TECHNICAL ARTICLE

Design of Architected Materials for ThermoelasticMacrostructures Using Level Set Method

LEI LI ,1,4 ZONGLIANG DU,1,2 and H. ALICIA KIM1,3

1.—Structural Engineering Department, University of California, San Diego, 9500 Gilman Drive,San Diego, CA 92093, USA. 2.—Department of Engineering Mechanics, Dalian University ofTechnology, Dalian 116023, Liaoning, P.R. China. 3.—Cardiff School of Engineering, CardiffUniversity, The Queen’s Buildings, 14-17 The Parade, Cardiff CF24 3AA, UK. 4.—e-mail:lel008@eng.ucsd.edu

A level set topology optimization method is introduced and used to designperiodic architected materials optimized for the maximum macrostructuralstiffness considering thermoelasticity. The design variables are defined at themicroscopic scale and updated by minimizing the total structural complianceinduced by mechanical and thermal expansion loads at the macroscopic scale.The two scales are coupled by the effective elasticity tensor calculated throughthe homogenization theory. A decomposition method is constructed to for-mulate several subproblems from the original optimization problem, enablingthe efficient solution of this otherwise computationally expensive problem,especially when the number of material subdomains is large. The proposedmethod is demonstrated through several numerical examples. It is shown thatthe macrostructural geometry and boundary conditions have a significantimpact on the optimized material designs when thermoelastic effects areconsidered. Porous material with well-designed microstructure is preferredover solid material when the thermal load is nonzero. Moreover, when a largernumber of material microstructures is allowed in optimization, the overallperformance is improved due to the expanded design space.

INTRODUCTION

Cellular materials composed of periodicallyrepeated porous unit cells with special microscalearchitectures have shown significant potential foruse in various engineering applications1,2 because oftheir tailorable properties, such as high stiffness-to-density ratio, high permeability, low thermal con-ductivity, and other exotic properties that areusually unattained at bulk scale.3,4 The propertiesof these architected materials not only rely on thebasic material constituents, but also stronglydepend on the spatial arrangement of void andsolids—that is, the configuration of the cellularmicroarchitecture. Thanks to the recently emergingadditive manufacturing (AM) technologies withimproving feature control at small scales, greatopportunities have opened up to fabricate archi-tected materials with unprecedentedly complexgeometries and features.5,6 As a result, purposefuldesign of materials with tailored microstructures to

achieve desired properties for specific applicationsat macroscale is attracting ever-increasingattention.

Before AM became an active research field, designof architected materials using topology optimizationhad already flourished, with the first attempt beingmade by Sigmund7, in which the homogenizedmaterial effective properties were optimized to thedesired values. Based on the idea of inverse homog-enization proposed in Ref. 7, topology optimizationhas been applied for architected material designs,aiming to (a) achieve extremal material properties,such as maximum bulk or shear moduli,8–12 mini-mum negative Poisson’s ratio,10 extremal thermalexpansion,13 maximum bandgaps14 and energy dis-sipation capacity;15 or (b) tailor material propertiesto target values, e.g., effective Young’s moduli,16

Poisson’s ratio;17–19 or (c) realize multifunctionality,such as simultaneous heat and electricity trans-portation,20 permeability and stiffness,21 conductiv-ity and stiffness,22 and viscoelastic damping and

JOM, Vol. 72, No. 4, 2020

https://doi.org/10.1007/s11837-020-04046-2� 2020 The Minerals, Metals & Materials Society

1734 (Published online February 10, 2020)

stiffness maximizations.23 Readers are referred toRefs. 24 and 25 and references therein for morecomprehensive reviews on this topic.

Topology optimization is considered ideal for AMdesign due to its freeform formulation, expandeddesign space, and flexibility.26 The first AM oftopologically optimized material can be traced backto Sigmund et al.,27 in which a microstructure withthe maximum piezoelectric charge coefficient wasfabricated by stereolithography. Later, Hollister28

used selective laser sintering to build an optimizedlattice scaffold for tissue engineering. With theadvances in AM, more topology-optimized archi-tected materials have been realized using variousprocessing techniques in recent years, For example,negative Poisson’s ratio by electron beam melting ofmetal,29 selective laser melting of nylon,30 anddirect ink writing of silicone;31 tunable thermalexpansion by AM of multimaterial photopoly-mer;32,33 stiffness by selective laser melting of Ti-Al6-V4 alloy;34,35 and high strength and low stiff-ness by electron beam melting of Co-Cr-Mo alloy.36

Despite the success of the abovementioned topol-ogy optimization studies in architected materialdesigns, it should be noted that, although thematerial microstructures are optimized to achievethe extremal or desirable material properties, theyare not necessarily efficient or optimal for use intheir composing macrostructure, since the macro-scopic performance was not considered in the mate-rial design process in those studies. The desiredmaterial properties required to attain the bestmacroscopic functional performance when used ina structure are usually unknown, not to mentionthat the macrostructural geometry and boundaryconditions vary in practical applications and cansignificantly affect the optimized material design.37

To address this deficiency of decoupled materialdesign at only the microscale, a hierarchical methodto optimize the material microstructure andmacrostructure simultaneously was proposed byRodrigues et al.38 and Coelho et al.39 using one,unified volume constraint. Later, Liu et al.40 pro-posed a porous anisotropic material with penaliza-tion (PAMP) method for concurrent material andstructural design with a single microstructuralpattern and two separate volume constraints forboth scales. Xia and Breitkopf41 used the FE2

approach for nonlinear multiscale analysis to closelycouple the microstructural properties to the macro-scopic strain field at each optimization iteration,and optimized the designs at both macro- andmicroscale using separate volume constraints.These multiscale design studies either considereduniform or a few40,42 material microstructures,which may not fully explore the design space, orelementwise material microstructures,39,41 whichmay be numerically expensive, especially when thenumber of macroelements is large. Sivapuramet al.43 introduced a generalized problem formula-tion to simultaneously design the structure and any

number of material microstructures with an arbi-trary number or type of constraints. In this formu-lation, the integrated multiscale optimizationproblem is decomposed into one macroscale opti-mization and multiple material microscale opti-mizations, which can then be efficiently solved bydistributed computing. Similar decomposition ideaswere adopted in more recent multiscale designstudies using the parameterized level setmethod.44,45 Based on these approaches, variousmultiscale topology optimization studies have beencarried out for both linear46–52 and nonlinear53–55

material properties.Note that a majority of the studies on multiscale

topology optimization have focused on problemswith only mechanical loads. However, in aerospaceengineering and advance manufacturing, manystructures are subjected to both mechanical andthermal loads simultaneously, e.g., the thermalprotection system of a space shuttle, supportingstructures for combustion of jet engines, and addi-tively manufactured parts with repeated materialheating and cooling. Therefore, how to designoptimum architected material for light weight, highthermal stiffness, and superior thermomechanicalproperties for these applications has become anactive research topic. With the consideration ofthermoelasticity, several topology optimizationstudies have been conducted56–60 to design struc-tures at macroscale, revealing that the thermalloads have a great impact on the optimized designand the volume fraction constraint can be inactivewhen the thermoelastic effect becomes prominent.Coupling the macroscopic thermal elastic propertieswith the microscopic material distribution, Denget al.61 and Yan et al.62 recently showed that, underboth mechanical and thermal loads, structurescomposed of porous materials with well-designedmicrostructures can exhibit substantially enhancedstructural stiffness. In these studies, however, onlya single material microstructure was considered inthe optimization, with limited macrostructuralgeometries, which might not have explored the fullpotential of the architected material design forthermoelastic effects.

In this study, a design method for architectedmaterials considering thermoelasticity is proposedbased on the multiscale level set topology optimiza-tion framework introduced by Sivapuram et al.43

The optimization problem is formulated in such away that the design variables are optimized at themicroscale by maximizing the total structural stiff-ness at the macroscale under the specified mechan-ical and thermal loading. The homogenizationmethod is then used to compute the materialeffective properties that bridge the two scales. Adecomposition approach is employed to efficientlysolve for multiple materials at various macroscopicregions simultaneously in a distributed but coupledmanner. Finally, numerical examples are presentedto obtain a new thermoelastic metamaterial. The

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method 1735

remainder of this manuscript is organized as fol-lows: ‘‘Formulation of Optimization Problem withThermoelasticity’’ section describes the problemformulation for material design with thermoelastic-ity. ‘‘Level Set Topology Optimization Method’’section discusses the multiscale level set topologyoptimization for solving the problem in ‘‘Formula-tion of Optimization Problem with Thermoelastic-ity’’ section, along with the sensitivity analysis.‘‘Numerical Examples’’ section presents a number ofoptimized microstructure topologies obtained usingthe proposed approach with corresponding analyses,followed by the conclusions in ‘‘Conclusion’’ section.

FORMULATION OF OPTIMIZATIONPROBLEM WITH THERMOELASTICITY

A schematic illustration of the architected mate-rial design for a macrostructure is shown in Fig. 1,

where the macrostructural domain X 2 Rd (d ¼ 2 or3) is composed of N subdomains (X ¼ [N

I¼1 XI withN = 2 in this figure). Each subdomain has a uniqueunderlying material architecture represented by theperiodically repeated unit cell YI with microstruc-ture pattern XYI

. The boundary of the macrostruc-ture (C ¼ @X) is partitioned into two disjointsegments such that C ¼ Cu [ Ct and Cu \ Ct ¼ ;.The macroscopic domain X is subjected to bodyforces b and temperature change DT, surface trac-tions �t are imposed on Ct, while the displacementsare fixed on Cu. The boundary value problem for thethermoelastic structure is given as

r � rþ b ¼ 0 in Xr � n ¼ �t on Ct

u ¼ 0 on Cu

8<

:ð1Þ

where u is the macroscale displacement field and n isthe outward normal to the structural boundary. rdenotes the stress tensor, which reads

r ¼ DH : eðuÞ � et� �

; ð2Þ

where eðuÞ ¼ rsu is the strain tensor with rsudenoting the symmetric gradient operator, et is thethermal strain, defined as et ¼ aHTDTe

0 for the plane-

stress case, where aHT is the effective thermalexpansion coefficient, e0

ij ¼ dij with i; j ¼ 1; 2 for the

two-dimensional (2D) case and DT ¼ T � Tref is theuniform temperature change in the structure with

respect to the reference temperature Tref ¼ 0. DH isthe effective elasticity tensor that correlates themacroscale and microscale.

The weak form of Eq. 1 can be expressed as

aðu; vÞ ¼ �aðDT; vÞ þ ‘ðvÞ; 8v 2 Uad; ð3Þwhere v is the macroscale virtual displacement thatsatisfies the kinematically admissible displacementUad ¼ v 2 H1ðXÞjv ¼ 0 on Cu

� �and

aðu; vÞ ¼ r

[NI¼1

XI

eðuÞ : DHI : eðvÞdX;

�aðDT; vÞ ¼ r

[NI¼1

XI

et : DHI : eðvÞdX

¼ r

[NI¼1

XI

aHTDTe0 : DH

I : eðvÞdX;

‘ðvÞ ¼ r

[NI¼1

XI

b � vdXþ rCt

�t � vdC:

ð4Þ

Assuming that the material unit cell is suffi-ciently smaller than the macrostructure, DH

I forunit cell YI can be calculated using the asymptotichomogenization theory63 as

DHI;ijkl ¼

1

YIj j rYI

Dpqrs �eðijÞpq � e�ðijÞpq

� ��eðklÞrs � e�ðklÞrs

� �dY;

i; j; k; l ¼ 1; 2; . . . ;dð Þ;ð5Þ

where DHI;ijkl and Dpqrs are the components of the

homogenized elasticity tensor DHI and the base

material elasticity tensor D, respectively, �eðijÞ isthe unit test strain tensor, and e�ðijÞ is the charac-teristic strain tensor with Y-periodicity that can bedetermined by solving the canonical equations inthe unit cell as

rYI

Dpqrs�eðijÞpq ers vðklÞ

� �dY ¼ r

YI

Dpqrse�pq vðijÞ� �

ers vðklÞ� �

dY;

8vðklÞ 2 Uv;

ð6Þ

where v denotes the characteristic displacementfields in the unit cell and v is the virtual displace-ment field satisfying the kinematically admissibledisplacement space Uv ¼ vjv is Y-periodicf g:

It has been proven mathematically that, for porousmaterial with a single base material, aHT ¼ aT, whereaT is the thermal expansion coefficient for the basematerial.61 Thus, we do not distinguish them in thisstudy. To allow the use of a fixed and uniform meshthroughout the optimization, the ersatz materialinterpolation64 is used in this work for an efficientapproximation of elements that are cut by the level set

Fig. 1. A macrostructure composed of architected materials.

Li, Du, and Kim1736

boundary, as this approach has been demonstrated tobe effective.43,65,66 As the elemental Young’s modulusis interpolated by the element area fraction, the sameaT is used for all the elements to avoid overpenaliza-tion of thermal stress in the cut elements.

The design objective is to find the optimummicrostructure for each subdomain such that theresulting macrostructure has the minimum struc-tural compliance induced by both mechanical andthermal loads with a specified amount of basematerial. Thus, the optimization problem is formu-lated at the macroscale level with regard to thedesign variables at the microscale level

MinimizeXY1

;XY2;...;XYN

f XY1;XY2

; . . . ;XYN

� �

¼ r

[NI¼1

XI

b � udXþ rCt

�t � udC

þ r

[NI¼1

XI

aTDTe0 : DH

I : eðuÞdX

subject to aðu; vÞ ¼ �aðDT; vÞ þ ‘ðvÞ; 8v 2 Uad

gI XYIð Þ ¼ r

XYI

dX� VfI YIj j � 0; I ¼ 1; 2; . . . ;N;

ð7Þ

where the objective f is defined as the totalmacrostructural compliance induced by mechanicaland thermal loads, gI is the volume fraction con-straint for YI unit cell, and VfI is the correspondingprescribed volume fraction constraint, while YIj j isthe volume of the unit cell YI.

LEVEL SET TOPOLOGY OPTIMIZATIONMETHOD

Hadamard’s boundary variation method is con-sidered for shape optimization, and the level setfunction is used to represent the boundary of amaterial microstructure implicitly,

/ðxÞ> 0 x 2 XY

/ðxÞ ¼ 0 x 2 CY

/ðxÞ< 0 x 2 Yn XY [ CYð Þ

8<

:; ð8Þ

where /ðxÞ is the level set function at x in the unitcell Y, XY is the material domain, and CY is itsboundary. The benefit of the level set method is thata structure with a smooth and well-defined bound-ary is always guaranteed, which is not easilyattained by traditional density-based methods.Thus, the optimized design can be directly addi-tively manufactured without post-processing, avoid-ing the possible resulting loss in performance. Thefollowing Hamilton–Jacobi equation67–69 is solved toupdate the boundary described by /:

@/ðx; tÞ@t

þ VnðxÞ r/ðx; tÞj j ¼ 0; ð9Þ

where Vn denotes the normal inward velocity of thestructural boundary point x and t is the pseudotime. Equation 9 can be written in discrete form as

/kþ1j ¼ /k

j � Dt r/kj

���

���Vn;j; ð10Þ

where k is the number of optimization iteration, Dtis the pseudo time step, and Vn;j is the normalvelocity at boundary point j. Here Vn;j are treated asthe design variables updated by optimization. As aperiodical reinitialization of / into the signeddistance function is typically needed to regularizethe level set function, in this study, / is reinitializedafter every update in Eq. 10.

Multiscale Optimization ProblemDecomposition

Instead of solving the general optimization prob-lem Eq. 7 directly, a variant of the sequential linearprogramming (SLP) method proposed in Ref. 70 isused, in which Eq. 7 are linearized about each

material design XkYI

at the current iteration k andthen decomposed into N subproblems43 as follows:

minimizeDXYI

@f

@XkYI

� DXkYI

subject to@gI

@XkYI

� DXkYI

� ��gkI ; I ¼ 1; 2; . . . ;N

DXYI� DXk

YI� D�XYI

;

ð11Þ

where DXkYI

is the update for the current microscale

design XkYI

within the bounds DXYIand D�XYI

, whichare further determined by the Courant–Friedrichs–Lewy (CFL) stability condition as detailed below. �gkIdenotes the constraint function change at the kthiteration. The decomposed subproblem for each unitcell can be solved independently and simultane-ously. It is therefore straightforward to distributethe computation and take advantage of parallelcomputing.43

The method proposed in Ref. 71 is reproducedherein to solve Eq. 11. For a specific unit cell, thematerial level set boundary is first discretized intonb points. The element area fractions are thencomputed and used to interpolate the equilibriumEqs. 3, 4, 5, and 6 to obtain the displacement field uand effective elasticity tensor DH. The subproblemin Eq. 11 can be further expressed in discrete formas (dropping the subscript I and superscript k forsimplicity)

@f

@XY� DXY �

Xnb

j¼1

Dt sf ;jlj� �

Vn;j ¼ DtSf � Vn;

@g

@XY� DXY �

Xnb

j¼1

Dt sg;jlj� �

Vn;j ¼ DtSg � Vn;

ð12Þ

where sf ;j and sg;j are the objective and constraintfunction sensitivities on the boundary point j,respectively, lj is the discrete boundary length forpoint j, Sf and Sg are vectors collecting all the

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method 1737

sensitivity and discrete boundary length multipli-cation, and Vn is the vector containing the boundary

point normal velocities Vn;j. By replacing DtVkn with

bk Skf þ kkSk

g

� �= Sk

f þ kkSkg

2,70 the actual optimiza-

tion problem to be solved at current iteration k canbe expressed by

minbk;kk

DtSkf � Vk

n bk; kk� �

s:t: DtSkg � Vk

n bk; kk� �

� ��gk

Vkn;min � Vk

n � Vkn;max:

ð13Þ

where Vkn;min and Vk

n;max are the minimum andmaximum velocities derived from the CFL condi-tion, respectively. The optimization problem Eq. 13is solved using the sequential least-squares SQP

(SLSQP) method, and the optimized bk and kk are

substituted back to DtVkn ¼

bk Skf þ kkSk

g

� �= Sk

f þ kkSkg

2to obtain the optimum

DtVkn at iteration k. DtVk

n is finally substituted backinto Eq. 10 to update the level set function. For allthe examples presented herein, all the boundarypoint movements at each iteration are limited to ahalf of a level set grid size.

Sensitivity Analysis

The shape sensitivities of the objective f andconstraint function gI are needed to solve thedecomposed subproblems in Eq. 13. Ignoring thebody forces in our problem, the compliance objectivesensitivity can be derived as

f 0 XYIð Þ

¼ � rX2XI

DH0

I;ijkleijðuÞeklðuÞ � 2aTDTDH0

I;ijkle0ijeklðuÞ

� �dX;

ð14Þwhere according to Eqs. 5 and 6, the shape sensi-tivity of DH

I;ijkl can be written as

DH0

I;ijkl XYIð Þ

¼ 1

YIj j rCYI

Dpqrs �eðijÞpq � e�ðijÞpq

� ��eðklÞrs � e�ðklÞrs

� �VndC:

ð15Þ

The shape sensitivity for the volume fractionconstraint can be easily written as

g0

I XYIð Þ ¼ r

CYI

VndC: ð16Þ

After calculating the sensitivities at Gauss points,the least-squares method is used to interpolate thesensitivities Sf and Sg at a boundary point.71

NUMERICAL EXAMPLES

Several numerical examples are presented in thissection to illustrate the design of materialmicrostructures for a macrostructure under bothmechanical and thermal loads. For the sake ofsimplicity, all involved quantities are dimensionlessand normalized, and the same mesh is used for thelevel set function and finite element analysis. Thedesign domains of all the macrostructures andmicrostructures are discretized by four-node quadri-lateral bilinear elements with size 1 9 1, and asquare unit cell with a 100 9 100 FE mesh is usedfor all the material microstructure design domains.It is assumed that the isotropic base material hasYoung’s modulus E ¼ 1, Poisson’s ratio m ¼ 0:3, andthermal expansion coefficient aT ¼ 0:001. Void ele-ments are assigned a weak Young’s modulus ofE ¼ 10�6. The least-squares interpolation radius isset as 2. The optimization is terminated if therelative change in the objective function betweentwo successive steps is less than 10�4.

Material Design for a Cantilever Beam

The first example is to design materialmicrostructures for a cantilever beam as shown inFig. 2a. The beam is fixed on the left edge, and adownward load F ¼ 1 is applied at the center of theright edge. The design domain experiences a uni-form temperature change DT, and the allowablevolume fraction of solid phase for the microstructureis set to be 40% of the total unit cell volume.Figure 2b shows the initial design used for themicrostructure domain.

The optimized material designs with DT ¼ 0 andvarious L�H are first generated, as shown inFig. 2c, d, and e, together with their optimizedeffective elasticity matrices. One can clearly seefrom Fig. 2 that the macrostructural geometry has abig impact on the optimized material microstruc-ture. When a long cantilever beam(L�H ¼ 80 � 20) is considered, material isdesigned with a larger stiffness in the horizontaldirection to improve the bending resistance, asshown in Fig. 2c. As the beam gets shorter, thematerial stiffness in the vertical direction graduallyincreases to resist shearing, as indicated in Fig. 2dand e. It is worth noting that the designs of thematerial microstructure with DT ¼ 0 are all ortho-tropic even though no orthotropy constraint isenforced.

Next, material designs for the same problemsetting but with nonzero temperature changes ofDT ¼ 200 and DT ¼ 500 are generated (Fig. 3),revealing distinct differences when compared withthe results obtained for DT ¼ 0 in Fig. 2. Theeffective elasticity matrices indicate that the opti-mized material designs become anisotropic withincreasing DT and anisotropic material can lead to a

Li, Du, and Kim1738

more minimum macrostructural compliance in thepresence of the thermal load. The material designsalso show reduced DH

11, DH12, DH

21, and DH22 values

with increasing DT. Examination of Eq. 4 revealsthat these four components are directly related tothe thermal expansion compliance, therefore reduc-ing DH

11, DH12, DH

21, and DH22 minimizes the thermal

compliance component. The DH12 and DH

21 values evenbecome negative when DT ¼ 500 (Fig. 3e and f).Figure 4 depicts the iterative histories of the

objective and constraint functions while obtainingthe design in Fig. 3b, along with the intermediatedesigns at various iterations. It is observed that thestructural compliance decreases rapidly during thefirst 100 iterations, due to the increasing porosity ofthe material design which reduces the thermalexpansion compliance. As the porosity increases,the mechanical load resistance of the design isweakened. The compliance then increases until thevolume fraction constraint becomes feasible ataround iteration 200. After that, the optimization

Fig. 2. Optimized material microstructures for the cantilever beam with different L� H under DT ¼ 0: (a) design domain, (b) initial design, (c)L� H ¼ 80� 20, (d) L� H ¼ 40� 40, and (e) L� H ¼ 20� 80 (left: unit cell; middle: 3 9 3 array; right: effective elasticity matrix).

Fig. 3. Optimized material microstructures for the cantilever beam with various L� H under DT ¼ 200 and DT ¼ 500: (a) L� H ¼ 80� 20,DT ¼ 200, (b) L� H ¼ 40� 40, DT ¼ 200, (c) L� H ¼ 20� 80, DT ¼ 200, (d) L� H ¼ 80� 20, DT ¼ 500, (e) L� H ¼ 40� 40, DT ¼ 500,and (f) L� H ¼ 20� 80, DT ¼ 500 (left: unit cell; middle: 3 9 3 array; right: effective elasticity matrix).

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method 1739

converges smoothly to an optimized design that hasa balanced resistance between the mechanical andthermal loads.

Note that the thermal loads are less influential inthe long cantilever beam L�H ¼ 80 � 20 evenwhen DT is high, comparing the designs in Fig. 3aand d versus Fig. 2c. This is because, for the longbeam, a small decrease in the material stiffness inhorizontal direction substantially reduces the beambending resistance, resulting in a large increase inthe mechanical component of the compliance. Inother words, the mechanical load is the dominantinfluence on the final design over the thermal loadin the long beam. As the beam gets shorter, thecompliance contribution from the mechanical loaddecreases while the thermal contribution becomesmore influential in Fig. 3b, c, e, and f. These resultsdemonstrate that the geometry of the macrostruc-tural design has a fundamental impact on thematerial design with nonzero thermal loads.

Fig. 4. Convergence history of the objective function and volumefraction constraint for cantilever beam with L� H ¼ 40� 40,DT ¼ 200.

Fig. 5. Optimized material microstructures for the cantilever beamwith four material subdomains and different DT values: (a) cantileverbeam subdomains, (b) DT ¼ 0, and (c) DT ¼ 500.

Table I. Elasticity matrix and volume fraction of the microstructure designs in Fig. 5

Materialsubdomain

Figure 5b Figure 5c

Effective elasticitymatrix

Volumefraction

Effective elasticitymatrix

Volumefraction

XY1 1:099 0:330 00:330 1:099 0

0 0 0:385

2

4

3

51 0:274 �0:073 0:013

�0:073 0:116 �0:0020:013 0:002 0:126

2

4

3

50.910

XY2 0:140 0:114 00:114 0:278 0

0 0 0:115

2

4

3

50.5 0:042 �0:028 �0:023

�0:028 0:080 0�0:023 0 0:068

2

4

3

50.5

XY3 0:140 0:114 00:114 0:278 0

0 0 0:115

2

4

3

50.5 0:036 �0:021 0:019

�0:021 0:066 0:0140:019 0:014 0:083

2

4

3

50.5

XY4 1:099 0:330 00:330 1:099 0

0 0 0:385

2

4

3

51 0:293 �0:064 0:029

�0:064 0:088 �0:0030:029 �0:003 0:110

2

4

3

50.865

Li, Du, and Kim1740

The proposed approach can be used to designmultiple material subdomains at different locationsof the macrostructure and with different volumefraction constraints. To demonstrate this, a can-tilever beam with L�H ¼ 80 � 50 and four subdo-mains is considered, as shown in Fig. 5a, in whichH1 ¼ H4 ¼ 10 and H2 ¼ H3 ¼ 15. The volume frac-tion constraints are set as 0.5 for X2 and X3, and 1for X1 and X4; i.e., no volume constraint is enforcedfor the top and bottom subdomains.

The optimized microstructures with DT ¼ 0 andDT ¼ 500 are shown in Fig. 5b and c, respectively,while the design details are presented in Table I.One can see from Fig. 5b that the optimized mate-rial designs are orthotropic and symmetric aboutthe central horizontal axis when DT ¼ 0. However,the orthotropy and symmetry are broken when thethermal load is nonzero, i.e., DT ¼ 500, in Fig. 5c.This is due to the fact that the thermal expansionload is a pressure-dependent load. The materialdesigns would be different in the tensile region ofthe top half of the beam versus the compressiveregion in the bottom half.

One can see that, although the allowable materialfor the top and bottom subdomains is 100%, porousmaterial designs are suggested for these two regionswhen DT ¼ 500. This is because the fully solidmaterial can produce large thermal expansion com-pliance compared with the porous material. Thus,instead of using up all the allowable material, aporous design is better in the presence of nonzerothermal loads. To illustrate these results, Fig. 5band c shows an array of 16 9 10 optimized materialunit cells. Note that, due to the assumption oflength scale separation in the homogenizationmethod, the optimum unit cell sizes should beinfinitely small. Thus, the selection of unit cell sizescan affect the final material design and its perfor-mance. Typically, the greater the number of unitcells considered, the better the obtained perfor-mance. When the unit cell sizes are sufficientlysmall, the optimized design and its performance willnot change significantly, as demonstrated in Ref. 52.

Additional parametric study is carried out toexamine the influence of the number of materialsubdomains and the temperature change on theoptimized macrostructural objective function. Tothis end, a L�H ¼ 80 � 60 cantilever beam com-posed of uniformly sized subdomains is considered,i.e., H1 ¼ H2 ¼ � � � ¼ Hnr in Fig. 5a, where nr indi-cates the number of subdomains. The results arepresented in Table II, revealing that, when thenumber of subdomain is set to 1, an increase of DTleads to a design with greater compliance due to theincreasing thermal compliance associated with theincreasing temperature. When DT ¼ 500, anincrease in nr leads to lower compliance. This isdue to the fact that, when a larger number ofsubdomains is considered in the optimization, alarger design space is explored by the optimizer,

hence an improved design performance can beexpected.

Material Design for a Simply Supported Beam

The second example is the material design for asimply supported beam in Fig. 6. The beam issubjected to DT, and three downward loads areapplied on the top surface of the beam. The allow-able volume fraction of the solid phase for thematerial microstructure is set to be 40% of the totalunit cell volume. The initial design for themicrostructure is the same as the one shown inFig. 2b.

The designs of the microstructures for the simplysupported beam under a single loading case, i.e.,F1 ¼ 1 and F2 ¼ 0, are first generated, and theresulting topologies for a range of DT and variousL�H are shown in Fig. 7, along with the corre-sponding effective elasticity matrices. One can seeby comparing Fig. 7a and d versus Figs. 2c and 3dthat, even when the macrostructure geometries andDT are the same, the macrostructural boundarycondition has a significant influence on the opti-mized material design. Once again, orthotropicmaterial designs are optimum for the simply sup-ported beam when DT ¼ 0, while anisotropicmicrostructures with lower DH

11, DH12, DH

21, and DH22

values are found to be optimum when DT ¼ 500.The L=H ratio again affects the material designs forboth DT ¼ 0 and DT ¼ 500 cases in terms of mate-rial horizontal and vertical stiffness. Therefore, the

Fig. 6. Simply supported beam with length L and height H.

Table II. Macrostructural compliance comparisonwith the optimized material microstructures withvarying number of material subdomains and DT,L�H ¼ 80 � 60

nr ¼ 1 DT ¼ 500

DT Compliance nr Compliance

0 81.88 1 282.26100 105.32 2 264.49200 152.75 3 246.42300 196.04 4 235.88400 244.16 5 227.03500 282.26 6 221.34

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method 1741

optimum macroscale and microscale designs aretightly coupled.

Next, the simply supported beam under threesymmetric loadings (i.e., F1 ¼ F2 ¼ 1) is optimized,and their results are presented in Fig. 8. ComparingFig. 7 with Fig. 8, it is seen that, for both DT ¼ 0and DT ¼ 500 cases, the material microstructureswith three loadings are generally different fromthose with one loading, and have higher stiffness in

both horizontal (D11) and vertical directions (D22). Itis interesting to notice from the right half of Fig. 8that, even with a nonzero temperature changeDT ¼ 500, the optimized materials are completelyor nearly orthotropic with three symmetric load-ings, but usually anisotropic with one loading. Thisexample demonstrates that the loading cases of themacrostructure may also affect the anisotropy of theoptimized material design even if DT is nonzero.

Fig. 7. Optimized material microstructures for the simply supported beam with different L� H under DT ¼ 0 and DT ¼ 500: (a)L� H ¼ 80� 20, DT ¼ 0, (b) L� H ¼ 80� 40, DT ¼ 0, (c) L� H ¼ 40� 40, DT ¼ 0, (d) L� H ¼ 80� 20, DT ¼ 500, (e) L� H ¼ 80� 40,DT ¼ 500, and (f) L� H ¼ 40� 40, DT ¼ 500 (left: unit cell; middle: 3 9 3 array; right: effective elasticity matrix).

Fig. 8. Optimized material microstructures for the three loadings of the simply supported beam with different L� H under DT ¼ 0 and DT ¼ 500:(a) L� H ¼ 80� 20, DT ¼ 0, (b) L� H ¼ 40� 40, DT ¼ 0, (c) L� H ¼ 40� 40, DT ¼ 0, (d) L� H ¼ 80� 20, DT ¼ 500, (e) L� H ¼ 80� 40,DT ¼ 500, and (f) L� H ¼ 40� 40, DT ¼ 500 (left: unit cell; middle: 3 9 3 array; right: effective elasticity matrix).

Li, Du, and Kim1742

CONCLUSION

A level set topology optimization method is intro-duced to directly design architected materialmicrostructures tailored for a macrostructure underboth mechanical and thermal loads, based on theidea of multiscale topology optimization. The designvariables are defined at the material scale andupdated by optimizing the objective function formu-lated at the structural macroscale. The homoge-nization method is used to calculate the effectiveelastic matrix that bridges these two scales, and theinverse homogenization method is used to deter-mine the optimum microstructures. The numericalresults demonstrate the effectiveness of the methodand show that the optimized material design for athermoelastic macrostructure highly depends on thegeometry, temperature change, mechanical loading,and boundary conditions of the macroscopic struc-tural design. In addition, the thermal loads cansignificantly influence the optimized materialdesign in terms of material orthotropy, symmetry,and porosity. When an increasing number of thematerial subdomains is considered, improved mate-rial designs with better overall performance areobtained. By incorporating the actual properties ofAM base materials (e.g., thermoplastics, metals,and alloys) and choosing reasonable unit cell sizes,the proposed method can generate optimized archi-tected material that is ready for AM. The currentwork can be naturally extended to simultaneouslydesign macrostructural geometry and microscalematerial for thermoelasticity, which will be investi-gated in our future work.

ACKNOWLEDGEMENTS

The presented work is supported by the NASATransformational Tools and Technologies Project(Contract No. NNX15AU22A) and DARPA (AwardNo. HR0011-16-2-0032).

REFERENCES

1. N.A. Fleck, V.S. Deshpande, and M.F. Ashby, Proc. R. Soc. AMath. Phys. Eng. Sci. 466, 2495 (2010).

2. L.J. Gibson and M.F. Ashby, Cellular Solids: Structure andProperties (Cambridge: Cambridge University Press, 1999).

3. X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte,E.B. Duoss, J.D. Kuntz, M.M. Biener, Q. Ge, J.A. Jackson,S.O. Kucheyev, N.X. Fang, and C.M. Spadaccini, Science344, 1373 (2014).

4. T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J.Lian, J.R. Greer, L. Valdevit, and W.B. Carter, Science 334,962 (2011).

5. N. Guo and M.C. Leu, Front. Mech. Eng. 8, 215 (2013).6. Q. Xu, Y. Lv, C. Dong, T.S. Sreeprased, A. Tian, H. Zhang, Y.

Tang, Z. Yu, and N. Li, Nanoscale 7, 10883 (2015).7. O. Sigmund, Int. J. Solids Struct. 31, 2313 (1994).8. X. Huang, A. Radman, and Y.M. Xie, Comput. Mater. Sci.

50, 1861 (2011).9. M.M. Neves, H. Rodrigues, and J.M. Guedes, Comput.

Struct. 76, 421 (2000).10. O. Sigmund, J. Mech. Phys. Solids 48, 397 (2000).11. L.V. Gibiansky and O. Sigmund, J. Mech. Phys. Solids 48,

461 (2000).

12. J. Gao, H. Li, L. Gao, and M. Xiao, Adv. Eng. Softw. 116, 89(2018).

13. O. Sigmund and S. Torquato, J. Mech. Phys. Solids 45, 1037(1997).

14. O. Sigmund and J.S. Jensen, Philos. Trans. R. Soc. A Math.Phys. Eng. Sci. 361, 1001 (2003).

15. R. Alberdi and K. Khandelwal, Struct. Multidiscip. Optim.59, 461 (2019).

16. X.Y. Yang, X. Huang, J.H. Rong, and Y.M. Xie, Comput.Mater. Sci. 67, 229 (2013).

17. Y. Wang, Z. Luo, N. Zhang, and Z. Kang, Comput. Mater.Sci. 87, 178 (2014).

18. O. Sigmund, Mech. Mater. 20, 351 (1995).19. F. Wang, O. Sigmund, and J.S. Jensen, J. Mech. Phys. So-

lids 69, 156 (2014).20. S. Torquato, S. Hyun, and A. Donev, Phys. Rev. Lett. 89,

266601 (2002).21. J.K. Guest and J.H. Prevost, Int. J. Solids Struct. 43, 7028

(2006).22. V.J. Challis, A.P. Roberts, and A.H. Wilkins, Int. J. Solids

Struct. 45, 4130 (2008).23. Y.M. Yi, S.H. Park, and S.K. Youn, Int. J. Solids Struct. 37,

4791 (2000).24. M. Osanov and J.K. Guest, Annu. Rev. Mater. Res. 46, 211

(2016).25. T.A. Schaedler and W.B. Carter, Annu. Rev. Mater. Res. 46,

187 (2016).26. J. Liu, A.T. Gaynor, S. Chen, Z. Kang, K. Suresh, A. Take-

zawa, L. Li, J. Kato, J. Tang, C.C.L. Wang, L. Cheng, X.Liang, and A.C. To, Struct. Multidiscip. Optim. 57, 2457(2018).

27. O. Sigmund, S. Torquato, and I.A. Aksay, J. Mater. Res. 13,1038 (1998).

28. S.J. Hollister, Nat. Mater. 4, 518 (2005).29. J. Schwerdtfeger, F. Wein, G. Leugering, R.F. Singer, C.

Korner, M. Stingl, and F. Schury, Adv. Mater. 23, 2650(2011).

30. E. Andreassen, B.S. Lazarov, and O. Sigmund, Mech. Mater.69, 1 (2014).

31. A. Clausen, F. Wang, J.S. Jensen, O. Sigmund, and J.A.Lewis, Adv. Mater. 27, 5523 (2015).

32. A. Takezawa, M. Kobashi, and M. Kitamura, APL Mater. 3,076103 (2015).

33. A. Takezawa and M. Kobashi, Compos. Part B Eng. 131, 21(2017).

34. D. Xiao, Y. Yang, X. Su, D. Wang, and J. Sun, Biomed.Mater. Eng. 23, 433 (2013).

35. C.Y. Lin, T. Wirtz, F. LaMarca, and S.J. Hollister, J.Biomed. Mater. Res. Part A 83, 272 (2007).

36. Y. Koizumi, A. Okazaki, A. Chiba, T. Kato, and A. Takeza-wa, Addit. Manuf. 12, 305 (2016).

37. X. Huang, S.W. Zhou, Y.M. Xie, and Q. Li, Comput. Mater.Sci. 67, 397 (2013).

38. H. Rodrigues, J.M. Guedes, and M.P. Bendsoe, Struct.Multidiscip. Optim. 24, 1 (2002).

39. P.G. Coelho, P.R. Fernandes, J.M. Guedes, and H.C. Ro-drigues, Struct. Multidiscip. Optim. 35, 107 (2008).

40. L. Liu, J. Yan, and G. Cheng, Comput. Struct. 86, 1417(2008).

41. L. Xia and P. Breitkopf, Comput. Methods Appl. Mech. Eng.278, 524 (2014).

42. W.M. Vicente, Z.H. Zuo, R. Pavanello, T.K.L. Calixto, R.Picelli, and Y.M. Xie, Comput. Methods Appl. Mech. Eng.301, 116 (2016).

43. R. Sivapuram, P.D. Dunning, and H.A. Kim, Struct. Mul-tidiscip. Optim. 54, 1267 (2016).

44. Y. Zhang, H. Li, M. Xiao, L. Gao, S. Chu, and J. Zhang,Struct. Multidiscip. Optim. 59, 1273 (2019).

45. Y. Zhang, M. Xiao, L. Gao, J. Gao, and H. Li, Mech. Syst.Signal Process. 135, 106369 (2020).

46. A. Abdulle and A. Di Blasio, Multiscale Model. Simul. 17,399 (2019).

Design of Architected Materials for Thermoelastic Macrostructures Using Level Set Method 1743

47. P.G. Donders, Homogenization Method for Topology Opt-mization of Structures Built with Lattice Materials. Doctoraldissertation, Ecole Polytechnique, Palaiseau, France, 2018.

48. C. Barbarosie and A.M. Toader, Mech. Adv. Mater. Struct.19, 290 (2012).

49. R. Lipton and M. Stuebner, Struct. Multidiscip. Optim. 33,351 (2007).

50. G. Allaire, P. Geoffroy-Donders, and O. Pantz, Comput.Math. Appl. 78, 2197 (2019).

51. A. Faure, G. Michailidis, R. Estevez, G. Parry, and G. Al-laire, in ECCOMAS Congress 2016—Proceedings of 7thEuropean Congress on Computational Methods in AppliedScience and Engineering, eds. by V.P.M. Papadrakakis, V.Papadopoulos, G. Stefanou (Crete Island, Greece, 2016), pp.3534–3545.

52. Z.H. Zuo, X. Huang, J.H. Rong, and Y.M. Xie, Mater. Des.51, 1023 (2013).

53. F. Fritzen, L. Xia, M. Leuschner, and P. Breitkopf, Int. J.Numer. Methods Eng. 106, 430 (2016).

54. J. Kato, D. Yachi, T. Kyoya, and K. Terada, Int. J. Numer.Methods Eng. 113, 1189 (2018).

55. P.B. Nakshatrala, D.A. Tortorelli, and K.B. Nakshatrala,Comput. Methods Appl. Mech. Eng. 261–262, 167 (2013).

56. H. Rodrigues and P. Fernandes, Int. J. Numer. MethodsEng. 38, 1951 (1995).

57. T. Gao and W. Zhang, Struct. Multidiscip. Optim. 42, 725(2010).

58. Q. Li, G.P. Steven, and Y.M. Xie, J. Therm. Stress. 24, 347(2001).

59. P. Pedersen and N.L. Pedersen, Struct. Multidiscip. Optim.42, 681 (2010).

60. Q. Xia and M.Y. Wang, Comput. Mech. 42, 837 (2008).61. J. Deng, J. Yan, and G. Cheng, Struct. Multidiscip. Optim.

47, 583 (2013).62. J. Yan, X. Guo, and G. Cheng, Comput. Mech. 57, 437 (2016).63. J. Guedes and N. Kikuchi, Comput. Methods Appl. Mech.

Eng. 83, 143 (1990).64. P.D. Dunning, H.A. Kim, and G. Mullineux, Finite Elem.

Anal. Des. 47, 933 (2011).65. H. Chung, O. Amir, and H.A. Kim, Comput. Methods Appl.

Mech. Eng. 361, 112735 (2020).66. S. Kambampati, R. Picelli, and H.A. Kim, in AIAA Scitech

2019 Forum (American Institute of Aeronautics and Astro-nautics, Reston, Virginia, 2019).

67. J.A. Sethian and A. Wiegmann, J. Comput. Phys. 163, 489(2000).

68. Y. Mei and X. Wang, Acta Mech. Sin. Xuebao 20, 507 (2004).69. G. Allaire, F. Jouve, and A.M. Toader, J. Comput. Phys. 194,

363 (2004).70. P.D. Dunning and H.A. Kim, Struct. Multidiscip. Optim. 51,

631 (2015).71. R. Picelli, S. Townsend, C. Brampton, J. Norato, and H.A.

Kim, Comput. Methods Appl. Mech. Eng. 329, 1 (2018).

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