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

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A multi-material level set-based topology optimization of flexoelectric
composites
Hamid Ghasemi, Harold S. Park, Timon Rabczuk

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Received date : 31 May 2017
Revised date : 30 November 2017
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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),
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A multi-material level set-based topology optimization of flexoelectric composites
Hamid Ghasemi2, Harold S. Park3, Timon Rabczuk1, 2
1

Duy Tan University, Institute of Research & Development, 3 Quang Trung, Danang, Viet Nam
Institute of Structural Mechanics, Bauhaus- Universität Weimar, Marienstraße 15, 99423 Weimar, Germany
3
Dep. of Mechanical Engineering, Boston University, Boston, MA 02215, USA
2

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

1. 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: ; ;
 


2
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


3
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 2 summarizes the discretized governing
equations of flexoelectricity, Section 3 contains the topology optimization based on the LSM,
Section 4 provides numerical examples, and Section 5 offers concluding remarks.

2. 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)


4

,

̅

,

0

(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

⋮

⋮

⋮

0

0

0

0

⋮

⋮

⋮

⋮

0

0

0

0

0

0

⋮

⋮

⋮

and

(12)

⋮

0
Moreover, , ,

⋮

⋮

0

can be written in matrix form as
1

0
0

1
0
κ
0

0

0

0

0

0
0
denotes Poisson’s ratio and

0

(13.a)

(13.b)
(13.c)

0

where

⋮

0 0
0

is the Young’s modulus.

(13.d)

6
blem
3. Levell Set Methood (LSM) aand optimizzation prob
3.1. LSM
M
Assumee Ω ⊂

⊂

(

2

3) whhere

is tthe entire structural
s
ddomain inclluding all

admissibble shapes, Ω . A singlee level set fu
function Φ
Φ

Ph
hase1:Φ
oundary:Φ
: Bo
Ph
hase2:Φ

is then defined
d
as

0 ∀ ∈ Ω ∖ Ω
0 ∀ ∈ Ω ∩
0 ∀ ∈ ∖ 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

.

0

(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)


7
t whole m

material dom
main into
Convenntional partiitioning of the

pphases,

,…,

, (including the

1 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

2

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
0

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
0 fo
orΦ
Φ



for

Δ

1
1 fo
orΔ
where Δ is the widtth of numeriical approxiimation.

Δ

Δ
Φ

Δ

Φ

(17)


8
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:

1



1



2

1

Φ

(18.a)

,


Φ
Φ

for three phases:

Φ


1

Φ
1

Φ

1





Φ

Φ



1

(18.b)
and
Φ

where 0

1

Φ

for four phases:

Φ

1

Φ



1

Φ

1 and

1



Φ
1


Φ


2

3

(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 0 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


9

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:
,
,
Subjectedto:


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
4. 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
0
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
0.

obtainedd by setting

Tab
ble-2 Properrties of activve 1 (

[16]), acttive 2, activve 3, passivee and void pphases

Y

P
Phase / color

μ /μ

A
Active 1 / blue

0.37

10
00

4.4 /

1

A
Active 2 / red

0.37

5
50

2.2 /

0.5

Acctive 3 / yellow
w

0.37

00
10

4.4 /

Paassive / green

0.37

1
10

V
Void
/ white

0.37

1
1

:
μ /μ
/ :



,

:



,

11

/

12.48

/

5
5.5

/

6.24

/

0

11

/

12.48

/

0

0

0
0.02

0.02

/

0

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 1 an
nd the passiive elastic pphases (Casee-1), and the non-piezooelectric Active 1 and

Case-2) accoording to T
Table-2. For both casess, the electroomechanical coupling
Active 2 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
1 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 1 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 1 and Active 2 leaad to the higgher

thann either the

single-pphase Activee 1 or Activee 2 counterpparts.

Fig.6.

andd

versus vvolume fracction of Actiive 1 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
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
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

6. 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 4 (a) and 6 (b). For all cases, the void
v
phase

is kept cconstant as 0.3
0

. The leength of the beam is 60

wheree

.

Fig. (100) includess the optim
mal topologies for diff
fferent beam
m aspect raatios of 4, 6 and 8
considerring

,

,

0
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 8 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 6 be smallerr than

whhen the beam
m aspect rattio is 4. 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
2.52

10

1.6
68

10

1.26

10

00037
0.0

0.00029

0
0.00072

99.14

6.73

13.63


17

mal topologiees of the bbeam with aaspect ratioos 4 (a), 6 (b) and 8 (c). In all
Fig.10. The optim
examplees:
60

0.28

0

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 1 and Activve 2 phases are considdered as nonn-piezoelecttric (

0 materialss (Case-1)
0)

and Acttive 1 as a nnon-piezoeleectric materrial and Acttive 3 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 2 (red), elastic
Fig.11. The optimaal topologyy for Case-11 composedd of Active 1 (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 4..

Fig.12. The optimaal topology for Case-2 ccomposed of
o Active 1 (blue), Actiive 3 (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 4..

5. 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 9. 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, 2 and

Ω
1, 2, 3, 4 is obtained by

(A11)


21


∑

Ω



∑

Ω




∑

Ω

1

(A12.b)

Φ

Φ

1

(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

1

for

Δ

Φ

Δ

(A14)

0otherwise
and

is calculated by


,
,

,

(A15)

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