Inverse design of a magneto-elastica for shape-morphing

Abstract: Slender magnetic elements provide a versatile platform for programmable shape-morphing under remote magnetic actuation. However, a general and physically interpretable framework for the inverse design of a `magneto-elastica' under prescribed boundary conditions remains lacking. In this work, we develop an explicit analytical formulation for the inverse design of a magneto-elastica based on the integral form of the moment equilibrium equations. This approach yields direct constraints on the admissible curvature and rotation fields, enabling a systematic characterization of the feasible design space. We identify the key dimensionless parameters that govern the competition between magnetic torques and elastic restoring moments and show that the applied boundary conditions are an essential ingredient. We obtain closed-form solutions for the beam tapering profiles required to generate desired actuated shapes in the cases of clamped--free and clamped--clamped configurations; in the latter case, this includes analytical expressions for the boundary reactions. The formulation recovers the classical inverse elastica in the absence of magnetic fields and reveals a linear scaling between curvature deviation and magnetic mismatch. A tessellation strategy based on stiffness tailoring is further proposed for the design of discretized morphing surfaces. The theoretical predictions are validated against discrete elastic rod simulations and experiments across representative geometries. This work establishes a consistent analytical framework for the inverse design of a magneto-elastica and provides new insight into magnetically-induced shape programming in slender structures.