Optimal design of functionally graded lattice structures using Hencky bar-grid model and topology optimization

Abstract

Presented herein is a novel design framework for obtaining the optimal design of functionally graded lattice (FGL) structures that involve using a physical discrete structural model called the Hencky bar-grid model (HBM) and topology optimization (TO). The continuous FGL structure is discretized by HBM comprising rigid bars, frictionless hinges, frictionless pulleys, elastic primary and secondary axial springs, and torsional springs. A penalty function is introduced to each of the HBM spring’s stiffnesses to model non-uniform material properties. The gradient-based TO method is applied to find the stiffest structure via minimizing the compliance or elastic strain energy by adjusting the HBM spring stiffnesses subjected to prescribed design constraints. The optimal design of FGL structures is constructed based on the optimal spring stiffnesses of the HBM. The proposed design framework is simple to implement and for obtaining optimal FGL structures as it involves a relatively small number of design variables such as the spring stiffnesses of each grid cell. As illustration of the HBM-TO method, some optimization problems of FGL structures are considered and their optimal solutions obtained. The solutions are shown to converge after a small number of iterations. A Python code is given in the Appendix for interested readers who wish to reproduce the results.

Full text

Vol.:(0123456789)
1 3
Structural and Multidisciplinary Optimization (2022) 65:276
https://doi.org/10.1007/s00158-022-03368-w
RESEARCH PAPER
Optimal design offunctionally graded lattice structures using Hencky
bar‑grid model andtopology optimization
Y.P.Zhang1 · C.M.Wang1· N.Challamel2· Y.M.Xie3· J.Yang3
Received: 19 February 2022 / Revised: 10 August 2022 / Accepted: 15 August 2022 / Published online: 16 September 2022
© The Author(s) 2022
Abstract
Presented herein is a novel design framework for obtaining the optimal design of functionally graded lattice (FGL) structures
that involve using a physical discrete structural model called the Hencky bar-grid model (HBM) and topology optimization
(TO). The continuous FGL structure is discretized by HBM comprising rigid bars, frictionless hinges, frictionless pulleys,
elastic primary and secondary axial springs, and torsional springs. A penalty function is introduced to each of the HBM
spring’s stiffnesses to model non-uniform material properties. The gradient-based TO method is applied to find the stiffest
structure via minimizing the compliance or elastic strain energy by adjusting the HBM spring stiffnesses subjected to pre-
scribed design constraints. The optimal design of FGL structures is constructed based on the optimal spring stiffnesses of
the HBM. The proposed design framework is simple to implement and for obtaining optimal FGL structures as it involves
a relatively small number of design variables such as the spring stiffnesses of each grid cell. As illustration of the HBM-TO
method, some optimization problems of FGL structures are considered and their optimal solutions obtained. The solutions
are shown to converge after a small number of iterations. A Python code is given in the Appendix for interested readers who
wish to reproduce the results.
Keywords Lattice structures· Functional grading· Hencky bar-grid model· Topology optimization
Abbreviations
FGL Functionally graded lattice
HBM Hencky bar-grid model
TO Topology optimization
GDM Greyscale density mapping
SIMP Solid isotropic material with penalization
GSO Ground structure optimization
OC Optimality criteria
FD Finite difference
1 Introduction
Additive manufacturing opens up new opportunities to fab-
ricate as-designed functionally graded lattice (FGL) struc-
tures by building up material layer-by-layer (Maconachie
etal. 2019). Lattice structures (also referred to as cellular
structures) have a clear network feature comprising basic
elements like solid struts or plates (Gibson 1989). Properly
designed FGL structures can achieve high performance
with great control in stiffness and strength, weight, energy
absorption, heat exchange and acoustic insulation (Zhang
etal. 2015; Ferro etal. 2022). Generally, elastic structures
with minimum compliance or minimum elastic strain energy
are the stiffest (Hassani and Hinton 1999; Kaveh etal. 2008;
Zhang etal. 2014). Topology optimization (TO) method is
commonly adopted to minimize the compliance or elastic
strain energy of FGL structures by iteratively updating the
material distribution subjected to loading, boundary condi-
tions and prescribed deflection or stress constraints (Panesar
etal. 2018; Liu etal. 2018; Li etal. 2018; Zhu etal. 2021).
Many design strategies have been proposed for obtain-
ing optimal designs of FGL structures with prescribed
mechanical properties. Most of the existing strategies may
Responsible Editor: Seonho Cho
* Y. P. Zhang
y.zhang16@uq.edu.au
1 School ofCivil Engineering, The University ofQueensland,
StLucia, QLD4072, Australia
2 Centre de Recherche, Université Bretagne Sud,
IRDL (CNRS UMR 6027), Rue de Saint Maudé,
BP92116,56321LorientCedex, France
3 School ofEngineering, RMIT University, Melbourne3001,
Australia
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Y.P.Zhang et al.
1 3
276 Page 2 of 23
be grouped into two categories. One category is the grey-
scale density mapping (GDM) strategy. This design strategy
first uses standard TO algorithms such as the solid isotropic
material with penalization (SIMP) method (Bendsøe 1989;
Zhou and Rozvany 1991; Mlejnek 1992; Sigmund 2001;
Andreassen etal. 2011; Wang etal. 2022), the evolutionary
structural optimization method (Querin etal. 1998, 2000;
Huang and Xie 2007), or the level set method (Wang etal.
2003; Challis and Guest 2009; Challis 2010; van Dijk etal.
2013; Azari Nejat etal. 2022; Lin etal. 2022) with finite
element discretization or Isogeometric Analysis (Guerder
etal. 2022) to determine a greyscale density solution of
material distribution. Next, FGL structures with a selected
type-based unit cell are established according to the obtained
material distribution. The other category is the ground struc-
ture optimization (GSO) strategy. In this strategy, physical
structures such as trusses, beams or frames are used to rep-
resent the FGL structures. Gradient-based and derivative
free methods have been used to optimize the cross-section
dimensions, node positions and the structural topology (or
element connectivity) of truss, beam and frame subjected
to prescribed design requirements such as volume ratio,
deflection and stress (Miguel etal. 2013; Nguyen etal. 2013;
Zhang etal. 2015; Han and Lu 2018). By comparing these
two main strategies, GDM is more efficient because it needs
to optimize a smaller number of design variables associ-
ated with each structural element. In contrast, GSO requires
the optimization of a much larger number of design vari-
ables including material properties, node positions and the
structure topology (or element connectivity) (Miguel etal.
2013). However, the GSO strategy provides a clearer physi-
cal structural configuration without the need of designing the
internal structure and the connectivity for each lattice unit
cell (Panesar etal. 2018).
In this paper, we propose a new strategy for the optimal
design of FGL structures. This new strategy involves using a
recently proposed lattice structure model called the Hencky
bar-grid model (HBM) (Zhang etal. 2021a, b, 2022). A
Hencky bar model is a type of bar-spring structural model
that was pioneered by Hencky (1921). Owing to its simple
and clear physical representation and great flexibility, it had
been recently revisited and expanded to study various static,
buckling and dynamic problems of different structural types
(Wang etal. 2017, 2020; Zhang etal. 2018c, b, a, d, 2019a,
b, 2022). HBM comprises bars and springs whose stiffnesses
may be adjusted to allow for different material property
distributions. The optimization of the spring stiffnesses is
performed by using a gradient-based TO method to mini-
mize the compliance while satisfying required constraints.
The proposed design framework combines the advantages
of both GDM and GSO strategies. Firstly, HBM is easy to
couple with standard TO algorithms such as SIMP to effi-
ciently obtain optimal designs involving a smaller number
of design variables such as the spring stiffnesses of each grid
cell. Secondly, HBM is a simple bar-spring model which
does not need detail design to the internal structure and con-
nectivity for every lattice unit cells.
The layout of this paper is as follows: Sect.2 presents
the optimization problem definition. The new version of
the HBM is presented in Sect.3. Section4 explains how to
select the value of the HBM spring stiffnesses for solving a
linear elasticity problem. The implementation of the HBM
for TO is described in Sect.5. Section6 illustrates exam-
ples of an FGL beam and an FGL plate that are optimally
designed through the proposed HBM-TO method. Section7
shows a summary of the proposed HBM-TO method and
Sect.8 gives the concluding remarks. Appendix A contains
a simple python code that was used to obtain the optimal
solutions presented in Sect.6.
2 Problem denition
Consider an elastic FGL structure with length
𝛼L
, width
L
, and a uniform thickness
h
, which is illustrated in Fig.1.
A non-uniform Young's modulus
E(x,y)
and a constant
Poisson’s ratio
𝜈
are assumed. The FGL structure is simply
supported along its longitudinal edges and is subjected to a
central line load
P
. Body forces are not considered.
The problem at hand is to determine the optimal design
of the cross-section of FGL structures composed by either
large or small lattice units for maximum stiffness (Kaveh
etal. 2008), i.e. having minimum compliance (Hassani and
Hinton 1999) and then design a structure made by function-
ally graded material based on the material distribution given
by the optimal FGL structures. Figure1b and c illustrates
optimal design examples that differ in terms of lattice cell
size. Figure1d illustrates an optimal design example of a
functionally graded solid structure.
3 Hencky bar‑grid model
Hencky (1921) proposed a discrete structural model com-
prising rigid bars connected by frictionless hinges and elas-
tic rotational springs for solving the elastic buckling prob-
lem of columns under a compressive axial load. Since then,
there has been much work done on the so-called Hencky
bar-chain/net/grid model (HBM) for analysis of all kinds of
structural forms from beams to frames to arches to plates
(Wang etal., 2020). Recently, Zhang etal. (2021b) extended
the HBM for solving plane elasticity problems which has the
ability to model 2D structures for the full range of Poisson’s
ratio. In contrast, some previous lattice models such as the
models formulated by Born and Karman (1912) and Hren-
nikoff (1941) can only model 2D structures for Poisson’s
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
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Page 3 of 23 276
ratio less than 1/3 (Zhang etal. 2021a; Challamel etal.
2022).
In this study, we develop a slightly different version of
HBM. A comparison of the new version of HBM and its pre-
decessor reported in Zhang etal. (2021b) is given in Fig.2.
The main difference between the newly proposed HBM and
its predecessor is the construction of the internal (i.e. not at
the edge or corner) rigid bars. The new HBM has twice the
number of rigid bars and springs. This change is essential,
as it allows the determination of local stiffness matrix for
each HBM grid cell (see Eq.(27)) which will be required
by the topology optimization algorithm developed for the
adoption of HBM.
Figure3 shows the structure representation, a unit rigid
bar-grid representation and a lattice structure representa-
tion of the new HBM. As shown in Fig.3a, the new version
of HBM discretize a continuum plane structure into a rigid
bar-grid system with a cell size

that are connected by fric-
tionless hinges. Within a grid cell, four primary axial springs
connect two rigid bars placed at each side of the grid cell
with stiffnesses
k
xx
and
kyy
in the x- and y- directions, respec-
tively, as shown in Fig.3b. Four secondary axial springs are
employed to model the Poisson effect with stiffnesses
kxy
and
kyx
in the y- and x- directions, respectively, as shown in
Fig.3b. Four torsional springs are installed at the four cor-
ners with stiffnesses
kS
as shown in Fig.3b. While primary
and secondary springs model the axial stiffness and the Pois-
son effect, respectively, torsional springs are employed to
resist in-plane shear forces. A lattice structure representation
related to HBM is shown in Fig.3c. It is worth noting that
the secondary axial springs of HBM are represented by some
curved bars. It is possible to use other structures to represent
the secondary axial springs.
For the sake of clarity, we shall demonstrate the topology
optimization of HBM by using the lattice structures com-
prising basic lattice cell as shown in Fig.3c. For a lattice cell
whose centre is located at
x
𝓁
=i+
1
2
and
y
𝓁
=j+
1
2
, its hori-
zontal and vertical axial bars have stiffnesses of
kxx
i+1
2
,j+
1
2
and
kyy
i+1
2
,j+
1
2
, respectively. Likewise, its curved bars located at the
corners and middle of the cell have stiffnesses of
kS
i+1
2
,j+
1
2
and
k
xy
i+1
2,j+1
2
(
or kyx
i+1
2,j+1
2
)
, respectively. We assume that the bars
in all lattice cells have a constant elastic modulus

E
and a
constant density
𝜌
but a varying cross-section area
A
i+1
2
,j+
1
2
.
Therefore, an HBM grid with stiffer springs results in a lat-
tice cell with a greater mass as shown in Fig.3d.
When a bar-grid cell as shown in Fig.3b is stretched or
shortened, the resulting axial forces
fx
,fy
are resisted by
both primary axial springs and secondary axial springs, i.e.
(1)
fx
i+1
2,j
=k
xx
i+1
2,j+1
2(ui+1,j−ui,j)+
1
2kxy
i+1
2,j+1
2
(
v
i,j
+
1
+v
i
+
1,j
+
1
−v
i,j
−v
i
+
1,j)
(2)
fy
i,j+1
2
=kyy
i+1
2,j+1
2(vi,j+1−vi,j)+
1
2kyx
i+1
2,j+
1
2
(u
i
+1,
j
+u
i
+1,
j
+1−u
i
,
j
−u
i
,
j
+1)
p
x
y
z
P
L
αL
x
y
P
L
αL
x
y
(a)
(b)
(c)
(d)
P
L
αL
x
y
Fig. 1 Design FGL structure subjected to a central line load a 3D
Geometry; b illustration of an optimal design with large lattice cells;
c illustration of an optimal design with small lattice cells; d illustra-
tion of a structure made by functionally graded material based on the
optimal FGL design
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Y.P.Zhang et al.
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276 Page 4 of 23
where u, v are the in-plane displacements in the x- and y-
directions at the joints, respectively. The subscripts i and j
indicate the location of corresponding joints in the model.
When the new version HBM cell undergoes in-plane
shearing, the elastic torsional springs are deformed due to
the change of angles at each corner. In this way, the lumped
torque TS for one bar-grid cell is given by
where θ is the change of angle at each corner of the bar-grid
cell. The superscripts a, b, c, and d denote the corner points
at the bar-grid cell (see Fig.3b).
(3)
fx
i+1
2,j+1
=k
xx
i+1
2,j+1
2(ui+1,j+1−ui,j+1)+
1
2kxy
i+1
2,j+
1
2
(
v
i
,
j
+1+v
i
+1,
j
+1−v
i
,
j
−v
i
+1,
j)
(4)
fy
i+1,j+1
2
=k
yy
i+1
2,j+1
2(vi+1,j+1−vi+1,j)+
1
2kyx
i+1
2,j+
1
2
(
u
i
+
1,j
+u
i
+
1,j
+
1
−u
i,j
−u
i,j
+
1)
(5)
TSa
i+1
2
,j+1
2
=k
S
i+1
2
,j+1
2
𝜃
a
i+1
2
,j+
1
2
(6)
TSb
i+1
2
,j+1
2
=k
S
i+1
2
,j+1
2
𝜃
b
i+1
2
,j+
1
2
(7)
TSc
i+1
2
,j+1
2
=k
S
i+1
2
,j+1
2
𝜃
c
i+1
2
,j+
1
2
(8)
TSd
i+1
2
,j+1
2
=k
S
i+1
2
,j+1
2
𝜃
d
i+1
2
,j+
1
2
Based on small angle approximation, θ may be expressed
in terms of the in-plane displacements u, v as
The elastic strain energy
U
i+1
2
,j+
1
2
including the contribu-
tion of the torsional energy stored in the deformed springs
for a bar-grid cell is
In view of Eqs. (1)–(13) the elastic strain energy
U
i+1
2
,j+
1
2
can be reformulated in terms of in-plane displacements u, v,
(9)
𝜃
a
i+1
2
,j+1
2
≈tan 𝜃a
i+1
2
,j+1
2
=1
2
(u
i,j+1
−u
i,j
𝓁+
v
i+1,j
−v
i,j
𝓁
)
(10)
𝜃
b
i+1
2
,j+1
2
≈tan 𝜃b
i+1
2
,j+1
2
=1
2
(u
i+1,j+1
−u
i+1,j
𝓁+
v
i+1,j
−v
i,j
𝓁
)
(11)
𝜃
c
i+1
2
,j+1
2
≈tan 𝜃c
i+1
2
,j+1
2
=1
2
(u
i+1,j+1
−u
i+1,j
𝓁+
v
i+1,j+1
−v
i,j+1
𝓁
)
(12)
𝜃
d
i+1
2
,j+1
2
≈tan 𝜃d
i+1
2
,j+1
2
=1
2(u
i,j+1
−u
i,j
𝓁+
v
i+1,j+1
−v
i,j+1
𝓁
)
(13)
U
i+1
2,j+1
2
=
1
2(ui+1,j−ui,j)fx
i+1
2,j
+
1
2(ui+1,j+1−ui,j+1)fx
i+1
2,j+1
+1
2(vi,j+1−vi,j)fy
i,j+1
2
+1
2(vi+1,j+1−vi+1,j)fy
i+1,j+
1
2
+1
2
𝜃a
i+1
2,j+1
2
TSa
i+1
2,j+1
2
+1
2
𝜃b
i+1
2,j+1
2
TSb
i+1
2,j+1
2
+1
2
𝜃c
i+1
2
,j+1
2
TSc
i+1
2
,j+1
2
+1
2
𝜃d
i+1
2
,j+1
2
TSd
i+1
2
,j+1
2
Fig. 2 Comparison of structure
representation between a prede-
cessor model and b new HBM
ll
l
l
i
j
0 1 2
0
1
2
ll
l
l
i
j
0 1 2
0
1
2
(a)
(b)
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
1 3
Page 5 of 23 276
A
l l l l l l l l l l l l l l l l
B
C
D
l
l
l
l
l
l
l
l
y
x
Fyy
Fyy Fyy Fyy+Fxy
F
yy
Fyy
Fyy
Fxy
Fxy
Fxy Fxy Fxy Fxy
Fyy+Fxy Fyy+Fxy
Fxy
Fxy Fxy
Fyy+Fxy
Fxy
Fxx
Fxx
Fxx+Fxy
Fxx
Fxx
Fxx+Fxy
bx
by
ui,j, vi,j ui+1,j, vi+1,j
ui+1,j+1, vi+1,j+1
ui,j+1, vi,j+1
torsional
spring
primary
axial
spring
secondary
axial spring
kS
frictionless
hinge
frictionless
pulley
kyy
kxx
kyx
kxy
l
l
ab
c
d
Fi+ ,j+
1
21
2
yyd Fi+ ,j+
1
21
2
yyc
Fi+ ,j+
1
21
2
yya Fi+ ,j+
1
21
2
yyb
Fi+ ,j+
1
21
2
xxb
Fi+ ,j+
1
21
2
xxc
Fi+ ,j+
1
21
2
xxd
Fi+ ,j+
1
21
2
xxa
Fi+ ,j+
1
21
2
xyd
Fi+ ,j+
1
21
2
xya
Fi+ ,j+
1
21
2
xya Fi+ ,j+
1
21
2
xyb
Fi+ ,j+
1
21
2
xyd Fi+ ,j+
1
21
2
xyc
Fi+ ,j+
1
21
2
xyc
Fi+ ,j+
1
21
2
xyb
primary
axial
spring
kyy
kxx
kyx
kxy
kS
torsional
spring
secondary
axial spring
l
l
HBM spring stiffness
(a)
(b) (c)
(d)
Fig. 3 New version of Hencky bar-grid model (HBM): a geometry and boundary conditions; b unit bar-grid cell representation; c lattice struc-
ture representation; d relation between HBM and lattice structure
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Y.P.Zhang et al.
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276 Page 6 of 23
Next, the work done by in-plane axial forces
Fxx
,
Fyy
, in-
plane shear force
Fxy
and the in-plane body forces
bx
and
by
lumped on the joints of each HBM cell as shown in Fig.3b
are given by
The total potential energy function is given by
where
Ω
is a set contains all the indexes of the bar-grid cells
for the HBM.
Now, the local stiffness matrix of an HBM cell is obtained
through a variational formulation, by taking
(14)
U
i+1
2,j+1
2
=
1
2k
xx
i+1
2,j+1
2[(ui+1,j−ui,j)2+(ui+1,j+1−ui,j+1)2
]
+1
2kyy
i+1
2,j+1
2[(vi,j+1−vi,j)2+(vi+1,j+1−vi+1,j)2]
+1
2kxy
i+1
2,j+1
2(ui+1,j−ui,j)(vi,j+1+vi+1,j+1−vi,j−vi+1,j)
+1
2kxy
i+1
2,j+1
2(ui+1,j+1−ui,j+1)(vi,j+1+vi+1,j+1−vi,j−vi+1,j
)
+1
2kyx
i+1
2,j+1
2(vi,j+1−vi,j)(ui+1,j+ui+1,j+1−ui,j−ui,j+1)
+1
2kyx
i+1
2,j+1
2(vi+1,j+1−vi+1,j)(ui+1,j+ui+1,j+1−ui,j−ui,j+1
)
+1
82kS
i+1
2,j+1
2[(ui,j+1−ui,j+vi+1,j−vi,j)2
+(ui+1,j+1−ui+1,j+vi+1,j−vi,j)2
+(ui+1,j+1−ui+1,j+vi+1,j+1−vi,j+1)2
+
(
ui,j+1−ui,j+vi+1,j+1−vi,j+1
)
2
]
(15)
W
i+1
2,j+1
2
=ui,j
(
Fxxa
i+1
2,j+1
2
+Fxya
i+1
2,j+1
2
+bxa
i+1
2,j+1
2
)
+vi,j(Fyya
i+1
2,j+1
2
+Fxya
i+1
2,j+1
2
+bya
i+1
2,j+1
2)
+ui+1,j(Fxxb
i+1
2,j+1
2
+Fxyb
i+1
2,j+1
2
+bxb
i+1
2,j+1
2)
+vi+1,j(Fyyb
i+1
2,j+1
2
+Fxyb
i+1
2,j+1
2
+byb
i+1
2,j+1
2)
+ui+1,j+1(Fxxc
i+1
2,j+1
2
+Fxyc
i+1
2,j+1
2
+bxc
i+1
2,j+1
2
)
+vi+1,j+1(Fyyc
i+1
2,j+1
2
+Fxyc
i+1
2,j+1
2
+byc
i+1
2,j+1
2
)
+ui,j+1(Fxxd
i+1
2,j+1
2
+Fxyd
i+1
2,j+1
2
+bxd
i+1
2,j+1
2)
+vi,j+1
(
Fyyd
i+1
2
,j+1
2
+Fxyd
i+1
2
,j+1
2
+byd
i+1
2
,j+1
2)
(16)
Π=
Ω
∑
i=0
Ω
∑
j=0(
Ui+1
2,j+1
2
−Wi+1
2,j+1
2
)
Based on Eq.(14), the discrete equations of the derivative
of the elastic strain energy
U
i+1
2
,j+
1
2
for an HBM cell
(
i
+
1
2
,j+
1
2
) are given by
(17)
𝛿Π=0
(18)
𝜕
Ui+1
2,j+1
2
𝜕ui,j
⇒−kxx
i+1
2,j+1
2ui+1,j−ui,j−1
2
kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2
vi,j+1+vi+1,j+1−vi,j−vi+1,j
−1
42kS
i+1
2,j+1
22ui,j+1−2ui,j+vi+1,
j
−v
i,j
+v
i
+
1,j
+
1
−v
i,j
+
1
(19)
𝜕
Ui+1
2,j+1
2
𝜕ui+1,j
⇒kxx
i+1
2,j+1
2ui+1,j−ui,j
+1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2
vi,j+1+vi+1,j+1−vi,j−vi+1,j
−1
42kS
i+1
2,j+1
22ui+1,j+1−2ui+1,j+vi+1,
j
−v
i
,
j
+v
i
+1,
j
+1−v
i
,
j
+1

(20)
𝜕
Ui+1
2,j+1
2
𝜕ui+1,j+1
⇒kxx
i+1
2,j+1
2ui+1,j+1−ui,j+1+1
2
kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2
vi,j+1+vi+1,j+1−vi,j−vi+1,j
+1
42kS
i+1
2,j+1
22ui+1,j+1−2ui+1,j+vi+1,
j
−v
i
,
j
+v
i
+1,
j
+1−v
i
,
j
+1

(21)
𝜕
Ui+1
2,j+1
2
𝜕ui,j+1
⇒−kxx
i+1
2,j+1
2ui+1,j+1−ui,j+1
−1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2
vi,j+1+vi+1,j+1−vi,j−vi+1,j
+1
42kS
i+1
2,j+1
22ui,j+1−2ui,j+vi+1,
j
−v
i
,
j
+v
i
+1,
j
+1−v
i
,
j
+1

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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
1 3
Page 7 of 23 276
and
Likewise, the discrete equations of the derivative of the exter-
nal work
W
i+1
2
,j+
1
2
for an HBM cell (
i
+
1
2
,j+
1
2
) are given by
(22)
𝜕
Ui+1
2,j+1
2
𝜕vi,j
⇒−kyy
i+1
2,j+1
2vi,j+1−vi,j−1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2

ui+1,j+ui+1,j+1−ui,j−ui,j+1−1
42kS
i+1
2,j+1
2

2v
i
+
1,j
−2v
i,j
+u
i,j
+
1
−u
i,j
+u
i
+
1,j
+
1
−u
i
+
1,j
(23)
𝜕
Ui+1
2,j+1
2
𝜕vi+1,j
⇒−kyy
i+1
2,j+1
2vi+1,j+1−vi+1,j−1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2

ui+1,j+ui+1,j+1−ui,j−ui,j+1+1
42kS
i+1
2,j+1
2

2v
i
+
1,j
−2v
i,j
+u
i,j
+
1
−u
i,j
+u
i
+
1,j
+
1
−u
i
+
1,j
(24)
𝜕
Ui+1
2,j+1
2
𝜕vi+1,j+1
⇒kyy
i+1
2,j+1
2vi+1,j+1−vi+1,j+1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2

ui+1,j+ui+1,j+1−ui,j−ui,j+1+1
42kS
i+1
2,j+1
2

2v
i
+1,
j
+1−2v
i
,
j
+1+u
i
,
j
+1−u
i
,
j
+u
i
+1,
j
+1−u
i
+1,
j
(25)
𝜕
Ui+1
2,j+1
2
𝜕vi,j+1
⇒kyy
i+1
2,j+1
2vi,j+1−vi,j+1
2kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2
ui+1,j+ui+1,j+1−ui,j−ui,j+1−1
42kS
i+1
2,j+1
2

2v
i
+1,
j
+1−2v
i
,
j
+1+u
i
,
j
+1−u
i
,
j
+u
i
+1,
j
+1−u
i
+1,
j
(26)
𝜕
Wi+1
2,j+1
2
𝜕ui,j
⇒Fxxa
i+1
2,j+1
2
+Fxya
i+1
2,j+1
2
+bxa
i+1
2,j+1
2
;
𝜕Wi+1
2,j+1
2
𝜕vi,j
⇒Fyya
i+1
2
,j+1
2
+Fxya
i+1
2
,j+1
2
+bya
i+1
2
,j+1
2
(27)
𝜕
Wi+1
2,j+1
2
𝜕ui+1,j
⇒Fxxb
i+1
2,j+1
2
+Fxyb
i+1
2,j+1
2
+bxb
i+1
2,j+1
2
;
𝜕Wi+1
2,j+1
2
𝜕vi+1,j
⇒Fyyb
i+1
2
,j+1
2
+Fxyb
i+1
2
,j+1
2
+byb
i+1
2
,j+1
2
(28)
𝜕
Wi+1
2,j+1
2
𝜕ui+1,j+1
⇒Fxxc
i+1
2,j+1
2
+Fxyc
i+1
2,j+1
2
+bxc
i+1
2,j+1
2
;
𝜕Wi+1
2,j+1
2
𝜕vi+1,j+1
⇒Fyyc
i+1
2
,j+1
2
+Fxyc
i+1
2
,j+1
2
+byc
i+1
2
,j+1
2
(29)
𝜕
Wi+1
2,j+1
2
𝜕ui,j+1
⇒Fxxd
i+1
2,j+1
2
+Fxyd
i+1
2,j+1
2
+bxd
i+1
2,j+1
2
;
𝜕Wi+1
2,j+1
2
𝜕v
i,j
+
1
⇒Fyyd
i+1
2,j+1
2
+Fxyd
i+1
2,j+1
2
+byd
i+1
2,j+
1
2
From Eqs. (18)–(29), the local stiffness matrix for a bar-
grid cell of the HBM is given by
where
with
and
k
xy
=1
2
(
kxy
i+1
2
,j+1
2
+kyx
i+1
2
,j+1
2)
;
(30)
[k]
i+1
2
,j+1
2
{u}
i+1
2
+j+1
2
={f}
i+1
2
,j+
1
2
(31)
[
k]i+1
2,j+1
2
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
axbc
xde−bf −d
ay−df −be d c
y
ax−
bf deb
ay−dc
y
be
axbc
xd
ay−df
sym.ax−b
ay
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(32)
a
x=kxx
+k
S
2𝓁
2;ay=kyy
+k
S
2𝓁
2
(33)
b
=k
xy
2
+kS
4𝓁2
(34)
swcx
=−k
xx
;c
y
=−k
yy
(35)
d
=k
xy
2
−kS
4𝓁
2
(36)
e=0
(37)
f=− k
S
2𝓁
2
(38)
{
u}i+1
2+j+1
2
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ui,j
vi,j
ui+1,j
vi+1,j
ui+1,j+1
vi+1,j+1
ui,j+1
vi,j+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
;{f}i+1
2+j+1
2
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
Fxxa
i+1
2,j+1
2
+Fxya
i+1
2,j+1
2
+bxa
i+1
2,j+1
2
Fyya
i+1
2,j+1
2
+Fxya
i+1
2,j+1
2
+bya
i+1
2,j+1
2
Fxxb
i+1
2,j+1
2
+Fxyb
i+1
2,j+1
2
+bxb
i+1
2,j+1
2
Fyyb
i+1
2,j+1
2
+Fxyb
i+1
2,j+1
2
+byb
i+1
2,j+1
2
Fxxc
i+1
2,j+1
2
+Fxyc
i+1
2,j+1
2
+bxc
i+1
2,j+1
2
Fyyc
i+1
2,j+1
2
+Fxyc
i+1
2,j+1
2
+byc
i+1
2,j+1
2
Fxxd
i+1
2,j+1
2
+Fxyd
i+1
2,j+1
2
+bxd
i+1
2,j+1
2
Fyyd
i+1
2
,j+1
2
+Fxyd
i+1
2
,j+1
2
+byd
i+1
2
,j+1
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
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Y.P.Zhang et al.
1 3
276 Page 8 of 23
In view of Eqs. (18)–(38) and by assembling the expres-
sions for all nodes in the system, the following linear system
of equation can be derived
where
and
The in-plane displacements of each joint
ui,j
and
vi,j
can
be calculated by solving Eq.(39). Next the axial forces
fx
i+1
2
,
j
and f
y
i,j+
1
2
and the torque
TS
1+1
2
,1+
1
2
are computed via the
obtained in-plane displacements
ui,j
and
vi,j
using Eqs.
(1)–(8).
4 Determination ofHBM spring stiness
In this section, we shall determine the spring stiffnesses
of HBM. For simplicity, it is assumed that each individual
HBM grid cell is homogeneous and isotropic. The inhomo-
geneous material properties of the structure are captured by
varying the material properties of different HBM cells. A
general version of the continuum equations of elastodynam-
ics (or Navier’s equations of elastodynamics) considering
inhomogeneous material properties and plane-strain can be
written as (Gurtin 1973)
and
(39)
[K]HBM{u}={F}HBM
(40)
[
K]HBM =
Ω
∑
i=0
Ω
∑
j=0
[k]i+1
2,j+
1
2
(41)
{
F}HBM =
Ω
∑
i=0
Ω
∑
j=0
{f}i+1
2,j+
1
2
(42)
{
u}=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
u0,0
u0,1
…
unx−1,ny−1
v0,0
v0,1
…
vnx−1,ny−1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(43)
∫
𝜕
𝜕x
[
(1−v)E
(1+v)(1−2v)
𝜕u
𝜕x+vE
(1+v)(1−2v)
𝜕v
𝜕y
]
+𝜕
𝜕y
[
E
2
(
1+v
)
𝜕u
𝜕y+E
2
(
1+v
)
𝜕v
𝜕x
]
dΩ=∫𝜌ud
Ω
By considering the local homogeneous properties within
the area of each HBM grid cell, i.e.
x
∈
[(
i−1
2)
𝓁,
(
i+1
2)
𝓁
]
,y∈
[(
j−1
2)
𝓁,
(
j+1
2)
𝓁
]
the general continuum equations of elastodynamics can be
simplified to
and
By expanding Eqs. (18) and (19) using Taylor’s series
following the procedures used in (Zhang etal. 2019c), we
obtain
and
By assuming the case of quasi-static, neglecting higher
order terms and comparing Eqs. (48) and (49) to the
(44)
∫
𝜕
𝜕y
[
(1−𝜈)E
(1+𝜈)(1−2𝜈)
𝜕v
𝜕y+𝜈E
(1+𝜈)(1−2𝜈)
𝜕u
𝜕x
]
+𝜕
𝜕x
[
E
2
(
1+𝜈
)
𝜕u
𝜕y+E
2
(
1+𝜈
)
𝜕v
𝜕x
]
dΩ=∫𝜌vd
Ω
(45)
E(x,y)=E
i+1
2,j+1
2
=E
e
;
𝜈
(x,y)=
𝜈i+1
2,j+
1
2
=𝜈e;
(i+1
2)𝓁
∫
(
i−1
2
)
𝓁
(j+1
2)𝓁
∫
(
j−1
2
)
𝓁
h
∫
0
dΩ=Ve
(46)
(
1−𝜈e
)
EeVe
(
1+𝜈e)(1−2𝜈e)
𝜕2u
𝜕x2+EeVe
2(1+𝜈e)
𝜕2
u
𝜕y2
+EeVe
2
(
1+𝜈
e)(
1−2𝜈
e)
𝜕2v
𝜕x𝜕y
=𝜌Veu
(47)
(
1−𝜈e
)
EeVe
(
1+𝜈e)(1−2𝜈e)
𝜕2v
𝜕y2+EeVe
2(1+𝜈e)
𝜕2
v
𝜕x
2
+EeVe
2
(
1+𝜈
e)(
1−2𝜈
e)
𝜕2u
𝜕x𝜕y
=𝜌Ve𝜈
(48)
2
kxx
i+1
2,j+1
2
𝓁2𝜕2u
𝜕x2+kS
i+1
2,j+1
2
𝜕2u
𝜕y2
+kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2𝓁2+kS
i+1
2,j+1
2

𝜕2v
𝜕
x
𝜕
y
+O

𝓁4

=0
(49)
2
kyy
i+1
2,j+1
2
𝓁2𝜕
2
v
𝜕y2+kS
i+1
2,j+1
2
𝜕
2
v
𝜕x2
+kxy
i+1
2,j+1
2
+kyx
i+1
2,j+1
2𝓁2+kS
i+1
2,j+1
2

𝜕2u
𝜕
x
𝜕
y
+O

𝓁4

=0
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
1 3
Page 9 of 23 276
continuum equation of motion Eqs. (46) and (47), it follows
that
A first-order central finite difference (FD) discretization
of Eqs. (43) and (44) results in
(50)
k
xx
i+1
2,j+1
2
=
(
1−𝜈e
)
EeVe
2
(
1+𝜈
e)(
1−2𝜈
e)
(51)
k
yy
i+1
2,j+1
2
=
(
1−𝜈e
)
EeVe
2
(
1+𝜈
e)(
1−2𝜈
e)
(52)
k
xy
i+1
2,j+1
2
=
𝜈
e
E
e
V
e
2
(
1+𝜈
e)(
1−2𝜈
e)
(53)
k
S
i+1
2,j+1
2
=EeVe𝓁
2
2
(
1+𝜈
e)
(54)
−
1
2

1−𝜈

EV
21+𝜈1−2𝜈i−1
2,j+1
2
+

1−𝜈

EV
21+𝜈1−2𝜈i−1
2,j−1
2

u
i,j−ui−1,j
2−1
2𝜈EV
21+𝜈1−2𝜈i−1
2,j+1
2
vi,j+1−vi,j+vi−1,j+1−vi−1,
j
22
+𝜈EV
21+𝜈1−2𝜈i−1
2,j−1
2
vi,j−vi,j−1+vi−1,j−vi−1,j−1
22
+1
21−𝜈EV
21+𝜈1−2𝜈i+1
2,j+1
2
+1−𝜈EV
21+𝜈1−2𝜈i+1
2,j−1
2
u
i+1,j−ui,j
2+1
2𝜈EV
21+𝜈1−2𝜈i+1
2,j+1
2
vi+1,j+1−vi+1,j+vi,j+1−vi,
j
22
+𝜈EV
21+𝜈1−2𝜈i+1
2,j−1
2
vi+1,j−vi+1,j−1+vi,j−vi,j−1
22
+EV
21+𝜈i+1
2,j+1
2
ui,j+1−ui,j+ui+1,j+1−ui+1,j
22
+EV
21+𝜈i−1
2,j+1
2
ui,j+1−ui,j+ui−1,j+1−ui−1,j
22
+EV
21+𝜈i+1
2,j+1
2
vi+1,j+1−vi,j+1+vi+1,j−vi,j
22
+EV
21+𝜈i−1
2,j+1
2
vi,j+1−vi−1,j+1+vi,j−vi−1,j
22
−EV
21+𝜈i+1
2,j−1
2
ui,j−ui,j−1+ui+1,j−ui+1,j−1
22
+EV
21+𝜈i−1
2,j−1
2
ui,j+1−ui,j+ui−1,j+1−ui−1,j
22
+EV
21+𝜈i+1
2,j−1
2
vi+1,j−vi,j+vi+1,j−1−vi,j−1
22
+EV
21+𝜈i−1
2,j−1
2
vi,j−vi−1,j+vi,j−1−vi−1,j−1
22
=1
4𝜌

Vi+1
2,j+1
2
+Vi−1
2,j+1
2
+Vi+1
2,j−1
2
+Vi−1
2,j−1
2

ui,j
The difference equation in y- direction is obtained from
Eq.(54) by swapping variables u, v and indices i, j. Now,
if we assemble four adjacent HBM grid cells like Fig.3b,
the difference equation of the middle node can be obtained
based on Eqs. (18)–(25) as follows:
The difference equation in y- direction is Eq.(55) by
swapping variables u, v and indices i, j. In view of Eqs.
(50)–(55), it can be seen that solving the linear algebraic
system given by the HBM proposed in this work is equiva-
lent to solving an inhomogeneous two-dimensional elasticity
problem using FD method.
Noting that the new HBM coincides with the old one
(Zhang etal. 2021b) when modelling homogeneous struc-
tures. So, it is possible to use the old HBM to model inho-
mogeneous structures, but the resulting energy functions
could not be readily applied to derive the local stiffness
matrix for each grid cell.
5 Topology optimization
The SIMP method is employed to optimize the HBM for-
mulated herein. The spring stiffnesses of each grid cell is
formulated as
(55)
−1
2

k
xx
i−1
2,j+1
2
+k
xx
i−1
2,j−1
2

ui,j−ui−1,j
−
1
4kxy
i−1
2,j+1
2vi,j+1−vi,j+vi−1,j+1−vi−1,j
+
kxy
i−1
2,j−1
2vi,j−vi,j−1+vi−1,j−vi−1,j−1
+
1
2kxx
i+1
2,j+1
2
+kxx
i+1
2,j−1
2ui+1,j−ui,j
+
1
4kxy
i+1
2,j+1
2vi+1,j+1−vi+1,j+vi,j+1−vi,j
+
kxy
i+1
2,j−1
2vi+1,j−vi+1,j−1+vi,j−vi,j−1
+
1
22kS
i+1
2,j+1
2ui,j+1−ui,j+ui+1,j+1−ui+1,j
+
kS
i−1
2,j+1
2ui,j+1−ui,j+ui−1,j+1−ui−1,j
+
kS
i+1
2,j+1
2vi+1,j+1−vi,j+1+vi+1,j−vi,j
+
kS
i−1
2,j+1
2vi,j+1−vi−1,j+1+vi,j−vi−1,j−1
2
2

kS
i+1
2,j−1
2ui,j−ui,j−1+ui+1,j−ui+1,j−1
+
kS
i−1
2,j−1
2ui,j+1−ui,j+ui−1,j+1−ui−1,j
+
kS
i+1
2,j−1
2vi+1,j−vi,j+vi+1,j−1−vi,j−1
+
kS
i−1
2,j−1
2

vi,j−vi−1,j+vi,j−1−vi−1,j−1

=0
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Y.P.Zhang et al.
1 3
276 Page 10 of 23
where
kxx
0
,k
yy
0
,kxy
0
,k
yx
0
and
kS
0
are the starting values of spring
stiffnesses for TO,
𝜌
is a non-dimensional variable intro-
duced for HBM-TO and
m
is the exponent for penalization.
In order to avoid singularity issues when solving the linear
system given in Eq.(39), all HBM spring stiffnesses are
constrained to have a minimum value of
𝜙min
k
0
at the void
areas. In view of Eqs. (56)–(61) and Eq.(14), the elastic
strain energy of a grid cell may be expressed as
where the starting value of elastic strain energy
U0
i+1
2
,j+
1
2
is
given by
(56)
kxx
i+1
2
,j+1
2
=𝜒 i+1
2
,j+1
2
k
xx
0
(57)
kyy
i+1
2
,j+1
2
=𝜒 i+1
2
,j+1
2
k
yy
0
(58)
kxy
i+1
2
,j+1
2
=𝜒 i+1
2,j+1
2
k
xy
0
(59)
kyx
i+1
2
,j+1
2
=𝜒 i+1
2,j+1
2
k
yx
0
(60)
kS
i+1
2
,j+1
2
=𝜒 i+1
2,j+1
2
k
S
0
(61)
𝜒
i+1
2,j+1
2
=𝜙min +𝜌
m
i+1
2
,j+1
2
(
1−𝜙min
)
(62)
U
i+1
2
,j+1
2
=𝜒 i+1
2
,j+1
2
U
0
i+1
2
,j+
1
2
(63)
U0
i+1
2,j+1
2
=
1
2k
xx
0
[
(ui+1,j−ui,j)2+(ui+1,j+1−ui,j+1)2
]
+1
2kyy
0[(vi,j+1−vi,j)2+(vi+1,j+1−vi+1,j)2]
+1
2kxy
0(ui+1,j−ui,j)(vi,j+1+vi+1,j+1−vi,j−vi+1,j)
+1
2kxy
0(ui+1,j+1−ui,j+1)(vi,j+1+vi+1,j+1−vi,j−vi+1,j
)
+1
2kyx
0(vi,j+1−vi,j)(ui+1,j+ui+1,j+1−ui,j−ui,j+1)
+1
2kyx
0(vi+1,j+1−vi+1,j)(ui+1,j+ui+1,j+1−ui,j−ui,j+1
)
+1
82kS
0[(ui,j+1−ui,j+vi+1,j−vi,j)2
+(ui+1,j+1−ui+1,j+vi+1,j−vi,j)2
+(ui+1,j+1−ui+1,j+vi+1,j+1−vi,j+1)2
+
(
ui,j+1−ui,j+vi+1,j+1−vi,j+1
)
2
]
In view of Eqs. (63) and (16), the total strain energy is
The TO problem for HBM can be posed as
where
Vf
is the prescribed volume fraction and
V(𝜌 )
and
V0
are the material volume and design volume, respectively.
There are various methods for solving the aforementioned
optimization problem such as Optimality Criteria (OC) (Yin
and Yang 2001), Sequential Linear Programming (Dunning
and Kim 2015), and Method of Moving Asymptotes (Svan-
berg 1987). The OC method is adopted herein due to its
efficiency in solving optimization problems with a single
objective function (Sigmund 1997).
Following the works of Bendsøe (1989), Sigmund (1997)
and Sigmund and Petersson (1998) and assuming a constant
change of design volume, the design variable
𝜌
i+1
2
,j+
1
2
is
updated as follows:
If
max (
0, 𝜌 i+1
2,j+1
2
−𝛿
)
≥𝜌 i+1
2,j+1
2

D𝛾
i+1
2
,j+
1
2
if
max
(0, 𝜌 i+1
2,j+1
2
−𝛿
)
< 𝜌 i+1
2,j+1
2

D𝛾
i+1
2
,j+1
2
≤min (1, 𝜌 i+1
2,j+1
2
+𝛿
)
if
𝜌
i+1
2,j+1
2

D𝛾
i+1
2
,j+1
2
>min
(
1, 𝜌 i+1
2,j+1
2
+𝛿
)
where δ is a positive value to limit the change of design vari-
able
𝜌
i+1
2
,j+
1
2
between two successive iterations and γ is a
numerical damping exponent. These parameters are intro-
duced to avoid drastic change of density between adjacent
cells that are impractical (i.e. porous structures).

D
i+1
2
,j+
1
2
is
an auxiliary variable defined as
where λ is a Lagrangian multiplier which can be obtained
through the bisection method. In view of Eq.(64), the HBM
grid cell sensitivity matrices
c
can be written as follows
(64)
U
(𝜌 )=
Ω
∑
i=0
Ω
∑
j=0
𝜒 i+1
2,j+1
2
(
𝜌 i+1
2,j+1
2
)
U0
i+1
2,j+1
2
.
(65)
min
𝜌
∈(0,1]
(
U(𝜌 )
)
, subject to: Eq.(39)and
V(𝜌 )
V
0
=Vf
(66)
𝜌
i+1
2
,j+1
2
←max
(
0, 𝜌 i+1
2
,j+1
2
−𝛿
)
(67)
𝜌
i+1
2,j+1
2
←𝜌 i+1
2,j+1
2

D
𝛾
i+1
2
,j+
1
2
(68)
𝜌
i+1
2
,j+1
2
←min
(
1, 𝜌 i+1
2
,j+1
2
+𝛿
)
(69)

D
i+1
2,j+1
2
=−
1
𝜆
𝜕Π
𝜕 𝜌 i+1
2
,j+1
2
=−𝜆−1ci+1
2+j+1
2
,
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
1 3
Page 11 of 23 276
In order to avoid potential “chess board patterns”, a sim-
ple filtering technique formulated by Andreassen etal.
(2011) is adopted by replacing the HBM grid cell sensitivity
matrices
c
i+1
2
+j+
1
2
with
where
𝜑 (i,j,k,l)
is a weight factor defined as
The k and l indicate the location of corresponding joints
in the model. Equations(71) and (72) indicate that the filter
is a local averaging operator whereby the HBM grid cell
sensitivity matrices
c
i+1
2
+j+
1
2
take on an average value of all
c
weighted by
𝜑
within a radius
r
. The filtered HBM grid cell
sensitivity matrices obtained through Eqs. (71) and (72) are
used when updating

D
i+1
2
,j+
1
2
in Eq.(69).
6 Results anddiscussions
In this section, we first validate the proposed HBM through
modeling continuous functionally graded structures. We
then apply the HBM-TO to determine the optimal design
of a beam and a plate FGL structures subjected to central
line loading. The considered boundary conditions include
loads and prescribed displacements. In the following exam-
ple problems, we assume the material has Poisson’s ratio of
0.3 and set the starting value for the elastic modulus
E0
as 1.
6.1 Validation ofHBM forfunctionally graded plane
bodies
In this section, plane-stress and plane-strain elasticity prob-
lems are considered that are confined in a square design
domain under the loading and displacements boundary con-
ditions illustrated in Fig.1. The functionally graded Young’s
modulus is assumed to be
The accuracy of the proposed HBM is assessed by com-
paring the predicted in-plane displacements u, v, with results
(70)
c
i+1
2+j+1
2
=𝜕U
𝜕 𝜌 i+1
2
,j+1
2
=−m𝜌 m−1
i+1
2,j+1
2
(
1−𝜙min
)
U0
i+1
2,j+
1
2
(71)
c
i+1
2+j+1
2
←1
∑
Ω
k∑
Ω
l
𝜑 (i,j,k,l)
Ω
�
k
Ω
�
l
𝜑 (i,j,k,l)ck+1
2,l+
1
2
(72)
𝜑
(i,j,k,l)=max
(
r−𝓁
√
(i−k)2+(j−l)2,0
)
(73)
E
(x,y)=2
[(
x
L−1
2
)2
+
(
y
L−1
2
)2]
obtained through the direct stiffness matrix method (Rao
2005). A general form of the local stiffness matrix for a unit
square FEM plane element and a square HBM cell with unit
bar length, can be expressed using Eq.(31). We assume iso-
tropic material properties within individual FEM elements
and HBM cells. The coefficients of the local stiffness matrix
are for plane-stress:
for plane-strain:
(74)
a
FEM =
1

1−𝜈2

1
2
−
𝜈
6

;aHBM =
1

1−𝜈2

3
4
−
𝜈
4

(75)
b
FEM =1

1−𝜈2

1
8+
𝜈
8

;bHBM =1
8

1−𝜈2

1
8+
𝜈
8

(76)
c
FEM =
1

1−𝜈2

−
1
4
−
𝜈
12

;cHBM =−
1
2

1−𝜈2

(77)
d
FEM =1

1−𝜈2

−1
8+3𝜈
8

;dHBM =1

1−𝜈2

−1
8+3𝜈
8

(78)
e
FEM =1

1−𝜈2

−1
4+
𝜈
12

;eHBM =
0
(79)
fFEM =
𝜈
6

1−𝜈2

;fHBM =1

1−𝜈2

−1
4+
𝜈
4

(80)
a
FEM =1
1+𝜈1−2𝜈

1
2−2𝜈
3

;a
HBM
=1

1+𝜈

1−2𝜈

3
4−𝜈

(81)
b
FEM =
1
8
(
1+𝜈
)(
1−2𝜈
)
;bHBM =
1
8
(
1+𝜈
)(
1−2𝜈
)
(82)
c
FEM =1
1+𝜈1−2𝜈

−1
4−
𝜈
2
;
c
HBM =1

1+𝜈

1−2𝜈

−1
2+
𝜈
2
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Y.P.Zhang et al.
1 3
276 Page 12 of 23
(83)
d
FEM =1
1+𝜈1−2𝜈

−1
8+
𝜈
2
;
d
HBM =1

1+𝜈

1−2𝜈

−1
8+
𝜈
2
(84)
e
FEM =1

1+𝜈

1−2𝜈

−1
4+
𝜈
3

;eHBM =
0
(85)
fFEM =
𝜈
6

1+𝜈

1−2𝜈

;fHBM =1

1+𝜈

1−2𝜈

−1
4+
𝜈
2

The abovementioned coefficients of the local FEM stiff-
ness matrix can be derived as detailed in Rao (2005). For
the plane-stress case, the coefficients of local HBM stiff-
ness matrix are obtained using the HBM spring stiffnesses
expression given by Zhang etal. (2021b). For the plane-
strain case, the coefficients of local HBM stiffness matrix
are computed by using Eqs. (50)–(53). It is interesting to
note that FEM and HBM have identical values for the coef-
ficients b and d which indicate that both FEM and HBM
have a similar physical interpretation of the Poisson effect.
In investigating the performance of HBM, several simula-
tions were run by adopting grid/mesh size
𝓁=
L
∕100
. For
Plane-Stress Problem
(a) FEM- (b) HBM-
(c) FEM- (d) HBM-
Fig. 4 A plane-stress body under a central point load. a horizontal displacement field u predicted by FEM; b horizontal displacement field u pre-
dicted by HBM; c vertical displacement field v predicted by FEM; d vertical displacement field v predicted by HBM
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
1 3
Page 13 of 23 276
each test, we only change the local stiffness matrix based on
the choice of FEM/HBM or plane-stress/plane-strain. Fig-
ure4a and b presents the horizontal displacement u fields
predicted by FEM and HBM, respectively, for the plane-
stress problem under a central point load. Figure4c and d
present the corresponding vertical displacement v fields.
Figure5a–d presents the in-plane displacements for the
plane-strain problem under a central line load. Based on the
results shown in Figs.4 and 5, it can be seen that the in-
plane displacements given by HBM agree well with those
obtained through FEM for both the plane-stress and plane-
strain problems. These results show that the proposed HBM
is a simple and robust physical structural model that handle
well elasticity problems for functionally graded structures.
Plane-Strain Problem
(a) FEM- (b) HBM-
(c) FEM- (d) HBM-
Fig. 5 A plane-strain body under a central line load. a horizontal displacement field u predicted by FEM; b horizontal displacement field u pre-
dicted by HBM; c vertical displacement field v predicted by FEM; d vertical displacement field v predicted by HBM
P
L
y
zx
L
L
2
Design domain
Fig. 6 Design domain, loading and boundary conditions for FGL
beams
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Y.P.Zhang et al.
1 3
276 Page 14 of 23
6.2 Optimal design ofFGL beams subjected
toacentral line load
The HBM-TO is first applied to the optimal design of an
FGL beam with a square cross-section (
𝛼=1
) under a cen-
tral line load. The bottom edges are fixed in y- direction.
The geometry and boundary conditions are shown in Fig.6.
The HBM-TO is carried out as follows.
1. Set the prescribed volume fraction
Vf=0.4
by initial-
izing all HBM design variable
𝜌
as 0.4;
2. Set parameters
𝛿=0.1
,
𝜙min =0.01
,
r=2
and
𝛾=0.5
;
3. Set the size of HBM grid
𝓁
as
L∕10
,
L∕20
or
L∕30
;
4. Select the exponent for penalization
m
as 0.5, 1 or 2;
5. Solve optimization problem defined by Eq.(65) and
using Eqs. (66)–(72);
6. The optimization process will stop when the change of
elastic strain energy
ΔU
is smaller than 0.01;
7. Finally, construct the FGL beam cross-section based on
the optimal
𝜒
.
Noting that the variation of the elastic modulus changes
the stiffness of the springs, thus the cell mass changes
accordingly. In actual design of FGL structures, the cross-
section area

A
, the elastic modulus

E
and the density
𝜌
of
lattice bars are dependent on the applied material.
Figure7 shows the optimal cross-section designs of FGL
structures obtained by HBM-TO. It can be seen that the solu-
tions are significantly affected by the size of the HBM grid
cell
𝓁
and the penalty exponent
m
. The cross-section mass
is more concentrated when
𝓁
is smaller and
m
is larger. In
contrast, the mass of cross-section becomes more distributed
when
𝓁
is larger and
m
is smaller. It can be seen from Fig.8
that the elastic strain energy converges after 25 iterations
and the solutions converge faster when
m
is smaller. The
converged elastic strain energy tends to be lower when
m
is smaller.
The optimally designed FGL beams can be classified into
two groups: “distributed” and “defined”; see also Fig.9. The
first group suggests the beams could be designed with func-
tionally graded properties such as a graded elastic modulus.
Fig. 7 Optimal designs of FGL
beams with a square cross-sec-
tion obtained by HBM-TO
=0.5 =1 =2
ℓ
=10
ℓ
=20
ℓ
=30
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
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The second group suggests that the beams could be designed
with a defined geometry. The observed results from Fig.9
show that the “distributed” FGL beams are stiffer than the
“defined” FGL beams. The former ones have a smoother
transition of mass which could help in preventing failure
initiations (Wang 1983; Mahamood etal. 2012). However,
the latter ones could be manufactured more readily.
Based on Eqs. (56)-(60), the variable
𝜒
describes the
distribution of the elastic modulus of the FGL beams. By
least-squares fitting
𝜒
from HBM-TO FGL beams using
𝓁
=
L
20
,m=
0.5
; and
𝓁
=
L
30
,m=
0.5
, one finds the fol-
lowing elastic modulus distribution function, i.e.
(86)
fb(x,y)=
⎧
⎪
⎨
⎪
⎩
0 if 
fb(x,y)<0
1 if 
fb(x,y)>1

fb(x,y)otherwise
;x∈[0, 1],y∈[0, 1
]
(a) ℓ=
10 (b) ℓ=
20 (c) ℓ=
30
Fig. 8 Convergence of elastic strain energy when optimizing FGL beams
Fig. 9 Two groups of HBM-TO
designed FGL beams
(a) Distributed (b) Defined
Fig. 10 Elastic modulus distribution for HBM-TO FGL beams with
square cross-section under a central line load
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Y.P.Zhang et al.
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276 Page 16 of 23
where
A contour plot showing the distribution given in Eq.(86)
is presented in Fig.10.
(87)

fb(x,y)=0.9 +0.9x−114xy
+6.7y−1.3x2+336.6x2y
−2248.6x2y2+618xy2−33.1y2
+0.8x3−445.3x3y+3261.3x3y2
−5684.9x3y3+3781.4x2y3
−938.9xy3+43.3y3−0.4x4
+222.7x4y−1630.6x4y2+2842.4x4y3
−1391.5x4y4+2783.0x3y4−1816.9x2y
4
+
425.4
xy
4
−
17.4
y
4
The functionally graded beams subjected to a central line
load with elastic modulus varying in x- and y-directions (as
shown in Figs.1d and 6) could be designed following the
distribution function given in Eq.(86) as follows
6.3 Optimal design ofFGL plate subjected
toacentral line load
For the second example, we apply HBM-TO to design the
cross-section of an FGL plate with a rectangular cross-sec-
tion under a central line load. The bottom edges are fixed in
the y- direction. The geometry and boundary conditions are
presented in Fig.11. In this example, the same parameters
of the previous example are used except
𝓁
=
L
10
or
L
20
in this
example.
Figure12 presents the optimal cross-section designs of
the FGL plates obtained from HBM-TO. Similar to what
observed in the previous example, results show that the
mass is more concentrated; thus, resulting in a more defined
geometry when
𝓁
is smaller and
m
is larger. It can be seen
from Fig.13 that the elastic strain energy converges after 80
iterations and the solutions with smaller
m
converge faster.
Similar to the HBM-TO designed FGL beams, the converged
elastic strain energy is lower when
m
is smaller. The results
of optimally designed FGL plates can be classified into
two groups. The first group suggests the FGL plates could
be designed with a more distributed elastic modulus and a
smoother transition of mass which could help in preventing
failure initiations (Wang 1983; Mahamood etal. 2012) as
shown in Fig.14a, while the other group (Fig.14b) suggests
(88)
E(x,y)=fb(x,y)E0
P
y
z
x
2LDesign domain
L
4L
Fig. 11 Design domain, loading and boundary conditions for FGL
plates
ℓ=10 ℓ=20
=0.5
=1
=2
Fig. 12 Optimal designs of FGL plate cross-section obtained by HBM-TO
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
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the plates could be designed with a specific shape which
could be manufactured more readily.
By least-squares fitting
𝜒
from HBM-TO FGL plates
using
𝓁
=
L
10
,m=
0.5
; and
𝓁
=
L
20
,m=
0.5
, one finds the
following elastic modulus distribution function, i.e.
where
A contour plot showing the distribution given in Eq.(89)
is presented in Fig.15.
The functionally graded plates subjected to a central line
load with elastic modulus varying along x- and y-directions
(as shown in Figs.1d and 11) could be designed following
the distribution function given in Eq.(89) as follows
(89)
fp(x,y)=
⎧
⎪
⎨
⎪
⎩
0 if 
fp(x,y)<0
1 if 
fp(x,y)>1

fp(x,y)otherwise
;x∈[0, 4],y∈[0, 1
]
(90)

fp(x,y)=0.896 −0.178x+2.007y
−26.706xy +0.359x2+33.812x2y
−155.349x2y2+115.226xy2−11.176y2
−0.157x3−13.567x3y+63.271x3y2
−90.855x3y3+220.799x2y3−156.353xy3
+14.341y3+0.020x4+1.696x4y−7.909x4y
2
+11.357x4x3−5.195x4y4+41.558x3y4
−100.542
x
2
y
4+69.703
xy
4−6.264
y
4
(91)
E(x,y)=fp(x,y)E0
7 Summary
Developed herein is a novel strategy to obtain optimal
designs of FGL structures. It involves the use of the Hencky
bar-grid model and topology optimization. The HBM for-
mulated herein extends a previous model with twice the
number of internal rigid bars and springs as shown in Fig.2.
The new HBM enables the determination of the local stiff-
ness matrix for each HBM grid cell by using the expressions
of the stiffnesses of elastic primary axial springs, elastic
secondary axial springs and torsional springs that are
employed to connect the rigid bars. A TO with filtering pro-
cesses is presented based on the obtained local stiffness
matrices for optimizing HBM. The objective is to minimize
the strain energy of an HBM subjected to boundary condi-
tions and equilibrium and compatibility constraints by vary-
ing spring stiffnesses within each grid cell. The obtained
HBM with optimal material property distribution is then
used to build FGL structures (see Fig.3c and d, for exam-
ple). We assume that the horizontal and vertical axial bars
have the stiffnesses of
kxx
i+1
2
,j+
1
2
and
kyy
i+1
2
,j+
1
2
, respectively.
Likewise, its curved bars located at the corners and middle
of the cell have the stiffnesses
kS
i+1
2
,j+
1
2
and
k
xy
i+1
2
,j+1
2(
orkyx
i+1
2
,j+1
2)
, respectively. The bars in all lattice cells
have a constant elastic modulus

E
and a constant density
𝜌
but a varying cross-section area

A
i+1
2
,j+
1
2
. Therefore, HBM
grid cells with larger spring stiffnesses corresponds to lattice
structures with thicker bars and thus more mass.
Fig. 13 Convergence of the
elastic strain energy when opti-
mizing the FGL plates
(a) ℓ=
10 (b)ℓ=
20
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Y.P.Zhang et al.
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276 Page 18 of 23
8 Concluding remarks
Based on the HBM-TO optimal designs of an FGL beam
and an FGL plate under a central line load, the following
observations may be drawn:
1. HBM-TO can efficiently find optimal FGL structures by
optimizing a small number of design variables such as
the spring stiffness of each grid cell;
2. HBM-TO does not need to perform extra optimization
for the structure and connectivity of each lattice unit
cell;
3. All optimal FGL structures studied herein can be effec-
tively obtained by HBM-TO with fewer than 80 itera-
tions;
4. The designed cross-section has more concentrated mass
and a more “defined” shape when using a smaller grid
cell size or a larger penalty exponent;
5. The “distributed” solutions are generally stiffer than the
“defined” ones and with a smoother transition of mass;
6. Polynomial functions that approximate the elastic mod-
ulus distribution for the optimal designed FGL beams
(Eqs. (88)) and FGL plates (Eqs. (91)) are presented for
the first time. These functions could be useful for fab-
rication, buckling and vibration analysis of these FGL
structures as shown in Fig.1d.
In this study, we only consider one type of lattice repre-
sentation as shown in Fig.3c. A more rigorous investigation
of the HBM lattice representation such as the effect of the
shape of the curved bars and the relation between the HBM
spring stiffness and the geometry of the unit lattice structure
could be performed in a future study (Cheng etal. 2019).
Owing to its physical representation, the HBM-TO can
account for local damage and local stiffening constrains
(Wang etal. 2009; Cui etal. 2011) as well as advanced
material constitutive behaviour (O’Brien 2008) by simply
adjusting the spring stiffnesses. Moreover, as a physical
model-based optimization framework, HBM-TO can be
applied to optimize micro- and nano-structures by setting
the grid length to the characteristic length of the intended
scale. Due to HBM having similar governing equations as
the ones given by standard FD method, the efficient alternat-
ing direction implicit method could be applied for dynamic
problems (Zhang etal. 2016, 2017). The proposed design
framework can be used for other lattice models (Wieghardt
1906; Born and Karman 1912; Hrennikoff 1941; McHenry
1943; Gazis etal. 1960; Suiker etal. 2001).
Appendix A
A Python 3 code is presented for solving the problems as
articulated in Sects.6.2 and 6.3. This code is highly inspired
by an open source code from Aage and Johansen (2013). The
open source libraries “numpy” (Harris etal. 2020), “scipy”
(Virtanen etal. 2020) and “matplotlib” (Hunter 2007) are
adopted in the following code.
(a) Distributed (b) Defined
Fig. 14 Two groups of HBM-TO designed FGL plates
Fig. 15 Elastic modulus distribution for HBM-TO FGL plates with
rectangular cross-section
𝛼
=
4
under a central line load
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Optimal design offunctionally graded lattice structures using Hencky bar‑grid model and…
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Y.P.Zhang et al.
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276 Page 20 of 23
Disclaimer: The authors do not guarantee that the code
is free from errors, and they shall not be liable in any event
caused by the use of the code.
Acknowledgements The authors wish to thank the review editor and
the two anonymous reviewers for their insightful suggestions and com-
ments that help to improve the paper.
Funding Open Access funding enabled and organized by CAUL and
its Member Institutions.
Declarations
Conflict of interest On behalf of all authors, the corresponding author
states that there is no conflict of interest.
Replication of results All important details have been presented in the
paper. The results obtained in this paper can be reproduced by using
the python code given in Appendix A.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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