Overview

This Mathematica framework constructs and optimizes the geometry of complex network manifolds. It models a physical network as a continuous surface manifold composed of cylindrical tubes (edges) that meet at polygonal junctions (vertices).

Using Mathematica's FEM package, the script finds the network's equilibrium 3D embedding by minimizing a tunable, physically-motivated energy functional. This makes it a powerful tool for research in:

1. Soft Matter & Biophysics: Modeling cytoskeletal networks, protein aggregates, or other physical polymer systems.

2. Materials Science: Simulating the structure of porous materials or foams.

3. Complex Systems: Finding minimal-energy embeddings for abstract networks with physical constraints.

The script models each tube as a quadrilateral mesh and seeks the lowest energy state by adjusting vertex coordinates, tube lengths, and local mesh geometry.

Prerequisites & Dependencies

Mathematica: Version 12.0 or newer. The code only relies on the built-in NDSolveFEM`` package of Mathematica.

QuadMeshNetworkPreset.nb:

This script sets up the topological and geometric characteristics for a physical network, including lists of nodes, links, their connectivities, and "stitching" information (e.g., which side and polarity of a sleeve boundary for each link is stitched to other links). The code can also automatically generate this information for a spatially embedded network (provided by a .swc file). The outputs are saved as .mx files in the preset/ directory.

preset/:

This directory contains pre-generated network presets using QuadMeshNetworkPreset.nb. These presets are used in other scripts, such as QuadMeshNetwork_trifur_script.nb.

QuadMeshNetworkScript.wl:

This is the main algorithm script - it provides all functions that will be used to model each tube as a quadrilateral mesh and seeks the lowest energy state by adjusting vertex coordinates, tube lengths, and local mesh geometry. These functions will be called and used in QuadMeshNetwork_trifur_script.nb and QuadMeshNetwork_bimodal_script.nb. Here is a breakdown of the workflow of these functions:

Workflow:

Run the script in the following sequence after executing your preset[].

Step 1: setupFEM[globalScale]

Generates a quad-meshed cylinder for each network edge.

globalScale: Controls mesh resolution (smaller value = higher resolution).

Defines Subscript[sqrtZScale, edgeN], the optimizable length scale for each tube.

Step 2: doAttribute[]

Calculates detailed geometric attributes, defining vertex polygon shapes and creating Interpolation functions (edgePathInterpolationList, edgePathInterpolationRotationList) to serve as guide paths for the tubes.

Step 3: doConstraint[]

Builds the symbolic energy functional (constraintsAll) by combining all physical components into a single expression for minimization.

Step 4: doInitialize[edgedλList, edgeSqrtZScaleList]

Generates a high-quality initial 3D embedding by placing the mesh coordinates along the interpolated guide paths, providing an excellent starting point for the optimizer.

edgedλList: Initial values for the conformal factors dλ.

edgeSqrtZScaleList: Initial values for the tube length scale factors.

Step 5: doFindMinimum[wfair, maxIterations, ...]

The main solver. Calls FindMinimum to find the coordinates and parameters that minimize the constraintsAll energy functional.

wfair0: Initial weight for bending energy (fairness).

maxIterations: Max iterations for the solver.

wSurface: Weight for surface area minimization.

wCircle: Weight to enforce circular shapes at network terminals.

wPathPerFair: Weight forcing tubes to adhere to their guide paths.

wIso: Weight for the local isometry/conformality constraint.

Step 6 (Optional): doChangeMode[...]

A utility to modify the physical model without redefining constraints.

confλ -> False: Enforces a stricter isometric model.

zScale -> False: Uses fixed-length tubes.

Interpreting the Output

The primary output is the findMinimumAll variable, which contains:

findMinimumAll[[1]]: The final, minimized value of the energy functional.

findMinimumAll[[2]]: A list of replacement rules (var -> value) with the optimized numerical values for all symbolic variables (coordinates, dλ, sqrtZScale).

QuadMeshNetwork_trifur_script.nb:

This script runs surface minimization for a trifurcation configuration. Execute the script line by line.

QuadMeshNetwork_bimodal_script.nb:

This script runs surface minimization for a bimodal-bifurcation configuration (i.e., bifurcation of two different link thicknesses). Execute the script line by line.

d[Lambda]/:

This directory contains the output files generated by QuadMeshNetwork_trifur_script.nb and QuadMeshNetwork_bimodal_script.nb.