Topology Optimization of Multiscale Structures Considering Local and Global Buckling Response Preprint

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Topology Optimization of Multiscale Structures Considering Local

and Global Buckling Response

Preprint

Christoﬀer Fyllgraf Christensen∗, Fengwen Wang, Ole Sigmund

Department of Civil and Mechanical Engineering, Technical University of Denmark, Nils Koppels All´e, Building 404, 2800 Kgs. Lyngby, Denmark

Abstract

Much work has been done in topology optimization of multiscale structures for maximum stiﬀness or minimum compliance design.

Such approaches date back to the original homogenization-based work by Bendsøe and Kikuchi from 1988, which lately has been

revived due to advances in manufacturing methods like additive manufacturing. Orthotropic microstructures locally oriented in

principal stress directions provide for highly eﬃcient stiﬀness optimal designs, whereas for the pure stiﬀness objective, porous

isotropic microstructures are sub-optimal and hence not useful. It has, however, been postulated and exempliﬁed that isotropic

microstructures (inﬁll) may enhance structural buckling stability but this has yet to be directly proven and optimized. In this work,

we optimize buckling stability of multiscale structures with isotropic porous inﬁll. To do this, we establish local density dependent

Willam-Warnke yield surfaces based on local buckling estimates from Bloch-Floquet-based cell analysis to predict local instability

of the homogenized materials. These local buckling-based stress constraints are combined with a global buckling criterion to obtain

topology optimized designs that take both local and global buckling stability into account. De-homogenized structures with small

and large cell sizes conﬁrm validity of the approach and demonstrate huge structural gains as well as time savings compared to

standard singlescale approaches.

Keywords: Topology Optimization, Multiscale Structure, Buckling Strength, Stability, Isotropic Microstructures, Stress

Constraint

1. Introduction

In recent years advances in additive manufacturing have facilitated the fabrication of multiscale or inﬁll structures

[1], which in turn has further promoted topology optimization of multiscale structures. The fact that structures with

tailored microstructures can be manufactured means that multiscale structures are no longer restricted to theoretical

research but can actually be utilized in real designs. A lot of research considering multiscale structural optimization

following the original work of Bendsøe and Kikuchi [2] has been done in recent years as reviewed by Wu, Sigmund

and Groen [3]. Work in [4] showed that multiscale designs can be projected to singlescale using an implicit geometry

description. This work was elaborated in [5, 6] for 2D problems and [7, 8] for 3D.

Previous multiscale structural optimization has mainly focused on compliance minimization without considering

structural stability. Recently, singlescale studies of buckling optimization has become more popular. Work in [9]

focused on the the use of linear buckling with the scope of solving large scale topology optimization. An extension

of this is the work in [10], where buckling resistance and local ductile failure constraints are combined. Furthermore,

the work in [11] considers singlescale buckling optimization, using the Topology Optimization of Binary Structures

(TOBS) method, subject to design dependent pressure loads. This method is limited to singlescale optimization due

to the binary nature of the TOBS method. Recent work in [12] focused on singlescale buckling optimization using

the level set method. Work on multiscale stability optimization is only sparsely covered in literature. Recent work

in [13] focuses on using ﬁlleted lattice structures considering a simpliﬁed local buckling formulation and yield stress

constraints for compliance optimization. The method lacks the opportunity to generate true void and solid due to the

∗Corresponding author.

Email address: chrify@mek.dtu.dk (Christoﬀer Fyllgraf Christensen)

arXiv:2210.11477v2 [cs.CE] 28 Apr 2023

Preprint - Topology Optimization of Multiscale Structures Considering Local and Global Buckling Response 2

(a) (b)

(c)

Figure 1: Comparison of HS and SIMP stiﬀness interpolations: (a) Stiﬀness interpolation with p=3 in SIMP, (b) Buckling load maximization

where the top half uses SIMP (BLF =36.26) and the bottom half uses HS (BLF =41.56), (c) Compliance minimization where the left half uses

SIMP (C=13.95 ·10−4) and the right half uses HS (C=13.29 ·10−4).

local stress constraints and strut radii bounds used in the microstructure. As a result microstructures appear in the

entire design domain even though global buckling is not considered and multiscale isotropic material is sub-optimal

for compliance optimized designs [14].

Earlier work [15] has shown that isotropic inﬁll may increase structural buckling resistance at the price of a small

stiﬀness reduction even with a predeﬁned constant inﬁll density. To fully explore the potential buckling resistance,

this study suggests to freely optimize the isotropic microstructure inﬁll densities in the entire design domain via

topology optimization. Previous studies have shown that microstructure buckling strength can be evaluated using

Bloch-Floquet wave theory [16, 17]. Based on buckling stability analysis of a chosen microstructure evaluated at

diﬀerent densities it is possible to formulate interpolation functions for local buckling stress constraints. Combining

this with a global buckling optimization provides a way of performing buckling optimization of multiscale structures.

In this work, the global buckling load is estimated using the linearized buckling analysis framework presented in

[18], as it increases computational eﬃciency, while producing suﬃciently accurate buckling estimates as discussed in

[9, 19]. Furthermore, the robust formulation by [20] and two-ﬁeld formulation by [21] are used to allow control over

the minimum density, thus allowing true void in the design.

We consider local and global buckling topology optimization of multiscale structures composed of a ﬁxed isotropic

triangular microstructure with spatially varying density. The stiﬀness of the isotropic triangular microstructure is very

close to the theoretical maximum provided by the Hashin-Shtrikman (HS) upper bound [22] as found in e.g. [23].

Thus, instead of performing a formal homogenization approach to compute the eﬀective properties of the triangular

microstructure, we can simply use the bulk and shear modulus provided by the HS upper bound to model its elastic

properties [14, 24, 25]. For pure single-load compliance minimization problems, the use of the isotropic (near-optimal)

microstructure is suboptimal and results in 0–1, non-composite solutions, c.f. [14]. For such problems, orthotropic

rank-2 or rectangular hole microstructures are optimal, however, such microstructures have low shear stiﬀness and

thus perform badly in terms of buckling stability [26]. For this reason, we here opt for the stiﬀness sub-optimal

but buckling-superior triangular microstructure. An added beneﬁt of the stiﬀness isotropic triangular microstructure

is that its buckling response is rather isotropic as well, thus making the establishing of a buckling “yield surface”

considerably simpler.

A comparison of the stiﬀness interpolations using HS and standard SIMP (p=3) interpolations is visualized in

Figure 1a. The diﬀerence between the two interpolation schemes is signiﬁcant for lower densities and explains the

advantage of using microstructures when considering buckling. Using the two schemes for pure compliance mini-

mization conﬁrms, contrary to optimally oriented rank-2 laminates or rectangular hole cells, that there is no advantage

Preprint - Topology Optimization of Multiscale Structures Considering Local and Global Buckling Response 3

Figure 2: Flowchart illustrating the work ﬂow of the method presented in this paper. The upper left box shows the main building blocks to set up the

homogenized optimization problem. The lower left box shows an example of a design domain. The center column show a optimized homogenized

design and its critical buckling mode. The second box from the right shows the de-homogenization method. Finally, the de-homogenized structure

with its critical buckling mode is shown in the right most box.

in using isotropic inﬁll for compliance (C), see Figure 1c, c.f. [14]. The small diﬀerence in compliances is simply a re-

sult of the diﬀerent stiﬀness interpolations acting in the intermediate density regions that occur due to the density ﬁlter.

However, for buckling load maximization large diﬀerences are visible in both design and performance as measured

by the Buckling Load Factor (BLF). The design obtained using SIMP (seen in the upper half of Figure 1b) is black

and white as intermediate densities are penalized. The multiscale design, based on the HS interpolation (seen in the

lower part of Figure 1b), shows a large region of intermediate densities. The explanation lies in the higher stiﬀness for

intermediate densities provided by the HS interpolation compared to SIMP as seen in Figure 1a. Comparing the per-

formance of the two designs shows a 14.62% improvement in global buckling stability for the multiscale design. This

indicates that isotropic microstructures are superior for buckling objectives. Of course this is only true if the structure

does not reach local buckling of the inﬁll before global buckling is reached. Therefore, local buckling stability must

be taken into account when optimizing against buckling using topology optimization of multiscale structures.

This paper will present the work ﬂow for performing topology optimization of multiscale structures while prevent-

ing buckling on both local and global scale. This is done according to the work ﬂow illustrated in Figure 2. The paper

is organized as follows: In Section 2 the optimization problem for multiscale structures is presented. This includes

a multiﬁeld method, interpolations and buckling stress constraint. Section 3 presents the results of two numerical

examples demonstrating the advantage of the method. Finally, Section 4 concludes the work presented in this paper.

2. Buckling Optimization of Multiscale Structures

This study aims to optimize global structural stability while preventing local microstructure buckling via buckling

stress constraints using a robust linearized buckling framework. The optimization problem using the robust formula-

Preprint - Topology Optimization of Multiscale Structures Considering Local and Global Buckling Response 4

tion [20] and a two-ﬁeld formulation that allows control over the minimum density [21] is written as

min

x,s: max

m



gλ(ρm)=JKS (γi(ρm))

JKS

0



,m∈ {e,b,d}, γi∈ B

s.t. : gc(ρe)=fTu(ρe)

C∗

e−1≤0,

:gs(ρm,˜

x, γi(ρm)) ≤0,m∈ {e,b,d}, γi∈ B

:gV(ρd)=Pjvjρd

V∗

dVΩ−1≤0,

:ρm

j=˜xj¯

˜sm

j,m∈ {e,b,d}

:xmin ≤xj≤1,∀j

: 0 ≤sj≤1.∀j

(1)

The meaning of each term in (1) will be explained over the next several pages and subsections. First, gλ(ρm) represents

the normalized aggregated inverse global stability objectives for m∈ {e,b,d}being the eroded,blue print and dilated

designs. The normalization factor JKS

0is simply the value of JKS (γi(ρm))for the initial design. gcand gVrepresent

structural compliance and volume constraints, respectively. gsis the local buckling stress constraint, which is depen-

dent on the global buckling load. The exact deﬁnition of gsis presented in the following subsections. vjand VΩare

the elemental and total structural volumes. The physical design ﬁeld ρmis constructed using a ﬁltered density ﬁeld

xand a void indicator ﬁeld ¯

˜sfollowing the method by [21]. This provides control over the minimum density while

allowing void regions in the design. The detailed design procedure will be presented in the following subsection.

The compliance measure fTudepends on the external load fand the displacements u(ρm) for the design, m. The

displacements are calculated by solving

K(ρm)u=f,(2)

where K(ρm) is the linear symmetric global stiﬀness matrix for the considered design ρm. The constant C∗

eis the

compliance upper limit for the eroded design. The maximum allowed volume fraction is enforced through the dilated

design using V∗

d. The value of V∗

dis continuously updated to ensure that the target volume V∗

bis enforced on the blue

print design. This strategy is used to eliminate numerical artifacts and stabilize convergence of the optimization

problem [20].

The critical buckling load is approximated using linearized buckling analysis (see [27]) based on the equilibrum

in (2), deﬁned as

(Gσ(ρm,u)−γiK(ρm))ϕi=0.(3)

where the eigenpairs (γi,ϕi) represent the eigenvalue and buckling mode vector, respectively. The actual Buckling

Load Factors (BLF) λiare recovered through the eigenvalues γias λi=−1/γi, ordered such that λi≤λi+1. The

global stress stiﬀness matrix Gσ(ρm,u) is built considering the design ρmand the current stress state resulting from

the displacements u(ρm). A Matlab implementation of (3) is presented in [18].

There exist as many eigenpairs (γi,ϕi),i∈ B0as there are DOFs in the system. Only a subset of eigenpairs

B ⊂ B0is included in the analysis. This subset should be large enough to capture all relevant buckling modes but not

so large that the problem becomes computationally infeasible ([28, 29]). The critical buckling load is determined by

the fundamental BLF λ1through fcr =λ1f.

The Kreisselmeier–Steinhauser (KS) function [30] approximates the minimum structural buckling factor by ag-

gregating the considered eigenvalues into one diﬀerentiable objective for each design min the robust formulation.

The version of the KS function used here diﬀers from the one in [30] by normalizing the eigenvalues. This provides

better scaling which improves stability of the optimization problem. The deﬁnition of the modiﬁed KS function is

JKS (γi(ρm))i∈B =γ0+γ0

Pln 



X

i∈B

P





γi(ρm)

γ0−1









,(4)

where Pis the aggregation parameter and γ0is equal to the fundamental eigenvalue γ1from the initial design. Here

(4) provides a smooth upper bound approximation of |γ1|=max

i∈B |γi|. The result is a smooth lower bound for λ1.

Preprint - Topology Optimization of Multiscale Structures Considering Local and Global Buckling Response 5

Figure 3: Illustration of how the physical design ﬁeld ρmis constructed from the density ﬁeld xand the void indicator ﬁeld s.

2.1. Multiﬁeld Method and Material Interpolation

The multiﬁeld method by [21] is used in this work to generate the physical design ﬁeld ρm. It provides control

over the minimum density of porous regions while also allowing fully void regions in the design. This is essential as

both the buckling resistance and buckling stress limit goes to zero for ρ→0. If nothing is done to deal with this issue

the optimization will always require some amount of material in every elements to comply with the constraint. This

essentially means that the method will not allow topology changes and is only useful for inﬁll optimization of already

known geometries. However, using the multiﬁeld method by [21] solves this issue as the stress limits are interpolated

using the density ﬁeld ˜

xwhile the stiﬀness interpolation is performed on the physical design ﬁeld ρm.

The construction procedure for the physical design ﬁeld ρmis illustrated in Figure 3. It shows how the indicator

ﬁeld, s, is ﬁltered through F(s) then projected using a Heaviside function Hm(˜

s) at three diﬀerent threshold levels.

The density ﬁeld is only ﬁltered through F(x). The product of these two ﬁelds form the physical design ﬁeld ρm.

The purpose of the density ﬁlter is to eliminate numerical artifacts such as checker boarding and make the design

mesh-independent [31]. The deﬁnition of the density ﬁlter is [31, 32]

˜yj=Pi∈Nj,iw(ni)yi

Pi∈Nj,iw(ni)⇒˜

y=F(y),y∈{x,s},(5)

where Nj,iis the set of neighborhood elements within the ﬁlter radii Rsand Rxfor the indicator ﬁeld sand density

ﬁeld xrespectively. In this work Rx=1.5∆xand Rs=4.5∆xis used throughout. The densities are weighted through

w(ni), where niare the element center points, i.e.

w(ni)=max 0,Rs/x− |ni−nj|.(6)

The smoothed Heaviside function Hm(˜

s) (see [20, 33]) is deﬁned as

˜s=Hm(˜

s)=tanh(βηm)+tanh (β(˜

s−ηm))

tanh(βηm)+tanh (β(1−ηm)),m∈ {e,b,d},(7)

where ηmis the threshold value. In this work ηb=0.5 and the eroded and dilated values are determined by ηb±∆η,

where ∆ηis speciﬁed explicitly for each of the numerical examples. The steepness of the projection is determined by

β. When βincreases the Heaviside function gets more and more non-linear. Therefore, a continuation method is used

on βimplying that it is initialized as β=2 and slowly increased throughout the optimization process to βmax =256 to

ensure a clear representation of void regions. The speciﬁc continuation scheme used in this work is stated in Section 3.

As discussed in the introduction, the stiﬀness of the considered triangular microstructure is very close to the

theoretical optimum provided by the HS bounds. Therefore, instead of performing numerical homogenization to

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TopologyOptimizationofMultiscaleStructuresConsideringLocalandGlobalBucklingResponsePreprintChristoerFyllgrafChristensen,FengwenWang,OleSigmundDepartmentofCivilandMechanicalEngineering,TechnicalUniversityofDenmark,NilsKoppelsAll´e,Building404,2800Kgs.Lyngby,DenmarkAbstractMuchworkhasbeendoneintopol...

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