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Article published in

Current Chinese Science, 2021, 1, 151-159

Lessons learnt from a national competition on

structural optimization and additive manufacturing

Yulin XIONG1, Dingwen BAO1, Xin YAN1,2, Tao XU1, Yi Min XIE1,*

1 Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001,

Australia

2 Centre of Architecture Research and Design, University of Chinese Academy of Sciences, Beijing 100190,

China

* Corresponding author. E-mail address: mike.xie@rmit.edu.au (Y.M. Xie).

Abstract

In a recent national competition on structural optimization and additive manufacturing in China, we won the first

prize with a highly efficient and elegant structural design. This paper describes important lessons learnt from the

competition. We used the bi-directional evolutionary structural optimization (BESO) technique for the

conceptional design and a smoothing technique to work out the details of the structural boundaries. This approach

was applied to the design of a hinge arm which must satisfy all the conditions set by the competition committee.

We carried out a nonlinear quasi-static loading simulation before the competition committee fabricated the design

using a stereolithography 3D printer. In the actual mechanical testing conducted by the national competition

committee, our design displayed superior performance to other entries. The insights gained from this award-

winning design will be useful for engineers and architects to employ topology optimization techniques to create

efficient and elegant structures in the future.

Keywords: Topology optimization; Bi-directional evolutionary structural optimization (BESO); Structural

optimization; Additive manufacturing; Load-bearing structural design; National competition;

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1. Introduction

As an advanced design technique, topology optimization has received much attention over the past three decades.

The aim of topology optimization is to find an optimal material layout to maximize the structural performance

while satisfying certain constraints. It is a useful tool for the conceptional design. The development of topology

optimization can be traced back to Michell’s work to minimize the weight of the truss layout [1]. Later researchers

proposed several methods for topology optimization of continuum structures, e.g., the solid isotropic material with

penalization (SIMP) method [2], the bi-directional evolutionary structural optimization (BESO) method [3], [4]

and the level set method (LSM) [5], [6], etc. By changing the prescribed objective, these techniques can be applied

to various disciplines, including biomechanics [7], thermodynamics [8], [9] acoustics [10] and optics [11], [12].

Conventional stiffness-based topology optimization is well developed and increasingly used for structural designs

in many fields, e.g., architecture [13], automotive [14] and aerospace industries [15]. It provides conceptional

designs in the load-bearing system to maximize structural stiffness while meeting the volume constraint. Some

optimized designs for engineering applications such as railway vehicles [16] and aircraft components [17]

demonstrate the effectiveness of topology optimization. However, there are still some difficulties in achieving

optimized designs in practice. Firstly, for the element-based topology optimization techniques, the results contain

zig-zag boundaries [18]. Thus, post-processing is necessary to determine the actual structural boundary [19], [20].

Secondly, topology-optimized designs are often accompanied by geometric complexity, which are hard to

manufacture by traditional machining. Additive manufacturing (AM) techniques provide a solution to this

problem [21]. Benefiting from its characteristics of the layer-by-layer formation of 3D components [22], AM can

be integrated with topology optimization to create highly efficient products with complex geometries in practical

applications.

In this work, we describe an efficient methodology for realizing optimized designs in practice, which makes the

topology optimization technique a usable and accessible technology. The BESO technique provides the

conceptional design, and the topology-optimized result is post-processed to obtain smooth structural boundaries.

The case study that we present considers the load-bearing structural design of a hinge arm. The design satisfies

all the conditions set by the third national competition on structural optimization and additive manufacturing,

which was organized by China Aerospace Science and Industry Corporation in Beijing in 2019. This competition

is one of the most prestigious structural optimization contests in China, attracting entries from a large number of

experienced aerospace engineers and researchers. In total there were 135 entries in the preliminary competition,

of which only 18 teams got into the final. All the designs must be verified through numerical simulations by

designers and confirmed experimentally by the competition committee. During the actual loading test in the final

competition, our design carried the highest load and won the first prize, which validated the effectiveness of our

design methodology. The insights gained from this award-winning design will be useful for engineers and

architects to use BESO to create elegant and efficient structures in the future [23].

The paper is organized as follows. In Section 2, the BESO technique for designing load-bearing structures is

introduced. In Section 3, the post-process technique for reconstructing the optimized result is proposed. In Section

4, the methodology is applied to the design of a hinge arm. In Section 5, the numerical simulation and loading test

are used for investigating the mechanical behaviour of the design, and the advantages of the topology-optimized

design are discussed. In Section 6, the main conclusions of this study are summarized.

2. Methodology for designing lightweight and load-bearing structure

2.1. BESO method

In this work, we use the BESO technique to create efficient load-bearing structures. As an important branch of

topology optimization, the BESO technique is based on the simple concept of gradually removing inefficient

material from a structure and adding material to the most needed locations at the same time [24]. It is widely

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recognized owing to its high-quality topology solutions [25], ease of understanding and implementation [26], and

excellent computational efficiency [27].

Considering the design domain where the material will be distributed. The whole domain is modelled as

elements that fill in the domain. We denote as the design variable of element . Since the material has only

linear elastic behaviour, the problem of maximizing the load-bearing capacity of structures is equivalent to

maximizing structural stiffness. With a given volume constraint , the structural stiffness is maximized when the

compliance is minimized [28]. The mathematical form of the optimization model can be expressed as:

*

Subject

11

Minimize: 22

or1

to (

,

:)

i min

V

x

dV

x

xx

=

=

=

TT

F u u Ku

(1)

where is the structural compliance, and are the global force vector, global displacement vector, and

stiffness matrix, respectively. Different from other topology optimization methods, the design variable of the

BESO technique is of discrete values. and indicate the solid and void element , where

to avoid singularities during finite element analysis (FEA) [29]. Using the adjoint method, the

compliance with regard to the change in the design variable can be found as [30]

10

( ) 1 ,

22

p

i

ii

C x p x

xx

−

= − = −

TT

i

i i i i

K

u u u K u

(2)

where , and denote the displacement vector of element , solid elemental stiffness matrix, and the penalty

exponent of material interpolation scheme [31], respectively. The sensitivity of element in the BESO technique

is defined as

1 ( ) 1 1 ,

22

pi

ii

i i i

SE

Cx x

p x x x

= − = =

T

i 0 i

u K u

(3)

where denote the stiffness matrix of element . It should be noted that the term

in the elemental

sensitivity is the elemental strain energy that carried out from FEA.

The raw sensitivity is processed to avoid mesh-dependent solutions and checkerboard patterns [32]. The and

are the filter radius and the distance between the elements and , respectively. An elemental sensitivity filter

scheme is used in each iteration, which is defined as follows.

,

ij j

j

iij

j

w

w

=

(4)

where is the weight factor of the filter scheme, which is formulated as

max( ).

ij min ij

w r r=−

(5)

The weight is independent of the elemental sensitivity and is calculated before the optimization process. To

improve the convergence of the BESO technique, the filtered sensitivity is further averaged with its historical

information [33]. Here, the sensitivity of the current iteration is simply averaged with that of the previous

iteration, as

1

1( ).

2

k k k

i i i

−

=+

(6)

4

BESO updates design from the initial full design and gradually reduces the structural volume in each iteration.

The volume of next iteration is determined by an evolutionary ratio and the current structural volume

as

1*

min( , (1 )).

kk

V V V er

+=−

(7)

It is noted that once reaches objective volume , it will remain constant. The evolutionary ratio is 1% in

this work. A threshold is employed for updating the design variables according to the and their

sensitivities. The update scheme of BESO is defined as

1

if ,

1 if ,

otherwise.

k

min i th

kk

ii th

k

i

xa

xa

x

+

=

(8)

The present scheme indicates that the design variables of solid elements are switched from 1 to if their

sensitivities are lower than the threshold, and the design variables of void elements are switched from to 1

if the sensitivities are higher than the threshold. The optimization algorithm is implemented by Python and linked

to Abaqus. The validity and convergence of the basic BESO computational framework have been analyzed and

verified in [34].

2.2. Smoothing technique for post-processing

The optimized results from BESO technique contain zig-zag boundaries, which may lead to the reduction of

structural performance and difficulty to manufacture. In order to obtain accurate structural boundaries, a

smoothing technique is proposed for reconstructing the element-based model [35-36].

As a well-known method for extraction of iso-surface, the marching cubes (MC) algorithm is integrated into the

smoothing technique to determine structural contour surface in the first step. The structural contour surface is

defined as a surface that represents points of a constant design variable (normally ) within a volume of

space. The algorithm traverses the design domain, taking eight adjacent points as an imaginary cube, then

determines the polygons that represent the part of the iso-surface within the cube. According to the states of eight

points, a total 256 () possible polygon configurations are defined in the algorithm. The polygon configurations

can be reduced to 23 patterns considering rotational symmetry. For computational efficiency reason, these patterns

are typically encoded and stored in a lookup table according to the states of the points in the cube. The vertex of

the generated polygons is placed along the cube’s edge by linearly interpolating the value of the points that are

connected by the edge [37]. These individual polygons are then merged into the iso-surface. From the perspective

of computer graphics, the MC algorithm transformed the element-based model into a polygon mesh model (i.e.

the iso-surface) without changing the structural topologies. This model type is widely used in computer graphics

to generate polygon mesh from element-based mesh, for its advantages of small storage space and easy

modification. In computational graphics, there are many modified versions of the MC algorithm for certain

requirements (such as the type of polygons). In this work, the basic MC algorithm is found to be quite effective.

In this work, the Laplacian surface editing technique [38] is employed to smooth the structure for the

determination of the actuary structural boundaries. The technique encodes vertices of the iso-surface from

Euclidean coordinates to Lagrange coordinates. It provides a representation of the surface mesh, where the

reconstruction of global coordinates always preserves the geometric details. Thus, the structural topology would

not be changed during surface editing. The Laplacian coordinate of a vertex is defined as

() ,

j

ii

j N i i

v

vd

=−

(9)

where and represent the neighbours of a vertex and neighbour set, respectively. And is the number

of the neighbour of . Surface editing operations can be efficiently and robustly applied to surface mesh with

Laplacian coordinates [38]. When the surface is edited, the Laplacian coordinates of vertices are decoded into

Euclidean coordinates. Laplacian smoothing is one of the most common algorithms for mesh denoising. It

repeatedly and simultaneously adjusts the coordinate of each vertex in the mesh to the geometric centre of its

neighbours [39].

5

()

1( ).

i j i

j N i i

d

−

(10)

Although the Laplacian smoothing algorithm is simple and efficient, it may produce an over smoothed result.

Some small structural features such as right angles and sharp boundaries might be lost during Laplacian smoothing.

An optional way is to convert the triangular mesh to a quadrangular mesh before Laplacian smoothing.

Quadrangular mesh could snap the sharp details, which is more suitable for mechanical models than triangular

mesh [40].

3. Application of the methodology to design a hinge arm

The case study that we consider the design of a hinge arm. The design conditions are set by the competition

committee. As shown in Fig. 1a, the hinge arm is fixed on the base by four connecting bolts, with an upward load

on its cantilever. The optimization goal is to maximize the load-bearing capacity of the structure while meeting

volume constraint (40 ml).

The structure is fabricated by the stereolithography apparatus printer RS4500. The material used in 3D printing is

photosensitive resin, and its Young’s modulus and Poisson’s ratio is 2510 MPa and 0.41, respectively. Since the

material has only linear elastic behaviour, the optimization objective can be equivalent to minimizing structural

compliance. There are some limitations of additive manufacturing that need to be considered in topology

optimization. It is required that the structural member size is larger than 1 mm, and the wall thickness is no smaller

than 0.5 mm. Besides, there should be no enclosed void in the structure.

Fig. 1b shows the requirements of the design domain, including non-design domain, bolts positions and loading

position provided by the organization. It is noted that the height and length of the design domain are not restricted

(red dot lines). Therefore, determining the appropriate design domain size for topology optimization is the key to

achieving the structural design with high performance.

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Fig. 1 (a) Boundary conditions and (b) requirements of the design domain.

With uncertain height and length of the design domain, several initial designs (Fig. 2a) were tested for finding an

appropriate design domain size. The design domain size of these models is changed according to the unreasonable

part in their results. Four tunnels are dug out in the initial design for installation of the bolts, and the left part is

devised for the installation of the loading tool and the connecting pin. The heights and length of these initial

designs are shown in Fig. 2a. The coarse mesh is used for determining the size of the design domain quickly.

Elements size is about . In order to simplify the finite element calculation, z-direction of

the upper surfaces of four bolt holes are fixed, and the inner surface of them are pinned to simulate the structure

being bolted to the base. The bottom face of the front of the structure is fixed in the z-direction as well to simulate

the supporting force provided by the base. The optimized results are shown in Fig. 2b. The top and left sides of

the optimized results are flat, which is unreasonable. These flat parts are caused by the limitation of the design

domain and indicate that the design domain should be increased in height and length.

Fig. 2 (a) Test design domains and (b) the optimized results.

The height, width and length of the final initial design is 95 mm, 60 mm and 165 mm, respectively. The height of

tunnels is 36 mm. A finer finite element mesh is employed to obtain an accurate optimal solution. Elements size

is about . It is noted that topology optimization based on a fine mesh might produce a

better solution, but more thin members. These thin members in the structure are easily broken during quasi-static

loading test. In order to improve the robustness and bearing capacity of the structure, the filter radius is set to 8

mm to obtain an optimized design with thicker members.

As shown in Fig. 3a, the BESO result is an arch shape supported by three cylinders. These cylinders extend into

six branches that connect to the two bolts holes in the middle, which are bolted to the base. External forces cause

an additional torsional load with the bolts in the middle as a rotational fulcrum. To balance the torsional load, the

BESO technique naturally produces a vertical cylinder at the front part of the structure. In this part, a balanced

torque can be generated by using the supporting force provided by the base. Besides, the front bolts are connected

to the cylinder, and they give tension to limit structural deformation in the length direction. The evolution history

of structural compliance and volume can be seen in Fig. 3b.

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Fig. 3 (a) The BESO result and (b) evolution histories of the compliance and the volume fraction.

Because the mesh result is composed of finite elements, such as triangles or quadrangles for shell and cubes or

tetrahedrons for solid, the BESO result is always a mesh with a coarse, irregular surface. The Grasshopper plugin

Ameba (http://ameba.xieym.com) developed by XIE Technologies has been used to solve the problem. The

smooth technique in Ameba has a strong ability to smooth the irregular surfaces. To place the loading bar and

bolts tightly and stably, some solid elements are fixed to retain the accurate geometry details during the smoothing

process. Figure 4 illustrates the process of smoothing. Once the form of design is finalized, it has been imported

into Abaqus for finite element analysis to get the accurate structural performance feedback which helps to re-test

and fine-tune the form to fix some structural defects (e.g. stress concentration caused by sharp boundaries) and

ensure the design has a better structural performance than unmodified one based on keeping the basic generated

geometry.

Fig. 4 (a) The original model, (b) the MC result and (c) the smoothed model.

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4. Experimental setup

In the paper, the smoothed BESO result is manufactured in photosensitive resin by a stereolithography apparatus

printer. The load-bearing capacity of the specimen was measured experimentally through a quasi-static loading

test. The competition committee provides both the 3D printer and the experimental machine. The experiment was

carried out on the compression test machine, as shown in Fig. 5. The base is made of 10 mm thick steel plate and

is fixed to the supporting frame by bolts on both sides. There are four bolt holes with a diameter of 6.5 mm in the

middle. These bolt holes are spaced 30 mm apart in width and 45 mm apart in length. Structures are fixed

underneath the base by four bolts. To avoid the torsion, the load is transferred from the loading cell to

the cantilever of the specimen through a connecting pin.

Fig. 5 The setup of the loading test.

During the test, the displacement and force are measured with a sampling frequency of 20 Hz. The force signals

are detected by the loading cell and collected by an oscilloscope. The test machine loaded downwards; the

specimen is subjected to an upward load correspondingly. The loading process will continue until the specimen

is destroyed. The competition committee provides these machines. It should be noted that in this study, only the

vertical reaction force of the structure is considered during loading. Because the loading time of the specimen

cannot be accurately estimated, the oscilloscope has a total duration of 100 seconds. It records the force curve

during the loading test, and the highest value of the curve is the maximum load that the specimen can bear (i.e.,

load-bearing capacity). The designs from 17 teams were tested on-site during the competition. The measurement

was obtained for ambient temperatures of 20 degrees.

5. Results and discussion

The experimental loading history of the award-winning design in the paper is detected by the loading cell on the

top is shown in Fig. 6a. The loading apparatus consists of a pin and a clamp connected to it. The testing force

curve and the displacement curve are blue and red, respectively. It should be noted that since the data is recorded

after the connecting pin has contacted the structure and started loading, a positive force can be observed in the

initial stage. The speed of the loading cell is 10 mm/min. As the loading process continues, the force gradually

rises until the structure is destroyed. The peak force is 1.920 kN, which appears at about 90 s. Then the force drops

rapidly, which indicates that the structure is completely broken and detached from the base. As shown in Fig. 6a,

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the horizontal section of the force curve (from 15s to 30s) may be caused by a slippage of the pin and clamp,

resulting in an underestimation of the structural load-bearing capacity.

Fig. 6 (a) Testing force and displacement curve and (b) simulation force and displacement curve of BESO

design.

A quasi-static nonlinear FEA on the same configuration of the quasi-static loading test is employed to verify the

proposed methodology. The material model used in the simulation is plastic rather than linear elastic. In this

case, the maximum stress of the material is set to 37 MPa at the strain of 3%. Then the material yields and

breaks at the strain of 9%. The smoothed model is meshed and simulated to investigate the response of the

structure. The connecting pin is loaded with the speed of 10 mm/min. Z-direction of the upper surfaces of four

bolt holes are fixed, and the inner surface of them are pinned. The bottom face of the front of the structure is

fixed in the z-direction to simulate the support provided by the base. The simulation loading history and

displacement of the BESO design is shown in Fig. 6b. The trend can be seen in the simulation that the force

increases until the structure broke. The peak force is 2.11 kN, which is larger than the test result. There is no

force platform during the simulation. These phenomena confirm that the structural load-bearing capacity is

underestimated in the test. In addition, In the loading test, when the structure is broken, the testing machine

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stops recording the force, so the curve after 90s is a horizontal line. In the simulation, however, after the

structure is broken, the remaining part still bears the load, so the force slightly increases.

The comparison of the broken structure and the unbroken structure is shown in Fig. 7. It can be observed that four

parts of the structure are destroyed at the same time. It indicates that the structure is of equal strength to the design.

It should be noticed that only one side of the middle bolts was broken, which is caused by the structural buckling.

It is recommended to introduce structural buckling constraint into the optimization in the future.

Fig. 7 The comparison between the unbroken and broken models.

6. Conclusions

In this paper, we present an effective topology optimization approach to maximizing the structural load-bearing

capacity and establish a procedure to achieve optimized structural designs. The bi-directional evolutionary

structural optimization (BESO) technique provides the conceptional design, and the smoothing technique is used

for determining the final structural boundary. This methodology is applied to the design of a hinge arm which

satisfy all the conditions set by the national competition on structural optimization and additive manufacturing.

The design is fabricated by a stereolithography 3D printer and tested by the competition committee. The

mechanical test validated the advantages of the methodology. In the loading test of the final competition, our

design carried the highest loading and won the first prize of the competition, clearly demonstrating the ability of

BESO to create efficient and elegant design. From the competition, we learnt important lessons for designing

structures with superior performance, e.g., the importance of defining an appropriate initial design domain, the

usage of a relatively large filter radius to create a robust design, and the crucial role played by the smoothing

technique to reduce stress concentrations in some regions. These insights will be useful for engineers and

architects to employ topology optimization techniques to create elegant and efficient structures.

Acknowledgements

We wish to thank the committee from China Aerospace Science and Industry Corporation for organizing the 3rd

national competition on structural optimization and additive manufacturing. We would like to thank team

members of the Centre for Innovative Structures and Materials at RMIT University for sharing ideas and providing

help. We also wish to acknowledge the technical support from Ameba software development team of Nanjing

Ameba Engineering Structure Optimization Research Institute. This project was supported by the National Natural

Science Foundation of China (51778283).

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MATERIALS

The material used in 3D printing is photosensitive resin, and its Young’s modulus is 2510 MPa and Poisson’s

ratio is 0.41. The material model used in the loading simulation is plastic, and the maximum stress of the

material is set to 37 MPa at the strain of 3%. The material breaks at the strain of 9%.

List of Abbreviations

SIMP: solid isotropic material with penalization

BESO: bi-directional evolutionary structural optimization

LSM: level set method

AM：additive manufacturing

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MC: marching cubes

FUNDING

This project was supported by the National Natural Science Foundation of China (51778283).

ETHICS APPROVAL AND CONSENT TO PARTICIPATE

None.

AVAILABILITY OF DATA AND MATERIALS

The data supporting the findings of the article is available from the authors.

CONSENT FOR PUBLICATION

There is no individuals’ data in this study.

CONFLICT OF INTEREST

There is no conflict of interest in the study.