Researchers use mathematical model to create shape-shifting objects

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Mother Nature does it best, but researchers at UT-Austin and Harvard may have found a way to recreate a diverse array of biological patterns and shapes through reverse engineering.

In a paper published last month in the Proceedings of the National Academy of Sciences, researchers attempted to model a technique that can grow any target shape from any starting shape. The purpose of this research was to understand the growth process of geometrically complex objects from nature and how they could be more easily molded or milled in manufacturing, according to the researchers.

Senior author of the study Lakshminarayanan Mahadevan and postdoctoral fellow Wim van Rees from Harvard University collaborated with Etienne Vouga, assistant professor of computer science at UT, to reverse-engineer different materials using algorithms and computational programming of the materials.

“Overall, our research combines our knowledge of the geometry and physics of slender shells with new mathematical algorithms and computations to create design rules for engineering shape,” Mahadevan said in a press release. “It paves the way for manufacturing advances in 4-D printing of shape-shifting optical and mechanical elements, soft robotics and tissue engineering.”

The researchers worked with different orientations of a bilayer, which Vouga defined as simply two thin surfaces glued together.

“Because they are glued together, it is impossible for both layers to get what they want. Instead, nature will compromise, and the bilayer will buckle into a pringle shape,” Vouga said. “Our research was about how to tame this process. How do you design the growth of the top layer (and) the bottom layer, so that when they are glued together and forced to compromise, they buckle into the desired 3-D shape?”

According to the researchers, most natural materials swell in response to moisture and shrink in response to heat, such as leaves curling up as they dry, which can alter the shape of materials.

“To apply our research to manufacturing, materials are needed that can grow a large amount in response to small stimuli,” Vouga said.

Using mathematical modeling and equations, the researchers found the connection between a bilayer and a single layer, allowing them to understand how the curvatures of a shape could be modelled by reverse-engineering them from a starting material, according to the researchers.

“The beauty of our solution is that it is very clean computationally,” Vouga said. “We derived closed-form formulas for how to map any desired 3-D target shape into growth patterns on the top and bottom layer of the bilayer so that, in practice, our algorithm runs in seconds.”

In order to verify that their research was correct, the researchers said they simulated the growth process for several examples including snapdragon flowers and the face of Max Planck, a founder of quantum physics. They wanted to check that once finished, the end shape looks like the target shape they had originally specified. While each simulation required several hours, they can be run faster on a supercomputing cluster.

The main limitation to this research is that the amount of growth required by certain shapes is too much for current manufacturing technologies, according to Vouga. For example, with the human hand, the fingers would have to come out of a flat starting surface which is too large of an increase in the starting material’s surface area.

“One exciting workaround we’re thinking about is how to modify the target shape by adding small corrugations, so that it still looks the same, but is a lot easier to manufacture,” Vouga said.

The researchers have already begun testing these ideas experimentally and are working on removing the limitations they currently face.

“One potential application (for this research) is in robotics,” Vouga said, “If we understand the relationship between growth and shape, we can create robots that move in predictable and complex ways without motors or other moving parts.”