* Corresponding author. Tel.: +65 6516 1228

E-mail address: mpelwf@nus.edu.sg (Wen Feng Lu)

Buckling optimization of Kagome lattice cores with free-form

trusses

Lei Zhanga, Stefanie Feihb, Stephen Daynesb, Yiqiang Wangc, Michael Yu Wangc, Jun Weib

and Wen Feng Lua,*

a Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1,

Singapore, 117575

b Joining Technology Group, Singapore Institute of Manufacturing Technology (SIMTech), 2

Fusionopolis Way, Singapore 138634

c Department of Mechanical and Aerospace Engineering, and Department of Electronic and

Computer Engineering of Hong Kong University of Science and Technology, Clear Water Bay,

Kowloon, Hong Kong

Published in Materials & Design, 145 (2018) 144–155

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Abstract

Lightweight lattice structures are an important class of cellular structures with high potential

for multi-functional applications. Considering load-bearing requirements, truss buckling is

one of the main failure mechanisms for low density and slender lattice structures. Critical

buckling load can be increased by modifying the profile of a truss. In this paper, we present a

shape design method to optimize the critical buckling load for lattice cores with free-form

trusses. The free-form truss is represented by Fourier series and implicit surfaces, having

smooth truss diameter variations and truss joints. The optimized truss profile is obtained by

solving a parametric shape optimization problem with Fourier series coefficients as design

variables. The method is used for designing optimized 1D columns and 3D Kagome lattice

structures as the cores of sandwich panels. The numerical results predict 26.8% and 20.4%

improvement of the critical buckling loads compared to their uniform counterparts of the

same mass, respectively. The optimized structures include complex smooth and curved

geometries that are well suited for additive manufacturing because of the greater design

freedom. Finally, the initial and optimized lattice cores are additively manufactured and

tested. The experimental results validate the effectiveness of the proposed method.

Keywords Additive manufacturing, Shape design, Buckling failure, Kagome lattice core,

Free-form truss

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1. Introduction

Over the past decades, lattice structures have received increasing attention due to their

superior specific stiffness and strength [1-3]. Despite their multi-functional applications,

many efforts have been conducted to design lattice structures for enhancing their mechanical

properties, e.g. compressive strength [4-6]. Elastic buckling and plastic yielding are two of

the main failure mechanisms of lightweight lattice structures, playing a crucial role on their

compressive strength. Typically, buckling failure may appear locally through core members

[6-10], or globally through the entire structure [5, 7, 11]. Since the critical buckling load of a

slender column decreases as a fourth power to column radius reduction, while the yielding

load deceases by only two orders, significant strength improvement can be achieved in the

range of low relative density by improving buckling resistance.

The optimal shape of a one-dimensional (1D) column under compression has been

investigated by a number of studies. For instance, [12, 13] solved this problem under

different boundary conditions using variational approaches, and the obtained optimal

eigenvalues were able to represent the theoretically maximum values. However, the

analytical optimal results included cross-sections of zero area that were undesirable because

they were not manufacturable. An alternative strategy to design the strongest column was to

adopt the constant strain energy approach [14-16]. In some subsequent studies, the finite

element method was employed to find the optimal column shape [15, 17], where the column

was discretized into beam elements with predefined cross-sections. The optimal shapes can

be obtained by iteratively changing the cross-sectional area of each element. All of these

approaches yielded similar optimal column shapes and improvements for critical buckling

loads.

The quest for the strongest truss frames or lattice structures is an extension of the

optimal 1D column problem. Buckling optimization of truss frames was investigated with

non-uniform truss members in [15, 17], in which the critical loads of truss frames were

improved without increasing the frame weight. In [18], the buckling constraint was taken into

consideration in design and optimization of truss frames. Additionally, some studies focused

on optimizing the sizes and angles of uniform truss members of lattice structures by taking

their local buckling failure modes into consideration [5, 7]. Gorguluarslan et al [19] improved

the critical buckling load of OctetFramed-based structures using uniform truss members by

individually optimizing the radii of the vertical, diagonal, and horizontal trusses. However,

4

non-uniform trusses are preferred for buckling optimization because they can produce

uniform specific strain energy due to the complex buckling mode shapes, which is claimed as

the optimal condition for maximizing critical buckling loads [14-16]. There are few studies

on designing 3D lattice structures with non-uniform truss cross-sections for improving their

critical buckling loads. One of the biggest challenges is the issue of multiple eigenvalues.

This issue is critical for 3D buckling optimization problems, because 3D structures with

symmetries are intrinsically multi-modal. In a relevant field, Amendola et al [20] investigated

the elastic and post-yield behavior of face-centered cubic lattices for seismic isolation devices

with truss members composed of two truncated bi-cones; however, more complex shape

variations were note investigated.

Fabrication of lattice structures with non-uniform truss profiles is a big challenge for

traditional manufacturing technologies. Recently, the advance of additive manufacturing

(AM) offers great design and fabrication flexibility for the lattice structures with high

geometrical complexity [21, 22]. Therein, the layer-by-layer fabrication processes make AM

well suited for fabricating structures with complex features such as lattices with non-uniform

trusses. To date, AM has been increasingly adopted to fabricate lattice structures for

improving structural stiffness and strength by controlling truss size, shape, and orientation

[10, 23], achieving target properties [20], and designing structures with a solid conformal

skin and lattice infill [11].

In this study, we present an effective design approach for improving the critical buckling

loads of sandwiched 3D Kagome lattice cores with free-form trusses. In this framework, the

shapes of the truss members are characterized by Fourier series (FSs) and implicit surfaces,

which guarantees smooth truss shape and the transitions at truss joints. In this regard, the

method transforms the original shape design problem into a parameter optimization problem.

In the design process, a truncated FS is applied since low-order terms strongly contribute to

smooth and slowly-varying shapes while high-order terms mainly formulate undesired local

sharp features.

Numerical design of both 1D columns and 3D Kagome lattice cores are carried out by

using the present approach. The optimized results include trusses with curved and non-

uniform sections, presenting significant improvement of their buckling critical loads. In

addition, both original and optimized structures of identical masses are fabricated via AM,

where the curved shapes of optimized trusses can be precisely captured. Experimental testing

are also conducted to evaluate their performance. Both numerical and experimental results

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demonstrate the effectiveness of the proposed method. Although only 1D columns and 3D

Kagome lattice cores are designed in this study, the present method is applicable for other

lattices of uniform trusses without any difficulties.

2. Optimization Methodology

2.1. Shape representation of free-form trusses with Fourier series

Typically, a uniform circular truss, or column, is defined by its sectional radius, which is

constant along the axial direction. A free-form truss on the other hand is defined by a

spatially dependent radius function, which is a linear combination of basis functions from a

complete function space [24]:

0

( ) ( )n n

n

r c

(1)

where ( )n is the nth order basis function with its associated coefficient nc , and is the

local coordinate collinear with the axial direction. Feasible candidates of the basis functions

{ ( )}n are among polynomials, FS or B-splines. The FS is adopted in this paper because of

its smoothness, symmetry, periodic properties and ability to flexibly describe modal shapes.

High-order terms in FS describe spatial features of high frequency, which usually do not exist

in engineering structures and therefore can be eliminated. Cutting off the first M orders of FS,

a free-form truss can be characterized as the radius function in terms of FS coefficients

{ , }n n by:

0

1

2 2( ) ( cos( ) sin( ))

2

M

n n

n

n nr

l l

(2)

where l is the length of the truss and 0 is the original diameter of the truss. For M = 0, the

above sequence degenerates into a constant, and therefore the trusses with uniform diameters

are obtained.

The 3D models of lattice structures with merged truss members are created based on an

implicit modelling approach for blending objects [25]. In a Cartesian coordinate system, the

implicit function of a single circular free-form truss, where z is in the axial direction, is

described by:

2 2 2( , , ) ( )f x y z x y r z (3)

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By applying rotating, cutting and blending operations, N trusses are blended to form a

3D lattice structure with a scalar field (x, y, z):

1

( , , ) exp( ( , , )) 1N

i i

i

x y z a f x y z

(4)

where fi(x, y, z) is the implicit function of the ith truss, and ai is the corresponding parameter

affecting the truss connection shape. By choosing appropriate ai values, smooth truss joint

sections are generated with minimal stress concentrations. Readers can refer to [26] for more

details of the above operations.

Figure 1. Generation of free-form lattice geometries with Fourier series: the free-form 1D columns

defined as2 2 2( ) ( ) 0f x, y,z y z r z with (a) ( ) 0.7 0.125cos( )r z z , (c)

( ) 0.7 0.125cos( )r z z , (e) ( ) 0.7 0.125cos( ) 0.25cos(2 )r z z z ; (b, d, f) the corresponding

3D Kagome lattice cores with core height h and internal angle θ.

Finally, in order to generate solid models for finite element analysis (FEA) and AM, a

zero isosurface of the scalar field is extracted to be the geometrical surface of the 3D lattice

structure. Figure 1 illustrates typical free-form trusses and the corresponding Kagome lattice

cores. Isosurface extraction commonly results in a polygon mesh that can be easily converted

into a triangular surface mesh to generate a stereolithography (STL) format file. The STL file

format is widely used in the fields of computer-aided design and AM, and therefore the

generated STL files are able to be imported into commercial softwares for FEA and AM. The

marching cubes algorithm is commonly used for isosurface extraction. However, this

algorithm may result in low quality surface meshes [26], and thus a mesh repair and cleaning

process is necessary to improve the mesh quality. In this paper, we use MeshLab [27] (an

open mesh processing software) to repair and clean the surface mesh if needed.

Compared to previously described work using discretized variable based methods [12-

18], the present FS-based implicit representation is highly beneficial for representing

complex truss shapes in a smooth manner using a limited number of design variables. In

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addition, this method is well suited for AM design because the smooth and exact model

surfaces can easily be converted to STL files.

2.2. Optimization formulation and design workflow

The buckling optimization problem aims to find the optimal shapes of truss members for

maximizing the critical buckling load of a lattice core subject to a given volumetric constraint.

This shape optimization problem is simplified to a parametric one by using the FS-based free-

form representation given in Eqs. (2) – (4), where the optimized shape Ω is determined by

design parameters { , }n n . To this end, the optimization is formulated by:

1,

( , ) ( 0,1,..., )Maximize

Subject to: ( ) 0

d 0

n n

n n

0 i i

J n M

+

V

K K v (5)

where the cost function is the first-order eigenvalue. The linear elastic buckling behavior is

governed by the eigenvalue equation in Eq. (5). Here,0K is the base state stiffness matrix

(consisting of the sum of the elastic stiffness, the initial stress stiffness and the load stiffness)

and K is the differential stiffness matrix consisting of the sum of the initial stress and load

stiffness due to perturbation stresses and loads, respectively [28]. The ith eigenvalue and the

associated eigenvector are i and

iv . V denotes the prescribed volume constraint. The

eigenvalue problem of the meshed structure is solved by Dassault Systèmes commercial

software package Abaqus, version 2016. Ideal geometry and perfect boundary conditions are

assumed in the FE analysis. Clamped-clamped boundary conditions are imposed by

constraining all degrees-of-freedom of the nodes located on the bottom face; similarly, the

movements in x- and y-directions of the nodes on the top face are constrained. It is worth

noting that the objective function refers to the lowest eigenvalue of the structure in each

iteration. Since the order of the eigenvalues may change during the optimization procedure,

the associated eigenvectors of 1 in each iteration may also be different.

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Figure 2. Optimization flowchart under Matlab-Abaqus connection framework.

The optimization problem is solved iteratively within an interconnected Matlab-Abaqus

framework. The whole iteration procedure runs in the Matlab environment, where Abaqus

acts as the FEA solver for the eigenvalue problem. The simulation flow consists of several

steps as outlined in Figure 2:

Step 1: Initialize design parameters for the optimization problem, as well as the necessary

parameters of optimizer, in controller component.

Step 2: Generate the geometry based on current parameters in Matlab-Geometry

component, which results in a STL file.

Step 3: Call Python script to run Abaqus for generating a volume mesh from the STL file

and solving the eigenvalue problem. The required results are exported into text

files.

Step 4: Read FEA results and update design parameters based on an optimization

algorithm in Matlab-Optimization component.

Step 5: Check the termination condition. If the size of a step or the change of the objective

function during a step is smaller than 10-3, the iterations terminate; otherwise go

back to Step 2.

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2.3. Numerical implementation

In this work, the 3D lattice core structures are modelled using solid continuum elements

in the FE analysis. Compared to most existing approaches with beam elements, e.g. [12-18],

the solid elements can precisely characterize the shapes of truss joints as well as diameter

variations, thus providing high computational accuracy. Furthermore, the solid model is more

suitable for transforming into a STL file for experimental validation.

In Step 2, the marching cubes isosurface extraction results in a triangular surface mesh

that is consistent with STL format. Based on the StereoLithography interface specification,

the STL file can be generated with the calculation of the facet normals of the triangles [29].

In Step 3, linear tetrahedron elements (C3D4) with an average element size of 0.05 mm

are generated from the STL surface mesh by the Abaqus in-built function ‘generateMesh’

[28].

For simplicity of implementation, we use a Matlab in-built function ‘fmincon’ in Step 4.

This function employs the interior point algorithm to find the local minimum of constrained

nonlinear multivariable functions [30, 31], making use of finite difference to estimate the

gradients of the objective and constraint functions. This method is suited for small-scale

problems, e.g. optimization of a lattice core of sandwich structures.

Eigenvalue multiplicity is a critical issue for buckling optimization [32-34]. Previous

approaches addressed this problem based on sensitivity analysis regarding critical buckling

load as objective function [17, 33, 34] or constraint [18, 32]. The finite difference approach

used in this study essentially do not take this issue into consideration. Although the proposed

method would result in sub-optimal solutions, significant improvements of buckling load are

obtained for lattice core structures in Section 4.

For guaranteeing the computational accuracy, the maximum order of FS, i.e. M in Eq.

(2), should be carefully determined. In theory, an infinite number of FS terms are able to

generate any arbitrary shapes in order to find the true global optimum. However, when

making use of the finite difference sensitivity approach, a large number of design variables

would result in significant increase of computational costs. In addition, from a geometrical

point of view, high-order terms in FS describe sharp features, hence leading to undesirable

stress concentrations. It is therefore necessary to select an appropriate number for the design

variables to guarantee an optimized solution providing a significantly improved objective

function while maintaining acceptable numerical costs.

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3. Experimental Methodology

3.1. Manufacturing details

Generally, any isotropic material, given a known Young’s modulus, may be utilized for

validation of the proposed optimization method. AM offers improved design flexibility for

realizing geometrically complex structures compared to traditionally subtractive

manufacturing techniques. For best printing accuracy, structures for this work were built by

MultiJet Printing technology (ProJet 5500X) with VisiJet CR-WT rigid plastic material in

ultra-high definition mode to achieve highest resolution (750 × 750 × 890 DPI with 29 m

layer thickness). In the printing process, wax was used as a support material and removed at

an elevated temperature of 65°C following print completion.

3.2. Material properties

VisiJet CR-WT is a thermoset material with ABS-like properties [4]. Preliminary

experimental tests indicate that the material properties are mostly independent of the building

direction with regard to printing orientation on the platform. Stiffness measurements are also

similar for tension and compression tests. Figure 3 shows a typical tensile stress-strain curve

of the material obtained from circular rods of 1.5 mm and 3 mm diameters. Strain was

measured with a contactless video extensometer method between two points, 25 mm apart. It

is noted that the material does not display a distinct linear elastic region but shows a smooth

decrease of tangent modulus with applied stress. For analytical purposes, the Young’s

modulus for the stress range of 2-10 MPa was determined as E = 2.14 GPa. This stress range

corresponds to the region of linear compression behavior for the buckling case studies

presented in this paper.

Table 1. Properties of candidate printable materials (k = 1.5).

Material Young’s

Modulus [GPa]

Yield strength,

0.2% [MPa]

Upper bound r/lb

ratio [-]

Inconel 205 700 0.019

Ti-6Al-4V 110 1080 0.032

VisiJet CR-WT 2.14 23.5 0.033

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Figure 3. Stress-strain characteristics of VisiJet CR-WT.

3.3. Geometry details

Geometric dimensions are selected to ensure the occurrence of elastic buckling rather

than plastic yielding. In this case, the following inequality constraint must be satisfied for any

slender structure, i.e. the Euler buckling strength is smaller than the plastic yielding strength

[8, 35]:

2 2

2( )( ) 14

s

ys b

Ek r

l

(6)

where k is in the range of 1-2 depending on the buckling boundary condition, Es/ys is the

Young’s modulus-to-yield strength ratio and r/lb the slenderness ratio of the truss. This

condition yields an upper bound of initial truss diameter related to material properties. As

shown in Table 1 assuming k = 1.5, polymeric materials such as VisiJet CR-WT have a

higher upper bound of truss radius for a given unsupported length compared to metallic alloys,

hence resulting in a larger design freedom within the linear elastic buckling range. Based on

the above calculation, the initial uniform truss dimensions for both 1D columns and 3D

Kagome lattice cores are determined.

3.4. Printing accuracy

The circular truss dimensions were measured with calipers for uniform columns,

resulting in an average measurement of 1.58 ± 0.04 mm, which is slightly larger (+5.3%) than

the stipulated diameter of 1.5 mm. The larger cross-section is attributed to inherent surface

roughness and small amounts of wax remaining on the surface.

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Figure 4. The (a) initial and (b) optimized columns, and the (c) initial and (d) optimized Kagome

lattice structures connected to face plates.

3.5. Compression test

In order to create close to clamped-clamped boundary conditions for both configurations,

trusses were joined to two rectangular face plates (see Figure 4). Preliminary experimental

studies on the single column were used to optimize the size and thickness of the face plates as

well as the fillet geometry for truss and face plate joints. For best imaging contrast, all single

column trusses were painted black, adding two white markers close to the face plates to be

captured (Figure 4 (a) and (b)). Since Kagome lattice structures have no vertical member, two

vertical prisms were printed at the center of the face plates for visible markers placement

(Figure 4 (c) and (d)).

Compression testing was performed on the initial and optimized 1D columns and the 3D

Kagome lattice structures, using an Instron 5982 universal test machine fitted with a 1 kN

load cell. Six specimens were tested for each structure and all specimens were deformed

under a displacement control of 0.65 mm/min. The compressive strain of columns and

Kagome lattice structures was measured in a contactless manner using an Instron AVE 2

video extensometer.

The Young’s modulus of the material, Es, may be calculated from the experimentally

measured stiffness for structures with uniform truss diameters. For single columns it is

calculated as:

13

cols

col

K lE

A (7)

where l is equal to the column’s unsupported length, Kcol is the measured stiffness from the

load-displacement curve and Acol is the cross-sectional area of the uniform column.

Similarly, for Kagome lattice core structures, we derive [9]:

4sin ( )

cells

EE

, with

Kag

cell

face

K hE

A (8)

where h is the height, KKag is the experimental stiffness measured from the load-displacement

curve, Aface is the top or bottom face area and is the relative density of the Kagome unit

cell.

4. Optimization Examples and Validation

4.1. Case study 1: Clamped-clamped column

Firstly, a 1D column is optimized by the proposed method. To fulfill elastic buckling

requirements according to Eq. (6), the geometry of the 1D column is defined with a length l =

24.4 mm and an initial uniform diameter of d = 1.5 mm. The volume constraint is set to V =

43 mm3.

Eliminating the sine terms of the FS because of geometric symmetries, the buckling

optimization problem of the clamped-clamped column is reformulated by:

1 ( ) ( 0,1,..., )Maximize

Subject to: ( ) 0

d 0

n

n

0 i i

J n M

+

V

α

K K v (9)

where { }n are the design variables. The upper and lower bounds of the first nine design

variables are (3, 0.4, 0.4, 0.3, 0.3, 0.1, 0.1, 0.1, 0.1) and (1, -0.3, -0.3, -0.2, -0.2, -0.1, -0.1, -

0.1, -0.1), respectively. The upper bound of α0 is twice of the initial diameter in order to

ensure a sufficiently large design space, while its lower bound is set to a positive value in

order to avoid the existence of zero cross-sections during the optimization process. Upper and

lower bounds are set closer to zero with increasing FS order because the contributions of the

FS coefficients decrease with increasing FS order.

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Figure 5. Optimization results of clamped-clamped buckling columns. (a) Undeformed and first

buckling mode of initial design (M = 0) and optimized designs with M = 2, 4, 6 and 8. The contour

plots show the displacement field and are scaled to 1.0. (b) Improvement for critical load and number

of iterations with respect to the FS orders involved.

Optimization results of various M values are shown in Figure 5 and Table 2. A

maximum improvement of 26.8% is achieved for 1D columns with a non-uniform diameter

compared to the original uniform column. Higher critical load improvement is achieved by

increasing the FS orders; however, from the 4th order on, the improvement rate for the critical

load decreases while the computational cost in terms of iterations grows rapidly (Figure 5 (b)).

To this end, an economic selection of M = 4 for our example which can result in 25.1%

improvement for the critical load. For better buckling resistance, users can further increase

the M-value and use the optimized result with M = 4 as an initial starting point for

optimization.

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Table 2. Optimized design parameters and the associated critical loads of the clamped-clamped 1D

columns with different FS orders.

Number of

orders M 0 1( , ,..., )M

Critical buckling

load [N]

0 (1.500) 36.30

2 (1.489 0.003 0.142) 43.75

4 (1.488 0.010 0.138 0.011 -0.055) 45.42

6 (1.478 0.038 0.158 -0.025 -0.081 0.024 0.036) 45.42

8 (1.481 -0.020 0.151 0.029 -0.072 0.004 0.024

0.016 -0.032 ) 46.02

The influence of initial design on the optimization results is investigated using another

initial design, i.e. αInit = (1.500, 0.35, 0), with α1 approaching to its upper bound. The results

show that the different initial design results in a similar optimized solution with an increased

number of iterations for convergence (see Figure 6). This may be attributed to the appropriate

selection of the upper and lower bounds for the design variables.

Figure 6. Optimization results of the clamped-clamped column with the initial design of αInit = (1.50,

0.35, 0). The improvements are based on the critical buckling load of the straight column with 1.5 mm

in diameter.

The optimal shape design of a slender clamped-clamped column against buckling failure

is one of the general benchmark studies of buckling optimization as discussed earlier. The

authors in [12] predicted and proved the theoretical maximum buckling load is 33% larger

than the critical buckling load of the uniform column. Similarly, the maximum critical load

improvements of methods developed by [17] (32.42% with 200 design variables) and [34]

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(32% with 50 design variables) are higher than those of our proposed method (26.8% with 8

orders FS). This discrepancy exists because the number of the FS orders is relatively small

(usually less than ten) compared to tens to hundreds of beam elements in the previously

derived solutions. The small number of the design variables in our work limits the design

space and results in sub-optimal solutions. While these earlier studies set a benchmark for the

possible design improvement, the methodologies may not be well-suited for AM due to the

added complexity of creating smooth truss joint connections and diameter variations from

discrete variable output.

Figure 7. Buckling modes (front and top views) of the (a) initial and (b) optimized column designs

with 8 orders of FS. The undeformed shapes (labelled by 0) and buckling modes I-VI are displayed

for both designs. Eigenvalues are displayed for each buckling mode. The contour plots show the

displacement field and is scaled to 1.0.

17

To further understand the improved design characteristics, the first six eigenvalues and

buckling modes of the initial and optimal designs with 8 orders of FS are presented in Figure

7. It is found that the buckling modes for a single column are intrinsically bimodal; each pair

of eigenvalues is quite close and the corresponding buckling modes are orthogonal. During

the optimization process, the first two buckling modes are enhanced to the detriment of

higher buckling modes (modes III-VI), and for the optimized column, the first and third

eigenvalues are now much closer.

The AM process generally introduces minimum size constraints due to printing

capability. Thin truss diameters, typically less than 1 mm, are more likely to fail during

printing/de-waxing and they can also be more sensitive to dimensional inaccuracies resulting

from the printer’s print resolution. In the present method, the minimum truss size constraint

can be easily introduced during the geometric modelling stage. If the updated radii violate the

size constraint, they are set equal to the prescribed minimum radius minr :

min( , , ) max( ( , , ), )cr x y z r x y z r (10)

where rc(x, y, z) and r(x, y, z) are the modified and original radius functions. The same

column optimization problem is analyzed again with a minimum size constraint, in which 4

orders of FS are involved. Optimized shapes with different minimum size constraints are

shown in Figure 8. It can be observed that the minimum size constraint limits the design

space; hence lower improvements of critical load are achieved. The critical load improvement

for the unconstrained problem represents the upper bound limit. Since the minimum radius of

the optimized shape without the size constraint is 0.52 mm, the minimum size constraint is

inactive when rmin < 0.52 mm (0.69r0), where r0 represents the radius of the initial uniform

column as shown in Figure 8 (b).

Figure 9 shows the experimental validation of the case study comparing the initial

buckling load and the optimized buckling load with 4 orders of FS and a minimum radius

constraint of rmin = 0.53 mm (0.7r0). The results firstly show that the experimental buckling

loads of the optimized columns are improved by an average of 15% for the test, while their

stiffness is 16% lower than that of uniform columns (see Table 3). The stiffness reduction is

attributed to the fact that the thinner column segments have significantly lower stiffness and

would weaken the stiffness of the whole structure. The experimental stiffness also results in

accurate calculations for the Young’s modulus, hence validating the displacement

measurement and the printing accuracy.

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Figure 8. Optimization results of clamped-clamped buckling columns with size constraint. (a) Shapes

and the buckling modes with various constraint values for minimum radius. The contour plots show

the displacement field and is scaled to 1.0. (b) Improvement for critical load as a function of minimum

size constraint.

Table 3. Experimental properties of initial and optimized 1D columns.

Critical load

[N]

Stiffness Kcol

[N/mm]

Young’s modulus from

stiffness [MPa], Eq. (7)

Initial 36.7 ± 2.9 165.6 ± 11.1 2286.0 ± 153.4

Optimized 42.2 ± 3.2 137.2 ± 8.3

Improvement 15% -16%

19

Figure 9. Load-displacement graphs of the initial and optimized 1D columns with minimal radius

constraint. Two typical buckling modes due to the experimental imperfections of the optimized

columns are highlighted.

Compared to the numerical results, a reduction of load improvement can be observed in

experiment. This is mainly caused by the imperfections of manufacturing and experimental

setup, including geometric imperfections and load misalignment. In addition, a change of

buckling modes from mode I/II to III/IV can also be observed for the optimized columns

during testing due to the imperfections, as shown in Figure 8. The change in buckling modes

was only observed for the optimized columns, because the difference of eigenvalues between

modes I/II and III/IV are very small and the modes are likely to switch if imperfections are

introduced. As the misalignment makes the ends of the columns tend to rotate instead of

being clamped, the clamped-clamped boundary condition cannot be accurately preformed. A

less distinct improvement for buckling load is expected due to the change of boundary

condition.

Analyzing the sensitivity of buckling behavior to manufacturing imperfections is a

critical issue, especially for slender and thin walled structures. The imperfections introduced

in manufacturing processes and experimental setup, such as initial geometric imperfections

and imperfect boundary conditions, can strongly influence buckling loads in experiments [36,

37]. Furthermore, the geometric imperfection introduced in FE analysis affect the buckling

strength as well [37-39]. The detailed study regarding to the effect of imperfections on

buckling behavior is beyond the scope of this paper.

4.2. Case Study 2: Kagome lattice core

Kagome lattices represent a specific lattice type that is found to be mechanically

efficient. A specific application is to construct sandwich structures. Kagome lattice is an ideal

20

core for sandwich structures because it offers high specific stiffness and strength due to the

near-ideal stretch-dominated behavior [8, 35, 40]. The ability to design and optimize 3D

lattice structures with joint connections like Kagome lattices in an efficient manner is a major

advantage of the developed method in this work.

In this example we aim to maximize the buckling strength of a sandwiched Kagome

lattice core, namely Kagome core enhanced by two face plates. The Kagome lattice core is

characterized by core height, truss internal angle and diameter (Figure 1). The height of the

core is fixed at h = 40 mm, with truss diameter of 1.53 mm and truss angle of 55° with

respect to the horizontal faces. The relative density of this unit cell is 33.32 10 . The

corresponding volume constraint is V = 274 mm3. Satisfying Eq. (6), the core fails by elastic

buckling with maximum truss stresses below yield stress of VisiJet CR-WT under

compression. A compressive load is applied to the reference point coupling the top face of

truss members, and in-plane movements and rotations are restricted representing clamped

boundary conditions at the top and bottom faces of the core. The given material is same as the

1D column example with a tangent modulus of E = 2.14 GPa for the given stress range as

experienced per truss. Since the geometry of Kagome lattice core is constructed based on

implicit surfaces and blended objects are introduced as per Eq. (4), three additional

parameters ai (i = 1, 2, 3) are required to describe the joint at the centre of the core. As the

three trusses of Kagome lattice core are considered identical in this work, we set a = a1 = a2 =

a3. The parameter is fixed by a = 12 based on preliminary stress analyses in order to avoid

stress concentration, and not further considered as a design variable in this case study. Finally,

sine terms of the FS are again omitted because of symmetry conditions for the core geometry.

The optimization problem is then the same as previously stated in Eq. (9). The upper and

lower bounds of the first nine design variables are (3, 0.2, 0.2, 0.2, 0.15, 0.15, 0.1, 0.05, 0.05)

and (0.8, -0.1, -0.1, -0.1, -0.1, -0.07, -0.07, -0.05, -0.05), respectively. Again, the bounds

become narrower with increasing FS order.

21

Figure 10. The optimization results of Kagome lattice structures. (a) The undeformed and the first

buckling mode of initial design (M = 0) and optimized designs with M = 2, 4, 6, and 8. The contour

plots show the displacement field and is scaled to 1.0. (b) Improvement for critical load and iterations

with respect to the orders involved.

In a similar manner to the first case study, results of optimization and convergence

studies are shown in Table 4 and Figure 10. An economic choice of the order of FS is again

M = 4 using the proposed method (Figure 10 (b)). It is noted that the variable α3 of some

optimized designs (e.g. M = 4, 6, 8) is close to its lower bound. In this case a reduced lower

bound for α3 may result in a higher critical load; however, since α3 corresponds to a high-

order term, a small difference in α3 will have little influence on the optimized results. To

demonstrate this, we re-ran the optimization of the Kagome lattice core with M = 4 and set

the lower bound of α3 to be -0.2. The optimized result is αOpt = (1.509, 0.035, -0.041, -0.126,

0.039), with α3 now converging away from its lower bound value. The critical buckling load

22

is 65.90 N, which is only slightly higher than the previous solution shown in Table 4.

Buckling modes and eigenvalues of initial and optimized design with 8 orders of FS are

compared in Figure 11. It is observed that the first three buckling modes are enhanced,

meanwhile sacrificing the eigenvalues for higher buckling modes. The optimization results

using another initial design, αInit = (1.1, -0.08, -0.08), with the design variables approaching

to the lower bounds are shown in Figure 12. Again, the initial design has a small influence on

the optimized solutions, while it affects the iteration numbers.

It is an interesting idea to generate the Kagome lattice core from the result of an

optimized column to minimize computational time. However, the generated Kagome lattice

core from the optimized clamped-clamped column has a significantly lower critical buckling

load (39.69 N) than the optimized one (59.72 N). This is because boundary conditions and

buckling modes of the clamped-clamped column and a single truss of the Kagome lattice core

are different. In addition, the truss joints also influence the optimization results.

Figure 11. Buckling modes and eigenvalues of (a) initial Kagome lattice core and (b) optimized

design with 8 orders of FS. The undeformed shape (labelled by 0) and buckling modes I-V are shown.

Eigenvalues are displayed for each buckling mode. The contour plots show the displacement field for

the respective buckling mode and is scaled to 1.0.

23

Figure 12. Optimization results of Kagome lattice structures with initial design of αInit = (1.1, -0.08, -

0.08). The improvements are based on the critical buckling load with straight trusses of 1.53 mm in

diameter.

Table 4. Optimized design parameters and critical loads of Kagome lattice structures with different

FS orders.

Number of orders

M 0 1( , ,..., )M

Critical buckling

load [N]

0 (1.530) 55.48

2 (1.519 0.039 -0.092) 56.17

4 (1.516 0.036 -0.019 -0.099 0.040) 65.02

6 (1.512 0.017 -0.006 -0.099 0.023 -0.001 -0.053) 65.16

8 (1.509 0.062 -0.062 -0.097 0.053 -0.028 -0.011

0.030 -0.025) 66.82

Figure 13 compares the experimental load-displacement graphs between the initial and

optimized Kagome lattice core structures with M = 8 and the corresponding deformations.

Both initial and optimized structures fail under buckling mode I/II as the deformed shapes of

experimental and numerical results in Figure 13. As numerically predicted, the optimized

Kagome lattice structures show a 24% improvement for the critical load during testing when

compared to the initial structures with the same volume and material properties (see Table 5).

The stiffness values of the initial Kagome core structures based on measured loads and

displacements agree with the analytical predictions (Eq. 8). The back-calculated Young’s

modulus of VisiJet CR-WT is slightly higher than that determined in section 3.2 (see Table 5).

It is noted that the experimental critical loads of the initial and optimized Kagome core

structures are higher than numerical predictions (see Table 4 and 5). This is attributed to the

higher back-calculated Young’s modulus and the slightly larger printed truss diameters, as

24

discussed in section 3. The good agreement between the experimental and optimization

results for deformed shapes and the improvement percentage of critical buckling loads

validates the effectiveness of the proposed buckling optimization method. It is worth noting

that the experimental results do not show the change of buckling modes compared to the

simulation results with perfect geometries and boundary conditions. This is because (1) the

Kagome core structures preform high stability in comparison with single columns; and (2) the

optimized Kagome core shows a large margin between the eigenvalues of modes I/II and III

(see Figure 11). Finally, the optimized Kagome lattice core structures have slightly smaller

stiffness than those with uniform trusses. This stiffness reduction is a result of the thinner

truss segments embedded in the optimized design. Based on buckling optimization results,

the smallest truss diameter of optimized Kagome lattice structure is 1.12 mm, which is 25%

less than the initial uniform value. Hence, the thin segments act as regions of lower stiffness

under static deformation.

Different failure locations of the initial and optimized Kagome lattice structures are

observed following peak load and localized truss buckling. As highlighted in Figure 13,

optimized Kagome lattice cores fractured around the thin truss segments, while failure

locations of initial Kagome lattice structures are found at truss locations with highest bending

deformation. As a result of the local thinning altering the failure locations, uniform Kagome

lattice structures also exhibit significantly higher elongation to failure compared with the

optimized ones.

Table 5. Experimental properties of initial and optimized Kagome lattice core structures.

Critical load

[N]

Stiffness KKag

[N/mm]

Young’s modulus from

stiffness [MPa], Eq. (8)

Initial 57.2 ± 4.6 169.3 ± 11.6 2195.4 ± 150.2

Optimized 71.2 ± 5.5 160.6 ± 15.0

Improvement 24% -5%

25

Figure 13. Load-displacement graphs of the initial and optimized Kagome lattice core structures.

5. Conclusions

In this paper, we optimize the critical buckling load for Kagome lattice core structures

with free-form trusses formulated by Fourier series. The initial and optimized structures are

fabricated by AM. Both numerical optimization and experimental testing on 1D columns and

3D sandwiched Kagome lattice core structures show that the critical buckling loads of the

optimized structures can be significantly improved. The good agreement between simulation

and experimental results validates the effectiveness of the proposed optimization method.

Improving the buckling load of lattice cores using the proposed method has the

following advantages: (1) it allows design and optimization of 3D lattice core structures with

smooth truss joints; (2) a limited number (usually less than 10) of design variables can lead to

moderate numerical costs; (3) the FS-based implicit geometric representation is well suited

for AM. In addition, any arbitrary kinds of lattice core structures and loading configurations

are supported. Scaling of the proposed approach to a large number of connected trusses is

theoretically possible, where sensitivity analysis-based approaches [16-19] are more preferred

for the computational accuracy and efficiency.

Low density lattice structures commonly have slender truss members and are dominated

by buckling failure. In such situations the presented buckling optimization method has shown

to be an effective means of improving their compressive strength. In the moderate density

range, however, plastic yielding dominates the failure instead of buckling. Hence our future

work will focus on strength optimization including both elastic buckling and plastic yielding

failures as well as identifying the optimum geometric parameters for the best failure

26

transition point. It is also envisaged that further nonlinear optimization is possible for

maximum energy uptake during compression loading (area under the load-displacement

curve) rather than improvement for peak load only.

Acknowledgments

This work is supported by a National University of Singapore (NUS) scholarship for Mr

Lei Zhang. The authors further acknowledge the support from the Agency for Science,

Technology and Research and the Engineering Research Council of Singapore through the

Additive Manufacturing Centre Initiative (SERC Grant no. 142 68 00088). The authors

would also like to thank SIMTech staff members Dr. Hongyi Yang and Mr Hengky Eng for

3D printing of the tested structures.

27

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