Geometry-Driven Design of Morphable Surface Structures Using Topology Optimization and Circle Packing

This work introduces a new computational framework for modeling and designing morphable surface structures based on an integrated approach that leverages circle packing for surface representation, conformal mapping to link local and global kinematics, and topology optimization for actuator design. The framework utilizes a unique strategy for employing optimized compliant actuators as the basic building blocks of the morphable surface. These actuators, designed as circular elements capable of modifying their radius and curvature, are optimized using level set topology optimization, considering both kinematic performance and structural stiffness. Circle packing is employed to represent the surface geometry, while conformal mapping guides the kinematic analysis, ensuring alignment between local actuator motions and desired global surface transformations. The design process involves mapping optimized component designs back onto the circle packing representation, facilitating coordinated control, and achieving harmony between local and global geometries. This leads to efficient actuation and enables precise control over the surface morphology. The effectiveness of the proposed framework is demonstrated through two numerical examples, showcasing its capability to design complex, morphable surfaces with potential applications in fields requiring dynamic shape adaptation.

Koebe–Andreev–Thurston circle packing theorem

Discrete conformal mappings

Conventional Level Set Method for Topology Optimization

Design Objective

Numerical Examples

Representative publications:

Journal:

1. Lingfeng Gao, Xiaoping Zhu, Masato Tanaka, Yuyang Song, Yuqing Zhou, Xianfeng David Gu, Shikui Chen*, “Geometry-Driven Design of Morphable Surface Structures Using Topology Optimization and Circle Packing”,Journal of Mechanical Design.

Conference:

1. Lingfeng Gao, Xiaoping Zhu, Masato Tanaka, Yuyang Song, Yuqing Zhou, Xianfeng David Gu, Shikui Chen*, “Geometry-Driven Design of Morphable Surface Structures Using Topology Optimization and Circle Packing”, the ASME 2024 International Design Engineering Technical Conferences &  Computers and Information in Engineering Conference (IDETC/CIE 2024), August 25-28, 2024, Washington, DC, USA.

Patents:

1. Systems and methods for forming multiple 3d structures from a circularly-packed network of structural elements.patent number 12,467,274.
2. Surface morphing technology using an interconnected network of circular compliant actuators.U.S. Patent number 20260044130A1.