PAPERS: Novel Applications of Design for AM

Four-Dimensional Printing for Freeform Surfaces: Design Optimization of Origami and Kirigami Structures

[+] Author and Article Information
Tsz-Ho Kwok

Department of Mechanical
and Automation Engineering,
The Chinese University of Hong Kong,
Hong Kong, China;
Epstein Department of Industrial
and Systems Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: tom.thkwok@gmail.com

Charlie C. L. Wang

Department of Mechanical
and Automation Engineering,
The Chinese University of Hong Kong,
Hong Kong, China
e-mail: cwang@mae.cuhk.edu.hk

Dongping Deng

Epstein Department of Industrial
and Systems Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: ddeng@usc.edu

Yunbo Zhang

Department of Mechanical
and Automation Engineering,
The Chinese University of Hong Kong,
Hong Kong, China;
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: will.yunbo.zhang@gmail.com

Yong Chen

Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA 90089 e-mail: yongchen@usc.edu

1Corresponding author.

Contributed by the Design for Manufacturing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 2, 2015; final manuscript received May 22, 2015; published online October 12, 2015. Assoc. Editor: Christopher Williams.

J. Mech. Des 137(11), 111413 (Oct 12, 2015) (10 pages) Paper No: MD-15-1174; doi: 10.1115/1.4031023 History: Received March 02, 2015; Revised May 22, 2015

A self-folding structure fabricated by additive manufacturing (AM) can be automatically folded into a demanding three-dimensional (3D) shape by actuation mechanisms such as heating. However, 3D surfaces can only be fabricated by self-folding structures when they are flattenable. Most generally, designed parts are not flattenable. To address the problem, we develop a shape optimization method to modify a nonflattenable surface into flattenable. The shape optimization framework is equipped with topological operators for adding interior/boundary cuts to further improve the flattenability. When inserting cuts, self-intersection is locally prevented on the flattened two-dimensional (2D) pieces. The total length of inserted cuts is also minimized to reduce artifacts on the finally folded 3D shape.

Copyright © 2015 by ASME
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Grahic Jump Location
Fig. 2

An illustration for the working principle of self-folding [4]. The self-folding structure is printed on a shrinking film with body and constraining layers (top). It is bended when heat is applied. The test case of a self-folded crane model is shown (bottom).

Grahic Jump Location
Fig. 3

For a triangle with vertices (v1 v2 v3) (left) deformed a shape in blue by an affine transformation T, we can extract the pure rotation matrix R from T using SVD

Grahic Jump Location
Fig. 4

Algorithm overview. The details of flattening and folding simulation are presented in Secs. 3.1 and 3.2, respectively. Our hybrid cutting insertion algorithm will be introduced in Sec. 4.

Grahic Jump Location
Fig. 1

For an input freeform surface to be fabricated by Origami-structure—the face model, our shape optimization can generate a 2D pattern for producing a self-folding structure by AM, which can be folded into a face model by heating (see the top row). Cuts can be automatically added onto the model to further improve the shape similarity of fabricated surface comparing to the input model. Adding too many cuts can generate unwanted artifacts on the surface of fabricated part—see the bottom-left corner for an example.

Grahic Jump Location
Fig. 5

Shape optimization: A highly nonflattenable surface—half sphere—is input to our system (left). The 2D pattern computed by our system (middle). The simulated 3D shape folded up from the 2D pattern (right).

Grahic Jump Location
Fig. 6

Different cuts are to be added at vertices with different local shapes. For an elliptic vertex, adding an interior cut will lead to self-intersection (top row). Therefore, boundary cuts are usually added to resolve the nonflattenable problem. The situation is reversed for a hyperbolic vertex. Boundary cuts lead to self-intersection while inserting an interior cut can resolve the problem (bottom row).

Grahic Jump Location
Fig. 7

For a highly nonflattenable input—the face model, only applying the shape optimization (without cuts) results in a shape with large shape approximation error—the nose and the mouth disappear. The overlaid polygon is the contour of the input shape. The result can be improved by adding an interior cut. A final result with high similarity as input can be obtained by our hybrid cut insertion algorithm, in which both the interior and boundary cuts are added.

Grahic Jump Location
Fig. 10

An example of car-body. Given the input model (top-left) that is not flattenable, the iterations of flattenability-based shape optimization can significantly improve the 3D shape folded from 2D patterns (see the first row, from M(1) to M(n)). The shape approximation error can be further improved by first inserting interior-only cuts and then adding both boundary and interior cuts (see the second row and the zoom windows). The corresponding 2D patterns are also shown at the bottom row.

Grahic Jump Location
Fig. 9

An example of flower blades. The input digital model shown in front and side view, and our framework can optimize it into a flattenable shape (top row). The mask images for body and constraining layers generated by our system (middle row). The 2D pattern is fabricated according to the mask images and then self-folded into the desired 3D shape after heating (bottom row). Multiple flower blades are put together to get the final product—a lamp.

Grahic Jump Location
Fig. 8

The mask images generated for fabricating the face model with only interior cut shown in Fig. 7. There are in total five layers, including two body layers, two constraining layers, and one layer of Shrinky-Dinks film in the center.

Grahic Jump Location
Fig. 11

An example of wave shape. (a) The fillets in the input model make the model nonflattenable. (b) The edges in the optimized shape are sharpened to improve its flattenability, and (c) the mask image that only includes the sharp edges. (d) The fabricated result.



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