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1. Computational Design and 3D Weaving of 2D-Printable Conformal Flexible Electronics using Harmonic Foliation Theory

   Ye, Qian ; Guo, Yang ; Gu, Xianfeng David ; Chen, Shikui ( August 2021 , Proceedings of the International Design Engineering Technical Conferences & Computers and Information in Engineering Conference)

   null (Ed.)

   This paper proposes a new way of designing and fabricating conformal flexible electronics on free-form surfaces, which can generate woven flexible electronics designs conforming to free-form 3D shapes with 2D printed electronic circuits. Utilizing our recently proposed foliation-based 3D weaving techniques, we can reap unprecedented advantages in conventional 2D electronic printing. The method is based on the foliation theory in differential geometry, which divides a surface into parallel leaves. Given a surface with circuit design, we first calculate a graph-value harmonic map and then create two sets of harmonic foliations perpendicular to each other. As the circuits are processed as the texture on the surface, they are separated and attached to each leaf. The warp and weft threads are then created and manually woven to reconstruct the surface and reconnect the circuits. Notably, The circuits are printed in 2D, which uniquely differentiates the proposed method from others. Compared with costly conformal 3D electronic printing methods requiring 5-axis CNC machines, our method is more reliable, more efficient, and economical. Moreover, the Harmonic foliation theory assures smoothness and orthogonality between every pair of woven yarns, which guarantees the precision of the flexible electronics woven on the surface. The proposed method provides an alternative solution to the design and physical realization of surface electronic textiles for various applications, including wearable electronics, sheet metal craft, architectural designs, and smart woven-composite parts with conformal sensors in the automotive and aerospace industry. The performance of the proposed method is depicted using two examples. 

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2. The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

   https://doi.org/10.1112/plms.12582  

   Viklund, Fredrik ; Wang, Yilin ( February 2024 , Proceedings of the London Mathematical Society)

   We study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined bywhenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

    

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3. Variational Level Set Method for Topology Optimization of Origami Fold Patterns

   https://doi.org/10.1115/1.4053925  

   Ye, Qian ; Gu, Xianfeng David ; Chen, Shikui ( August 2022 , Journal of Mechanical Design)

   Abstract With specific fold patterns, a 2D flat origami can be converted into a complex 3D structure under an external driving force. Origami inspires the engineering design of many self-assembled and re-configurable devices. This work aims to apply the level set-based topology optimization to the generative design of origami structures. The origami mechanism is simulated using thin shell models where the deformation on the surface and the deformation in the normal direction can be simplified and well captured. Moreover, the fold pattern is implicitly represented by the boundaries of the level set function. The folding topology is optimized by minimizing a new multiobjective function that balances kinematic performance with structural stiffness and geometric requirements. Besides regular straight folds, our proposed model can mimic crease patterns with curved folds. With the folding curves implicitly represented, the curvature flow is utilized to control the complexity of the folds generated. The performance of the proposed method is demonstrated by the computer generation and physical validation of two thin shell origami designs. 

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4. A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing

   https://doi.org/10.3934/dcds.2021077  

   Mooney, Connor ; Yang, Yang ( January 2021 , Discrete & Continuous Dynamical Systems)

   We give a proof by foliation that the cones over \begin{document}$ \mathbb{S}^k \times \mathbb{S}^l $\end{document} minimize parametric elliptic functionals for each \begin{document}$ k, \, l \geq 1 $\end{document}. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

    

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5. Photonic crystallization of two-dimensional MoS 2 for stretchable photodetectors

   https://doi.org/10.1039/c9nr02173f  

   Kim, Richard Hahnkee ; Leem, Juyoung ; Muratore, Christopher ; Nam, SungWoo ; Rao, Rahul ; Jawaid, Ali ; Durstock, Michael ; McConney, Michael ; Drummy, Lawrence ; Rai, Rachel ; et al ( July 2019 , Nanoscale)

   Low temperature synthesis of high quality two-dimensional (2D) materials directly on flexible substrates remains a fundamental limitation towards scalable realization of robust flexible electronics possessing the unique physical properties of atomically thin structures. Herein, we describe room temperature sputtering of uniform, stoichiometric amorphous MoS 2 and subsequent large area (>6.25 cm 2 ) photonic crystallization of 5 nm 2H-MoS 2 films in air to enable direct, scalable fabrication of ultrathin 2D photodetectors on stretchable polydimethylsiloxane (PDMS) substrates. The lateral photodetector devices demonstrate an average responsivity of 2.52 μW A −1 and a minimum response time of 120 ms under 515.6 nm illumination. Additionally, the surface wrinkled, or buckled, PDMS substrate with conformal MoS 2 retained the photoconductive behavior at tensile strains as high as 5.72% and over 1000 stretching cycles. The results indicate that the photonic crystallization method provides a significant advancement in incorporating high quality semiconducting 2D materials applied directly on polymer substrates for wearable and flexible electronic systems. 

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