An immersed boundary hierarchical B-spline method for flexoelectricity

Abstract

This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

Computational Modeling of Flexoelectricity—A Review

“…Note that the equipotential has not been used in previous works on flexoelectricity using IGA. It should be remarked that recently Codony et al [46] proposed the immersed boundary method on hierarchical B-spline for flexoelectricity problems with higher efficiency in terms of complex domain shapes smf boundary conditions handling. Moreover, the non-local conditions on non-smooth boundary (which we have neglected) was also treated.…”

“…Alternatively, the Argyris triangular element can also be used as a remedy to the C 1 requirement, as in the work of Yvonnet and Liu [45]. More recently, in a similar spirit to IGA approach, Codony et al [46] proposed the immersed boundary method on hierarchical B-Spline for flexoelectricity problems with higher efficiency. Besides, flexoelectric effect and the associated damage behavior were also approached with peridynamics method by Roy and Roy [47].…”

A C0 Interior Penalty Finite Element Method for Flexoelectricity

“…We consider the model in [7], where flexoelectricity is ruled by the following set of PDEs and boundary conditions:…”

“…C is the elasticity tensor, that depends on the Young modulus E and the Poisson ratio ν, h is the strain-gradient tensor, defined as h i jk mn = l 2 C i j m δ kn with the internal length scale parameter l, e and μ are the tensors of piezoelectric and flexoelectric coefficients and κ contains the dielectricity constants, see appendix B in [7] for detailed definitions.…”

“…However, in a more general context, defining a NURBS approximation with C 1 continuity in a whole domain with complex shape may not be straightforward, and the numerical integration of the resulting NURBS may be, again, very expensive [22]. An efficient alternative for complex domains is the immersed B-Spline method proposed in [7]. It considers B-Spline approximations based on a background regular grid, with an embedded domain.…”