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Abstract This paper 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. 

Abstract This paper presents a new computational framework for the co-optimization and co-control of morphable surface structures using topology optimization and circle-packing algorithms. The proposed approach integrates the design of optimized compliant components and the system-level control of the overall surface morphology. By representing the surface shape using circle packing and leveraging conformal mapping, the framework enables smooth deformation between 2D and 3D shapes while maintaining local geometry and global morphology. The morphing surface design problem is recast as designing circular compliant actuators using level-set topology optimization with displacements and stiffness objectives. The optimized component designs are then mapped back onto the circle packing representation for coordinated control of the surface morphology. This integrated approach ensures compatibility between local and global geometries and enables efficient actuation of the morphable surface. The effectiveness of the proposed framework is demonstrated through numerical examples and physical prototypes, showcasing its ability to design and control complex morphable surfaces with applications in various fields. The co-optimization and co-control capabilities of the framework are verified, highlighting its potential for realizing advanced morphable structures with optimized geometries and coordinated actuation. This integrated approach goes beyond conventional methods by considering both local component geometry and global system morphology and enabling coordinated control of the morphable surface. The general nature of our approach makes it applicable to a wide range of problems involving the design and control of morphable structures with complex, adaptive geometries.