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 QuadMeshNetworktrifurscript.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 QuadMeshNetworktrifurscript.nb and QuadMeshNetworkbimodalscript.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).

QuadMeshNetworktrifurscript.nb:

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

QuadMeshNetworkbimodalscript.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 QuadMeshNetworktrifurscript.nb and QuadMeshNetworkbimodalscript.nb.