Research Papers: Design Automation

Discovering Sequenced Origami Folding Through Nonlinear Mechanics and Topology Optimization

[+] Author and Article Information
Andrew S. Gillman

UES, Inc.,
Beavercreek, OH 45432;
Materials and Manufacturing Directorate,
Air Force Research Laboratory,
Wright-Patterson AFB, OH 45433

Kazuko Fuchi

Applied Mechanics Division,
University of Dayton Research Institute,
Dayton, OH 45469;
Air Force Research Laboratory,
Wright-Patterson AFB, OH 45433

Philip R. Buskohl

Materials and Manufacturing Directorate,
Air Force Research Laboratory,
Wright-Patterson AFB, OH 45433

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 31, 2018; final manuscript received October 6, 2018; published online January 11, 2019. Assoc. Editor: James K. Guest.

J. Mech. Des 141(4), 041401 (Jan 11, 2019) (11 pages) Paper No: MD-18-1084; doi: 10.1115/1.4041782 History: Received January 31, 2018; Revised October 06, 2018

Origami folding provides a novel method to transform two-dimensional (2D) sheets into complex functional structures. However, the enormity of the foldable design space necessitates development of algorithms to efficiently discover new origami fold patterns with specific performance objectives. To address this challenge, this work combines a recently developed efficient modified truss finite element model with a ground structure-based topology optimization framework. A nonlinear mechanics model is required to model the sequenced motion and large folding common in the actuation of origami structures. These highly nonlinear motions limit the ability to define convex objective functions, and parallelizable evolutionary optimization algorithms for traversing nonconvex origami design problems are developed and considered. The ability of this framework to discover fold topologies that maximize targeted actuation is verified for the well-known “Chomper” and “Square Twist” patterns. A simple twist-based design is also discovered using the verified framework. Through these case studies, the role of critical points and bifurcations emanating from sequenced deformation mechanisms (including interplay of folding, facet bending, and stretching) on design optimization is analyzed. In addition, the performance of both gradient and evolutionary optimization algorithms are explored, and genetic algorithms (GAs) consistently yield solutions with better performance given the apparent nonconvexity of the response-design space.

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Fig. 1

Schematic of base origami truss element with rotational hinge defined by adjoining triangles. The internal energy of the truss element connecting the nodes X1 and X2 captures both the axial elongation and the amount of folding as determined from the positions of the nodes defining the adjoining triangle facets, X3 and X4. The external energy is comprised of the applied nodal forces, F1 and F2.

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Fig. 2

Flow chart of topology optimization framework

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Fig. 3

Illustration of GA crossover operators for 8 fold line ground structure: (a) ground structure (left) and parent topologies with design vector chromosomes (right), (b) single point crossover and resulting children topologies, and (c) uniform (random) crossover and resulting children topologies. Here, αi = 0 refers to a soft, foldable segment, while αi = 1 designates a stiff, nonfoldable segment.

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Fig. 4

Schematic of HPC implementation of genetic algorithm: Master-Worker Pool. Np is the total number of computer processes.

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Fig. 5

Optimization problem setup for “Chomper” problem. Filled triangles represent fixed boundary conditions.

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Fig. 6

Optimization results for “Chomper” problem for stiffness ratios EA/Gstiff = 101 and Gstiff/Gsoft = 103: (a) evolution of objective function. Optimal fold pattern and actuation are shown for patterns discovered by (b) GA, and (c) gradient-based (MMA, SQP, and IP) methods.

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Fig. 7

Optimization results for “Chomper” problem for stiffness ratios EA/Gstiff = 103 and Gstiff/Gsoft = 103: (a) evolution of objective function. Optimal fold pattern and actuation are shown for patterns discovered by (b) GA, and (c) gradient-based (MMA, SQP, and IP) methods.

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Fig. 8

Analysis of diamond “Chomper” fold pattern for stiffness ratios EA/Gstiff = 103 and Gstiff/Gsoft = 103: (a) Bifurcating equilibrium paths for diamond fold pattern. (b) Total stretching and bending/folding internal energy as function of actuation displacement. (c) Schematic of bifurcated fold paths corresponding to (a).

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Fig. 9

Optimization of “Chomper” fold pattern for increasing amount of actuation: (a) objective function value for initial and final design of SQP optimization runs versus total input actuation. Design 1 data point is colored green to indicate discovery of desired optimal solution, while designs II and III highlight undesirable designs. (b) Initial and (c) final designs of select gradient optimization runs (colored circular markers in (a)) for varied amount of actuation. The dash length is proportional to the degree of fold softness, i.e., lines with longest dashes are softest achievable fold.

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Fig. 10

Analysis and optimization of simple twist actuator with stiffness ratios EA/Gstiff = 103 and Gstiff/Gsoft = 103: (a) objective function evolution of SQP (square markers) and GA (circular markers) runs. Select fold patterns (designs 1, 2, and 3) are loaded completely given optimal design at partial actuation. Inset image shows schematic of problem setup. (b) Fold patterns obtained from optimization with varied actuation. (c) Deformed shape for complete actuation of fold patterns in (b).

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Fig. 11

Schematic of “Square Twist” problem: (a) “Square twist” fold pattern. (b) Actuated flat-foldable configuration of (a). (c) Optimization problem setup for discovering square twist actuation for ground structure containing 176 design variables (candidate fold lines). The stiffness ratios are set to EA/Gstiff = 104 and Gstiff/Gsoft = 103 in this study.

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Fig. 12

Objective function evolution for “Square Twist” problem (see Fig. 11) solved with two sets of GA algorithms (nongradient algorithm) and multiple gradient-based algorithms (MMA, SQP, IP)

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Fig. 13

Local minima solutions of “Square Twist” problem for (a) SQP, (b) MMA, (c) IP, and (d) GA (see minimum values of objective function for respective objective function value lines, in Fig. 12. For each optimization algorithm, design variables are shown in (leftmost) continuous and (middle) thresholded form, where the dash length is proportional to the degree of fold softness, i.e. lines with longest dashes are softest achievable fold.

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Fig. 14

Local minima solutions of “Square Twist” problem for two sets of crossover/constraint algorithms in GA: (a) optimal fold and actuation for GA algorithm set 1, (b) result of GA algorithm set 2 (Fig. 13(d) result), (c) actuation of pattern (b) with optimal bottom half mirrored



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