Framework for flexoelectric structures with cyclic symmetry
Monica Dingle1, David Codony2,1, Irene Arias1,3
1Laboratori de Càlcul Numèric (LaCàN), Universitat Politècnica de Catalunya (UPC), Campus Nord UPC-C2,E-08034,Barcelona,Spain; 2College of Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA; 3Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), 08034 Barcelona, Spain
Continuum modelling of fexoelectricity, i.e. the ability of some dielectric materials to polarize under non-homogeneous deformations and viceversa, has been a historically challenging task due to the high-order nature of the PDE that describe their behaviour. Given the fact that the equations are a system of 4th order coupled PDE, we require high order continuity of the functional space used to describe the approximation of the solution. In order to face such intricacies, we use a complex computational framework which features an immersed hierarchical B-Spline approach [1], where the domain boundary is embedded within a Cartesian mesh. Then, the B-Spline basis which discretizes the unknowns is built on top of this. It is already known that such B-Spline bases present a Cartesian structure which makes them extremely useful when modelling structures that present orthogonal translational symmetry. By exploiting such symmetry, we are able to reduce the computational domain to one single periodic unit cell to which we apply generalised periodic boundary conditions [2,3]. Here in this work, we extend this methodology in order to apply the latter to structures with cyclic symmetry, that is, structures which are constructed by repeating a pattern around a fixed axis. To do so, we have rewritten the flexoelectric problem in a different curvilinear coordinate system which allows us to use the same computational framework, opening up to the possibility of exploiting other types of symmetries that might aid the design of flexoelectric metamaterials and devices.
References
[1] D. Codony, O. Marco, S. Fernández-Méndez, and I. Arias, “An immersed boundary hierarchical b-spline method for flexoelectricity,” Computer Methods in Applied Mechanics and Engineering, vol. 354, pp. 750–782, 2019.
[2] A. Mocci, J. Barceló-Mercader, D. Codony, and I. Arias, “Geometrically polarized architected dielectrics with apparent piezoelectricity,” Journal of the Mechanics and Physics of Solids, vol. 157, p. 104643, 2021.
[3] J. Barceló-Mercader, D. Codony, S. Fernández-Méndez, and I. Arias, “Weak enforcement of interface continuity and generalized periodicity in high-order electromechanical problems,” International Journal for Numerical Methods in Engineering, vol. 123, no. 4, pp. 901–923, 2022.
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