Topology optimization of flexoelectric structures

Nanthakumar, S.S.; Zhuang, Xiaoying; Park, Harold S.; Rabczuk, Timon

doi:10.1016/j.jmps.2017.05.010

Cited by 107 publications

(51 citation statements)

References 68 publications

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“…The equations are a coupled system of 4th-order partial differential equations, which renders analytical solutions difficult to obtain and precludes the use of conventional C 0 finite elements. Several numerical alternatives have been proposed in the literature, based on smooth approximations with at least C 1 continuity [22][23][24][25][26][27][28] or on mixed formulations [29,30]. The first self-consistent numerical solution of the linear flexoelectric problem was provided by Abdollahi et.…”

Section: Introductionmentioning

confidence: 99%

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An immersed boundary hierarchical B-spline method for flexoelectricity

Codony

Marco

Fernández‐Méndez

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

“…This approach was also used by Deng [30]. Another alternative is the isogeometric approach, which has been used to perform topology optimization on 2D flexoelectric cantilever beams [25][26][27]. More recently, the C 1 triangular Argyris element was used by Yvonnet et.…”

Section: Introductionmentioning

confidence: 99%

An immersed boundary hierarchical B-spline method for flexoelectricity

Codony

Marco

Fernández‐Méndez

et al. 2019

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

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“…Finally, we anticipate that the staggered formulation presented here may have applicability to a different class of electromechanical coupling in soft materials that has recently emerged, that of flexoelectricity [48,49,50]. In computational formulations of flexoelectricity, all approaches to-date have also followed a monolithic formulation involving complex electromechanical coupling tensors [23,51,52,53,54]. It is possible that staggered formulations following the approach proposed here may be similarly effective for problems involving flexoelectricity; such investigations are currently underway.…”

Section: Discussionmentioning

confidence: 94%

A staggered explicit–implicit finite element formulation for electroactive polymers

Seifi

Park

2018

Computer Methods in Applied Mechanics and Engineering

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Electroactive polymers such as dielectric elastomers (DEs) have attracted significant attention in recent years. Computational techniques to solve the coupled electromechanical system of equations for this class of materials have universally centered around fully coupled monolithic formulations, which while generating good accuracy requires significant computational expense. However, this has significantly hindered the ability to solve large scale, fully three-dimensional problems involving complex deformations and electromechanical instabilities of DEs. In this work, we provide theoretical basis for the effectiveness and accuracy of staggered explicit-implicit finite element formulations for this class of electromechanically coupled materials, and elicit the simplicity of the resulting staggered formulation. We demonstrate the stability and accuracy of the staggered approach by solving complex electromechanically coupled problems involving electroactive polymers, where we focus on problems involving electromechanical instabilities such as creasing, wrinkling, and bursting drops. In all examples, essentially identical results to the fully monolithic solution are obtained, showing the accuracy of the staggered approach at a significantly reduced computational cost.

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“…For instance, this flexoelectricity coupling has been identified in polymers [8], biological membranes [9], and, recently its role in the bone remodeling process has been evidenced [10,11]. The design of mesostructure producing this effect is a new trend in topological optimization [12] since the mastery of this effect makes it possible to consider the development of new sensors, energy harvesting devices, and even the development of polymer actuators. This last application arouses a strong interest in the field of soft robotics, where the optimization of this coupling would allow the control of the geometry of polymers via an electric field [13].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law

2020

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We determine the different symmetry classes of bi-dimensional flexoelectric tensors. Using the harmonic decomposition method, we show that there are six symmetry classes. We also provide the matrix representations of the flexoelectric tensor and of the complete flexoelectric law, for each symmetry class.Symmetry 2020, 12, 674 2 of 29 concern this aspect of the project. Furthermore, this task has to be undertaken by considering the final application, hence by trying to give a physical content to invariants.Piezoelectric and flexoelectric effects can be introduced via the following constitutive equation [1,5]:where p is the polarization, q the electric field, ε ∼ the strain tensor, and η the strain-gradient tensor.The second-order tensor S ∼ , the third-order tensor P , and the fourth-order tensor F ≈ describe the dielectric effect, the direct piezoelectric and flexoelectric effects [5], respectively. It is important to note that in centrosymmetric materials, the third-order tensor P vanishes. This means that piezoelectricity exists only in non-centrosymmetric materials.The vector space of flexoelectric tensors is denoted by Flex. For the optimal design of flexoelectric structures, the understanding of the algebraic structure of Flex is mandatory. Important issues concerning Flex include the following questions:

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Topology optimization of flexoelectric structures

Cited by 107 publications

References 68 publications

An immersed boundary hierarchical B-spline method for flexoelectricity

An immersed boundary hierarchical B-spline method for flexoelectricity

A staggered explicit–implicit finite element formulation for electroactive polymers

Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law

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