare mesh and time-step sensitivity against reference solutions
• extract displacement, force, convergence and energy histories

The case is the first benchmark presented in the beamFoam paper.

Prerequisites

• A compiled beamFoam installation
• A sourced, supported OpenFOAM.com environment
• gnuplot with the pdfcairo terminal for generating the supplied plots
• ParaView for visualising the deformed beam

The default case uses OpenFOAM's implicit Newmark scheme, a time step of 0.001 s, and 20 beam segments.

Problem Description

A cantilever column with a square cross-section is clamped at its lower end and subjected to a sudden constant force at its upper end. The force acts from t = 0 s until the simulation ends at t = 1 s, producing large three-dimensional deformation and vibration.

The benchmark compares the one-dimensional beamFoam model against detailed three-dimensional solids4Foam and ABAQUS solutions.

The first bending period is approximately 1 s. The paper recommends resolving this period with at least 20 to 50 points, corresponding to a time step of approximately 0.02 s or smaller.

Case Setup

Beam Geometry and Material

The beam definition is stored in constant/beamProperties:

beamModel coupledTotalLagNewtonRaphsonBeam;

beams
(
    beam_0
    {
        crossSectionModel rectangle;

        rectangleCrossSectionModelDict
        {
            b 0.2;
            h 0.2;
            nx 1;
            ny 1;
        }

        length      2;
        nSegments   20;
        E           15.293e6;
        G           5.882e6;
        rho         1000;
    }
);

createBeamMesh initially creates the beam along the global x-axis. The Allrun script then rotates beam_0 by 90 degrees about the negative y-axis:

setInitialPositionBeam \
    -cellZone beam_0 \
    -translate '(0 0 0)' \
    -rotateAngle '((0 -1 0) 90)'

Boundary Conditions and Loading

The left patch represents the clamped lower end:

• 0/W: fixedValue displacement
• 0/Theta: fixedValue rotation

The right patch represents the loaded upper end:

• 0/W: forceBeamDisplacementNR, reading constant/timeVsForce
• 0/Theta: momentBeamRotationNR, reading the zero-moment history in constant/timeVsMoment

The applied force history is:

(
    (0 (2000 2000 0))
    (1 (2000 2000 0))
)

Time Integration and Output

system/fvSchemes selects the implicit Newmark method for the first- and second-time derivatives. system/controlDict defines deltaT, the end time, write frequency and the following function objects:

• beamDisplacements: displacement history at the right patch
• beamForcesMoments: force and moment history at the left patch
• beamConvergenceData: nonlinear convergence history
• beamEnergyData: internal, kinetic and total energy histories

Running the Tutorial

From this tutorial directory:

./Allclean
./Allrun

The script performs:

1. createBeamMesh
2. setInitialPositionBeam
3. beamFoam
4. gnuplot allPlots.gnuplot

The main generated files are:

• log.createBeamMesh
• log.setInitialPositionBeam
• log.beamFoam
• displacementPlot.pdf
• energyPlot.pdf
• postProcessing/0/beamDisplacements_right.dat
• postProcessing/0/beamEnergyData.dat

Post-Processing

To inspect the deformed column:

touch case.foam
paraview case.foam

Apply Warp By Vector using pointW. The upper end should undergo a large oscillatory displacement and can fall below its initial base level during the motion.

displacementPlot.pdf shows the displacement magnitude at the loaded end. energyPlot.pdf shows the internal, kinetic and total energy histories.

Expected Results

The paper reports that:

• approximately four Newton iterations are required per time step
• even a five-segment beam gives a response close to the reference results
• the 20-segment result converges closely to solids4Foam and ABAQUS
• Newmark captures the oscillatory response more accurately than backward Euler for a given time step
• time steps of 0.01 s and 0.001 s closely match the reference response

Reference displacement histories from solids4Foam and ABAQUS are supplied in solidPointDisplacement_pointDisp.dat and cantilever_abaqus_50percent.txt.

The execution times reported in the paper were obtained on an Apple M1 Pro and should be treated as comparative reference values rather than acceptance criteria:

Mesh Sensitivity Study

Run:

./runSweep.sh

The script runs meshes with 5, 10, 20 and 40 segments. It writes:

• timing_summary.txt
• displacement histories under dispResults/
• dispPlotVaryingMesh.pdf

The script modifies nSegments in constant/beamProperties and leaves the last tested value in place. Restore the default value of 20 before running the standard case again.

Time-Step Sensitivity Study

Keep the Newmark scheme selected in system/fvSchemes, then run:

./runTimeSweep.sh

The script tests deltaT = 0.1, 0.05, 0.01 and 0.001 s. It writes:

• dt_timing_summary.txt
• displacement histories under dispResultsTime/
• dispPlotVaryingDT.pdf

The script modifies deltaT and writeInterval in system/controlDict and leaves the last tested values in place.

To investigate backward Euler, change both defaults in system/fvSchemes from Newmark to Euler before running the sweep. Coarse Euler time steps introduce substantial numerical damping.

Troubleshooting

• If createBeamMesh or beamFoam is not found, source OpenFOAM and rebuild beamFoam.
• If the beam remains horizontal, inspect log.setInitialPositionBeam for errors.
• If plotting fails, confirm that gnuplot supports the pdfcairo terminal. The solver results remain available under postProcessing/.
• Run ./Allclean before repeating the standard case after a sensitivity study.

References

• Bali, S., Taran, A., Tuković, Ž., Pakrashi, V., and Cardiff, P. (2025). beamFoam: A Cell-Centred Finite Volume Solver for Nonlinear Geometrically-Exact Beams in OpenFOAM. OpenFOAM Journal, 5, 180-210.
• Cardiff, P., et al. (2025). A Jacobian-Free Newton-Krylov Method for Cell-Centred Finite Volume Solid Mechanics.