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

Abstract

We present a formulation for the weak enforcement of continuity conditions atmaterialinterfacesinhigh-orderproblemsbymeansofNitsche’smethod,whichis particularly suited for unfitted discretizations. This formulation is extendedto impose generalized periodicity conditions at the unit cell boundaries of peri-odic structures. The formulation is derived for flexoelectricity, a high-orderelectromechanical coupling between strain gradient and electric field, mathe-matically modeled as a coupled system of fourth-order PDEs. The design of flex-oelectricdevicesrequiresthesolutionofhigh-orderboundaryvalueproblemsoncomplex material architectures, including general multimaterial arrangements.ThiscanbeefficientlyachievedwithanimmersedboundaryB-splinesapproach.Furthermore, the design of flexoelectric metamaterials also involves the anal-ysis of periodic unit cells with generalized periodicity conditions. Optimalhigh-order convergence rates are obtained with an unfitted B-spline approxi-mation, confirming the reliability of the method. The numerical simulationsillustrate the usefulness of the proposed approach toward the design of func-tional electromechanical multimaterial devices and metamaterials harnessingthe flexoelectric effect.