Weak enforcement of interface continuity and generalized periodicity in high-order electromechanical problems

We present a formulation for the weak enforcement of continuity conditions at material interfaces in high-order problems by means of Nitsche's method, which is particularly suited for unfitted discretizations. This formulation is extended to impose generalized periodicity conditions at the unit cell boundaries of periodic structures. The formulation is derived for flexoelectricity, a high-order electromechanical coupling between strain gradient and electric field, mathematically modeled as a coupled system of fourth-order PDEs. The design of flexoelectric devices requires the solution of high-order boundary value problems on complex material architectures, including general multimaterial arrangements. This can be efficiently achieved with an immersed boundary B-splines approach. Furthermore, the design of flexoelectric metamaterials also involves the analysis of periodic unit cells with generalized periodicity conditions. Optimal high-order convergence rates are obtained with an unfitted B-spline approximation, confirming the reliability of the method. The numerical simulations illustrate the usefulness of the proposed approach toward the design of functional electromechanical multimaterial devices and metamaterials harnessing the flexoelectric effect.