### A Level-Set Based IGA Formulation for Topology Optimization of Flexoelectric Materials

H. Ghasemi, H.S. Park and T. Rabczuk

Accepted for publication in *Computer Methods in Applied Mechanics and Engineering* 2016

#### Abstract

This paper presents a design methodology based on a combination of isogeometric analysis (IGA), level set and point wise density mapping techniques
for topology optimization of a continuum considering piezoelectric and flexoelectric effects. The fourth order partial differential equations (PDEs)
of flexoelectricity, which require at least C^{1} continuous approximations, are discretized by using Non-Uniform Rational B-spline (NURBS). The point
wise density mapping technique with consistent derivatives is directly used in the weak form of the governing equations. The boundary of the design
domain is clearly and implicitly represented by a level set function. The accuracy of the IGA model is confirmed through numerical examples including
a cantilever beam under a point load and a truncated pyramid under compression with different electrical boundary conditions. Finally, we provide
numerical examples demonstrating the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization.

### Topology Optimization of Piezoelectric Nanostructures

S.S. Nanthakumar, T. Lahmer, X. Zhuang, H.S. Park and T. Rabczuk

*Journal of the Mechanics and Physics of Solids* 2016; 94:316-335

#### Abstract

We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the
energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a
linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are
found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our
results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which
enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.
This paper is available in PDF form
.

### Surface Effects on the Piezoelectricity of ZnO Nanowires

S. Dai and H.S. Park

*Journal of the Mechanics and Physics of Solids* 2013; 61:385-397

#### Abstract

We utilize classical molecular dynamics to study surface effects on the piezoelectric properties of ZnO nanowires as calculated under uniaxial loading.
An important point to our work is that we have utilized two types of surface treatments, those of charge compensation and surface passivation, to
eliminate the polarization divergence that otherwise occurs due to the polar (0001) surfaces of ZnO. In doing so, we find that if appropriate
surface treatments are utilized, the elastic modulus and the piezoelectric properties for ZnO nanowires having a variety of axial and surface
orientations are all reduced as compared to the bulk value as a result of polarization-reduction in the polar [0001] direction. The reduction in
effective piezoelectric constant is found to be independent of the expansion or contraction of the polar (0001) surface in response to surface
stresses. Instead, the surface polarization and thus effective piezoelectric constant is substantially reduced due to a reduction in the bond
length of the Zn-O dimer closest to the polar (0001) surface. Furthermore, depending on the nanowire axial orientation, we find in the
absence of surface treatment that the piezoelectric properties of ZnO are either effectively lost due to unphysical transformations from the wurtzite
to non-piezoelectric d-BCT phases, or also become smaller with decreasing nanowire size. The overall implication of this study is that if
enhancement of the piezoelectric properties of ZnO is desired, then continued miniaturization of square or nearly square cross section
ZnO wires to the nanometer scale is not likely to achieve this result.
This paper is available in PDF form
.

### Surface Piezoelectricity, Size-effects in Nanostructures and the Emergence of Piezoelectricity in Non-piezoelectric Materials

S. Dai, M. Gharbi, P. Sharma and H.S. Park

*Journal of Applied Physics* 2011; 110:104305

#### Abstract

In this work, using a combination of a theoretical framework and atomistic calculations, we highlight the concept of surface
piezoelectricity that can be used to interpret the piezoelectricity of nanostructures. Focusing on three specific material systems
(ZnO, SrTiO_{3} and BaTiO_{3}), we discuss the renormalization of apparent piezoelectric behavior at small scales. In a rather interesting
interplay of symmetry and surface effects, we show that nanostructures of certain non-piezoelectric materials may also exhibit
piezoelectric behavior. Finally, for the case of ZnO, using a comparison with first principles calculations, we also comment on the
fidelity of the widely-used core-shell interatomic potentials to capture non-bulk electro-mechanical response.
This paper is available in PDF form
.

### A New Multiscale Formulation for the Electromechanical Behavior of Nanomaterials

H.S. Park, M. Devel and Z. Wang

*Computer Methods in Applied Mechanics and Engineering* 2011; 200:2447-2457

#### Abstract

We present a new multiscale, finite deformation, electromechanical formulation to capture the response of surface-dominated nanomaterials
to externally applied electric fields. To do so, we develop and discretize a total energy that combines both mechanical and electrostatic
terms, where the mechanical potential energy is derived from any standard interatomic atomistic potential, and where the electrostatic
potential energy is derived using a Gaussian-dipole approach. By utilizing Cauchy-Born kinematics, we derive both the bulk and surface
electrostatic Piola-Kirchoff stresses that are required to evaluate the resulting electromechanical finite element equilibrium equations,
where the surface Piola-Kirchoff stress enables us to capture the non-bulk electric field-driven polarization of atoms near the surfaces
of nanomaterials. Because we minimize a total energy, the present formulation has distinct advantages as compared to previous approaches,
where in particular, only one governing equation is required to be solved. This is in contrast to previous approaches which require
either the staggered or monolithic solution of both the mechanical and electrostatic equations, along with coupling terms that link
the two domains. The present approach thus leads to a significant reduction in computational expense both in terms of fewer equations
to solve and also in eliminating the need to remesh either the mechanical or electrostatic domains due to being based on a total Lagrangian
formulation. Though the approach can apply to three-dimensional cases, we concentrate in this paper on the one-dimensional case. We first
derive the necessary formulas, then give numerical examples to validate the proposed approach in comparison to fully atomistic
electromechanical calculations.
This paper is available in PDF form
.

### Piezoelectric Constants for ZnO Calculated Using Classical Polarizable Core-Shell Potentials

S. Dai, M.L. Dunn and H.S. Park

*Nanotechnology* 2010; 21:445707

#### Abstract

We demonstrate the feasibility of using classical atomistic simulations, i.e. molecular dynamics and molecular statics, to study the
piezoelectric properties of ZnO using core-shell interatomic potentials. We accomplish this by reporting piezoelectric constants for
ZnO as calculated using two different classical interatomic core-shell potentials, that originally proposed by
Binks et al., and that proposed by Nyberg et al. We demonstrate that the
classical core-shell potentials are able to qualitatively reproduce the piezoelectric constants as compared to benchmark \emph{ab initio}
calculations. We further demonstrate that while the presence of the shell is required to capture the electron polarization effects that
control the clamped ion part of the piezoelectric constant, the major shortcoming of the classical potentials is a significant
underprediction of the clamped ion term as compared to previous ab initio results. However, the present results suggest
that overall, these classical core-shell potentials are sufficiently accurate to be utilized for large scale atomistic
simulations of the piezoelectric response of ZnO nanostructures.
This paper is available in PDF form
.