Research Papers

A Novel Method to Design and Optimize Flat-Foldable Origami Structures Through a Genetic Algorithm

[+] Author and Article Information
Daniel A. McAdams

Department of Mechanical Engineering,
Texas A&M University,
3123 TAMU,
College Station, TX 77843-3123
e-mail: dmcadams@tamu.edu

Wei Li

Department of Mechanical Engineering,
Texas A&M University,
3123 TAMU,
College Station, TX 77843-3123
e-mail: liwei.meen@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF Computing AND INFORMATION Science IN ENGINEERING. Manuscript received October 23, 2013; final manuscript received January 15, 2014; published online June 2, 2014. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 14(3), 031008 (Jun 02, 2014) (10 pages) Paper No: JCISE-13-1227; doi: 10.1115/1.4026509 History: Received October 23, 2013; Revised January 15, 2014

As advantages of foldable or deployable structures have been established, origami artists and engineers have started to study the engineering applications of origami structures. Methods of computational origami design that serve different types of origami have already been developed. However, most of the existing design methods focus on automatically deriving the crease pattern to realize a given folded finished shape, without actually designing the finished shape itself. To include final shape design into the computational origami design and optimization process, this paper presents a genetic algorithm that aims to develop origami structures featuring optimal geometric, functional, and foldability properties. In accordance with origami, the genetic algorithm is adapted both in the aspects the individual encoding method and the evolutionary operators. To compliment the Genetic algorithms (GA), a new origami crease pattern representation scheme is created. The crease pattern is analogous to the ice-cracks on a frozen lake surface, where each crack is equivalent to a crease and each forking point to a vertex. Thus to form the creases and vertices in an “ice-cracking”-like origami crease pattern, we pick one vertex as the starting location, and let the rest of the creases and vertices grow in the same manner that cracks extend and fork form in ice. In this research, the GA encodes the geometric information of forming the creases and vertices according to the development sequence through the ice-cracking process. Meanwhile, we adapt the evolutionary operators and introduce auxiliary mechanisms for the GA, so as to balance the preservation of both elitism and diversity and accelerate the emergence of optimal design outcomes through the evolutionary design process.

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Harbin, R., 1997, Secrets of Origami: The Japanese Art of Paper Folding, 3rd ed., Dover Publications, Mineola, NY.
Gantes, C. J., 2001, Deployable Structures: Analysis and Design, WIT Press, Billerica, MA.
Demaine, E. D., Demaine, M. L., and Ku, J., 2011, “Folding Any Orthogonal Maze,” Fifth International Meeting of Origami Science, Mathematics, and Education.
Demaine, E. D., and O'Rourke, J., 2007, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, Cambridge, UK.
Hull, T., 2006, Project Origami: Activities for Exploring Mathematics, A.K. Peters, Natick, MA.
Lang, R. J., 1996, “A Computational Algorithm for Origami Design,” 12th Annual ACM Symposium on Computational Geometry, Philadelphia, PA, p. 8.
Lang, R. J., 2003, Origami Design Secrets: Mathematical Methods for an Ancient Art, A.K. Peters, Natick, MA.
Tachi, T., 2006, “3D Origami Design Based on Tucking Molecule,” Fourth International Meeting of Origami Science, Mathematics, and Education, Pasadena, CA, pp. 259–272.
Tachi, T., 2010, “Origamizing Polyhedral Surfaces,” IEEE Trans. Visualization Comput. Graphics, 16, pp. 298–311. [CrossRef]
Mitchell, M., 1996, An Introduction to Genetic Algorithms. MIT Press, Cambridge, MA.
Gallagher, K., and Sambridge, M., 1994, “Genetic Algorithms: A Powerful Tool for Large-Scale Nonlinear Optimization Problems,” Comput. Geosci., 20, pp. 1229–1236. [CrossRef]
Mitani, J., 2011, “A Method for Designing Crease Patterns for Flat-Foldable Origami With Numerical Optimization,” J. Geom. Graphics, 15, pp. 195–201.
O'Rourke, J., 2011, How to Fold It: The Mathematics of Linkages, Origami, and Polyhedra, Cambridge University Press, Cambridge, UK.
Li, W., and McAdams, D. A., 2013, “A Novel Pixelated Multicellular Representation for Origami Structures That Innovates Computational Design and Control,” ASME 2013 International Design Engineering Technical Conferences (IDETC) and Computers and Information in Engineering Conference (CIE), Portland, OR, Aug. 4–7, ASME Paper No. DETC2013-13231, p. V06BT07A041. [CrossRef]
Poma, F., 2009, On The Flat-Foldability Of A Crease Pattern. Available online at: http://poisson.phc.unipi.it/~poma/Ffcp.pdf.
Schneider, J., 2004, Flat-Foldability of Origami Crease Patterns, Available online at: http://www.organicorigami.com/thrackle/class/hon394/papers/SchneiderFlatFoldability.pdf.
MathWorks, Digital Modulation.
Jensenm, M. T., 2003, “Reducing the Run-Time Complexity of Multiobjective EAs: The NSGA-II and Other Algorithms,” IEEE Trans. Evol. Comput.,7, pp. 503–515. [CrossRef]
Zitzler, E., and Thiele, L., 1999, “Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach,” IEEE Trans. Evol. Comput., 3, pp. 257–271. [CrossRef]
Knowles, J., and Corne, D., 1999, “The Pareto Archived Evolution Strategy: A New Baseline Algorithm for Pareto Multiobjective Optimisation,” CEC 99. Proceedings of the 1999 Congress on Evolutionary Computation, Washington, DC, pp. 98–105.
Konak, A., Coit, D. W., and Smith, A. E., 2006, “Multi-Objective Optimization Using Genetic Algorithms: A Tutorial,” Reliab. Eng. Syst. Saf., 91, pp. 992–1007. [CrossRef]
Goldberg, D. E., 1989, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Professional, Upper Saddle River, NJ.


Grahic Jump Location
Fig. 1

(a) A crease pattern represented by the locations of the vertices and the creases that link in between; (b) the mutation on one bit in the genetic code for the first intuitive geometric representation of (a) results in an invalid crease pattern with a breakage; (c) the mutation on 1 bit in the genetic code for the second intuitive geometric representation of (a) results in an invalid crease pattern due to the missing of one crease

Grahic Jump Location
Fig. 2

The steps taken by ice-cracking to derive the vertices and creases through a systematic sequence. (a) The full crease pattern with circled numbers labeling the vertices and diamonded numbers labeling the creases; (b) the initialization step that locates the first vertex; (c) the first forking step that gets the #2 vertex as well as the #1 crease; (d) the second forking step that gets the #3 vertex and the #6 crease; (e) the resolution step that gets the #4 vertex and two creases; (f) the resultant crease pattern without the MV-assignment after the finalization step.

Grahic Jump Location
Fig. 3

The four steps—initialization, forking, resolution and finalization—of ice-cracking. The arrows show how the different steps can be arranged, with initialization being the first step and finalization the last step.

Grahic Jump Location
Fig. 4

Illustrative patterns. (a) Initialization step; (b) forking step; (c) the second forking step for getting a pinwheel pattern; (d) resolution step; (e) the resolution step for getting the #4 vertex in a pinwheel pattern.

Grahic Jump Location
Fig. 5

(a) A crease pattern design (normalized fitness of 0.0324) found through the GA for the problem in Sec. 4, where vertices are labeled by numbers in circles, and creases by numbers in diamonds; (b) the intermediate folded state; (c) the corresponding flat-folded state profile that has an area of about 0.2501. Five vertices are still on the boundary of the profile.

Grahic Jump Location
Fig. 8

Fold a square sheet into a bowl. (a) A square sheet with a crease pattern, where solid lines are valleys and dashed-dotted lines are mountains. The crease pattern and the crease folding angles are supposed to be designed by GA with ice-cracking. (b) The sheet of (a) is folded into a bowl/dish shape that can hold water. The shape is required to be rigid-foldable, so that every face keeps flat through the folding.




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