Inverse Kirigami Design

Constrained optimization framework for inverse kirigami design

If you use this code in your work, please cite the following paper:

G. P. T. Choi, L. H. Dudte, and L. Mahadevan, "Programming shape using kirigami tessellations." Nature Materials, 18(9), 999-1004, 2019.

Copyright (c) 2019-2025, Gary P. T. Choi, Levi H. Dudte, L. Mahadevan

https://www.math.cuhk.edu.hk/~ptchoi

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Main programs:

• MAIN_quad.m: producing 2D generalized quad kirigami patterns
• MAIN_triangle.m: producing 2D generalized triangle kirigami patterns
• MAIN_hexagon.m: producing 2D generalized hexagon kirigami patterns
• MAIN_islamicquad.m: producing 2D generalized ancient Islamic quad kirigami patterns
• MAIN_islamictriangle.m: producing 2D generalized ancient Islamic triangle kirigami patterns
• MAIN_quad_3D.m: producing 3D generalized quad kirigami patterns
• MAIN_triangle_3D.m: producing 3D generalized triangle kirigami patterns

In each main program, the following parameters can be changed:

• width: the number of columns of unit cells in the kirigami pattern
• height: the number of rows of unit cells in the kirigami pattern
• shape_name: load a pre-defined target shape in the library
• (for 2D) initial_map_type: the type of the initial map for the optimization in the deployed space (1: standard, 2: rescaled, 3: conformal, 4: Teichmuller)
• (for 3D) initial_map_type: the type of the initial map for the optimization in the deployed space (1: standard, 2: rescaled, 3: analytic, 4: Teichmuller)
• (only for quad and triangle) fix_contracted_boundary_shape: enforce the desired contracted pattern to be rectangular (0: no, 1: yes)
• (only for quad and triangle) rectangular_ratio: further specify the width-to-height ratio if the contracted pattern is required to be rectangular (0: no/not applicable, other positive number: the prescribed ratio)
• (only for hexagon) fix_regular_shape: enforce all angles to be $2\pi/3$ (0: no, 1: yes)

Remarks:

• Some precomputed examples can be found in the results and results_3D folders
  • All parameters used are encoded in the file names:
    • (For quad/triangle) unitcelltype_shapename_width_height_initialmaptype_fixcontractedboundary_rectangularratio_contracted/deployed
    • (For hexagon) unitcelltype_shapename_width_height_initialmaptype_fixregularshape_contracted/deployed
    • (For ancient Islamic) unitcelltype_shapename_width_height_initialmaptype_contracted/deployed
• To create other user-defined target shapes, see the functions in the shapes and shapes_3D folders
• If the optimization terminates prematurely or does not produce a desirable pattern, please try:
  • Changing the algorithm used in fmincon (sqp or interior-point)
  • Changing other fmincon parameters (MaxFunctionEvaluations, MaxIterations, ConstraintTolerance, StepTolerance)
  • Using other types of initial map (standard, rescaled, conformal/analytic, Teichmuller)
  • Changing the kirigami pattern size (width, height)

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Dependencies:

• Schwarz-Christoffel Toolbox (for conformal initial map)

• Rectangular Conformal Map (for Teichmuller initial map)

• DistMesh (for Teichmuller initial map)

• Kabsch Algorithm (for rotating the resulting contracted pattern)

• point2trimesh (for shape matching constraint in 3D)

• AxelRot (for 3D rotation of tiles)