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

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.

A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.