Research Papers

GDFE: Geometry-Driven Finite Element for Four-Dimensional Printing

[+] Author and Article Information
Tsz-Ho Kwok

Department of Mechanical,
Industrial and Aerospace Engineering,
Concordia University,
Montreal, QC H3G 1M8, Canada
e-mail: tszho.kwok@concordia.ca

Yong Chen

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

Manuscript received April 24, 2017; final manuscript received July 2, 2017; published online September 13, 2017. Assoc. Editor: Zhijian J. Pei.

J. Manuf. Sci. Eng 139(11), 111006 (Sep 13, 2017) (8 pages) Paper No: MANU-17-1281; doi: 10.1115/1.4037429 History: Received April 24, 2017; Revised July 02, 2017

Four-dimensional (4D) printing is a new category of printing that expands the fabrication process to include time as the fourth dimension, and its simulation and planning need to take time into consideration as well. The common tool for estimating the behavior of a deformable object is the finite element method (FEM). However, there are various sources of deformation in 4D printing, e.g., hardware and material settings. To model the behavior by FEM, a complete understanding of the process is needed and a mathematical model should be established for the structure–property–process relationship. However, the relationship is usually complicated, which requires different kinds of testing to formulate such models due to the process complexity. With the insight that the characteristic of shape change is the primary focus in 4D printing, this paper introduces geometry-driven finite element (GDFE) to simplify the modeling process by inducing deformation behavior from a few physical experiments. The principle of GDFE is based on the relationship between material structure and shape transformation. Accordingly, a deformation simulation can be developed for 4D printing by applying the principles to the GDFEs. The GDFE framework provides an intuitive and effective way to enable simulation and planning for 4D printing even when a complete mathematical model of new material is not available yet. The use of the GDFE framework for some applications is also presented in this paper.

Copyright © 2017 by ASME
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Fig. 1

Difference between a crease fold and a thick fold. Thickness (t) and fold width (L) are negligible (e.g., paper material) in crease fold, but not in thick fold.

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Fig. 2

The datasets in Table 1 are plotted by the fold angle against the design parameters, and a linear fitting finds K = 0.3706

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Fig. 3

When an active material contracts in one side by a shrinkage ratio R with another side constrained by a passive material, it self-transforms to an arc under heat with a bending angle α

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Fig. 4

Similar to FEM, the GDFE framework subdivides the design domain into a set of GDFEs, so that the deformation principle is applied locally in each element

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Fig. 5

The final shape of a C-GDFE can be computed by linear contraction on one side while another side is constrained. It is used to compute the target shape and optimize the current shape to it.

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Fig. 6

The GDFE framework is validated by comparing the simulated bending angle α with the calculated angle α̃ by the analytical model in Eq. (1). Different fold widths L are tested, and the results show good matches.

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Fig. 7

The test cases (courtesy: Deng and Chen [9]) in (a) and (c) are reconstructed by the GDFE simulator in (b) and (d). The drawing lines are just used to facilitate alignment during fabrication without any effect on the deformation.

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Fig. 8

Comparison between the physical fabrication and the simulation result by GDFE on two freeform surfaces

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Fig. 9

By inputting a series of couples (Li, ri), the corresponding letters can be generated. L is the length, and r is the radius for the circular arc (r = 0 means straight). Positive value of r means folding in anticlockwise direction, and negative value folds in clockwise.

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Fig. 10

A hand-sketched letter S is reproduced by 4D printing. Increasing the levels of polyline fitting on the tangent graphs can improve the shape approximation.

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Fig. 11

All the results here are approximating a same circle, but different results are obtained with different properties of hardware (minimum fabrication width) and material (contraction ratio)




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