Optimizing remanent magnetization in magnetorheological elastomers under external permanent magnet actuation

The magneto-mechanical coupling governing the response of magnetorheological elastomers (MREs) requires computational tools for their design and optimization. The existing frameworks are customarily based on ideal boundary value problems that optimize MREs under ideal, non-realistic homogeneous magnetic sources. We present an approach that addresses these limitations by solving strongly coupled magneto-mechanical partial differential equations within the optimization loop, explicitly accounting for external permanent magnets and their interaction with the deformable elastomer. The approach combines a coupled magneto-elastic boundary value problem with an auxiliary mesh-motion problem, which allows the free-space to deform consistently with the MRE geometry. The direction of the remanent magnetization in the MRE is represented by continuous design fields, updated through a gradient-based optimizer with adjoint sensitivities and filtering. A series of benchmark problems demonstrate the framework: robustness with respect to the initial guess; symmetry of the solution under reversed polarity; influence of magnet placement; transition between external-field and self-interaction dominated actuation; and adaptation to opposite objectives in a pull-push actuator. The results highlight how the explicit modeling of the free-space and the magnetic source leads to robust and physically consistent designs, providing a foundation for advanced MRE-based actuators.