A dynamically reprogrammable metasurface with self-evolving shape morphing

Dynamic shape-morphing soft materials systems are ubiquitous in living organisms; they are also of rapidly increasing relevance to emerging technologies in soft machines, flexible electronics, and smart medicines. Soft matter equipped with responsive components can switch between designed shapes or structures, but cannot support the types of dynamic morphing capabilities needed to reproduce natural, continuous processes of interest for many applications. Challenges lie in the development of schemes to reprogram target shapes post fabrication, especially when complexities associated with the operating physics and disturbances from the environment can prohibit the use of deterministic theoretical models to guide inverse design and control strategies. Here, we present a mechanical metasurface constructed from a matrix of filamentary metal traces, driven by reprogrammable, distributed Lorentz forces that follow from passage of electrical currents in the presence of a static magnetic field. The resulting system demonstrates complex, dynamic morphing capabilities with response times within 0.1 s. Implementing an in-situ stereo-imaging feedback strategy with a digitally controlled actuation scheme guided by an optimization algorithm, yields surfaces that can self-evolve into a wide range of 3-dimensional (3D) target shapes with high precision, including an ability to morph against extrinsic or intrinsic perturbations. These concepts support a data-driven approach to the design of dynamic, soft matter, with many unique characteristics.


Results
Soft matter that can dynamically reconfigure their shapes upon interactions with environment or perceptions of information is thriving 33 . Pioneering studies rely on an exploitation of responsive materials or material configurations to create active structures that shift their shapes in response to external stimuli [34][35][36][37][38] . Smart materials (e.g., liquid crystal elastomers 11,[13][14][15][16][17]39,40 , shape memory polymers 41 , hydrogels 10,12,24 , and others 25 ) and multimaterial structures 11,26 enable large structural deformation but face challenges in implementing fast control to refined structures. The design of shape-morphing process usually requires prerequisite modeling effort to be programmed into the fabrication process, and is therefore hard to reprogram on-the-fly (e.g., 3D printing 11,27 , magnetization 19,42 , laser or wafer-jet cutting 29,30,43 , mechanical buckling 28 ). The desire to swiftly shift shapes among large number of configurations post fabrication invites the investigations on programmable stimulus (e.g., temperature 13,44 , magnetic field 20 , electric current 22,23 ). However, limitations remain in the accessible design space and the real-time inverse design because of the challenges in establishing analytical solutions or barriers in high computational costs due to the complexity arising from nonlinearity or high dimensionality. Also, existing computer-aided methods usually leave the inclusion of imperfections, damages, or the coupling between the system with the unforeseen environment. Incorporating instant feedback is necessary for the morphing process to see the deployment scheme to precisely account for specific, multifunctional, or time-varying requirements 45 . The time constraints and the complexity in actuation, feedback, or modeling all contribute to a prolonged programming cycle that limits the possible shapes or shape responses to remain discrete and quasi-static.
Here, we demonstrate a dynamically reprogrammable mechanical metasurface with a closed-loop 3-dimensional (3D) shape control, based on a digital, fast, and precise Lorentz force actuation scheme. The metasurface takes the form of interconnected, 4 serpentine-shaped beams that consist of a thin conductive layer of gold (Au, thickness ℎ Au = 300 nm) encapsulated by polyimide (PI, thickness ℎ PI = 7.5 μm, width PI = 160 μm) (see Methods section 'Sample fabrication', Supplementary Note S1, and deformations (u = {ui}, where ui is the displacement of the i th node) of the sample in a magnetic field B aligned with its diagonal, enabling a large set of accessible 3D shapes from the same precursor structure. The unusual structure and material design further enables the system to adopt an approximate, linearized model, such that the nodal displacement response to the input voltages follows, where the coupling matrix = { } fully characterizes the electro-magneto-mechanical system. Fig. 1c [11][12][13][14].
In addition to the abstract, implicit shapes, the reprogrammable metasurface demonstrates an ability to reproduce dynamic processes in nature that involve a temporal series of complex shapes, provided with the inversely designed current distributions.   The experiment-driven process works as a physical simulation to accommodate pronounced nonlinearity without a significant increase in the computational cost. Fig. 4a introduces a 2×2 sample (L = W = 25.0 mm) consisting of serpentine beams with the relative arc length reduced morphing into the same target shape in Fig. 3b. Centered in the same magnetic setup, the sample exhibits an amplified non-linear mechanical behavior in response to input voltages (Supplementary Note S13 and Supplementary Fig.   36). The model-driven approach based on the linear-system assumption results in an absolute maximum error of ~8%. The experimental-driven approach achieves more accurate morphing result in ~20 iterations with absolute errors below 1%.
Guided by the experiment-driven process, the metasurface can also self-adjust to morph against unknown perturbations. Fig. 4b-d shows three representative cases in which a 4×4 sample morphs with perturbed magnetic field, external mechanical load, and 8 intrinsic damage, respectively. In all cases, the model-driven approach following the original inverse design results in absolute maximum errors of ~8-10%. In comparison, the experiment-driven approach adapts the shape to reach the target with absolute errors below ~3% that is comparable with that of an intact sample (~2%) (Supplementary Video 6). The significantly boosted accuracy level demonstrates a 'self-sustained' morphing ability enabled by the experiment-driven process.
The adaptive, self-evolving metasurface platform delivers a semi-real-time morphing scheme to learn the continuously evolving surface of a real object in-time. In this experiment, a duplicated stereo-imaging setup measures the displacement of a 4×4 array of markers (with inter-spacing a0 = 15 mm) on the palm ( Supplementary Fig. 37a). The optimization acts directly to minimize the displacement difference between the 16 markers and their corresponding nodes of a 4×4 sample. Given continuity, the gradient-descent process takes the last morphing result as the initial state for the next morphing task. This differential method (with the target descent ( ) ~0.08) requires only ≤3 iterations (~20 s) to reach the optimum. Fig. 5a shows representative frames from a video recording of hand making eight gestures with different fingers moving (see Supplementary Fig. 37b, c and Supplementary Video 7 for complete results of all gestures). All morphing results agree with the target with absolute errors below 2%.
In addition to self-evolving to optimize shapes, the metasurface can self-evolve to optimize functions. Setting multiple target functions drives the optimization towards emergent multifunctionality, with an ability to decouple naturally coupled functions. on a receiving screen (Supplementary Fig. 38a) and II) achieve the target displacement of its central node. The optimization takes a hybrid strategy combining the model-driven 9 and experiment-driven processes (Supplementary Note S14). While the voltages control the reflected beam paths, a top camera provides an imaging feedback of the distances between the beam spots on the screen. The model-driven process predicts the difference between the central nodal displacement and the target. The total loss takes a linear combination of the two errors (Supplementary Note S14, Supplementary Fig. 38b). Fig.   5d shows the self-evolving results of three optical configurations with distinctive incident beam angles. Fig. 5e shows that the metasurface can morph to overlap the laser spots on the receiving screen with a range of possible shapes (Supplementary Fig. 39a). By  Fig.   29b, c and Supplementary Note S12).

Data availability
All data are contained within the manuscript. Raw data are available from the corresponding authors upon reasonable request.

Code availability
The codes that support the findings of this study are available from the corresponding authors upon reasonable request

Competing interests
The authors declare no competing interests.  Port voltages define the current density distribution (J) in the sample and hence control the local Lorentz force actuation. c, Finite element analysis (FEA) provide a linear-model approximation of the nodal displacement in response to the input voltages for the 4×4 sample. Experimental characterization using a side camera agrees with the FEA prediction. d, FEA and experimental results of a 4×4 and 8×8 sample morphing into four target implicit shape shifting processes with control on instantaneous velocity and acceleration of the dynamics. Scale bars, 5 mm.  Figure   Fig Exp.
e Multi-view stereo Flow diagram of the model-driven inverse design approach (top, blue) and an experiment-driven self-evolving process enabled by an in-situ 3D-imaging feedback and a gradient-descent based optimization algorithm (bottom, red). b, Target implicit shapes and optical images of the experiment-driven morphing results of a 4×4 sample. c, 3D reconstructed surfaces overlaid with contour plots of the minimized errors (e) and d, histograms of the minimized errors for model-driven and experiment-driven outputs. Scale bars, 5 mm.