Mohammad Abobala

Maximality and Minimality of Ideals In Some Neutrosophic Rings

Keywords: Neutrosophic ring, refined neutrosophic ring, maximal ideal, minimal ideal, AH-ideal. 3. Ideals in Neutrosophic rings Remark 3

This paper studied the condition of maximal and minimal ideals in neutrosophic ring theory.

Since every neutrosophic ring R(I) can be understood as Then each subset of R(I) has the form ; P,S are two subsets of R. We call P the real part, S the neutrosophic part of M.

An important question arises here. This question is:

When M is a neutrosophic ideal of R(I)?. In other words, what conditions on the real part P and neutrosophic part S which make M be an ideal?.

The following theorem clarifies the necessary and sufficient condition to answer the previous question.

Theorem 3.2:

Let R(I) be a neutrosophic ring, be any subset of R(I), then M is a neutrosophic ideal if and only if the following conditions are true:

(a) P is an ideal on R.

(b) P is contained in S.

(c) S is an ideal of R.

Proof:

Firstly, we assume that (a),(b), and (c) are true, we have: is a subgroup of , that is because if we find . Now, suppose that we have by the assumption, we regard that thus , which means that M is a neutrosophic ideal of R(I).

Conversely, we suppose that is a neutrosophic ideal of R(I). Let be two arbitrary elements in P, and be two arbitrary elements in S, we have , by

using the assumption we have M as an ideal, hence thus thus are two subgroups of .

For every thus this means that are ideals in the classical ring R. Now, we prove that P is contained in S. We have , that is because R(I) has a unity 1. On the other hand, we can write and that is because M is an ideal of R(I), hence , thus , by regarding that is an arbitrary element of P, we get that

The previous theorem ensures that each ideal is an AH-ideal, since P,S are supposed to be classical ideals of R.

Example 3.3:

Let be the ring of integers, be the corresponding neutrosophic ring, we have:

(a) , are three ideals of R, with

(b) is an ideal of R(I). (c) is not a neutrosophic ideal, that is because is not contained in S.

Example 3.4:

Let be the corresponding neutrosophic ring. Consider the set . We have M as an ideal of R(I), that is because Theorem 3.5:

The following theorem determines the form of maximal ideals in R(I).

Let R(I) be a neutrosophic ring, be an ideal of R(I), then M is maximal if and only if P is maximal in R with or .

Proof:

Suppose that M is maximal of R(I), let be any ideal of R(I) with the property then and by the assumption of the maximality of M, we find that , this implies that which means that P is maximal in R or . On the other hand and P is maximal, thus . Since , hence the only non trivial maximal ideal is , with P as a maximal ideal in R.

The converse is clear.

Theorem 3.6:

The following theorem describes minimal ideals in R(I).

Let R(I) be a neutrosophic ring, be an ideal of R(I), then M is minimal if and only if S is minimal in R and .

Proof:

Suppose that M is minimal of R(I), let be any ideal of R(I) with the property then and by the assumption of the minimality of M, we find that , this implies that which means that are minimal in R. On the other hand and S is minimal, thus . Since is a sub-ideal of P+SI, hence .

The converse is clear.

Remark 3.7:

According to Theorem 5.1 and Theorem 6.1, we get a full description of the structure of maximal and minimal ideals in the neutrosophic ring R(I).

(a) Non trivial Maximal ideals in R(I) has the form {P+RI}, where P is maximal in R.

(b) Non trivial minimal ideals have the form {{0}+SI} where S is minimal in R.

Example 3.8:

Let Z(I) be the neutrosophic ring of integers, non trivial maximal ideals in Z(I) are

, where p is any prime number.

Ideals in refined neutrosophic rings Remark 4.1:

Since every refined neutrosophic ring R( ) can be understood as

Then each subset of has the form ; are two subsets of R.

An important question arises here. This question is:

When M is a refined neutrosophic ideal of ?. In other words, what conditions on which make M an ideal?.

The following theorem clarifies the necessary and sufficient condition to answer the previous question.

Theorem 4.2:

Consider the following: be a refined neutrosophic ring, be a subset of . M is an ideal of if and only if (a) are ideals on R .

Proof:

Suppose that M is an ideal, then we have for every are two elements of are subgroups of .

For every , thus , thus are ideals of R.

On the other hand, we have that is because is an arbitrary element in P.

Also,

That is because is an arbitrary element in S.

For the converse, we suppose that Where P is any maximal ideal of R.

Proof:

We assume that M is a maximal ideal, and hence , we have , or . This implies that should be maximal; but we have that hence .

The converse is clear.

Theorem 4.6:

Consider the following: be a refined neutrosophic ring, be any non trivial minimal ideal of M has the following form:

Where P is any minimal ideal of R.

Proof:

The proof is similar to Theorem 5.2.

Theorem 4 . 5 :

 45 (a) and (b) are true, we have (M,+) as a subgroup of . Let be the refined neutrosophic ring of integers modulo 20, we have . Consider the following: be a refined neutrosophic ring, be any non trivial maximal ideal of M has the following form:

Example 4. 7 :

 7 (a) Consider the refined neutrosophic ring of integers modulo 8, we have is a minimal ideal of . Hence is a minimal ideal of . (b) is maximal in . Hence is maximal in


X Chen

J Yvonnet
email: julien.yvonnet@univ-paris-est.fr

H S Park

S Yao

Enhanced converse exoelectricity in piezoelectric composites by coupling topology optimization with homogenization

We demonstrate that large apparent converse flexoelectric properties can be obtained in piezoelectric composites using theoretical approaches. To do so, we first present a numerical homogenization method accounting for all electromechanical terms related to strain and electric field gradient. We then evaluate the coefficients of the model by numerical simulations on periodic piezoelectric composites. After combining the homogenization approach with topology optimization to enhance the converse properties of the composite, we present numerical results that reveal that the apparent converse flexoelectric coefficients, as well as those associated with the higher order coupling terms involving the electric field gradient, are of the same order as the direct flexoelectric properties of the local constituents. These results suggest that both converse and higher order electromechanical coupling effects may contribute strongly to the flexoelectric response and properties of piezoelectric composites.

Finally, we show that it is theoretically possible to obtain optimized designs of composites with apparent converse flexoelectric properties 1-2 orders of magnitude larger than ones obtained with naive guess designs.

I. INTRODUCTION

Direct flexoelectricity is a phenomenon whereby an electrical polarization can be induced by generating a strain gradient, or an inhomogeneous deformation on the sample. This so-called direct flexoelectric effect has been widely studied in ferroelectric materials [START_REF] Cross | Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients[END_REF] and complex oxide ceramics [START_REF] Zhu | Experimental studies of the converse flexoelectric effect induced by inhomogeneous electric field in a barium strontium titanate composition[END_REF][START_REF] Ma | Flexoelectric polarization of barium strontium titanate in the paraelectric state[END_REF][START_REF] Zubko | Strain-gradient induced polarization in SrTiO3 single crystals[END_REF][START_REF] Ma | Strain-gradient induced electric polarization in lead zirconate titanate ceramics[END_REF] as the effects are much larger in these materials, but more recently in several polymers [START_REF] Chu | Flexoelectricity in several thermoplastic and thermosetting polymers[END_REF][START_REF] Zubko | Flexoelectric effect in solids[END_REF] as well as in biological membranes [START_REF] Deng | Flexoelectricity in soft materials and biological membranes[END_REF] . Kogan 9 formulated the first phenomenological theory of flexelectricity and estimated the range of values for flexoelectric coefficients.

Tagantsev [START_REF] Tagantsev | Theory of flexoelectric effect in crystals[END_REF] developed a microscopic theory for the bulk contributions, using the rigid-ion approximation as well as a phenomenological description. Promising applications of direct flexoelectricity have been studied, like the possibility of building a piezoelectric composite with nonpiezoelectric materials [START_REF] Sharma | Piezoelectric thin-film superlattices without using piezoelectric materials[END_REF][START_REF] Fousek | Possible piezoelectric composites based on the flexoelectric effect[END_REF][START_REF] Bhaskar | A flexoelectric microelectromechanical system on silicon[END_REF][START_REF] Maranganti | Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Greens function solutions and embedded inclusions[END_REF][START_REF] Sharma | On the possibility of piezoelectric nanocomposites without using piezoelectric materials[END_REF][START_REF] Majdoub | Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect[END_REF] , energy harvesters [START_REF] Deng | Nanoscale flexoelectric energy harvesting[END_REF][START_REF] Liang | Flexoelectric energy harvesters based on timoshenko laminated beam theory[END_REF] or new field-gradient-based sensors [START_REF] Huang | Flexoelectric strain gradient detection using ba0. 64sr0. 36tio3 for sensing[END_REF] . In [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] , it was demonstrated that piezocomposites with designed microstructures could exhibit apparent enhanced flexoelectric properties of the same order as oxide ceramics or ferroelectrics.

In contrast to direct flexoelectricity, the converse flexoelectric effect describes a mechanical strain induced by an electric field gradient. Studies on converse flexoelectricity have only emerged very recently and remain limited [START_REF] Zhu | Experimental studies of the converse flexoelectric effect induced by inhomogeneous electric field in a barium strontium titanate composition[END_REF][START_REF] Wang | Converse flexoelectricity around ferroelectric domain walls[END_REF][START_REF] Tian | Size-dependent direct and converse flexoelectricity around a micro-hole[END_REF][START_REF] Abdollahi | Converse flexoelectricity yields large piezoresponse force microscopy signals in non-piezoelectric materials[END_REF][START_REF] Zhang | Converse flexoelectricity with relative permittivity gradient[END_REF][START_REF] Shen | Converse flexoelectric effect in comb electrode piezoelectric microbeam[END_REF][START_REF] Shu | Converse flexoelectric coefficient f 1212 in bulk ba0. 67sr0. 33tio3[END_REF][START_REF] Shu | Relationship between direct and converse flexoelectric coefficients[END_REF] . As noted in [START_REF] Wang | Converse flexoelectricity around ferroelectric domain walls[END_REF] , understanding and modeling of the converse flexoelectric effect may not only help understanding unexplained enhanced piezoelectricity in dielectrics and ferroelectrics, but also in designing and developing new electromechanical devices. The inverse and converse flexoelectric effects have been experimentally demonstrated by applying a voltage to a capacitor and measuring its bending [START_REF] Bhaskar | A flexoelectric microelectromechanical system on silicon[END_REF][START_REF] Bursian | Changes in curvature of a ferroelectric film due to polarization[END_REF] , and by applying a voltage to a truncated pyramid so as to generate an inhomogeneous electric field, thus allowing the sample to deform [START_REF] Cross | Flexoelectric effects: Charge separation in insulating solids subjected to elastic strain gradients[END_REF][START_REF] Zhu | Experimental studies of the converse flexoelectric effect induced by inhomogeneous electric field in a barium strontium titanate composition[END_REF][START_REF] Zhu | Piezoelectric composite based on the enhanced flexoelectric effects[END_REF] . Fu et al. [START_REF] Zhu | Experimental studies of the converse flexoelectric effect induced by inhomogeneous electric field in a barium strontium titanate composition[END_REF] , reported experimental observations of the converse flexoelectric effect in a Ba 0.67 Sr 0.33 TiO 3 (BST) trapezoidal ceramic block under an inhomogeneous electric field. In [START_REF] Shu | Converse flexoelectric coefficient f 1212 in bulk ba0. 67sr0. 33tio3[END_REF] , Shu et al. measured converse flexoelectric coefficients in BST. Wang et al. [START_REF] Wang | Converse flexoelectricity around ferroelectric domain walls[END_REF] showed the important role of converse flexoelectricity on asymmetric structures surrounding domain walls in PbTiO 3 .

Abdollahi et al. [START_REF] Abdollahi | Converse flexoelectricity yields large piezoresponse force microscopy signals in non-piezoelectric materials[END_REF] demonstrated a large effective piezoelectric response in non-piezoelectric materials such as SrTiO 3 and demonstrated that converse flexoelectricity may have non-negligible effects in thin films. In [START_REF] Zhang | Converse flexoelectricity with relative permittivity gradient[END_REF] , converse flexoelectric effects were generated by the design permittivity gradient with BST powder and a substrate. Shen and Chen [START_REF] Shen | Converse flexoelectric effect in comb electrode piezoelectric microbeam[END_REF] demonstrated the converse flexoelectric effect in a lead zirconate titanate micro beam. Tian et al. [START_REF] Tian | Size-dependent direct and converse flexoelectricity around a micro-hole[END_REF] provided explicit solutions for physical fields around a micro hole with simultaneous consideration of the strain gradient elasticity, direct flexoelectricity and converse piezoelectricity. Shu et al. [START_REF] Shu | Relationship between direct and converse flexoelectric coefficients[END_REF] provided relationships between converse and direct flexoelectric coefficients. Mawassy et al. [START_REF] Mawassy | A variational approach of homogenization of piezoelectric composites towards piezoelectric and flexoelectric effective media[END_REF] developed an extended flexoelectric framework involving electric field gradient coupling terms and proposed a finite element framework for their evaluation. Finally, Wang et al. [START_REF] Wang | Flexoelectricity in solids: Progress, challenges, and perspectives[END_REF] conducted an extensive survey on theoretical and experimental approach to determine the direct and converse flexoelectric values in several ferroelectric oxides.

Therefore, two key issues related to flexoelectricity we attempt to resolve in the present work are: first, to enable calculations of the converse flexoelectric effect for general situations, and second, to determine the importance of higher order electric field gradient effects on the flexoelectric response. Therefore, in this paper, we employ a homogenization method to predict the apparent converse flexoelectric properties of piezoelectric composites. Following our previous work [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] where the homogenized direct flexoelectric properties were provided, we propose here an extended effective model accounting for all coupling terms between strain, electric field, strain gradient and electric field gradient. We provide expressions for all of the coupling tensors in a fully anisotropic context, and demonstrate via numerical examples that these coupling terms, as well as the converse flexoelectric effect, are of the same order as the direct flexoelectric constants of the constituent materials. Finally, we combine this model with topology optimization to obtain tailored microstructures with converse flexoelectric properties that exceed those obtained with naive guesses.

II. AN EXTENDED FLEXOELECTRIC MODEL

We define the total energy density W for an electromechanical system where all couplings between strains ε, stress σ , electric field E, strain gradient ∇ε and electric field gradient ∇E are taken into account:

W = 1 2 C i jkl ε i j ε kl - 1 2 α i j E i E j -e i jk E i ε jk + 1 2 G i jklmn ∇ε i jk ∇ε lmn + F i jkl E i ∇ε jkl + M i jklm ε i j ∇ε klm -T i jk E i ∇E jk -K i jkl ε i j ∇E kl - 1 2 L i jkl ∇E i j ∇E kl -H i jklm ∇E i j ∇ε klm (1)
In Eq. ( 1), C, α and e denote the effective forth-order elastic, second-order dielectric and third-order piezoelectric tensors, respectively. The term F denotes the effective fourth-order flex-oelectric tensor, while M and G correspond to higher-order strain gradient elastic tensors (see e.g. [START_REF] Yvonnet | Computational second-order homogenization of materials with effective anisotropic strain gradient behavior[END_REF] ). The term K is the so-called converse flexoelectric tensor.

We note that there are several new coupling tensors in the above energy density expression, whose interpretation is as follows: T denotes the relation between an additional polarization (electric field) and an electric field gradient; L denotes the relation between a polarization gradient (or electric field gradient) and an electric field gradient and H denotes the relation between a polarization gradient (or electric field gradient) and a strain gradient. A similar expression has also been provided in [START_REF] Mawassy | A variational approach of homogenization of piezoelectric composites towards piezoelectric and flexoelectric effective media[END_REF] .

The effective stress tensor σ , effective electric displacement d, effective hyperstress tensor S and hyper-electric displacement P associated with energy density function (1) are defined as:

σ i j = ∂W ∂ ε i j , d i = - ∂W ∂ E i , S i jk = ∂W ∂ ∇ε i jk , P i j = - ∂W ∂ ∇E i j (2)
The corresponding expressions for the stress σ , the electric displacement d, the hyperstress S and hyper electric displacement P are provided by:

σ i j =C i jkl ε kl -e ki j E k + M i jklm ∇ε klm -K i jkl ∇E kl (3)
d i =e i jk ε jk + α i j E j -F i jkl ∇ε jkl + T i jk ∇E jk (4)
S i jk =M lmi jk ε lm + F li jk E l + G i jklmn ∇ε lmn + H lmi jk ∇E lm (5)
P i j =K kli j ε kl + T ki j E k -H i jklm ∇ε klm + L i jkl ∇E kl (6)
The corresponding equilibrium equations relating these quantities are given by σ i j, j -S i jk, jk = 0 (7)

d i,i -P i j,i j = 0 (8)
A complete description of boundary conditions for such model can be found in [START_REF] Sidhardh | Exact solutions for flexoelectric response in elastic dielectric nanobeams considering generalized constitutive gradient theories[END_REF] .

III. MICROSCALE EQUATIONS

σ i j, j = 0, in Ω (9) d i,i = 0, in Ω (10)
with:

σ i j = C i jkl ε kl -e ki j E k (11)
d i = e i jk ε jk + α i j E j (12)
where

ε i j = 1 2 (u i, j + u j,i ).
Assuming that interfaces between the different material phases are denoted collectively by Γ, we assume perfectly bonded interfacial conditions, i.e.

[[σ i j n j ]] = 0, [[u i ]] = 0 on Γ (13) [[d i n i ]] = 0, [[φ ]] = 0 on Γ (14)
where [[.]] denotes the jump accross Γ. We define the macro quantities as (.) ≡ . = 1 V Ω (.)dV with V the volume (area in 2D) of Ω. The effective electric field can be computed by prescribing the following electric quadratic boundary conditions over the RVE (see [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] ):

φ = -E i x i - 1 2 ∇E i j x i x j + φ on ∂ Ω ( 15
)
where φ is the electric potential such that E i = -φ ,i and φ is either zero or a periodic fluctuation on ∂ Ω. Mechanical quadratic boundary condition are introduced to prescribe an effective strain and strain gradient [START_REF] Gologanu | Recent extensions of gurson's model for porous ductile metals[END_REF][START_REF] Forest | Mechanics of generalized continua: construction by homogenizaton[END_REF] :

u i = ε i j x j + 1 2 g i jk x j x k + ũi , on ∂ Ω ( 16
)
where u i is a displacement vector related to strain through ε i j = u i, j + u j,i and

g i jk = ∇ε i jk + ∇ε ik j -∇ε jki ( 17
)
and ũi is either zero or periodic on ∂ Ω. Eqs. ( 9)-( 10) are completed with the boundary conditions ( 15)-( 16). These equations are here solved by the finite element method (see details in [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] ).

It is worth noting that in the case of a homogeneous domain, using ( 15) and ( 16) to compute ε i j and E i and introducing them in Eqs. ( 9)-( 10) does not lead to non-vanishing right-hand terms. This is a well-known issue in strain gradient homogenization problem, which can cause a dependence to the number of unit cells within the RVE and to non-vanishing higher-order properties in the case of materials which do not have local gradient effects (see a discussion in [START_REF] Yvonnet | Computational second-order homogenization of materials with effective anisotropic strain gradient behavior[END_REF] ). In our previous work [START_REF] Chen | Topology optimization of flexoelectric composites using computational homogenization[END_REF] , we have introduced appropriate body forces to balance these non-equilibrated terms. It has also been shown in the context of purely mechanical gradient effects that such procedure is consistent with asymptotic homogenization [START_REF] Monchiet | Strain-gradient homogenization: a bridge between asymptotic expansion and quadratic boundary condition methods[END_REF] . However, it has also been discussed in [START_REF] Yvonnet | Computational second-order homogenization of materials with effective anisotropic strain gradient behavior[END_REF] that such body forces can lead to spurious over predicted effective gradient properties when one of the phase has very low properties. For this reason, we did not adopt this approach in the present paper, which focuses on the development of the homogenization model and on the topology optimization problem. One potential solution to address the above-mentioned issues could be the use of Lagrange multipliers to enforce homogeneous strain gradient and electric field gradients within a homogeneous RVE, to extend the method proposed in [START_REF] Kaczmarczyk | Scale transition and enforcement of rve boundary conditions in second-order computational homogenization[END_REF] .

IV. EFFECTIVE TENSORS

The explicit expressions for tensors C, α, e, G, F and M can be found e.g. in [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] . Following the procedure described in the same reference, the expressions of the new coupling terms including the converse flexoelectric tensor K are provided by:

T i jk = B 0 ipq C pqrs B 1 rs jk -h 0 ip e pqr B 1 qr jk -B 0 ipq e pqr h 1 r jk -h 0 ip α pq h 1 q jk (18)
K i jkl = -A 0 i jpq C pqrs B 1 rskl + D 0 i jp e pqr B 1 qrkl + A 0 i jpq e pqr h 1 rkl + D 0 i jp α pq h 1 qkl ( 19
)
L i jkl = B 1 i jpq C pqrs B 1 rskl -2h 1 i jp e pqr B 1 qrkl -h 1 i jp α pq h 1 qkl ( 20
)
H i jklm = -B 1 i jpq C pqrs A 1 rsklm + B 1 i jpq E pqr D 1 rklm + h 1 i jp e pqr A 1 qrklm + h 1 i jp α pq D 1 qklm (21)
Above, the fields B 0 , B 1 , h 0 , h 1 , D 0 , D 1 , A 0 and A 1 are local fields which are obtained by solving the RVE problem (9)-( 16) by finite elements. The definition for these tensors can be found in [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] and are provided for convenience in supplementary material.

The strong strain or electric field localizations within the medium due to the heterogeneities may lead to a violation of the small perturbation assumption in realistic applications. In that case, extensions to nonlinear formulations of flexoelectricity are available in the literature (see e.g. [START_REF] Liu | An energy formulation of continuum magneto-electro-elasticity with applications[END_REF][START_REF] Yvonnet | A numerical framework for modeling flexoelectricity and maxwell stress in soft dielectrics at finite strains[END_REF] ). However, in the nonlinear case, the effective properties depend on the local fields and identifying the related models can be challenging. Even though this task is out of the scope of this paper, possible strategies for this purpose could rely on data-driven approaches based on artificial intelligence, such as in [START_REF] Lu | A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites[END_REF][START_REF] Le | Computational homogenization of nonlinear elastic materials using neural networks[END_REF] .

V. SIMP TOPOLOGY OPTIMIZATION FOR CONVERSE FLEXOELECTRIC

COMPOSITES

In this section, we formulate the topology optimization problem to maximize the absolute values of the converse flexoelectric tensor components in (19). First, the periodic unit cell is discretized with a regular mesh of N e 4-node quadrilateral finite elements. We define the inclusion material density ρ e in each element e, e = 1, 2, ..., N e such that ρ e = 1 is associated with the inclusion phase and ρ = 0 is associated with the matrix phase. The topology optimization is formulated as follows:

Maximize :

|K i jkl (ρ)| subject : KU = F : ∑ N e
e=1 ρ e v e /(∑ N e e=1 v e ) = f 0 ≤ ρ e ≤ 1, e = 1, 2, ..., N e (22) where KU = F is the discrete system obtained when discretizing Eqs. ( 9)-( 16) by the Finite Elmeent Method (see details e.g. in [START_REF] Yvonnet | Apparent flexoelectricity due to heterogeneous piezoelectricity[END_REF] ). Above, v e is the volume of an element e and f is the inclusion volume fraction.

We use the SIMP method [START_REF] Bendsøe | Optimal shape design as a material distribution problem[END_REF][START_REF] Rozvany | Generalized shape optimization without homogenization[END_REF][START_REF] Bendsøe | Material interpolation schemes in topology optimization[END_REF] to solve the problem. In this framework, the local material properties are interpolated with respect to the local density in a continuous manner, using penalty exponents to enforce local densities to converge to values close to 0 or 1. For composites made of two phases, we use the following expression:

[C i jkl (ρ)] = ρ pc [C 2 i jkl ] + (1 -ρ pc )[C 1 i jkl ] [α i j (ρ)] = ρ pa [α 2 i j ] + (1 -ρ pa )[α 1 i j ] [e ki j (ρ)] = ρ pe [e 2 ki j ] + (1 -ρ pe )[E 2 ki j ] (23)
where the superscript 1 and 2 are associated with matrix and inclusion phase, respectively and pc, pa and pe are penalty exponents. In the numerical examples, these values are chosen as

pc = pa = pe = 3.
The above problem (22) requires evaluating the gradient of the objective function with respect to the local densities (subsequently referred to as sensitivities). The adjoint method has been widely used for sensitivity analysis of gradient-based optimization algorithm [START_REF] Komkov | Design Sensitivity Analysis of Structural Systems[END_REF][START_REF] Bendsøe | Topology Optimization Theory, Methods[END_REF] , and is also employed here. The corresponding Lagrangian function for the optimization problem ( 22) is formed by introducing an adjoint vector λ as:

L =K i jkl + λ • (KU -F) (24)
where KU -F = 0 holds for arbitrary adjoint vectors λ. Differentiating the Lagrangian function L with respect to the design variable ρ, we have:

∂ L ∂ ρ = ∂ K i jkl ∂ ρ + λ • ∂ (KU -F) ∂ ρ (25)
The detailed expression can be found following the procedure described in our previous work on the topology optimization of direct flexoelectric properties [START_REF] Chen | Topology optimization of flexoelectric composites using computational homogenization[END_REF] . The optimization problem ( 22) is solved by the Conservative Convex Separable Approximations (CCSA) optimizer [START_REF] Svanberg | A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations[END_REF] based on the adjoint sensitivity.

VI. NUMERICAL INVESTIGATIONS A. Composite with piezoelectric phase

In this section, we investigate through numerical simulations the significance of the converse flexoelectric and other higher order electromechanical coupling terms in a piezoelectric composite.

The geometry of the RVE is depicted in Fig. 1 (b), and the triangular inclusion is chosen so as to increase the strain and polarization gradient effects. The characteristic size of the RVE is = 1 mm.

The position of points A, B and C in

     (GPa) (26)
[α 1 ] =   2.079 0 0 4.065   (C • m -2 ) (27)
[e 1 ] =   -2.120582 -2.120582 0 0 0 0   (nC • m -1 •V -1 ) (28)
The properties of the inclusion phase are defined with respect to the angle θ according to

α 2 i j = R ip R jq α 1 pq (29)
e 2 i jk = R ip R jq R kr e 1 pqr ( 30
)
C 2 i jkl = R ip R jq R kr R ls C 1 pqrs (31)
with

R =   cos(θ ) -sin(θ ) sin(θ ) cos(θ )   (32)
In Fig. 2a, we compute the evolution of the components of the converse flexoelectric tensor K with respect to the mismatch angle θ . The values are normalized with respect to the flexoelectric component F 1221 of PZT to evaluate their significance. We can notice that the components K 1111 , erties and only plays a negligible role here. Therefore, these results demonstrate that the higher order electromechanical terms H and T can make a significant contribution to the electromechanical response of piezoelectric composites. More specifically, the coefficients T i jk are associated with additional polarization/electric displacement induced by the electric field gradient, and thus characterize the importance of these additional effects on the flexoelectric behavior. Taking these new terms into account in the modeling and simulation of flexoelectric structures may help to design new flexoelectric-based sensors and actuators based on the mechanical and electrical gradient effects.

K

B. Topology optimization of ceramic/ceramic piezoelectric composite

Having established that the converse flexoelectric effect makes a significant contribution to the overall flexoelectric response of the PZT/PZT composites, we now perform topology optimization to determine topologies that maximize the converse flexoelectric contributions. We thus consider the topology optimization of a two-phase composite made of piezoelectric phases. Each phase is made with PZT (lead zirconium titanate ceramics) as in the previous example. Here, the crystal lattice is oriented by a mismatch angle of θ = π in the inclusion phase. Then via (31)- (32), the properties of the inclusion phase can be obtained as

[C 2 ] = [C 1 ] given by (26), [α 2 ] = [α 1 ]
given by (27) and

[e 2 ] =   2.120582 2.120582 0 0 0 0   (nC • m -1 •V -1 ). (33)
We perform the topology optimization of the inclusion shape with respect to the converse flex- which represents a significant improvement as compared to the reference triangular solutions by a factor of 1.89 for the components K 1111 , K 2211 and K 1212 . From Fig. 3, we can see that the three optimized unit cells obtained by K 1111 , K 2211 and K 1212 have similar topologies. Finally, we note that the optimized microstructures are similar to the ones obtained by optimizing the direct flexoelectric constants F 1221 and F 2112 for the PZT/PZT case [START_REF] Chen | Topology optimization of flexoelectric composites using computational homogenization[END_REF] .

C. Topology optimization of ceramic/doped piezoelectric polymer composite

In this example, we replace the stiff PZT inclusion with a soft, dielectric, polymer inclusion (polyvinylidene fluoride, PVDF). The elastic, piezoelectric and dielectric properties for the polymer are given below. In comparison to the PZT properties, all of the polymer properties are 1-2 orders in magnitude lower than for PZT. The material parameters of matrix PZT are expressed in ( 26)-( 28) [START_REF] Pettermann | A comprehensive unit cell model: a study of coupled effects in piezoelectric 1-3 composites[END_REF][START_REF] Brenner | Numerical computation of the response of piezoelectric composites using fourier transform[END_REF] , while the material properties of PVDF are described in ( 34)-( 36) [START_REF] Ramadan | A review of piezoelectric polymers as functional materials for electromechanical transducers[END_REF] .

     (GPa) (34)
[α 2 ] =   0.025 0 0 0.084   (C • m -2 ) ( 35
)

[e 2 ] =   0.1272 0.0873 0 0 0 0   (nC • m -1 •V -1 ) (36)
We perform topology optimization of the PVDF inclusion with respect to the converse flexo- for the PZT/Polymer composites with triangular inclusion.

We obtained six different optimized unit cells, and a significant improvement can be found as compared to the reference triangular solutions. The optimal absolute values are respectively. In the different cases, we can note that the optimized geometry favors the localization of these fields near the interfaces, which may be expected due to the problem being one of a soft inclusion within a stiff matrix.

K 1111 = 0.3420 × 10 -3 C.m -1 , K 2211 = 0.2054 × 10 -3 C.m -1 , K 1212 = 0.0923 × 10 -3 C.m -1 , K 2222 = 0.1218 × 10 -3 C.m -1 , K 2212 = 0.3267 × 10 -3 C.m -1 and K 1211 = 0.0821 × 10 -3 C.m -1 ,

VII. CONCLUSIONS

We have proposed an extended flexoelectric model which takes into account not only converse flexoelectric effects, but also all other higher order electromechanical coupling terms. A homogenization procedure has been introduced to evaluate numerically these apparent properties in piezoelectric composites. Then, we have combined this model with topology optimization to design microstructures with enhanced converse flexoelectric properties. The numerical investigations revealed that the apparent converse flexoelectric coefficients in a composite made of periodic triangular inclusions have the same order of magnitude as the direct flexoelectric properties of the local constituents. Furthermore, we showed that the other higher order coupling terms, i.e. that relate the electric field to an applied electric field gradient and the strain gradient (bending) to the electric field gradient have non-negligible values as compared to the flexoelectric coefficients. Fi- nally, we show that optimized designs can lead to effective converse flexoelectric properties which can be improved by 1-2 orders of magnitude as compared to guess designs for ceramics/ceramics or polymer/ceramics composites.

VIII. SUPPLEMENTARY MATERIAL

See the supplementary material for further details on Finite Element discretization of the microscale RVE problem and numerical calculation of effective flexoelectric properties.

FIG. 1 :

 1 FIG. 1: (a) Periodic piezoelectric structure; (b) Representative Volume Element (RVE). Both matrix phases (phase 1) and inclusion (phase 2) are made of the same piezoelectric material, but rotated by a mismatch angle θ .

Fig. 1 ( 50 [

 150 b) are defined according to A = {-a; a}, B = {a; 0}, C = {-a; -a} with a = √ 0.8 /2 and corresponds to a volume fraction of inclusions equal to f = 0.4. Each phase is made of PZT (lead zirconium titanate ceramics), but the main orientation of the crystal in both phase is rotated by a mismatch angle θ ∈ [0, 2π] to create a heterogeneity. The matrix and inclusion phases are denoted by the superscripts 1 and 2, respectively, in Fig. 1 (b)and in the following. The mechanical, dielectric and piezoelectric properties of the PZT matrix phase are given in this 2D configuration by49,

AngleFIG. 2 :

 2 FIG. 2: Evolution of the components of the effective converse flexoelectric tensor: K (a) and of higher order electromechanical coupling term T (b), L (c) and H (d) with respect to the mismatch angle in the piezoelectric composite with triangular inclusions

FIG. 3 :

 3 FIG. 3: Optimal topology for K for the PZT/PZT composite: (a) K 1111 ; (b) K 2211 ; (c) K 1212 .

FIG. 4 :

 4 FIG. 4: Topology optimization process with respect to normalized flexoelectric components and volume fractions for the PZT/PZT composite: (a) K 1111 ; (b) K 2211 ; (c) K 1212 .

  electric coefficients K 1111 , K 2211 , K 1212 , K 2222 , K 2212 and K 1211 . To ensure that these results can be compared against the previous PZT/PZT results, we set the volume fraction of the PVDF inclusion to be f = 0.4 for all cases. Similarly, the initial guess is set by ρ e = 0.4, e = 1, 2, ..., Ne = 6400. The final optimal unit cells of the converse flexoelectric coefficients K 1111 , K 2211 , K 1212 , K 2222 , K 2212 and K 1211 are shown in Fig. 5. In these figures, the cyan and black colors refer to the inclusion PVDF and matrix PZT, respectively. The reference solutions calculated by a triangular PVDF inclusion as in Fig. 1 (b) are shown for each case. The reference values obtained are K 1111 = 0.0432 × 10 -3 C.m -1 , K 2211 = 0.0139 × 10 -3 C.m -1 , K 1212 = 0.0073 × 10 -3 C.m -1 , K 2222 = 0.0262 × 10 -3 C.m -1 , K 2212 = 0.0033 × 10 -3 C.m -1 and K 1211 = 0.0004 × 10 -3 C.m -1

FIG. 5 :

 5 FIG. 5: Optimal topology for K for the PZT/PVDF composite: (a) K 1111 ; (b) K 2211 ; (c) K 1212 ; (d) K 2222 ; (e) K 2212 ; (f) K 1211 .

  FIG. 6: (a) Electric field gradient component ∇E 22 and (b) strain component ε 22 within the PZT-PVDF microstructure corresponding to the optimized coefficient K 2222 .

S. Kogan, "Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals," Soviet Physics-Solid State 5(10) (1964).

IX. ACKNOWLEDGEMENTS X.C & S.Y gratefully acknowledge the financial support from the National Key R&D Program of China (No.2016YFB1200602-33), and X.C acknowledges the funding supported by the Fundamental Research Funds from the Central Universities of Central South University.

X. DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.