Axial mechanical properties and robust optimization of foam-filled hierarchical structures.pdf

Composite Structures 289 (2022) 115501
Available online 24 March 2022
0263-8223/© 2022 Elsevier Ltd. All rights reserved.
Axial mechanical properties and robust optimization of foam-filled
hierarchical structures
Xiang Xu a
, Yong Zhang b,*
, Jianguang Fang c
, Xinbo Chen a,*
, Zhe Liu a
, Yanan Xu a,d
,
Yunkai Gao a
a
School of Automotive Studies, Tongji University, Shanghai 201804, China
b
College of Mechanical Engineering and Automation, Huaqiao University, Xiamen 361021, China
c
School of Civil and Environmental Engineering, University of Technology Sydney, Sydney, NSW 2007, Australia
d
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
A R T I C L E I N F O
Keywords:
Crashworthiness
Thin-walled structures
Hierarchical structures
Robust optimization
Foam-filled structures
A B S T R A C T
In this study, two novel foam-filled hierarchical structures are proposed, namely foam-filled square hierarchical
tubes (FSHT) and foam-filled circular hierarchical tubes (FCHT). First, dynamic FEA models of the foam-filled
hierarchical tubes are established by LS-DYNA and validated against experimental data. The mechanical be
haviors of the foam-filled hierarchical tubes under axial load are investigated. The results show that novel foam-
filled hierarchical tubes have better crashworthiness performances than conventional foam-filled square tubes
(FST) and foam-filled circular tubes (FCT). Then, the theoretical solution of energy absorption is developed, and
its accuracy is validated. Next, the design parameters (foam density and thin-wall thicknesses) of FSHT and FCHT
are analyzed by orthogonal array design, and the effects of design variables on crashworthiness are studied.
Finally, to obtain the optimal design parameters of FSHT and FCHT, the robust optimization method considering
the manufacturing uncertainty is implemented by employing the interval uncertainty model (IUM) and the
feedforward neural network (FNN). Structures obtained with the proposed robust optimization method are of
more stable performance compared with the traditional deterministic optimization counterparts. The findings
provide a hybrid design combining hierarchical tubes with foam fillers with superior crashworthiness perfor
mance for energy-absorbing devices, and the robust optimization method can be used to design efficient light
weight crashworthy structures.
1. Introduction
In recent years, lightweight thin-walled structures have been widely
used in energy-absorbing devices in the field of transportation. Essen
tially, structural energy-absorbing is a dynamic behavior related to time
history. The external kinetic energy is transformed into the plastic en
ergy of the thin-walled structures when the collision occurs, to reduce
the impact damage for the passengers[1–5]. Over the past two decades,
a large number of experimental [6–8], numerical[9–12], and theoretical
methods [13–15] have been explored to analyze the crushing mechan
ical behavior of thin-walled structures.
One of the methods to improve crashworthiness is to design effective
cell configuration of thin-walled structures[16–19]. It is found that the
hierarchical structures have superior crashworthiness. For example,
Fang et al. [20] introduced structural hierarchy by replacing the sides of
hexagons with smaller cells to study their crushing characteristics under
out-of-plane loading. The results show that this hierarchical design can
improve energy absorption, and increasing the order and number of
hexagons can further exploit this advantage. Sun et al. [21] and Zhang
et al. [22] constructed a fractal-appearing hexagonal hierarchical
structure by iteratively replacing each three-edge vertex of a base hex
agonal network with a smaller regular hexagon up to second order,
which presents that the energy absorption of the 2nd order configura
tion is nearly twice as high as that of the 0th order. Fan and co-authors
[23,24] found that hierarchical configuration design can greatly
improve the crushing energy absorption characteristics of thin-walled
tubes by enhancing the plastic bending moment. Zhang et al. [25–27]
conducted an in-depth analysis of different types of edge and vertex
hierarchical structures by experiments, numerical simulation, and
theoretical derivation. The excellent energy absorption capacity of
* Corresponding authors.
E-mail addresses: zhangyong@hqu.edu.cn (Y. Zhang), chenxinbo@tongji.edu.cn (X. Chen).
Contents lists available at ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct
https://doi.org/10.1016/j.compstruct.2022.115501
Received 3 February 2021; Received in revised form 14 January 2022; Accepted 21 March 2022

2
crashworthiness structures can be obtained by setting different wall
thicknesses and hierarchies in different locations.
Another way to improve the crashworthiness is to add various fillers
in thin-walled structures[28,29]. Metal foam has attracted attention as
an internal filler for additional energy-absorbing parts. For example,
Goel [30] analyzed the crashworthiness performances of the single,
double, and multi-cell foam-filled tubes. It was found that the aluminum
foam filler can effectively affect the deformation and energy absorption,
and better crashworthiness performance could be achieved by changing
the configuration of the thin-walled structure and the foam. Meanwhile,
Altin et al. [31] proposed different multi-cell foam-filled tubes and
compared the crashworthiness of different filled structures by the FE
analysis. The results show that the crashworthiness potential of the
foam-filled structures is higher than that of their original counterparts.
There are also some studies on different types of filling methods. Fang
et al. [32] investigated the dynamic mechanical characteristics of
functionally graded foam-filled thin-walled structures, the results show
that the functionally graded foam structure absorbs more energy than
the uniform foam under the same mass. Aktay et al. [33] introduced a
series of extruded polystyrene foam-filled structures and studied their
energy absorption by experimental and numerical methods. Different
foam filling methods have a significant influence on crashworthiness.
The energy absorption potential of foam-filled multi-cell tubes seems to
be more susceptible to different cross-sectional configurations and foam
distribution. Consequently, it is necessary to design the cross-sectional
configuration and filling mode of foam-filled structures.
Considering the excellent crashworthiness of novel hierarchical
structures and foam-filled structures, the combination of both is worth
studying. To our best knowledge, there is little research on the inter
action effect between the foam and the hierarchical tubular structure.
Therefore, this study proposes and analyzes novel foam-filled square
hierarchical tubular structures (FSHT) and foam-filled circular hierar
chical tubular structures (FCHT).
Meanwhile, optimization design is an effective way to improve the
crashworthiness performance of foam-filled structures, and related
research has been carried out year after year. For example, Gao et al.
Fig. 1. Square hierarchical structures (SHT).
Fig. 2. Cross-section configuration of foam-filled hierarchical tubes.
X. Xu et al.

3
[34] used the Kriging model and the multi-objective particle swarm
optimization algorithm to achieve the optimization of the foam-filled
ellipse tube. Altin et al. [35] investigated the foam filling schemes for
crashworthiness optimization of thin-walled multi-cell circular tubes. Bi
et al. [36] optimized the single and multi-cell hexagonal columns filled
with foams to maximize specific energy absorption. It is enlightening
that the above optimization method does not consider the uncertainty of
design variables, and its optimization results may not be acceptable
under uncertain real conditions. Therefore, this study will carry out
robust optimization of uncertain parameters for foam-filled hierarchical
structures.
The main framework of this study is included as follows. The models
of the foam-filled hierarchical structures and crashworthiness indicators
are introduced in Section 2. The mechanical responses of the foam-filled
hierarchical structures are discussed and analyzed in Section 3. Section 4
is the robust optimization of the foam-filled hierarchical structures. The
main conclusions are described in Section 5.
2. Modelling of foam-filled hierarchical structures
2.1. Geometrical features and FEA models
As shown in Fig. 1, a novel square hierarchy was presented in Zhang
et al. [37] and Fan et al. [38]. Different mechanical properties of the
square hierarchical tubes (SHT) can be obtained by adjusting the hier
archy design factor N. The hierarchy design factor N represents the
number of smaller cells arranged along the original edge (original square
tubes (ST) can be regarded as a first-order configuration). By filling the
foam inside the hierarchical tubes, the foam-filled hierarchical tubes are
formed. The foam-filled square tube (FST) and the foam-filled square
hierarchical tubes (FSHT) with N = 5 ∼ 7 are presented in Fig. 2.
Similarly, the hierarchical configuration is developed to circular tubes
(CT) in this study, the foam-filled circular tube (FCT) and the foam-filled
circular hierarchical tubes (FCHT) with N = 5 ∼ 7 are presented in
Fig. 2. The square hierarchical tubes and circular hierarchical tubes
(CHT) have the same cell number and side length when N is the same.
Louter and Linner are the outer and inner side lengths of FSHT, respectively.
Here,
{
4Louter = πdouter
4Linner = πdinner
(1)
where douter and dinner are the outer and inner diameters of FCHT,
respectively. The width Louter is 60 mm, and the wall thickness of the
single-cell square (circular) tube is t0 = 2mm. In addition, the different
local thicknesses of the foam-filled hierarchical structures can be
represented byT1,T2, andT3, as shown in Fig. 2.
In this study, the material of all structures is aluminum alloy
AA6061. The main material parameters are as follows [26]: Young’s
modulus E = 68GPa; Poison’s ratio v = 0.33; density ρ = 2700kg/m3
;
Initial yield stress σy = 96.8MPa and ultimate stress σu = 182.6MPa. The
commercial software LS-DYNA is employed as the solver. The consti
tutive behavior of thin-walled structures is modeled with MAT_24. The
aluminum foam material is modeled with the MAT _154. The yield cri
terion is postulated as follows[39–41]:
ϕ =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1
1 + (α/3)
2
(
σ3
e + α2σ2
m
)
√
− Y (2)
ϕ and Y are the yield surface and yield stress, respectively. σe is the
von Mises effective stress and σm is the mean stress. Besides, α is the
parameter that determines the shape of the yield surface, as shown in Eq.
(3).
α2
=
9
(
1 − 2vp
)
2
(
1 + vp
) (3)
vp is the plastic coefficient, vp = 0 and α = 2.12 are used herein [42].
The strain hardening rule of the foam material model is defined as
follows:
Y = σp + γ
̂
ε
εD
+ α2ln
(
1
1 − (̂
ε/εD)
β
)
(4)
where ̂
ε is the equivalent plastic strain,σp,α2,γ,εD, and β are the ma
terial parameters dependent upon foam density, as follows:
(
σp, α2, γ, 1
/
β, Ep
)
= C0 + C1
(
ρf
ρf0
)k
(5)
εD = − ln
(
ρf
ρf0
)
(6)
whereC0, C1 and k are material constants and given in Table 1. ρf and
ρf0 are the foam density and base material density, respectively. All
foam-filled structures have the same mass by allocating different foam
densities and thin-wall thicknesses, the main geometric and mechanical
parameters of materials also as shown in Table 2.
The height h of the foam-filled hierarchical structures is 200 mm.
Fig. 3 shows the loading conditions of filled structures. The solid ele
ments of foam with a dimension of 2mm × 2mm × 2mm and the shell
elements of thin-walled tubes with a dimension of 1mm × 1mm are
acceptable to balance the numerical accuracy and computational cost
according to the mesh convergence study. In addition, the contact
definition between the foam-filled hierarchical tube and the rigid wall is
set as an automatic surface-to-surface contact, the contact definition
between the foam and the inner wall is also the automatic surface-to-
surface contact, and an automatic single-surface contact is applied to
avoid collapse penetration. Based on modeling experience, the dynamic
and static friction coefficients are 0.2 and 0.3, respectively [43]. Finally,
the finite element analysis (FEA) models of FSHT and FCHT are shown in
Fig. 4.
To test the effectiveness of the FEA model, three experiment cases
(Fig. 5 (a): SHT[25]; Fig. 5 (b): CHT; Fig. 5 (c): FCT[44]) were used to
carry out compression analysis. These experimental specimens were
fabricated by the wire electrical discharge machining technique from
aluminum alloy blocks. Detailed geometric parameters are shown in
Table 3.
The setting parameters of all simulation models are close to the
experimental conditions as much as possible. Fig. 5 shows the defor
mation characteristics of the experimental and FEA models. It can be
found that the FEA deformation results are in good agreement with the
experiments. Moreover, the crushing force curves of FEA are basically
consistent with that of this experiment. Therefore, the FEA method
Table 1
Main material parameters of foam[41].
Parameter σp (MPa) α2 (MPa) 1/β γ (MPa) Ep (MPa)
C0(MPa) 0 0 0.22 0 0
C1(MPa) 720 140 320 42 0.33 × 106
k 2.33 0.45 4.66 1.42 2.45
Table 2
Parameters of foam-filled hierarchical structures.
Group Wall thickness (mm) Foam density (kg/m3
) Total Mass (kg)
FST 2.00 200.00 0.3984
FSHTN = 5 0.83 549.11 0.3984
FSHTN = 6 0.80 443.83 0.3984
FSHTN = 7 0.78 387.10 0.3984
FCT 2.00 153.51 0.3984
FCHTN = 5 0.76 429.37 0.3984
FCHTN = 6 0.73 347.10 0.3984
FCHTN = 7 0.71 301.95 0.3984
X. Xu et al.

4
established in this study is effective and can be used for subsequent
analysis and optimization.
2.2. Evaluation of crashworthiness
In this study, to explore the crashworthiness of the foam-filled hi
erarchical structures under the external load, four crashworthiness in
dicators are applied to evaluate the structural crashworthiness [45–47].
Specific energy absorption (SEA) as the energy absorption per unit mass
is applied to evaluate the energy absorption efficiency. To evaluate the
hazard of crushing, the peck crushing force (PCF) is used to evaluate the
maximum crushing load. Besides, to evaluate the loading efficiency and
carry capacity, crush load efficiency (CLE) and mean crushing force (Fm)
are considered. The mathematical expression of the above crashwor
thiness indicators are as follows:
SEA =
EA
mass
=
∫dis
0
F(x)dx
mass
(7)
Fm =
∫dis
0
F(x)dx
dis
(8)
CLE =
Fm
PCF
(9)
EA is the total amount of energy absorbed during the crushing pro
cess, dis is the effective crushing displacement and F(x) is the crushing
force. The crashworthiness analysis in this study adopts the crushing
velocity of 15 m/s.
3. Mechanical properties of foam-filled hierarchical structures
3.1. Crashworthiness performances
For the foam-filled hierarchical structure, the energy absorption is
mainly composed of three components (including the foam, the thin-
walled tube, and the interaction effects). The energy absorption con
tributions of these components should be studied in depth. To address
this issue, the foam filler effect on the energy absorption performances of
FSHT and FCHT is investigated in this Section. The total energy ab
sorption of FSHT and FCHT are presented in Fig. 6 and Fig. 7.
All foam-filled tubes have the same compression displacement (120
mm) and mass and. By the comparative analysis, it can be found that the
energy absorbed of the foam-filled hierarchical tube is larger than the
sum of the thin-walled tube and foam. Recent studies also have shown
that foam has a positive effect on the energy absorption capacity of thin-
walled structures [48]. In Fig. 6 and Fig. 7, the pink shadow area rep
resents the energy absorption induced by the interaction effect, and the
shadow area increases with the compression displacement.
Fig. 8 shows the percentage contribution of each component. The
energy absorption ranking according to the contribution percentage is:
thin-walled tube > foam > interaction effect. For the contribution of an
interaction effect, Fig. 8 (i) presents that all FSHT (FCHT) have a higher
contribution of interaction effect than FST (FCT). The maximum
contribution ratio of FSHT is 10.54 %, that of FCHT is 6.97 %, and that of
FST and FCT are 7.82 % and 4.32 %, respectively. This result shows that
foam-filled hierarchical tubes have a higher interaction effect than
foam-filled single tubes, suggesting that foam-filled hierarchical tubes
have a higher energy absorption potential than conventional foam-filled
tubes.
Fig. 3. Loading conditions.
Fig. 4. FEA models.
X. Xu et al.

5
3.2. Theoretical analysis of foam-filled hierarchical structures
The theoretical derivation is helpful to analyze the crushing force of
novel hybrid structures more efficiently. In this study, the crushing force
of foam-filled hierarchical structures is mainly composed of the crushing
force Ftube
m of the thin-wall tubes, the crushing force F
foam
m of the foam and
the interaction effect force Finteraction
m between the thin-wall structures
and the foam. Therefore, the mean crushing force Fm of foam-filled hi
erarchical structures is defined as[48]:
Fm = Ftube
m + Ffoam
m + Finteraction
m (10)
For the thin-wall tubes, the sum of total energy dissipation Etube
total ac
cording to the simplified super folding element theory is [49]:
Etube
total = 2HFtube
m (11)
where Etube
total is the energy absorption in one crushing fold and 2H is
the wavelength. Total energy absorption Etube
total of the specific angle
element is mainly composed of bending energy Etube
b and membrane
energyEtube
m . Here, the bending energy Etube
b can be solved by Eq. (12).
Etube
b =
∑3
i=1
∑m
j=1
λiM0bj =
∑m
j=1
2πM0bj (12)
The main parameters are as follows: M0 = σ0t2
/4 is the fully plastic
moment; σ0 =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
σyσu/(1 + nh)
√
is the flow stress and nh is the power law
hardening index; Besides, bj and m are the width and the panel numbers,
respectively; λ is deformation rotation angles at the plastic hinge. The
thin-walled tube of FSHT can be divided into three parts (part a, part b,
and part c, as shown in Fig. 9 (a)). For part a, the deformation mode is a
symmetric fold as shown in Fig. 10 (a). Therefore, the membrane energy
dissipation Etube
m a of part a can be expressed as:
Etube
m a = 8M0H2
/
t (13)
Part b is one three-panel element, and its deformation mode is shown
in Fig. 10 (b). The member energy Etube
m b is.
Fig. 5. The results of FEA and experimental test[25,44].
Table 3
Geometric parameters of experiment samples.
Test samples Wall thickness Louter(douter) Linner(dinner) Foam density ρf
SHT 0.6 mm 64 mm 21 mm
CHT 0.8 mm 50 mm 35 mm
FCT 1 mm 60 mm 280 kg/m3
X. Xu et al.

6
Etube
m b =
12.4M0H2
t
(14)
Moreover, the deformation mode of part c is shown in Fig. 10 (c), and
its member energy Etube
m c is expanded as.
Etube
m c =
16M0H2
t
(15)
Therefore, the crushing force Ftube FSHT
m of thin-walled tubes for FSHT
can be expressed as follows:
Ftube FSHT
m =
Etube
total
2H
=
∑m
i=12πM0b + 4Etube
m a + (8N − 16)Etube
m b + 4Etube
m c
2H
(16)
∑
m
i=1
2πM0b = 24πM0L(N − 1)
/
N (17)
According to the stationary condition, the H can be solved as follows:
∂Ftube FSHT
m
∂H
= 0 (18)
H =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
3πL(N − 1)tN FSHT
(12.4N − 12.8)N
√
(19)
Therefore, the mean crushing force Ftube FSHT
m can be expressed as
follows:
Ftube FSHT
m = 2σ0t1.5
N FSHT
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
3πL(N − 1)(12.4N − 12.8)/N
√
(20)
where tN FSHT is the wall thickness of FSHT with different N, it can be
expressed as:
tN FSHT =
Nt0
3(N − 1)
(21)
Finally,
Ftube FSHT
m = 2σ0
(
Nt0
3(N − 1)
)1.5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
3πL(N − 1)(12.4N − 12.8)/N
√
(22)
In addition, the thin-walled tube of FCHT can be divided into two
components (circular tube and 2-arcs joint element). The membrane
energy dissipation Etube
m arc of the circular tube under a full fold formation
is:
Etube
m arc = 2πσ0tH2
(23)
Here, the arc wall is replaced approximatively by a straight wall, as
shown in Fig. 11. Whereupon, 2-arc joint element (part d and part e) can
be regarded as a T-shape element.
Therefore, the crushing force Ftube FCHT
m of thin-walled tubes for FCHT
can be expressed as follows:
Ftube FCHT
m =
2πM0Lc + 2Etube
m arc + (4N − 4)Etube
m b
2H
(24)
Similarly, according to Eq. (18), that is:
H =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(2NLπ − 2Lπ + NL)tN FCHT
(6.2N − 4.2)N
√
(25)
Ftube FCHT
m = 2σ0t1.5
N FCHT
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(2NLπ − 2Lπ + NL)(6.2N − 4.2)/N
√
(26)
where tN FCHT is the wall thickness of the FCHT with different N,
which can be expressed as:
Fig. 6. Energy absorption of FSHT.
X. Xu et al.

7
tN FCHT =
Nπt0
2πN − 2π + N
(27)
Finally,
Ftube FCHT
m = 2σ0
(
Nπt0
2πN − 2π + N
)1.5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(2NLπ − 2Lπ + NL)(6.2N − 4.2)/N
√
(28)
In addition, the mean crushing force Ffoam
m of the foam can be solved
as:
{
Ffoam
m FSHT = σf (Louter(N − 2)/N )
2
Ffoam
m FCHT = σf (2Louter(N − 2)/N )
2
/
π
(29)
The interaction effect force Finteraction
m can be expressed:
Fig. 7. Energy absorption of FCHT.
Fig. 8. Energy absorption contribution.
X. Xu et al.

8
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Finteraction
m FSHT = Cavg H
̅̅̅̅̅̅̅̅̅
σ0σf
√
(Louter(N − 2)/N)
Nt0
3(N − 1)
Finteraction
m FCHT = Cavg C
̅̅̅̅̅̅̅̅̅
σ0σf
√
((4Louter(N − 2)/N )/π )
Nπt0
2πN − 2π + N
(30)
where Cavg H = 8.71Dc − 0.07 [50] and Cavg C = 2.78Dc +1.29 [51],Dc
is the relative displacement. Finally, the Fm FSHT of FSHT and the Fm FCHT
of FCHT are obtained according to Eq. (10).
Fm FSHT = 2σ0
(
Nt0
3(N − 1)
)1.5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅
3πLouter(N − 1)(12.4N − 12.8)/N
√
+ σf (Louter(N − 2)/N)2
+ Cavg H
̅̅̅̅̅̅̅̅̅
σ0σf
√
(Louter(N − 2)/N)
Nt0
3(N − 1)
(31)
Fm FCHT = 2σ0
(
Nπt0
2πN − 2π +N
)1.5 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Louter(2Nπ − 2π +N)(6.2N − 4.2)/N
√
+
σf
π
(
2Louter(N − 2)
N
)2
+CavgC
̅̅̅̅̅̅̅̅̅
σ0σf
√
(
4Louter(N − 2)
Nπ
)
Nπt0
2πN − 2π +N
(32)
In the process of dynamic crushing, the dynamic enhancing factor kd
and the effective crushing coefficient γe need to be considered. γe is 0.7
and kd is 1.1 by referring to relevant studies [50,52] in this study.
Therefore, the modified theoretical formula of mean crushing force
(Fm FSHT or Fm FCHT) can be written as:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Fm FSHT =
kd
γe
Fm FSHT
Fm FCHT =
kd
γe
Fm FCHT
(33)
Furthermore, the theoretical expression of energy absorption for
FSHT and FCHT can be expressed as:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
EAFSHT =
∫ dis
0
Fm FSHT dx
EAFCHT =
∫ dis
0
Fm FCHT dx
(34)
To illustrate the accuracy of the theoretical derivation model, Fig. 12
shows the comparison of EA between theoretical analysis and FEA with
different N. The data results clearly show that the maximum errors be
tween the theoretical solution and FEA for FSHT and FCHT are 5.2 %
and 5.98%, respectively. The difference between the theoretical analysis
and FEA may be caused by uncertain interaction effects, uncertain dy
namic effects and uncertain local mechanical characteristics, etc. Here,
the theoretical derivation models still show a good agreement with FEA
models. Through the above comparative analysis, the theoretical deri
vation models are of satisfactory accuracy to predict the energy ab
sorption of FSHT and FCHT.
3.3. Selecting best potential configuration
To select the most potential structural configuration from FSHT and
FCHT, the main crashworthiness indicators will be compared in this
Section. Table 4 shows all FEA results with three crashworthiness in
dicators. It is worth noting that: although the conventional foam-filled
single tubes (FST or FCT) have a lowerPCF, their SEA and CLE are also
lower than the corresponding foam-filled hierarchical tubes (FSHT or
FCHT). Besides, FSHT has a higher energy absorption capacity (SEA)
than FCHT under the same mass.
To obtain the crushing behaviors of all structures more intuitively,
the deformation modes are presented in Fig. 13. Apparently, FSHT and
Fig. 9. Representative constituent parts.
Fig. 10. Deformation modes of different angle elements.
Fig. 11. Angular element with arc wall.
X. Xu et al.

9
FCHT have more folding waves than FST and FCT. Relevant studies have
shown that the greater the number of stable folds of thin-walled struc
tures, the greater its potential for absorbing plastic deformation energy.
It is worth noting that the folding deformation mode of FSHT shows a
non-progressive and unstable deformation mode whenN = 7, indicating
that the increase in the number of hierarchical cells will not always
develop towards the optimal crashworthiness performance. Therefore, it
is necessary to comprehensively consider different indicators to evaluate
the structural crashworthiness performances, and to select the best po
tential configuration.
Because of the advantages of the TOPSIS in multi-criteria decision
making (MCDM)[53,54], this study will use the TOPSIS method to
evaluate the crashworthiness performances and select the most potential
configuration[55,56]. The main steps of this method are as follows.
Step 1: Define the weight w’
j of j-th criterion by the subjective weight
∊j and the objective weightwj. The original decision matrix X is as fol
lows.
X =
[
xij
]
s×c
=
⎡
⎣
x11 x12 ⋯ x1c
⋮ ⋮ ⋱ ⋮
xs1 xs2 ⋯ xsc
⎤
⎦ (35)
Fig. 12. The results of theoretical analysis and FEA: (a) FSHT; (b) FCHT.
Table 4
FEA results of foam-filled hierarchical structures.
ID Groups SEA (kJ/kg) PCF (kN) CLE
1 FST 11.58 67.21 0.57
2 FSHT N = 5 24.10 94.57 0.85
3 FSHT N = 6 23.04 91.77 0.83
4 FSHT N = 7 23.08 96.70 0.79
5 FCT 11.45 62.30 0.61
6 FCHT N = 5 18.85 78.89 0.79
7 FCHT N = 6 19.04 77.12 0.82
8 FCHT N = 7 19.95 76.93 0.86
Fig. 13. Deformation modes of foam-filled hierarchical structures.
X. Xu et al.

10
where xij is the value of the i-th alternative with respect to the j-th
criterion, i = 1, 2, ⋯, s and j = 1, 2, ⋯, c (s = 8 and c = 3 in this study
referring to Table 4). Then, the decision matrix (35) can be normalized
using Eq. (36).
qij =
xij
(
x1j + ⋯ + xsj
) (36)
The information entropy of the j-th criterion is calculated as:
Δe = −
1
ln(s)
∑s
i=1
qij⋅lnqij (37)
The criterion with higher information entropy Δe has higher varia
tion. The weight through deviation degree dj is:
dj = 1 − Δe (38)
The objective weight wj of the crashworthiness criterion is calculated
as:
wj =
dj
(d1 + ⋯ + dc)
(39)
The final weight w’
j can be defined by the subjective weight ∊j and the
objective weightwj..
w’
j =
∊j.wj
(∊1.w1 + ⋯ + ∊c.wc)
,
(
∑
c
g=1
∊j = 1
)
(40)
Step 2: Normalized decision matrix can be defined by using the
vector normalization method, and the normalized value rij and the
matrix Nm×c are expressed as follows.
rij =
xij
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(
x2
1j + ⋯ + x2
sj
)
√ (41)
Ns×c =
[
rij
]
s×c
(42)
The weighted normalized decision matrix V can be calculated by
establishing the diagonal matrix w’
c×c with elementw’
j ..
V = Ns×c⋅w’
c×c =
(
vij
)
s×c
(43)
Step 3: When J’
and J are the subsets of negative and positive criteria,
respectively. The negative ideal solution I−
and the positive ideal solu
tion I+
of alternatives are calculated as:
I−
=
{(
maxvij|j ∈ J
)
;
(
minvij|j ∈J’
) }
=
(
v−
1 , v−
2 , ⋯, v−
c
)
(44)
I+
=
{(
minvij|j ∈ J
)
;
(
maxvij|j ∈J’
) }
=
(
v+
1 , v+
2 , ⋯, v+
c
)
(45)
Step 4: Finally, the distance between the positive ideal solution d+
i
and the negative ideal solution d−
i of each alternative can be calculated
by Eq. (46) and Eq. (47). Then, the closeness coefficient of each alter
native Ci is calculated by Eq. (48) [25,55]:
d+
i =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
∑c
j=1
(
vij − v+
j
)2
√
(46)
d−
i =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
∑c
j=1
(
vij − v−
j
)2
√
(47)
Ci =
d−
i
(d−
i + d+
i )
(48)
The SEA and PCF are contradictory indicators and are considered to
be of equal importance in this study, so two subjective weights (∊SEA =
0.4,∊PCF = 0.4, ∊CLE = 0.2 and∊SEA = 0.35,∊PCF = 0.35, ∊CLE = 0.3)are
considered for theSEA, PCF andCLE. The result of multi-criteria decision
making is shown in Table 5. The crashworthiness ranking scheme is
arranged according to the closeness coefficientCi. This ranking indicates
that higher Ci has greater priority to be chosen. The ranking result shows
that the FSHT with N = 5 and the FCHT with N = 7 can be regarded as
the most potential structural configuration for foam-filled square hier
archical tubes and foam-filled circular hierarchical tubes, respectively.
3.4. Parameter sensitivity analysis
To further explore the effects of independent geometric parameters
on the crashworthiness performances, a group of specimens with three
levels of T1,T2,T3, and ρf have been modeled and investigated (T1,T2,
and T3 are presented in Fig. 2, and ρf is the foam density). Table 6 lists
the specific values for all factor levels. Table 7 is the orthogonal array
(L9(34
)) of four factors with three levels. The FSHT with N = 5 and the
FCHT with N = 7 are analyzed in this Section.
Take the FSHT as an example, Fig. 14 (a)–(d) plot the variations of
theSEA,PCF,Fm, and CLE against the four parameters, respectively. The
curve of the SEA is perceived to have higher slopes obviously when T3
varies, and T3 has a relatively high effect on the SEA than T1, T2 andρf . In
addition, it can be found that all parameters have positive influences on
the PCF andFm, both rising with the increment of parameters. Compar
atively, T1, T2 and ρf have a relatively small effect on the PCF andFm.
Fig. 14 (d) shows that the CLE presents a significant growing trend when
T2 increases from the first level to the second level, and then the CLE
decreased when T2 is at the third level. In the three horizontal design
spaces, T2 had a more significant effect on the CLE than T1, T3 andρf . For
the FCHT (Fig. 15), the crashworthiness indicators at different param
eter levels are consistent with FSHT. It is concluded from the above
results that the analyzed parameters have different effects on foam-filled
Table 5
Ranking results.
Groups ∊SEA = 0.4, ∊PCF = 0.4, ∊CLE =
0.2
∊SEA = 0.35, ∊PCF = 0.35,
∊CLE = 0.3
Ci Ranking Ci Ranking
FST 0.1580 8 0.1570 8
FSHT N ¼ 5 0.8312 1 0.8321 1
FSHT N = 6 0.8185 2 0.8193 2
FSHT N = 7 0.7985 3 0.7981 3
FCT 0.1782 7 0.1778 7
FCHT N = 5 0.5832 6 0.5855 6
FCHT N = 6 0.5999 5 0.6030 5
FCHT N ¼ 7 0.6689 4 0.6722 4
Table 6
Three-level design for foam-filled hierarchical structures.
Level 1 Level 2 Level 3
T1(mm) 0.4 0.8 1.2
T2(mm) 0.4 0.8 1.2
T3(mm) 0.4 0.8 1.2
ρf (kg/m3
) 150 350 550
Table 7
Orthogonal array (L9(34
)) of four factors with three levels.
ID T1 T2 T3 ρf
1 1 1 1 1
2 1 2 2 2
3 1 3 3 3
4 2 1 2 3
5 2 2 3 1
6 2 3 1 2
7 3 1 3 2
8 3 2 1 3
9 3 3 2 1
X. Xu et al.

11
hierarchical tubes, and the effect of T3 on the energy absorption capacity
and peak crushing force is more obvious. Therefore, in the robust opti
mization design stage of foam-filled hierarchical tubes, the significant
effect of T3 on crashworthiness should be considered.
4. Multi-objective robust optimization
4.1. Optimization model
Usually, the engineering optimization problem needs to consider
many objectives under the actual working conditions. The conventional
deterministic multi-objective optimization model is:
⎧
⎨
⎩
min F =
{
f1(x), ⋯, fq(x)
}
q = 1, 2, ⋯r
s.t. Gi(x) ≤ 0, i = 1, 2, ⋯l
xmin ≤ x ≤ xmax
(49)
where
{
f1(x), ⋯, fq(x)
}
are the objective functions and r is the
number of objectives. Gi(x) is the i-th constraint condition and l is the
number of constraints. In addition, xmin and xmax are the lower and upper
limits of the design space, respectively. Different from the conventional
deterministic optimization, the uncertainties of design variables need to
be considered during actual processing. Stochastic probability models
are often used to construct uncertain models, but when the distribution
of design variables is unknown due to the lack of test samples, the in
terval uncertainty optimization method is a better choice. Therefore, the
interval uncertainty model (IUM) is employed in this study. The interval
value xI
can be expressed as[57]:
xI
=
[
xIL
, xIU
]
(50)
where xIL
and xIU
are the lower and upper bounds of xI
, respectively.
The interval length xID
= xIU
− xIL
can be regarded the deviation range,
Fig. 14. Effects of the independent parameters for FSHT.
Fig. 15. Effects of the independent parameters for FCHT.
X. Xu et al.

12
which reflects the uncertainty degree of xI
. The interval radius of xI
is
xIR
= (xIU
− xIL
)/2, which can be used to represent the manufacturing
tolerance accuracy in structural design[58]. If xIR
= xID
= 0, xI
de
generates to a definite real number without interval uncertainty.
Therefore, the deterministic multi-objective optimization can be trans
formed into the interval uncertainty problem, as follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
min F =
{
f1
(
xI
)
, ⋯, fq
(
xI
) }
q = 1, 2, ⋯r
s.t. Gi
(
xI
)
≤ 0, i = 1, 2, ⋯l
xIL
≤ xNV
≤ xIU
xmin ≤ x ≤ xmax
(51)
where the superscript NV represents the nominal value without un
certainty. The nominal value xNV
can be regarded as the intermediate
value of the interval variable, i.e.xNV
= (xIL
+ xIU
)/2, so xIL
≤ xNV
≤ xIU
can be written asxNV
− xIR
≤ xNV
≤ xNV
+ xIR
. It should be noted that
when the design parameters are considered as interval values with a
specified deviation range, their responses are also interval values. For a
multi-objective robust optimization model, Eq. (51) can be expressed as:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min F =
{
fNV
1 + β1fID
1 , ⋯, fNV
q + βqfID
q
}
q = 1, 2, ⋯r
s.t. Gi
(
xI
)
≤ 0, i = 1, 2, ⋯l
xIL
≤ xNV
≤ xIU
xmin ≤ x ≤ xmax
(52)
where fID
q represents the deviation range of objective value, and β is
the weight factor. In the robust optimization model, the objective
function is the weighted sum of the nominal value and its uncertainty
deviation range.
The novel foam-filled hierarchical structures with superior crash
worthiness should be designed to reduce initial impact damage and
absorb more energy. In this study, the maximum SEA and minimum PCF
are considered as the main crashworthiness optimization objectives. The
main design variables include foam density (ρf ) and wall thickness (T1,
Fig. 16. The flowchart of multi-objective robust optimization.
Fig. 17. Accuracy analysis of different approximate models.
X. Xu et al.

13
T2, T3). The deterministic mathematic model of the foam-filled hierar
chical structures is formulated as:
⎧
⎨
⎩
min F = { − SEA, PCF}
s.t. 0.4 mm ≤ T1, T2, T3 ≤ 1.2 mm
150 kg/m3
≤ ρf ≤ 550 kg
/
m3
(53)
The multi-objective robust optimization of the foam-filled hierar
chical structures is formulated as:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min F =
{
− SEANV
+ βSEASEAID
, PCFNV
+ βPCFPCFID
}
s.t. 0.4 mm ≤ T1, T2, T3 ≤ 1.2 mm
150 kg
/
m3
≤ ρf ≤ 550 kg
/
m3
Tj − TIR
j ≤ Tj ≤ Tj + TIR
j j = 1, 2, 3
ρf − ρf
IR
≤ ρf ≤ ρf + ρf
IR
(54)
where TID
j and ρf
ID
are the deviation ranges of Tj andρf , respectively.
The deviation range of different design variables can be defined ac
cording to needs. In addition, it is necessary to limit the maximum peak
force (Pmax) due to the serious damage induced by excessive peak
crushing force. Therefore, the optimization expression can also be
written as follows:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
min F =
{
− SEANV
, SEAID
+ PCFID
}
s.t. PCFNV
− Pmax ≤ 0
0.4 mm ≤ T1, T2, T3 ≤ 1.2 mm
150 kg
/
m3
≤ ρf ≤ 550 kg
/
m3
Tj − TIR
j ≤ Tj ≤ Tj + TIR
j j = 1, 2, 3
ρf − ρf
IR
≤ ρf ≤ ρf + ρf
IR
(55)
4.2. Optimization process
To obtain a robust design for novel foam-filled hierarchical struc
tures, Fig. 16 presents a flowchart to depict the corresponding optimi
zation steps. The main steps of multi-objective robust optimization are
as follows:
Step 1: Define the optimization problem, including output responses,
design variables, and their design space.
Step 2: Using sampling technology to select training points in the
design space and establish the design of experiments (DOE) by
Fig. 18. Accuracy analysis of approximate models (FNNs).
Fig. 19. Deterministic optimization results.
Fig. 20. Robust optimization results from Eq. (54).
X. Xu et al.

14
numerical simulation (LS-DYNA). It is worth noting that this step
does not consider uncertainty, so the method of establishing an
approximate response model is similar to the conventional deter
ministic optimization process. The acceptable deterministic
approximate model can be obtained by the approximate model al
gorithm. The approximate model trained in this step is used to
characterize the nominal response (fNV
).
Step 3: Determine the possible deviation radius xIR
of design vari
ables. Using sampling technology to select training points in the
design space and establish the DOE. The genetic algorithm (GA) is
used to calculate the upper and lower bounds of responses for each
sample point, in which the objective function comes from the
approximate model established in the second step. Then, the
acceptable approximate model of lower and upper responses (fIL
and
fIU
) can be obtained by the approximate model algorithm in this step.
Step 4: The approximate models established in the second and third
steps are combined as the output response of the optimization model.
The objective function is the weighted sum of nominal value fNV
and
its uncertainty deviation range fID
(fID
= fIU
− fIL
). The optimization
model is calculated by non-dominated genetic algorithm II (NSGA-II)
Table 8
Optimization results at the ideal points.
Cases Groups T1 (mm) T2 (mm) T3 (mm) ρf (kg/m3
) SEANV
(J/g) SEAID
(J/g) PCFNV
(kN) PCFID
(kN)
TIR
j = 0.05 mm, ρIR
f = 25 kg/m3 FSHT
(deterministic)
0.44 0.55 1.19 169 20.83 5.19 54.43 58.02
FSHT
(robust)
0.52 1.13 0.95 195 25.42 0.08 100.3 15.81
FCHT
(deterministic)
0.52 1.02 0.85 160 23.82 6.04 83.73 68.36
FCHT
(robust)
0.55 1.19 1.05 156 27.47 0.15 110.3 0.76
TIR
j = 0.1 mm, ρIR
f = 50 kg/m3 FSHT
(deterministic)
0.44 0.55 1.19 169 20.83 5.19 54.43 58.02
FSHT
(robust)
0.45 1.11 0.72 549 23.73 1.04 122.2 13.62
FCHT
(deterministic)
0.52 1.02 0.85 160 23.82 6.04 83.73 68.36
FCHT
(robust)
0.52 0.58 1.16 549 23.27 1.18 139.8 11.35
Fig. 21. Robust optimization results from Eq. (55).
Table 9
Optimization results with PCF ≤ 120kN.
Cases Groups T1 (mm) T2 (mm) T3 (mm) ρf (kg/m3
) SEANV
(J/g) SEAID
(J/g) PCFNV
(kN) PCFID
(kN)
TIR
j = 0.05 mm, ρIR
f = 25 kg/m3 FSHT
(deterministic)
1.11 0.85 1.15 547 26.41 4.58 119.9 23.09
FSHT
(robust)
1.06 0.89 1.16 549 26.29 2.01 119.9 15.03
FCHT
(deterministic)
0.57 0.96 1.19 168 29.61 9.48 118.3 60.27
FCHT
(robust)
0.64 0.91 1.19 151 29.54 4.20 119.8 30.86
TIR
j = 0.1 mm, ρIR
f = 50 kg/m3 FSHT
(deterministic)
1.11 0.85 1.15 547 26.41 4.58 119.9 23.09
FSHT
(robust)
1.06 0.87 1.18 549 26.08 2.09 119.7 15.21
FCHT
(deterministic)
0.57 0.96 1.19 168 29.61 9.48 118.3 60.27
FCHT
(robust)
0.60 1.06 1.18 153 29.12 2.75 119.9 35.08
X. Xu et al.

15
to obtain the Pareto front. The maximum iteration of the multi-
objective solver is 100, the crossover rate is 0.9, and the distribu
tion index of crossover and mutation is 20 in this study.
Step 5: According to different design requirements, the optimal so
lution can be selected in the Pareto front.
4.3. Approximate model
To reduce the expensive computational cost, approximate models are
often used in optimization models. Relevant studies show that the
feedforward neural network (FNN)[59], support vector regression (SVR)
[60], and Kriging[61] all have good approximation abilities for
nonlinear problems. Different approximation models are discussed for
crashworthiness performances of foam-filled hierarchical tubes in this
study to determine the appropriate approximation model. The radial
basis function (RBF) neural network is chosen as the FNN in this study
[62–64]. Totally 40 samples (10 samples are used for accuracy test) are
generated by optimal Latin hypercube sampling (OLHS) in step 2 of the
optimization flowchart. The determination coefficientR2
, the relative
maximum error (RM), and the relative average error (Ra) are the accu
racy indicators, which can be expressed as follows:
R2
= 1 −
∑ne
i=1(yi − ̂
yi)2
∑ne
i=1(yi − yi)2
(56)
Ra = (1/ne)
∑ne
i=1
(|yi − ̂
yi|/|yi| ) (57)
RM = max
i∈(1,⋯,ne)
(|yi − ̂
yi|/|yi| ) (58)
where yi denotes the response value from FEA, ̂
yi is the corre
sponding predictive value, yi is the mean ofyi, and ne is the number of
examination points. As shown in Fig. 17 (c), the determination co
efficients of all approximation models are close to 1, which indicates
that FNN, SVR, and Kriging are of good prediction ability for the
crashworthiness performances. The maximum errors of FNN and SVR,
Kriging are less than 6% (in Fig. 17 (b)), and FNN has lower error than
SVR and Kriging (in Fig. 17 (a)). Therefore, it is believed that the FNN
approximate model is acceptable for the crashworthiness design of
foam-filled hierarchical tubes.
4.4. Optimization results and discussion
For the FSHT with N = 5 and the FCHT with N = 7, two uncertainty
cases (case1:TIR
j = 0.05mm,ρIR
f = 25kg/m3
; case2:TIR
j = 0.1mm, ρIR
f =
50kg/m3
) are defined for optimization design in this study. To obtain
acceptable approximate models, totally 50 samples (10 samples are used
for accuracy test) are generated by OLHS in step 2 of the optimization
flowchart, totally 4000 samples (500 samples are used for accuracy test)
are generated by OLHS in step 3 of the optimization flowchart. The
accuracy assessment of all FNN approximate models is presented in
Fig. 18. The determination coefficients of all FNNs are close to 1, and the
maximum error is less than 5%. Therefore, it is believed that the FNN
approximate models of foam-filled hierarchical tubes are reliable for
optimization design. The deterministic Pareto front sets of FSHT and
FCHT are obtained in Fig. 19. It is worth noting that the two objectives
(SEA and PCF) conflict with each other. The points with higher SEA and
the lower PCF are located in the lower left of Pareto. For the optimiza
tion problem without considering the uncertainty of design parameters,
the best energy absorption configuration can be chosen from FSHT when
the PCF is limited to 92.5kN, and FCHT has more potential energy ab
sorption configuration when the PCF is greater than 92.5kN. The Pareto
sets provide a variety of optimal solutions for different design needs.
Fig. 20 is the Pareto front sets of FSHT and FCHT under two uncertain
conditions calculated by Eq. (54), the weight factors βSEA and βPCF are set
to 1. In this study, the minimum distance selection method (MDSM) is
used to choose an ideal point in the Pareto set[65]. As the most ideal
solution, the point selected by MDSM can usually provide appropriate
trade-offs and overall best conditions in the Pareto space.
Table 8 is the optimization results at the ideal points. By comparing
the results of the robust optimization and deterministic optimization, it
is found that the robust optimization under the two uncertainty cases
can obtain a lower deviation range of response. Moreover, assuming the
peak force does not exceed 120kN, the optimization results can be
calculated by Eq. (55), as shown in Fig. 21. Compared with the deter
ministic optimization, the robust optimization can obtain lowerSEA, but
the deviation range SEAID
(PCFID
) is smaller, as shown in Table 9. The
results of the robust optimization slightly reduce SEA, but significantly
improve the stability performance against the uncertainty of design
variables. For example, when TIR
j = 0.1mm and ρIR
f = 50kg/m3
, the SEAs
of FSHT and FCHT in the deterministic optimization are 26.41 J/g and
29.61 J/g, respectively, and the SEAs of FSHT and FCHT in the robust
optimization are 26.08 J/g and 29.12 J/g, respectively. However, for the
deviation range SEAID
of FSHT and FCHT, the values in the deterministic
optimization are 4.58 J/g and 9.48 J/g, respectively, but the values in
the robust optimization are 2.09 J/g and 2.75 J/g, respectively. These
results imply that the robust optimization proposed in this study can
significantly improve the robustness performance than the deterministic
optimization under uncertainty. The robust optimization method can
provide a more reasonable optimization scheme for foam-filled hierar
chical structures when the machining accuracy of design parameters is
uncertain.
5. Conclusions
Two novel foam-filled hierarchical tubes (FSHT and FCHT) are
proposed to improve structural crashworthiness in this study. Based on
the above results from the numerical, theoretical, and optimization
models, main conclusions can be drawn as follows:
(1) Through the crashworthiness comparison of different foam-filled
structures, it is found that novel foam-filled hierarchical struc
tures have more prominent crashworthiness potential than con
ventional foam-filled square structures and circular foam-filled
structures.
(2) Theoretical analysis of novel foam-filled hierarchical structures is
carried out. The theoretical model of energy absorption is
derived. The maximum errors of FSHT and FCHT are 5.2% and
5.98%, respectively, indicating that the theoretical solution has
satisfactory accuracy.
(3) The FSHT with N = 5 and the FCHT with N = 7 can be selected as
the most potential structural configuration according to multi-
criteria decision making. From the orthogonal array design, the
effect of T3 on the energy absorption capacity and the peak
crushing force is more evident thanT1,T2, andρf .
(4) Moreover, the multi-objective robust optimization is performed
by IUM and FNN, and it is observed that the robust optimization
result has a lower indicator fluctuation range than the deter
ministic optimization. This optimization method applies to the
uncertainty optimization of design parameters induced by
manufacturing.
On the whole, through the crashworthiness analysis and robustness
optimization of foam-filled hierarchical structures under axial load, a
filler hybrid structure design with superior crashworthiness is provided.
The effective crashworthiness design and robust optimization way can
be used to guide the production of lightweight protective structures.
X. Xu et al.

16
CRediT authorship contribution statement
Xiang Xu: Methodology, Investigation, Software, Writing – original
draft, Writing – review & editing. Yong Zhang: Funding acquisition,
Supervision, Writing – review & editing. Jianguang Fang: Methodol
ogy, Supervision, Writing – review & editing. Xinbo Chen: Funding
acquisition, Supervision, Methodology, Writing – review & editing. Zhe
Liu: Conceptualization, Investigation, Writing – original draft. Yanan
Xu: Methodology, Investigation, Writing – review & editing. Yunkai
Gao: Supervision, Conceptualization, Investigation, Methodology.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation
of China (52075188, 51805123), the Program for New Century Excel
lent Talents in Fujian Province University, and the Youth Innovation
Fund of Xiamen City (3502Z20206003). The authors would like to ex
press their sincere thanks for providing research funding. The authors
would also like to express their sincere thanks to the anonymous re
viewers for their valuable suggestions, which are very helpful to
improve our work.
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