A multi-material level set-based topology optimization of flexoelectric composites

Accepted Manuscript A multi-material level set-based topology optimization of flexoelectric composites Hamid Ghasemi, Harold S Park, Timon Rabczuk PII: DOI: Reference: S0045-7825(17)30756-9 https://doi.org/10.1016/j.cma.2017.12.005 CMA 11697 To appear in: Comput Methods Appl Mech Engrg Received date : 31 May 2017 Revised date : 30 November 2017 Accepted date : December 2017 Please cite this article as: H Ghasemi, H.S Park, T Rabczuk, A multi-material level set-based topology optimization of flexoelectric composites, Comput Methods Appl Mech Engrg (2017), https://doi.org/10.1016/j.cma.2017.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain A multi-material level set-based topology optimization of flexoelectric composites Hamid Ghasemi2, Harold S Park3, Timon Rabczuk1, Duy Tan University, Institute of Research & Development, Quang Trung, Danang, Viet Nam Institute of Structural Mechanics, Bauhaus- Universität Weimar, Marienstraße 15, 99423 Weimar, Germany Dep of Mechanical Engineering, Boston University, Boston, MA 02215, USA Abstract We present a computational design methodology for topology optimization of multi-materialbased flexoelectric composites The methodology extends our recently proposed design methodology for a single flexoelectric material We adopt the multi-phase vector level set (LS) model which easily copes with various numbers of phases, efficiently satisfies multiple constraints and intrinsically avoids overlap or vacuum among different phases We extend the point wise density mapping technique for multi-material design and use the B-spline elements to discretize the partial differential equations (PDEs) of flexoelectricity The dependence of the objective function on the design variables is incorporated using the adjoint technique The obtained design sensitivities are used in the Hamilton–Jacobi (H-J) equation to update the LS function We provide numerical examples for two, three and four phase flexoelectric composites to demonstrate the flexibility of the model as well as the significant enhancement in electromechanical coupling coefficient that can be obtained using multi-material topology optimization for flexoelectric composites Keywords: Topology optimization, Flexoelectricity, Level set, Multi-material, B-spline elements Introduction In dielectric crystals with non-centrosymmetric crystal structure such as quartz and ZnO, electrical polarization is generated upon the application of uniform mechanical strain This property of certain materials, which is known as piezoelectricity, is caused by relative displacements between the centers of oppositely charged ions Details about the governing equations of piezoelectricity are available in [1-3] Corresponding Authors: E-Mail: hamid.ghasemi@uni-weimar.de; parkhs@bu.edu; timon.rabczuk@uni-weimar.de   When the mechanical strain is applied non-uniformly, the inversion symmetry of a dielectric unit cell can be broken locally Thus all dielectric materials, including those with centrosymmetric crystal structures, can produce an electrical polarization This phenomenon is known as the flexoelectric effect, where the gradient of mechanical strain can induce electrical polarization in a dielectric solid Readers are referred to [4, 5] and references therein for more details Micro-Nano electromechanical sensors and actuators made from piezoelectric or flexoelectric materials are increasingly used in applications such as implanted biomedical systems [6], environmental monitoring [7] and structural health monitoring [8] These sensors and actuators are structurally simpler, provide high power density, and allow a broader range of material choice; however, their efficiency is usually low [9] Conventional flexoelectric ceramics or single crystals are usually brittle and therefore susceptible to fracture In contrast, flexoelectric polymers are flexible but exhibit weaker flexoelectric performance Moreover, in a single flexoelectric structure, zones with high strain gradients contribute more to electrical energy generation Thus, the efficiency of a sensor or an actuator fabricated entirely from a single flexoelectric material might be suboptimal More interestingly, there exist significant opportunities to design piezoelectric composites without using piezoelectric constitutive materials while reaching piezoelectric performance that rivals that seen in highly piezoelectric materials [4] Therefore, there are significant opportunities in being able to design multi-phase flexoelectric composites to bridge the gap between high flexoelectric performance and poor structural properties Topology optimization is a powerful approach that determines the best material distribution within the design domain The present authors have already presented a computational framework for topology optimization of single material flexoelectric micro and nanostructures to enhance their energy conversion efficiency [10, 11] The present research however, exploits the capabilities of topology optimization for the systematic design of a multi-phase micro and nano sensors and actuators made from different active and passive materials Contributions on piezoelectric structure design are often restricted by the optimal design of the host structure with fixed piezoelectric elements [12] or optimal design of piezoelectric elements with the given structure [13, 14] Studies on multi-material design of piezoelectric structures are relatively rare In fact, available works on multi-material topology optimization mostly employ Isotropic Material with Penalization (SIMP) technique [15] Furthermore, we are not aware of any previous work studying the optimization of multi-material flexoelectric composites By use of the level set method, this work provides a new perspective on simultaneous topology optimization of the elastic, flexoelectric and void phases within the design domain such that multi-material flexoelectric composites can be designed The remainder of this paper is organized thus: Section summarizes the discretized governing equations of flexoelectricity, Section contains the topology optimization based on the LSM, Section provides numerical examples, and Section offers concluding remarks A summary of the governing equations and discretization A summary of the governing equations of the flexoelectricity is presented in this section More details are available in [10, 16-18] and references therein Accounting for the flexoelectricity, the enthalpy density, , can be written as , where , , is the fourth-order elasticity tensor, tensor of piezoelectricity, is the mechanical strain, is the electric field, direct and converse effects) flexoelectric tensor and ) and physical ( is the third-order is the fourth-order total (including both is the second-order dielectric tensor / ), higher-order ( The different stresses / electric displacements including the usual ( (1) , / / ) ones are then defined through the following relations and (2) and , , (3) , and (4) , thus , (5) , , (6) , which are the governing equations of the flexoelectricity By imposing boundary conditions and integration over the domain, Ω, the total electrical enthalpy is , Using Hamilton’s principle, we finally have Ω (7) , ̅ , (8) which is the weak form of the governing equations of the flexoelectricity In Eq (8) mechanical displacements, Ω is the is the electric potential, ̅ is the prescribed mechanical tractions is the surface charge density Γ and Γ are boundaries of Ω corresponding to mechanical and tractions and electric displacements, respectively Using B-spline basis functions, where the superscripts and , ∑ ∑ , ∑ ∑ , and , we approximate , , , , and fields as , (9.a) , (9.b) denote nodal parameters at the mesh control points, mechanical and electrical fields, respectively The discrete system of Eq (8) is eventually expressed as (10) where ∑ ∑ ∑ ∑ Ω (11.a) Ω (11.b) Ω (11.c) Ω ∑ (11.e) ∑ (11.f) In Eqs (11.a-f), the subscript, , in Ω , Γ Ω ⋃ Ω Moreover, , (11.d) and Γ denotes the finite element where contain the spatial derivatives of the B-spline basis functions The second derivatives of the basis functions, , are obtained by Eq (12) 5 0 ⋮ ⋮ ⋮ , ⋮ ⋮ , ⋮ ⋮ ⋮ ⋮ 0 0 ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋮ ⋮ ⋮ and (12) ⋮ Moreover, , , ⋮ ⋮ can be written in matrix form as 0 κ 0 0 0 denotes Poisson’s ratio and (13.a) (13.b) (13.c) where ⋮ 0 is the Young’s modulus (13.d) blem Levell Set Methood (LSM) aand optimizzation prob 3.1 LSM M Assumee Ω ⊂ ⊂ ( 3) whhere is tthe entire structural s ddomain inclluding all admissibble shapes, Ω A singlee level set fu function Φ Φ Ph hase 1: Φ oundary: Φ : Bo Ph hase 2: Φ is then defined d as ∀ ∈ Ω ∖ Ω ∀ ∈ Ω ∩ ∀ ∈ ∖ Ω as schem matically shhown in Fig (1.a) We uuse B-splinee basis functtions, , , (14) , to define Φ accordinng to Φ where , , ∑ ∑ , , , (15) , arre the numbber of basis functions inn the orthoggonal directtions and , denotes correspoonding nodal values of the LS A As shown inn Fig (1.b)), the zero iso-surface of Φ implicittly representts the designn boundary Γ The level set function is dynam mically upddated at eacch time step by solvingg the Hamiltton-Jacobi (H-J) paartial differeential equatiion | Φ| in whichh (16) is the noormal component of thee velocity veector ( is the unnit outward normal to tthe boundarry Γ The fiield ) and | | deteermines geoometric mottion of the boundarry Γ and is chosen baseed on the deesign sensitiivity of the oobjective fuunction Φ iis initiated as a siggned distancce function and the abbove H-J eqquation is ssolved by aan explicit first-order upwind scheme [199] Fig.1 L Level set fuunction (a), boundary representati r ion with levvel set (b) and densityy mapping techniquue (c) t whole m material dom main into Convenntional partiitioning of the pphases, ,…, , (including the LS functionns where eaach one reppresents a diistinct mateerial phase void phhase) using [20] inttroduces a rrange of coomputationaal challenges: 1) numerrical difficuulties to maaintain the “partitioon conditionns” D ⋃ Ø , ⋂ annd and 2) complexity aassociated with a hhigh numbeer of level sset functionns To remove these shhortcomings, we follow w [21] and adopt thhe vector LS S approach [22] where a number oof domain into n level sset functionn partitions the t design o overlapping regions, combinaations of thhe zero-leveel sets Ω 1, … , zero-levvel sets of tthese functiions Ω :Φ 1, … , , obbtained by different ) In this schheme the innterior regioons of the can ooverlap Thuus, each pooint ∈ belongs to one andd only one material phhase which essentially satisfies thee partition cconditions [21] We willl focus on exxamining fllexoelectric compositess using up too two level set functionns Fig (2) illustratees four matterial phasees defined bby two levell-set functioons Φ andd Φ For thhe case of three phhases (incluuding the vooid phase), Φ determinnes the soliid and the vvoid phases while Φ distinguuishes differrent solid maaterial phases Fig.2 F Four materiaal phases , , , are repressented by tw wo level-set functions Φ and Φ Φ Φ Φ … Φ We connsider the veector level-sset function Φ Φ … Φ where Φ aand the vectoor Heavisidde function iss a smooth approximattion of the Heaviside functionn defined byy 0 fo or Φ Φ for Δ 1 fo or Δ where Δ is the widtth of numeriical approxiimation Δ Δ Φ Δ Φ (17) Using point-wise mapping to control an element-wise constant phase density distribution (as represented in Fig (1.c) for a single material phase), we define Φ for two phases: 2 Φ (18.a) , Φ Φ for three phases: Φ Φ Φ Φ Φ (18.b) and Φ where Φ for four phases: Φ Φ Φ and Φ Φ 2 (18.c) is the center of a finite element These element densities are embedded in the electromechanical problem to obtain effective material properties where Eqs (13.a-d) define materials and , ∑ for two phases (19.a) , ∑ for three phases (19.b) , ∑ for four phases (19.c) , , , Superscript represents properties of the bulk for the void phase contain appropriately small values to avoid singularity of the stiffness matrix Assuming where 1, … , , the volume integrals of some functional over a material domain can then be defined as (20) where is a matrix containing all vectors of 1, … , the LS function Φ ) contains related design variables, points Each vector , (associated with defined on the mesh of control 3.2 Optimization problem , is defined as The electromechanical coupling coefficient, (21) where and are the electrical and mechanical (or strain) energies, respectively By extending and in Eq (21) and defining the objective function, the inverse of we have , where , and , , , as (22) Eventually, in its general form the optimization problem can be summarized as Eq (23) and Table-1: Minimize: , , Subjected to: Ω 1, … , (23) Table-1 Summary of the optimization problem Inputs Initial nodal values of the level set functions, Design variables Nodal values of the level set functions, Design constraints Volume of the material phases, outputs Optimum distributions of material phases, where Material properties , Solver settings & parameters , 1, … , where System of coupled governing equations , is the total volume of the material phase in each optimization iteration and is the corresponding given volume To satisfy the volume constraints, we use the augmented Lagrangian method combining the properties of the Lagrangian (the second term in Eq (24.a)) and the quadratic penalty functions (the third term in Eq (24.a)) It seeks the solution by replacing the original constrained problem by a sequence of unconstrained sub-problems through estimating explicit Lagrangian multipliers 10 at each step to avoid the ill-conditioning that is inherent in the quadratic penalty function (see [23] for more details) Following [23], we define ∑ and Λ are parameters in iteration which are updated according to the following scheme , Λ where then (24.a) ∈ 0,1 is a fixed parameter Λ (24.b) and Λ start with appropriately chosen initial values; that approximately minimizes will be found and Λ are subsequently updated and the process is repeated until the solution converges The classical Lagrangian objective function is obtained by discarding the last term of Eq (24.a) The normal velocity in Eq (16) is chosen as a descent direction for the Lagrangian according to 1, … , (25) where different terms of Eq (25) are derived in Appendix A The flowchart of the entire optimization process is presented in Fig.3 11 Fig.3 The flowchhart of the ooptimizationn process Numerical exam mples We perfform a suite of examplees of multi-m material beam ms with lineear elastic m material propperties and under 2D D plane straain conditionns In all folllowing exaamples, we aassume a 60 beam diiscretized byy 48 15 cantilever 12 qquadratic B-spline elem ments, unlesss otherwisee specified The beam is subjected to a doownward pooint load off 100 att the top of the free edgge while oppen circuit electricaal boundaryy conditions are imposeed as shownn in Fig (4.aa) We investigate two, three and four phaase composiite beams A All models aare discretizzed by quaddratic B-spline elementss (see Fig (4.b)) w where red doots representt control poiints (see [100] for more ddetails) 12 Fig.4 Loadinng and bounndary condittions (a), discretizationn (b) material propperties of thee active (pieezoelectric oor flexoelectric), passivve (elastic) Table-2 includes m An active nnon-piezoeleectric materrial experiennces pure fl flexoelectriccity and is and voiid phases A obtainedd by setting Tab ble-2 Properrties of activve ( [16]), acttive 2, activve 3, passivee and void pphases Y P Phase / color μ /μ A Active / blue 0.37 10 00 4.4 / A Active / red 0.37 50 2.2 / 0.5 Acctive / yellow w 0.37 00 10 4.4 / Paassive / green 0.37 10 V Void / white 0.37 1 : μ /μ / : , : , 11 / 12.48 / 5.5 / 6.24 / 11 / 12.48 / 0 0.02 0.02 / 0 0.0 0089 , / : / / : / / 0.0089 / , wo phase com mposite 4.1 Tw In this section, we w assume the beam m is made from: the non-piezooelectric (i.e setting 0) Active an nd the passiive elastic pphases (Casee-1), and the non-piezooelectric Active and Case-2) accoording to T Table-2 For both casess, the electroomechanical coupling Active phases (C coefficieent, , is m measured foor various coompositions of constituuent phases, while the nnormalized electrom mechanical coupling ccoefficient ( ) is obbtained by normalizing the casees by the electrom mechanical ccoupling coeefficient of tthe beam w with 100% Active A matterial Fig (55) belongs to the Case-1 and it is observabble that, by combining c thhe passive aand the activve phases a higher than thee single-phaase counterppart can be obtained; hhowever, theere is a poinnt where thhe result is 13 material on the one hhand increaases optimal In fact, more soft passive m subsequuently decreaases , which , butt on the othher hand, it produces hiigher strain and strain gradients, which gives g rise too higher and Thus, in the t optimal material coombination there is a tradeofff between thhese two connflicting efffects Fig.5 versus voluume fractionn of Active for Case-1 w the opttimized topoologies are ppresented as well One Fig (6) represents tthe results foor Case-2 while can obseerve that anyy combinatiion of the Active and Active leaad to the higgher thann either the single-pphase Activee or Activee counterpparts Fig.6 andd versus vvolume fracction of Actiive for Casse-2 4.2 Thrree phase ccomposite Let us aassume the nnon-piezoellectric activee, passive aand void phaases (Activee 1, Passive and Void in Tablle-2) Fig (7.a-e) incclude the optimal toopologies A As mentionned in Tabble-2, the flexoeleectric phase ( ) is show wn in blue, elastic ( ) in green annd void ( ) in white coolors The solid phhase in Fig (7.a) only iincludes thee flexoelecttric phase (zzero elastic phase); whhile in Fig 14 (7.b) Ω 0.3 Ω 0.56 , are connsidered as Ω volume cconstraints 0.14 , and annd accordinng to Eq (118) We wriite these connstraints in compact foorm as 0.56 , 0.14 , 0.3 in which w 0.42 , 0.28 , 0.3 in Figg (7.c), , (7.d) annd , We also set , , 0.14 4 , 0.56 , 0.3 are calculated , , , , 0.28 , 0.4 42 , 0.3 in F Fig (7.e) 0.7 Fig.7 The T optimaal topologies for the flexoelectrric beam cconsidering 0.56 (b b), In aall insets 0.42 (c)), 0.28 in Fig (d) and 0.14 (a), (e) where 0.3 The flexooelectric phhase is show wn in blue, elastic in green annd void in white w colors Browsinng Fig (7) ffrom the topp towards thhe bottom, oone can visuually find thhat the elasttic (green) phase inncreases, the flexoelecttric (blue) phase p decreaases and thee void (whiite) remainss constant Furtherm more, becauuse of the laarger strain gradients around a the pperimeter, thhe flexoelecctric phase 15 ( of the beaam, whereaas the elastiic material is in the concenttrates on thhe outside (perimeter) interior A rigorrous scrutinny of the vvolume connstraints fulffillment as well as thhe objectivee function minimizzation is prresented in Fig (8) The graphs belong b to F Fig (7.e) annd illustratee how the volumess and the objective o funnction convverge precissely and sm moothly tow wards the sppecified or minimuum values , Fig.8 P Phases voluumes and objective o fuunction verssus iteration for 0.14 , 0.56 , 0.3 , (correspoonding to Fiig.7(e)) To find how the elaastic phase impacts thee efficiency of the device, of Fig (7) ( and the normalized results, is meeasured for each inset , (by the sollid beam wiith 100% fleexoelectric pphase) are presenteed in Fig (99.a) The vooid phase is constant (0 0.3 ) in all cases and the t solid maaterial can have diffferent combbinations off the flexoellectric and eelastic phasees For the bulk elasticc structure, is zeero since theere is no acttive materiaal When 14% flexoelecctric phase iis added 0.000 022 ( 5.57) and ffor 28% flexxoelectric, is 0.00 0037 ( by increeasing the flexoelectriic phase to 0.42% nott only doess decreasees to the vallue of further reduction in i 0.0 00033 ( i.e becomes 9.14) Inteerestingly, not inncrease but it instead 8.1) Furtther increasiing the flexooelectric phhase yields 0.0001 16 ( 3.98) for thee flexoelecttric device with 70% flexoeleectric and 300% void phaases We repeeat the probblem by meaasuring o the beam with the same length aand the aspeect ratio of of The similar trennd is observ rved as shown in Fig (9.b) We observe thhat by combbining the 16 a phasees a higherr electromechanical cooefficient thhan the sinngle-phase passive and the active counterppart can be obtained Fig.9 T The normalized electrom mechanical coupling cooefficient, , versus volume v fracttion of the flexoeleectric phase for the beam m with aspeect ratio of (a) and (b) For all cases, the void v phase is kept cconstant as 0.3 The leength of the beam is 60 wheree Fig (100) includess the optim mal topologies for diff fferent beam m aspect raatios of 4, and considerring , , 0.28 , 0.42 ,, 0.3 The resuults are preesented in Table-3 For compparing resullts, it shoulld be notedd that the fflexoelectricc size effecct and the volume ratio of thee flexoelectrric material are contraddictory The former causes the highhest for the beam m with the aspect ratioo of thouggh there is lless active m material to generate g eleectricity in for the beam with the aspect compariison with thhe smaller asspect ratio bbeams The llatter makess ratio of be smallerr than whhen the beam m aspect rattio is It is obvious thaat for the solid beams, larger asspect ratio leeads to largeer Table-33 (see [110]) and for differennt beam asppect ratios Aspectt ratio 2.52 10 1.6 68 10 1.26 10 00037 0.0 0.00029 0.00072 99.14 6.73 13.63 17 mal topologiees of the bbeam with aaspect ratioos (a), (b) and (c) In all Fig.10 The optim examplees: 60 0.28 0.3 annd wheree The leength of the beam is 4.3 Fou ur phase coomposite Here, we w consider tthe beam made m from foour phases, as presenteed in Table-22, through ttwo cases: Active and Activve phases are considdered as nonn-piezoelecttric ( materialss (Case-1) 0) and Acttive as a nnon-piezoeleectric materrial and Acttive as a ppure piezoelectric ( / 0) materiall without anny flexoeleectric properties (Case 2) In bothh cases, theere are also void and elastic pphases andd , , , 0.21, 0.28, 0.21, 0.3 aare set as volume constraints Fig (111) and Fig (12) show optimal toppologies forr Case-1 andd Case-2, reespectively For each case thee history of the t objectivve function aand volume constraints are presentted separatelly 18 Active (red), elastic Fig.11 The optimaal topologyy for Case-11 composedd of Active (blue), A (green) and hole ((white) phaases wherre 60 , , , 0.21, 0.28, 0.21, 0.3 are seet as four equality e dessign constraaints The leength of the beam is aand its aspect ratio is Fig.12 The optimaal topology for Case-2 ccomposed of o Active (blue), Actiive (yellow w), elastic (green) and hole ((white) phaases wherre 60 , , , 0.21, 0.28, 0.21, 0.3 are seet as four equality e dessign constraaints The leength of the beam is aand its aspect ratio is Conccluding rem marks The B-sspline elem ments whichh were succcessfully im mplemented to model fflexoelectricc effect in dielectriic materials are combinned with thee vector levvel set technnique, with tthe goal of eenhancing 19 the electromechanical performance of multi-phase micro and nano sensors and actuators made from different active (flexoelectric and piezoelectric) and passive (elastic) materials The numerical examples show the capabilities of the model to design two, three and four phase micro sensors with the optimal electromechanical coupling coefficient defined by where and are the electrical and mechanical energies, respectively For the two phase composite made from the active and passive phases, our results show that at the optimal volume fractions of constituents, the normalized electromechanical coupling coefficient ( ) is 2.5 times larger than what that obtained from a beam made purely from the active material For the three phase composite case (made from active and passive materials as well as holes), is increased by a factor of The results demonstrate the competing effects of increasing volume fraction of the soft passive material in the composite, which on the one hand decreases and on the other hand, increases by increasing by increasing , by producing higher strain gradients Thus, in the optimal materials combination there is a tradeoff between these two competing effects Future work will focus on studies on numerical stability, updating procedure, geometry mapping and regularization One crucial aspect of the method is the determination of the Lagrange multipliers to minimize the objective function while the multiple equality volume constraints are also precisely fulfilled It is possible that an optimality criteria method would better treat this kind of constraint by means of the move limit and the damping factor; however, the LS function is susceptible to becoming too flat or too steep, both of which may give rise to convergence issues Acknowledgments: Hamid Ghasemi and Timon Rabczuk gratefully acknowledge the financial support by European Research Council for COMBAT project (Grant number 615132) Harold Park acknowledges the support of the Mechanical Engineering department at Boston University Appendix A: Sensitivity analysis The coupled system of equations in a single global residual form is expressed as ∗ ∗ , , , (A1) 20 where ∗ and ∗ are residuals that must be simultaneously satisfied; where and are solution (i.e displacement and electric potential) fields The objective function then takes the , form constraints, , We calculate the sensitivity of the objective function, and , in Eq (25) with respect to and , and volume Using the chain-rule we have 1, … , where (A2) and the term inside the brackets is obtained by differentiating Eq (A1) as (A3) into the definition of By substituting we obtain Ω Ω (A4) Having obtained , one can write (A5) where ∑ Ω (A6) ∑ Ω ∑ (A7) Ω ∑ (A8) Ω (A9) and ∑ where with 1, … , is the number of level set functions and , and , , and , , (A10) are obtained according to Eq (19) One can also obtain the last term of Eq (A2) as Ω For the case of four material phases, for 1, and Ω 1, 2, 3, is obtained by (A11) 21 ∑ Ω ∑ Ω ∑ Ω (A12.b) Φ Φ (A12.a) Φ Φ ∑ Ω Φ Φ (A12.c) Φ Φ (A12.d) and for the case of three material phases, one can write Ω where Φ ∑ Ω ∑ Ω Φ (A13.a) Φ Φ (A13.b) Φ Φ (A13.c) is the approximate Dirac delta function defined as Φ for Δ Φ Δ (A14) otherwise and is calculated by , , , (A15) References: 1S Nanthakumar, T Lahmer, X Zhuang, G Zi, T Rabczuk, Detection of material interfaces using a regularized level set method in piezoelectric structures, Inverse Problems in Science and Engineering 24(1) (2016) 153–176 2S S Nanthakumar, T Lahmer, T Rabczuk, Detection of multiple flaws in piezoelectric structures using XFEM and Level sets, Computer Methods in Applied Mechanics and Engineering 275 (2014) 98-112 3S S Nanthakumar, T Lahmer, T Rabczuk, Detection of flaws in piezoelectric structures using XFEM, International Journal of Numerical Methods in Engineering 96(6) (2013) 373–389 4N.D Sharma, R Maranganti, P Sharma, On the possibility of piezoelectric nanocomposites without using piezoelectric materials, Journal of the Mechanics and Physics of Solids 55 (2007) 2328–2350 5P V Yudin, A K Tagantsev, TOPICAL REVIEW: Fundamentals of flexoelectricity in solids, Nanotechnology 24 (2013) 432001 (36pp), doi:10.1088/09574484/24/43/432001 22 6H Cao, V Leung, C Chow, H Chan, Enabling technologies for wireless body area networks: A survey and outlook, IEEE Communications Magazine 47(12) (2009) 84-93 7D Sanders, Environmental sensors and networks of sensors, Sensor Review 28 (2008) 273-274 88- S Kim, S Pakzad, D Culler, J Demmel, G Fenves, S Glaser, M Turon, Health monitoring of civil infrastructures using wireless sensor networks, Proceedings of the Sixth International Symposium on Information Processing in Sensor Networks (2007) 254 –263 9Z L Wang, W Wu, Nanotechnology-enabled energy harvesting for self-powered micro-/nanosystems, Angewandte Chemie International Edition 51 (2012) 11641–11903 10H Ghasemi, H S Park, T Rabczuk, A level-set based IGA formulation for topology optimization of flexoelectric materials, Comput Methods Appl Mech Engrg 313 (2017) 239-258 11S.S Nanthakumar, X Zhuang, H.S Park, T Rabczuk, Topology optimization of flexoelectric structures, Journal of the Mechanics and Physics of Solids, 105 (2017) 217234 12E Silva, S Nishiwaki, N Kikuchi, Topology optimization design of flextensional actuators, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 47 (2000) 657–671 13Q Nguyen, L Tong, Voltage and evolutionary piezoelectric actuator design optimization for static shape control of smart plate structures, Materials and Design 28 (2007) 387–399 14A Takezawa, M Kitamura, S L Vatanabe, Design methodology of piezoelectric energy-harvesting skin using topology optimization, Struct Multidisc Optim 49 (2014) 281-297 15GIN Rozvany, M Zhou, T Birker, Generalized shape optimization without homogenization, Struct Multidisc Optim (1992) 250–254 16Amir Abdollahi, Christian Peco, Daniel Millan, Marino Arroyo, Irene Arias, Computational evaluation of the flexoelectric effect in dielectric solids, JOURNAL OF APPLIED PHYSICS 116 (2014) 093502 17M S Majdoub, P Sharma, T Cagin, Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, PHYSICAL REVIEW B 79 (2009) 119904 (E) 18S Shen, S Hu, A theory of flexoelectricity with surface effect for elastic dielectrics, J of the mechanics and physics of solids 58 (2010) 665-677 19Michael Yu Wang, Xiaoming Wang, Dongming Guo, A level set method for structural topology optimization, Comput Methods Appl Mech Engrg 192 (2003) 227246 20H K Zhao, B Merriman, S Osher, L Wang, Capturing the behavior of bubbles and drops using the variational level set approach, J Comput Phys 143 (1998) 495–518 21M Wang, X Wang, ‘‘Color’’ level sets: a multi-phase method for structural topology optimization with multiple materials, Comput Methods Appl Mech Engrg 193 (2004) 469–496 22L.A Vese, T.F Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int J Comput Vision 50 (3) (2002) 271–293 23 23J Luo, Z Luo, L Chen, L Tong, MY Wang, A semi implicit level set method for structural shape and topology optimization, J Comput Phys 227 (2008) 5561-5581  ... Voltage and evolutionary piezoelectric actuator design optimization for static shape control of smart plate structures, Materials and Design 28 (2007) 387–399 1 4A Takezawa, M Kitamura, S L Vatanabe,... research however, exploits the capabilities of topology optimization for the systematic design of a multi-phase micro and nano sensors and actuators made from different active and passive materials.. .A multi-material level set-based topology optimization of flexoelectric composites Hamid Ghasemi2, Harold S Park3, Timon Rabczuk1, Duy Tan University, Institute of Research & Development, Quang