SDQ_meshes

This repository contains the code used for the publication Fabrication-aware strip-decomposable quadrilateral meshes, by Ioanna Mitropoulou, Amir Vaxman, Olga Diamanti, Benjamin Dillenburger, published in Computer-Aided Design, Volume 168, 2024, doi.

In particular, the code implements;

• The vector field optimization method presented in that paper, where two transversal 2-fields are optimized jointly and integrated in two coupled strip networks (publication Section 4, see vector_field.h).
• The overlay of the two networks into a Strip-Decomposable Quad (SDQ) mesh (publication Section 4.4, see quad_mesh.h).
• Strip-based editing operations to fix topologic defects of the strip networks (publication Section 5, see quad_mesh.h and strips.h).
• Tracing paths on the strip network (publication Section 6.1, see paths.h and paths_tracing.h)

The dependencies of this code include directional, libhedra, and libigl.

For questions feel free to contact the author Ioanna Mitropoulou: [email protected]

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⚠️ Important Note ⚠️

Following the recent cleanup and update to the latest libraries, a bug has been identified that affects high-genus geometries. This issue impacts the integration step required for seamless parametrization and is currently under investigation. Detailed information about this bug can be found in the following issue. As a result, the integration step will not work on complex models, such as those in data/batwing and data/costa_minimal_surface_Complex. I am actively working on resolving this issue.

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The logic

What is an SDQ mesh? It is the overlay of two orthogonal and topologically coupled strip networks, such that each network discretizes its counterpart.

An SDQ mesh (right) is the overlay of two transversal topologically-coupled strip networks.

In this configuration, edges can be two-colored, i.e., we assign a color to every edge so that no two neighboring edges sharing a quad have the same color. Consequently, all vertices have even valences, and all faces are quadrilaterals, except for boundary faces. An SDQ mesh is therefore a quad mesh that can be unambiguously decomposed into two independent discrete strip networks that share the shame topology and discretization.

Applications

The SDQ mesh representation generated with this code has been used to create the print paths layout for the fabrication of a full-scale 2.0-meter tall thin shell geometry for the project Fluid Forms. The print paths are aligned to principal curvature directions, and the rationale behind this choice has been described in the publication: (link to be added).

Fluid forms full scale prototype fabricated with nonplanar paths.

In addition, the code was used to generate the layout for the creation of the prototypes presented in the paper Non-planar 3D Printing of Double Shells (link to be added), presented in RobARCH 2024.

Various prototypes presented in RobARCH.

Costa minimal surface prototype fabricated with paths aligned to principal curvature directions.

While SDQ meshes have been applied to nonplanar 3D printing, their underlying representation, which consists of two sets of orthogonal strips can be useful in numerous other fabrication scenarios. Examples of such scenarios include woven structures, surfaces from sheets in crossing directions, warp-weft fabrication, custom rebar, cable-net structures, and the design of secondary transversal structural elements on structural strip patterns.

Examples of applications of SDQ meshes. Left: Minima Maxima world expo pavilion, by TheVeryMany, Marc Fornes. Middle: Bertoia diamond chair, by Harry Bertoia. Right: NEST HiLo roof - Full-scale construction prototype, by Block Research Group

Installation

This code has been tested on Windows 10 and 11. To install on Windows download the repository and its submodules:

git clone --recurse-submodules https://github.com/ioannaMitropoulou/SDQ_meshes.git

Then, go to the SDQ_meshes folder, open a shell and call:

mkdir build
cd build
cmake ..

Note that Matlab needs to be installed for the build to work. The code has been developed with Matlab 2021b.

Information for usage

Input command line arguments

One command line argument is required that points to the data folder where the inputs are saved. In that folder, there should be:

• a mesh.obj (input triangle mesh, with a good quality triangulation, scaled to fit within the unit sphere),
• and a scale_factor.txt (with a single float number that determines the scale of the parametrization). For examples of these, see the existing folders in the data folder.

Data and output folders

A library of examples can be found in the data folder. The types of geometries that work well with our method are relatively smooth and simple geometries with open boundaries, such as the ones included in the examples.

When running each example, the results are visualized in the viewer. At the same time, an output folder created automatically in its folder, where all the output information is saved in various formats (mostly in .txt), and are available for further analysis.

Workflow

The following workflow must be followed to generate an SDQ mesh:

• Select parameters of input constraints. For example, the 'Align with' parameter allows to select between alignment with principal curvature directions (default option and most common scenario), boundaries, or constraints-from-file. If constraints-from-file are selected, then these should be given into a file named fvf_constraints.txt that is positioned into the data folder. Each line should contain the constraint for a face in the format: [fi x y z weight], where fi is the face index, x y z describe the constraint vector that is tangent on fi, and weight is the weight of that constraint. The example data\curvy_surface contains an fvf_constraints.txt that will be loaded if the option Aligned with > ConstraintsFromFile is selected.

• Initialize vector field (key 'D'). Initializes the vector field and does a first direct solve of the energy.

• Optimize vector field (key 'V'). Runs the iterative vector field optimization process.

• Integrate 2x2 field. Runs the seamless parametrization routine, and the results are displayed as texture on the original mesh.

• Generate quad mesh. Once the quad mesh has been generated various new menus related to its editing, partitioning and evaluation are activated.

• To view the separatrices of the singularities or to partition from singularities, you can find the buttons on the Partitioning menu.

• Once the partitions have been generated, paths subdividing each strip can also be traced from the same panel. These are, by default, aligned with the blue direction.

In many cases, singularities are nicely aligned by default, leading to a good partitioning out of the box. Then the workflow is complete at this point. In more complex shapes, however, singulartities might not be aligned, leading to winding strips that would be unfeasible for fabrication. In those cases, the editing operations can help solve the strip-based topological problems to create a good partitioning.

Note that once a vector field has been optimized, or a quad mesh has been generated, the current state can be serialized (key 'Q') and deserialized using the buttons on the Utilities menu.

The steps of the workflow can also be seen in the complementary videos from the paper submission (Appendix A, supplementary data). Note that the UI has small differences because it was cleaned up after the submission.

Grasshopper visualization

If you would like to visualize the results within the Rhino-Grasshopper environment, you can open the grasshopper/vizualization&debugging.gh file. All the results that are saved in .txt format on the output folder can be loaded using python scripts. I always use grasshopper as a visualization and debugging tool in this way, because it allows me to better inspect the geometry, highlight, name and show/hide things interactively to find what I'm looking for.