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 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.

Mathematical and computational modeling of flexoelectricity

“…domains composed by multiple materials with different properties, can be used to achieve a specific functionality of an electromechanical device. The paradigmatic example consists on cantilevers bimorphs for piezoelectric transduction 5,73,74 , but one can also think of flexoelectricity-based devices with multimaterial stacks 46,75,76 or structured materials with geometrically polarized cavities 2,77-79 and even conducting/insulating inclusions 80,81 .…”

“…In the context of discrete flexoelectricity boundary value problems, high-order interface conditions must be considered 81 . Such conditions correspond to high-order continuity of the state variables and high-order equilibrium, (i) across the interface I between two adjacents subdomains, as well as (ii) at the sharp regions S of the interfaces, shared by multiple -possibly more than two-subdomains (see Fig.…”

“…An elegant way of enforcing highorder generalized-periodic conditions in a discrete boundary value problem consists on simply constructing a high-order generalized-periodic approximation space 82 . Alternatively, one can resort to enforcing generalized periodicity conditions (76) weakly 81 .…”

Computational modeling and rational design of flexoelectric metamaterials and devices

“…By its symmetry, it is not geometrically polarized along the horizontal direction, and hence it should not exhibit effective piezoelectricity when loaded horizontally. To quantify this effect, we considered the standard sandwich device and rotated the lattice microstructure relative to the device axis for designs A to D. We implemented this by appropriate rotation of the generalized periodic boundary conditions [65] (details will be discussed in later section 3.3.5). We represented the effective piezoelectric coefficients h and d, suitably normalized, in polar plots where continuous/dashed lines refer to positive/negative values, Fig.…”

Mathematical and computational modeling of flexoelectricity at mesoscopic and atomistic scales

“…This paper uses the computational machinery developed in Barceló-Mercader et al (2021a) in order to characterize the e ective piezoelectric coe cient of non-piezoelectric metamaterials working under compression. Di erent bending-dominated lattices, geometrical orientations, constituents thickness and exoelectric materials are considered in order to assess the throughput sensitivity.…”