+1 Disclination Buckling: A PyMembrane Tutorial#

initial mesh

Figure: Snapshots of a Monte Carlo simulation of an open +1 disclination that shows buckling into a conical shape. (Left) the initial flat configuration; (Right) the relaxed buckled configuration. The colour bar represents the local mean curvature of the mesh.#

Introduction#

This example studies out-of-plane buckling of an open +1 disclination, a standard thin-sheet mechanics problem. In the triangulated mesh, the central vertex has five neighbours instead of six, which introduces the elastic frustration that drives buckling into a cone-like shape.

What This Example Demonstrates#

All three packaged disclination examples use the same packaged input meshes from InputFiles.zip and the same elastic forces: harmonic stretching, limit protection, and dihedral bending. They differ only in the time-stepping or sampling method:

  • pymembrane.examples.disclination uses a Brownian vertex-move integrator.

  • pymembrane.examples.disclination_mc uses a Monte Carlo vertex-move integrator with the documented annealing schedule.

  • pymembrane.examples.disclination_verlet uses a velocity-Verlet integrator.

Each version writes initial mesh.vtk and pentagon_t*.vtk snapshots. The Monte Carlo version also writes final_mesh.vtk. --quick keeps the same mesh, force model, and integrator choice, but reduces the number of steps and snapshots.

The packaged module pymembrane.examples.hybrid_mc_bd demonstrates a conservative hybrid workflow on the same physical problem. In each cycle it first performs Brownian dynamics relaxation with Mesh>Brownian>vertex>move and then performs Monte Carlo sampling with Mesh>MonteCarlo>vertex>move before writing a VTK snapshot. --quick reduces the number of hybrid cycles and the MD/MC step counts, but does not change the mesh, forces, or integrator names.

How to Run#

Brownian version:

python -m pymembrane.examples.disclination --quick

Monte Carlo version:

python -m pymembrane.examples.disclination_mc --quick

Velocity-Verlet version:

python -m pymembrane.examples.disclination_verlet --quick

Hybrid Brownian + Monte Carlo version:

python -m pymembrane.examples.hybrid_mc_bd --quick

All three commands also support --output-dir:

python -m pymembrane.examples.disclination --quick --output-dir results

The hybrid example supports the same output-directory option:

python -m pymembrane.examples.hybrid_mc_bd --quick --output-dir results

Inputs#

  • packaged disclination meshes from InputFiles.zip

  • default mesh size parameter: N=14

  • output directory: current working directory unless --output-dir is used

Model Ingredients#

  • mesh: open +1 disclination from docs/examples/01_disclination

  • forces: Mesh>Harmonic, Mesh>Limit, Mesh>Bending>Dihedral

  • integrators:

    Brownian: Mesh>Brownian>vertex>move

    Monte Carlo: Mesh>MonteCarlo>vertex>move

    Velocity-Verlet: Mesh>VelocityVerlet>vertex>move

    Hybrid: alternating Mesh>Brownian>vertex>move and Mesh>MonteCarlo>vertex>move

Expected Output#

  • initial mesh.vtk

  • pentagon_t0.vtk

  • additional pentagon_t*.vtk snapshots

  • final_mesh.vtk for the Monte Carlo version

  • initial_mesh.vtk and hybrid_t*.vtk for the hybrid example

  • output format: legacy ASCII .vtk

Quick Mode#

Quick mode keeps the same physical setup but reduces snapshot counts and the number of MD or MC steps.

The source versions of these examples are kept under docs/examples:

  • docs/examples/01_disclination/Brownian/disclination.py

  • docs/examples/01_disclination/MC/disclination.py

  • docs/examples/01_disclination/Verlet/disclination.py

The installed pymembrane.examples versions include the input data required to run each example directly after installation.

How to visualize the result#

Open the VTK files in ParaView to inspect the development of the conical shape. Comparing the three integrator variants is useful for understanding how the same physical setup relaxes under different update rules.

References#

[Seung88]

Seung, H. S., & Nelson, D. R. (1988). Microstructure of two-dimensional disclinations. Physical Review A, 38(2), 1005.

[Nelson87]

Nelson, D. R. (1987). Order, frustration, and defects in liquids and glasses. Physical Review B, 36(10), 5788.