A C0 Interior Penalty Finite Element Method for Flexoelectricity

Ventura, Jordi; Codony, David; Fernández‐Méndez, Sonia

doi:10.1007/s10915-021-01613-w

Cited by 14 publications

(19 citation statements)

References 23 publications

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“…Moreover, as a FE method, numerical integration is straight-forward, prescribed values of the solution can be directly enforced setting nodal values, material interfaces do not need any special treatment as long as the mesh fits to the interface, and meshes can be refined where needed to capture features in the solution. In particular, in the presence of boundary layers in the electric field, an anisotropic FE mesh can be considered to refine along the boundary only in the orthogonal direction, as shown in the flexoelectric beam simulations in [21]. As expected for a C0-IPM formulation for a fourth-order PDE problem [4,5,12], the convergence rates are close to the convergence rates of standard FEM in second-order PDEs.…”

Section: Introductionmentioning

confidence: 57%

“…Flexoelectricity is a two-way electromechanical coupling, present in all dielectrics, that is relevant only at small scales [20], and can be modelled by a set of fourth-order Partial Differential Equations (PDEs) with proper boundary conditions. There are several approaches in the literature for the solution of fourth-order PDEs and, in particular, for the solution of flexoelectricity problems, such as mixed finite element methods [16,10], meshless methods [1], C 1 approximations on regular grids with embedded domains [13,18,8,22,17,9] or C 0 interior penalty finite element methods (C0-IPM) [21].…”

Section: Introductionmentioning

confidence: 99%

“…In [21] the authors derive a new formulation for the solution of the flexoelectricity problems in finite domains, based on the C0-IPM method initially proposed for biharmonic equations [4,5,11]. Its good performance is experimentally tested with synthetic problems and with realistic beam problems.…”

Section: Introductionmentioning

confidence: 99%

“…As expected for a C0-IPM formulation for a fourth-order PDE problem [4,5,12], the convergence rates are close to the convergence rates of standard FEM in second-order PDEs. More precisely, the convergence tests in [21] exhibit rates between p and p + 1 when a p-th degree FE approximation is used, for p = 3, 4.…”

Section: Introductionmentioning

confidence: 99%

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C0-IPM with Generalised Periodicity and Application to Flexoelectricity-Based 2D Metamaterials

2022

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We propose a methodology to solve high-order PDE boundary value problems with generalised periodicity, in the framework of the C 0 interior penalty method. The method is developed for the analysis of flexoelectricitybased metamaterial unit cells, formalising the corresponding problem statement and weak form, and giving details on the implementation of the local and macro conditions for generalised periodicity. Numerical examples demonstrate the high-order convergence of the method and its applicability in realistic problem settings.

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Section: Introductionmentioning

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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C0-IPM with Generalised Periodicity and Application to Flexoelectricity-Based 2D Metamaterials

2022

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“…They suffer however from stability issues and from cumbersome modeldependent implementations, as well as from a higher computational cost due to the large number of additional unknowns. 𝐶 0 penalty methods also consider standard C0 finite element approximations and impose the required continuity across elements weakly (Ventura et al, 2021).…”

Section: State Of the Artmentioning

confidence: 99%

Mathematical and computational modeling of flexoelectricity at fixed and moving interfaces, fracture surfaces and contact

Barceló Mercader

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(English) Flexoelectricity is a two-way coupling between strain gradient and polarisation or strain and polarization-gradient. Harnessing flexoelectricity as a functional property requires gradient engineering. This is a major step from the uniform-field configurations of piezoelectric devices, since piezoelectricity couples strains and polarisation. Gradients can be generated by non-uniform deformation, such as bending and torsion, or through non-uniform material distributions, electrode configurations and complex geometries. Gradient engineering thus requires accurate quantitative modelling tools capable of efficiently dealing with all these elements with high-physical fidelity in order to build engineering tools for the design of flexoelectric devices. From a mathematical point of view, flexoelectricity is modelled as a system of coupled high-order PDEs. This poses important challenges to computationally solving boundary value problems in general multi-material samples with complex geometries and electrode configurations.In this thesis, a theoretical and computational model for flexoelectricity in the presence of interfaces has been developed and implemented in the context of the hierarchical B-spline immersed boundary approach. This framework has been used to study physical material interfaces, as well as fictitious interfaces such as generalised periodicity unit cell boundaries. The former has been used in multimaterial symmetry-breaking arrangements up-scaling flexoelectricity in electromechanical devices. For the latter case, an elegant and efficient implementation making use of the periodicity of the B-spline bases functions has been derived and used for the design and analysis of flexoelectric architected periodic lattice metamaterials. Moving interfaces such as cracks and ferroelectric domain boundaries have also been studied by coupling phase-field models for fracture and ferroelectric microstructure accounting for flexoelectricity. Finally, flexoelectricity has been shown as a plausible cause for the asymmetric tribology observed in ferroelectrics in tight collaboration with experimentalists. (Español) La flexoelectricidad es un acoplamiento bidireccional entre el gradiente de tensión y la polarización o entre la tensión y el gradiente de polarización. Aprovechar la flexoelectricidad como una propiedad funcional requiere ingeniería de gradientes. Este es un gran paso desde las configuraciones de campo uniforme de los dispositivos piezoeléctricos, ya que la piezoelectricidad acopla tensiones y polarización. Los gradientes pueden generarse por deformación no uniforme, como flexión y torsión, o mediante distribuciones de materiales no uniformes, configuraciones de electrodos y geometrías complejas. Por lo tanto, la ingeniería de gradientes requiere herramientas precisas de modelado cuantitativo capaces de manejar de manera eficiente todos estos elementos con alta fidelidad física para construir herramientas de ingeniería para el diseño de dispositivos flexoeléctricos. Desde un punto de vista matemático, la flexoelectricidad se modela como un sistema de EDPs acopladas de alto orden. Esto plantea desafíos importantes para resolver computacionalmente problemas de condición de frontera en muestras generales de múltiples materiales con geometrías complejas y configuraciones de electrodos. En esta tesis, se ha desarrollado e implementado un modelo teórico y computacional para flexoelectricidad en presencia de interfaces en el contexto del enfoque de B-spline jerárquicas en problemas de condición de frontera. Este marco se ha utilizado para estudiar interfaces de materiales físicos, así como interfaces ficticias como los límites de celdas de unidades de periodicidad generalizadas. El primero se ha utilizado en arreglos multimateriales que rompen la simetría aumentando la flexoelectricidad en dispositivos electromecánicos. Para el último caso, se ha derivado y utilizado una implementación elegante y eficiente que hace uso de la periodicidad de las funciones de las bases B-spline para el diseño y análisis de metamateriales flexoeléctricos con arquitectura periódica. Las interfaces móviles, como las grietas y los límites de los dominios ferroeléctricos, también se han estudiado mediante el acoplamiento de modelos de campo de fase para fracturas y microestructuras ferroeléctricas que representan la flexoelectricidad. Finalmente, se ha demostrado que la flexoelectricidad es una causa plausible de la tribología asimétrica observada en los ferroeléctricos en estrecha colaboración con experimentalistas.

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

Barceló-Mercader

Codony

Fernández‐Méndez

et al. 2021

Numerical Meth Engineering

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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.

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A C0 Interior Penalty Finite Element Method for Flexoelectricity

Cited by 14 publications

References 23 publications

C0-IPM with Generalised Periodicity and Application to Flexoelectricity-Based 2D Metamaterials

C0-IPM with Generalised Periodicity and Application to Flexoelectricity-Based 2D Metamaterials

Mathematical and computational modeling of flexoelectricity at fixed and moving interfaces, fracture surfaces and contact

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

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