Controlling the maximum first principal stress in topology optimization

Abstract

Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion. In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint. The present methodology will be useful for designing concrete structures.

Full text

RESEARCH PAPER
Controlling the maximum first principal stress in topology
optimization
Anbang Chen
1
&Kun Cai
1
&Zi-Long Zhao
1
&Yiyi Zhou
2
&Liang Xia
3
&Yi Min Xie
1
Received: 8 March 2020 /Revised: 28 June 2020 /Accepte d: 21 July 2 020
#Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion.
In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be
controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing
with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-
norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization
problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design
obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering
the FPS constraint. The present methodology will be useful for designing concrete structures.
Keywords Topology optimization .Stress constraints .BESO .First principal stress
1 Introduction
Structural optimization has undergone rapid development in
the past few decades. Schmit (1960) proposed the idea of
designing minimum cost systems by mathematical program-
ming techniques. Bendsøe and Kikuchi (1988)introducedthis
idea in topology optimization. Several optimization tech-
niques, such as the homogenization method (Bendsøe 1989),
the solid isotropic material with penalization (SIMP) method
(Bendsøe 1995), the level-set method (Wang et al. 2003), the
evolutionary structural optimization (ESO) method (Xie and
Steven 1993,1997), and the bi-directional evolutionary
structural optimization (BESO) method (Huang and Xie
2007b,2010), have been developed and used widely in mul-
tiple disciplines.
The ESO and BESO methods have been used for solving
topology optimization problems in many areas of structural
engineering. These problems include structural frequency op-
timization (Xie and Steven 1994), minimizing structural vol-
ume with a displacement or compliance constraint (Liang
et al. 2000), structural complexity control in topology optimi-
zation (Zhao et al. 2020a; Xiong et al. 2020), topology opti-
mization for energy absorption structures (Huang et al. 2007),
design of periodic structures (Huang and Xie 2008), geomet-
rical and material nonlinearity problems (Huang and Xie
2007a), stiffness optimization of structures with multiple ma-
terials (Huang and Xie 2009), maximizing the fracture resis-
tance of quasi-brittle composites (Xia et al. 2018a), stress
minimization designs (Xia et al. 2018b), biomechanical mor-
phogenesis (Zhao et al. 2018,2020b), stiffness maximization
of structures with von Mises constraints (Fan et al. 2019), and
diverse and competitive designs (Xie et al. 2019;Yangetal.
2019;Heetal.2020).
Among all the issues mentioned above, stress-based topol-
ogy optimization has been recognized as a challenging one.
Three main challenges and corresponding remedy strategies in
stress-based topology optimization have been summarized by
Le et al. (2010). The first challenge, the “singularity”problem,
Responsible Editor: YoonYoung Kim
*Kun Cai
kun.cai@rmit.edu.au
*Yi Min Xie
mike.xie@rmit.edu.au
1
Centre for Innovative Structures and Materials, School of
Engineering, RMIT University, Melbourne 3001, Australia
2
School of Civil Engineering and Architecture, Changzhou Instituteof
Technology, Changzhou 21300, China
3
State Key Laboratory of Digital Manufacturing Equipment and
Technology, Huazhong University of Science and Technology,
Wuhan 430074, China
Structural and Multidisciplinary Optimization
https://doi.org/10.1007/s00158-020-02701-5

is encountered when optimizing truss layouts using the
density-based methods, where elements with low densities
could have high-stress values (Kirsch 1990; Cheng and Guo
1997; Rozvany 2001). It can be avoided by using discrete
topology optimization methods. The second challenge is in-
duced by the local nature of stress constraints, which results in
a large number of constraints (Duysinx and Bendsøe 1998). In
general, to overcome the difficulty, the local stress measure is
transferred into a global one by the p-norm or the
Kreisselmeier–Steinhauser (KS) functions (Yang and Chen
1996; Duysinx and Sigmund 1998). The third challenge is
highly nonlinear stress behavior (Le et al. 2011). The stress
distribution is much sensitive to even subtle topological vari-
ations, particularly in the critical regions with high-stress con-
centration. The stress levels of elements are sensitive to the
changes in the topological variables of their neighbors, espe-
cially in critical regions such as sharp and reentrant corners.
To deal with it,the filtering technique can be used to produce a
mesh-independent solution (Le et al. 2010), where both sen-
sitivity numbers (which is used to rank elements) and design
variables are smoothed.
In previous works, almost all material failure constraints
were based on the global p-norm stress, the elasto-plasticity,
the local stress constraints, or the von Mises yield criterion
(Luo et al. 2013b; Amir 2017; Xia et al. 2018b; Herfelt et al.
2019; Fan et al. 2019;Blachowskietal.2020). These criteria
can be used to predict the failure of metal materials. However,
in civil engineering, building materials behave different
strengths in tension and compression. For instance, concrete
and cement have high compressive strength but low tensile
strength (Cai 2011; Luo et al. 2013a). Tensile membrane and
fibrous materials, e.g., cloth, steel cables, and nylon ropes,
have high tensile strength but very low compressive strength
due to buckling (Cai et al. 2014). From a practical point of
view, the different behaviors of materials in terms of tension
or compression criterion could generate a unique optimization
design (Duysinx et al. 2008). Therefore controlling the prin-
cipal stress is imperative in practical engineering applications.
Despite the large quantity of existing research, stress-
constrained optimization considering the maximum principal
stress remains an open issue, where nonlinear stress behavior
and topological oscillation would be induced during the form-
finding process. To this end, finer finite element mesh, smaller
evolution ratio (which is used to add or remove elements), and
larger filter radius are commended.
In this study, a BESO-based approach is developed for
controlling the maximum first principal stress (FPS) in a
compliance minimization problem (BESO-FPS). The pa-
per is organized as below. In Section 2, the new approach
is developed and integrated into the BESO technique. In
Section 3, typical numerical examples are used to demon-
strate the effectiveness of the approach. In Section 4, con-
clusions are summarized.
2Methodology
2.1 Optimization problem formulation
In topology optimization, structural analysis is performed by
the finite element method, the whole design domain is first
discretized into Nfinite elements, and the pseudo density of
each element is treated as a design variable (DV). All the DVs
form into a vector, i.e., x=(x
1
,x
2
,…,x
N
)
T
. By using the BESO
method, each DV has binary values of either 1 (presence) for
solid element or 0 (absence) for void element, i.e.,
xi¼1presenceðÞ
0 absenceðÞ
;i¼1;2;…;N:ð1Þ
The element stiffness can be linked to the associated topol-
ogy DV in a linear manner that element effective stiffness
matrix is
ki¼xik0;ð2Þ
where k
0
is the stiffness matrix of a solid element.
For saving computational cost, in general, a fixed finite
element mesh is used in structural analysis throughout the
optimization. x
i
of void elements is set to be a very small
positive value, e.g., ρ
min
= 0.001, to avoid singularity of the
global stiffness.
Accordingly, the effective stress vector of the ith element
yields
σi¼kiBiui
xi
¼k0Biui;ð3Þ
where u
i
is the nodal displacement vector of the element and
B
i
is the geometric matrix associated with the strain and the
displacement. Therefore, the effective stress state of the ele-
ment depends on the displacement field of structure but does
not depend on the related DVs.
For a design domain-containing linear elastic material,
which needs to limit FPS, the topology optimization problem
for minimizing the structural compliance can be defined as
Min C¼1
2UTKU
s:t:KU ¼F
VxðÞ¼∑xivi−Vf¼0
0<σmax
1¼max σi;1ji¼1;2;…;N

≤σ*
1
x¼xi∈ρmin;1
fgj
i¼1;2;…;N
fg
8
>
>
>
>
>
<
>
>
>
>
>
:
;ð4Þ
where Cis the structural compliance and K,U,andFare the
global stiffness matrix, the global displacement, and load vec-
tors, respectively. “v
i
”is the volume of the ith element, V(x)is
the total volume of structure in the ith iteration, and V
f
is the
material volume required in design. σ
1
max
,σ
i,1
,andσ
1
*
are the
maximum FPS in structure, FPS of the ith element, and the
critical value of FPS, respectively.
<. Chen et al.

The objective function in (4) can be modified for stress
constraint by introducing a Lagrangian multiplier λ,i.e.,
f1¼Cþλ⋅σmax
1−σ*
1

:ð5Þ
f
1
is equal to the original objective function if σ
1
max
=σ
1
*
.
Otherwise, λ=0if σ
1
max
<σ
1
*
, which means the stress con-
straint is already satisfied, or λ→∞if σ
1
max
>σ
1
*
, which
means σ
1
max
should be minimized in order to satisfy the con-
straint in additional iterations to find a proper value of λ.
For a given σ
1
*
in the BESO method, the update of vari-
ables depends only on element-wisely determined sensitivity
numbers. Therefore, minimizing f
1
is equivalent to minimiz-
ing f
2
; the limiting value for principal stress can be removed
from the formulation in (5), which will be equivalently re-
placed with
f2¼Cþλ⋅σmax
1:ð6Þ
However, element stress is a local quantity. When directly
considering the local constraint in optimization, the computa-
tional cost is high on sensitivity evaluation no matter using the
direct method or the adjoint method (Tortorelli and Haber
1989). To overcome this difficulty, two classic global stress
measure methods can be used, p-norm, or the KS function
(Yang and Chen 1996). In this study, the p-norm scheme is
adopted, and therefore, the design objective function is mod-
ified by replacing the σ
1
max
with σ
pn
,i.e.,
f3¼Cþλ⋅σpn;ð7Þ
where
σpn ¼∑N
i¼1σp
i;1

1=p;ð8Þ
where positive pis the order of norm. When ptends to infinity,
σ
pn
=σ
1
max
(Le et al. 2010).
In civil engineering, materials like concrete are mainly kept
under compression. Hence, another aspect of FPS is that neg-
ative FPS in the design domain will be neglected. Equation (8)
is rewritten as
σpn ¼∑N
i¼1Hσi;1

σp
i;1
hi
1=p;ð9aÞ
where H(*) is the Heaviside step function, i.e.,
Hσi;1

¼1ifσi;1≥0
0ifσi;1<0:
ð9bÞ
2.2 Sensitivity analysis
In this work, a BESO with the maximum first principal stress
constraint (BESO-FPS) method is adopted to solve the
topology optimization in (4). The sensitivity analysis of the
principal stress constraint with respect to DVs is derived with
the following steps
∂f3
∂xj
¼∂C
∂xj
þλ⋅∂σpn
∂xj
;ð10Þ
where ∂C
∂xjcould be easily achieved by using the adjoint method
∂C
∂xj
¼−1
2uT
jk0ðÞ
juj;ð11Þ
where k0ðÞ
jdenotes the jth element stiffness matrix with solid
material.
For simplicity, in a two-dimensional structure, a stress ma-
trix can be diagonalized, as
MTσiM¼ac
−ca

σxx σxy
σyx σyy

a−c
ca

¼Λ
¼σ10
0σ2

;ð12Þ
where M¼a−c
ðcaÞand a
2
+c
2
=1.Therefore, σ
1
=
a
2
σ
xx
+c
2
σ
yy
+2acσ
xy
,i.e.,
σi;1¼a2
ic2
i2aici

σxx
σyy
σxy
0
@1
A¼AT
iσi¼AT
iD0Biui:ð13Þ
According to the definitionin (9), the derivative of σ
pn
with
respect to the jth DV is
∂σpn
∂xj
¼σ1−p
pn ∑N
i¼1Hσi;1

σp−1
i;1
∂σi;1
∂xj

;ð14Þ
and according to (13), we have
∂σi;1
∂σi
¼AT
i:ð15Þ
Therefore,
∂σi;1
∂xj
¼∂σi;1
∂σi
⋅∂σi
∂xj
¼AT
i
∂σi
∂xj
:ð16Þ
By substituting (3)into(16), the derivative can be further
written as
∂σi;1
∂xj
¼AT
iD0BiLi
∂U
∂xj
;ð17Þ
where the matrix L
i
gathers the nodal displacements of
the ith element from the global displacement vector sat-
isfying u
i
=L
i
U.
Controlling the maximum first principal stress in topology optimization

By differentiating the discretized equilibrium equation of
the structure, we have
∂K
∂xj
UþK∂U
∂xj
¼∂F
∂xj
¼0;ð18aÞ
∂U
∂xj
¼−K−1⋅∂K
∂xj
U:ð18bÞ
Substituting (18b)into(17), leads to
∂σi;1
∂xj
¼−AT
iD0BiLiK−1∂K
∂xj
U:ð19Þ
The following equation can be obtained by substituting
(19)into(14), i.e.,
∂σpn
∂xj
¼−σ1−p
pn ∑N
i¼1Hσi;1

σp−1
i;1AT
iD0BiLi

K−1∂K
∂xj
U:ð20Þ
Calculation of K
−1
in (20) can be replaced with a solution
of the following adjoint problem
Kμ¼∑N
i¼1Hσi;1

σp−1
i;1D0BiLi
ðÞ
TAi:ð21Þ
Hence, the sensitivity of σ
pn
can be further simplified as
∂σpn
∂xj
¼−σ1−p
pn μT∂K
∂xj
U:ð22Þ
The pseudo-load on the right-hand side of the adjoint prob-
lem (21) is assembled in analogy to body forces. Recalling the
effective stiffness model in (2), the derivative of the global
stiffness matrix with respect to the jth DV is calculated as
∂K
∂xj
¼∑N
i¼1LT
i
∂ki
∂xj
Li¼LT
jk0
jLj:ð23Þ
By substituting (23)into(22), the sensitivity of the p-norm
global stress measure can be eventually evaluated as
∂σpn
∂xj
¼−σ1−p
pn μT
jk0ðÞ
juj;ð24Þ
where μ
j
is the vector of adjoint nodal values of the jth
element.
Substituting (11)and(24)into(10), the final sensitivity of
objective function is
∂f3
∂xj
¼−1
2uT
jk0ðÞ
juj−λ⋅σ1−p
pn μT
jk0ðÞ
juj:ð25Þ
The value of λcan be determined by internal iterations,
e.g., the external point penalty method (Fan et al. 2019).
For the ease of implementing the calculation, λincreases
iteratively in an exponential manner as
λlþ1ðÞ
¼λlðÞþ2l;if σmax
1>σ*
1:ð26aÞ
where lis the iterations number, and λ
(l)
is the Lagrangian
multiplier in the lth iteration.
Once the stress constraint is satisfied, λis determined using
the bisection method to ensure that
σ*
1−Δ

≤σmax
1≤σ*
1:ð26bÞ
where Δis a small portion of σ
1
*
, e.g., 1 %. With the update of
λ, the above procedure of the determination of Lagrangian
multiplier is repeated until a prescribed maximum iteration
number in case σ
1
max
still violates the requirement of (26b),
either higher than σ
1
*
or lower than (σ
1
*
−Δ).
Finally, the sensitivity number used in the BESO method
can be defined using the element sensitivity from (25), i.e.,
αj¼−xj
∂f3
∂xj
¼xj
1
2uT
jk0ðÞ
jujþλ⋅σ1−p
pn μT
jk0ðÞ
juj

when xj¼1
0whenxj¼0
8
<
::
ð27Þ
This equation indicates that sensitivity numbers are evalu-
ated only for solid elements (x
j
= 1) and are set as zero directly
for void elements (x
j
=0).
Before updating DVs, the above raw sensitivity is
smoothed by using the filtering technique in order to produce
a mesh-independent solution (Sigmund and Petersson 1998;
Bourdin 2001; Luo and Bao 2019). A simplified element sen-
sitivity filter scheme without losing accuracy is here employed
as follows.
e
αj¼∑nel
i¼1ωijαi
∑nel
i¼1ωij
;ð28aÞ
ωij ¼max 0;rmin−rij

;ð28bÞ
where nel denotes the number of neighboring elements, r
ij
is
the distance between the centroids of the central element jand
its neighboring elements i,ω
ij
is the weight function (0 ≤ω
ij
<
1), and r
min
is the filter radius. It is noted that ω
ij
is independent
of the element sensitivities and is pre-calculated. Furthermore,
in order to achieve a convergent solution, the smoothed ele-
ment sensitivity e
αjis averaged with their historical informa-
tion (Huang and Xie 2007b).
b
αj¼e
α
kðÞ
jþe
α
kþ1ðÞ
j
2;ð29Þ
where e
αkðÞ
jand e
αkþ1ðÞ
jare the element sensitivity in the kth and
(k+ 1)th iteration, respectively. The evolution of the material
volume is expressed by
Vkþ1ðÞ
¼VkðÞ 1−ERðÞ;ð30Þ
<. Chen et al.

where ER is the evolutionary volume ratio. Once the material
volume reaches the target volume fraction, the volume of the
structure will be constant for the remaining iterations as
Vkþ1ðÞ
¼Vf:ð31Þ
Then all the elements, both solid and void elements, are
sorted based on their sensitivity numbers. The DVs are up-
dated as
xkþ1ðÞ
j¼
0if
b
αj≤b
α
th
del
1if
b
αj>b
α
th
add
xkðÞ
jotherwise
8
>
>
<
>
>
:
;ð32Þ
where b
αth
del and b
αth
add are the threshold sensitivity numbers for
removing and adding elements, respectively. They can be de-
termined by the bisection algorithm (Huang and Xie 2010).
It is noted that due to the highly nonlinear stress behavior,
filtering DVs is recommended to stabilize the optimization
procedure by using the same filter scheme mentioned above,
where the filter radius could be different (Le et al. 2010;Xia
et al. 2018b; Fan et al. 2019).
The optimization stops when the stress constraint, the vol-
ume constraint, and the convergence criterion are satisfied.
The convergence criterion is defined in terms of the change
of the objective value:
error ¼
j∑
M
i¼1
Ck−iþ1−∑
M
i¼1
Ck−M−iþ1j
∑
M
i¼1
Ck−iþ1
≤τ;ð33Þ
where kis the current iteration number, τis an allowable
convergence tolerance, and Mis an integer number.
Typically, we adopt M= 5, which implies that the change of
the mean compliance over the last 10 iterations is small
enough, e.g., τ=10
−6
(Huang and Xie 2010).
2.3 Overall update of BESO-FPS algorithm
To solve the optimization described in (4) using the BESO-
FPS method, the following essential steps are required, i.e.,
Step 1 Discretize the initial design domain with specific
boundary conditions;
Step 2 Assign parameters in optimization, including volume
fraction V
f
, evolution ratio ER,filterradiusr
min
and
norm parameter p;
Step 3 Conduct structural analysis for obtaining strain and
stress fields. Find the elements with the maximum
FPS (in the design domain);
Step 4 Solve the value of Lagrangian multiplier λwith re-
spect to the stress constraint;
Step 5 Calculate the sensitivities of the elements in the de-
sign domain;
Step 6 Modify the sensitivities by using its history informa-
tion and filter scheme;
Step 7 Update the DVs by adding and/or deleting elements;
Step 8 Return to Step 4 if the stress constraint is not satisfied;
Step 9 Return to Step 3 if the volume constraint or the con-
vergence criterion is not satisfied;
Step 10 Stop for post-processing.
The flowchart of the proposed method is given in Fig. 1.
3 Numerical examples
In this section, four examples are presented for illustrating the
performance of the proposed method. The unit square bilinear
elements with sizes of 1 m × 1 m are used to discretize the
plane stress design domain in all examples. The material is set
to be linearly elastic. Young’s modulus and Poisson’sratioof
material are 20 GPa and 0.3, respectively. For comparison of
the structural performances, the basic parameters involved in
the BESO-FPS method are set to be constant; e.g., the filter
radius r
sen
for sensitivity is triple of the element size, and the
filter radius r
DV
for DVs is quadruple of the finite element
size. Furthermore, the optimal designs generated by the basic
BESO method with and without von Mises constraint (Fan
et al. 2019) will be presented and will be compared with the
optimal design by the present BESO-FPS method.
3.1 Effect of the stress norm parameter p
In Fig. 2, a simply-supported deep beam structure is shown.
Since the design domain in Fig. 2is optimal by the BESO
method with controlling the maximum von Mises stress (Fan
et al. 2019), we test the feasibility of the present approach and
briefly discuss the effect of the norm parameter pusing the
similar example. In this example, the design domain contains
43,200 unit square elements. V
f
= 50%, ER = 0.5%, and σ*
1=
3.00 MPa.
There are 11 solutions shown in Fig. 3including a BESO
solution (without stress constraint), a stiffness design with von
Mises stress constraint, i.e., BESO-VM solution, and nine
stiffness designs with the maximum FPS constraint with using
different values of p(BESO-FPS solutions).
In the BESO design, the maximum FPS approaches
6.12 MPa, which is higher than that in the BESO-VM solu-
tion, and much higher than those in the BESO-FPS solutions.
In the BESO solution, the tip of the crack is not smoothed well
after several updates of the material layout. This is because the
BESO solution is obtained by a stiffness-oriented algorithm.
A small number of elements with high local stresses do not
Controlling the maximum first principal stress in topology optimization

influence the global layout of materials. In the BESO-VM
solution, the crack is smoothed very well, and corresponding-
ly, the structure has two more merits except for lower stress
levels in structure when compared with BESO solution. One is
that the structural compliance (objective function) increases
6% in this example. The other is that the material layout be-
comes simpler, e.g., the shoulders of the optimal structure
connect with the middle part by a few straight rods.
In a stiffness design, the material will be fully used when
they are under pure tension or compression. If the material has
an upper limit of tensile stress, the material layout in the struc-
ture will be different from that in the basic BESO design.
Among the nine BESO-FPS solutions, the crack is well
smoothed when p> 3. Actually, the solution with respect to
p= 3 is not a feasible solution due to 3.743 MPa > σ*
1
= 3.00 MPa. For the solution of p= 4, structural compliance
is much higher than that of the BESO solution. When p=5,
the solution is very similar to the BESO-VM solution, e.g., the
well-smoothed crack, low-stress level, and simple topology.
However, the BESO-FPS solution has higher structural com-
pliance than the BESO-VM solution. When p> 5, the struc-
tural compliance is controlled to be equal or lower than that of
the BESO-VM solution. Meanwhile, the structure has a better
topology when pis higher for this structure.
Here, we provide several of BESO-FPS solutions with re-
spect to different values of pmainly because this problem
confuse scholars who adopted the p-norm scheme to over-
come local stress constraint in optimization (Holmberg et al.
2013; Takezawa et al. 2014; Picelli et al. 2018; Xia et al.
2018b; Liu et al. 2019). Two reasons lead to this difficulty.
Fig. 1 Flowchart of the BESO-
FPS method.
Fig. 2 Design domain of a 360-m × 120-m-deep beam having a crack at
the center of bottom with a depth of 30 m and thickness of 1 m (Xia et al.
2018b; Fan et al. 2019). Symmetrically, a uniformly distributed force of
400 kN/m is applied to the gray area at the top of the beam.
<. Chen et al.

One is that the stress distribution is sensitive to material layout
in structure and fluctuates seriously if material layout varies.
The other is that most scholars prefer the integer-order p-norm
function and the higher integer order p-norm function may
lead to lower accuracy of the solution value of σ
pn
.Hence,
p< 10 is generally suggested. In this example, a perturbation
of pin a small range is considered in the design. Theoretically,
the p-norm solution approaches that of the original design
problem with local stress constraints (Fig. 4). Here, a higher
value of pis recommendable.
3.2 Feasibility of strength design
In this example, a clamped–clamped deep beam structure
(Fig. 5) is optimal by the present method to examine the
influence of the boundary conditions. The structure has sizes
of 400 m × 100 m, and a uniform load of 8000kN/m is applied
in the middle of the top edge. We test V
f
=40%,ER=1%,and
p=20.
One can predict that the allowable value of FPS (σ*
1)
should have a lower boundary in this structure. If the value
of σ*
1is too low, the feasible solution does not exist. In this
example, the bisection algorithm is used to find the minimal
value of σ*
1of the design domain with a given boundary
condition.
To measure the feasibility of the solution with the FPS
constraint, two cases of boundary schemes are considered.
In case I, the two vertical edges are partly fixed from top to
Fig. 3 Optimal topologies
attaching FPS fields in different
cases. “BESO”means basic
BESO solution. “BESO-VM”
represents BESO solution
considering von Mises stress
constraint (Fan et al. 2019).
“BESO-FPS”means the solutions
of the present approach. Above
each topology, the digit in blue
font represents unified
compliance of structure. The digit
in red font, e.g., “σmax
1=6.120”,
means maximum FPS in the
optimal structure. “p=3”means
norm parameter pin BESO-FPS
solution.
Fig. 5 Two cases of different boundary conditions for the same clamped–
clamped deep beam structure. A uniform load of 8000 kN/mis applied in
the middle of the upper edge. In Case I, the width of fixation varies from
20 to 100 m with an increment of20 m (i.e., nis an integer). In case II, the
width of the fixation is 20 m.
Fig. 4 Variation of the compliances of the optimal structures with p.The
compliance ratio is the normalized compliance of the subsequent design
to the initial design.
Controlling the maximum first principal stress in topology optimization

bottom. In case II, the two vertical edges are fixed symmetri-
cally with the same width of 5 m.
Case I: Different length
In Fig. 6, two groups of solutions are shown for com-
parison of their structural properties. When n=1, the top
20 m of the two vertical sides are fixed. Hence, the path
of force transferring starts from the middle of topside to
the two vertical edges in both BESO and BESO-FPS so-
lutions. In the BESO solution, the maximum FPS is
4.044 MPa, which is obviously higher than that, i.e.,
2.271 MPa, in the BESO-FPS solution. The maximum
stress appears at the two upper corners of the structure
in the BESO solution. The two arms in the BESO solution
are thicker with a lower slope than those in the BESO-
FPS solution. A higher slope means a lower level of axial
internal force. Hence, the two arms in the BESO-FPS
design have a lower stress level. As the two arms in the
BESO-FPS design are slimmer, the deflection of the ex-
ternal load is higher, which is verified by the larger value
of c= 1.15 (> 1.00 in the BESO solution).
When n= 2, 40 m of the upper part of the two vertical sides
is fixed. The two solutions are different from those when n=
1. And the stress level is reduced; e.g., maximum FPS is
3.47 MPa < 4.044 MPa when n= 1. The maximum FPS ap-
pears at the two upper corners in the BESO design, while in
the middle of the bottom in the BESO-FPS solution. In the
two designs, the boundaries between the compressive and
tensile parts are very clear.
When n= 3, the two solutions have a slight difference at the
two upper corners and the bottom center. Obviously, a slight
difference means the objective functions are similar. But, the
stress level is reduced significantly, e.g., from 3.278 to
2.094 MPa.
When n= 4, the two solutions are quite different. The ma-
terial in the BESO design is mainly under tension, rather than
under compression as in the BESO-FPS solution. The struc-
tural compliance of the BESO-FPS solution is 21% higher
than that of the BESO solution. The stress level decreases
from 3.242 to 1.563 MPa.
When n= 5, the two designs have more materials under
compression than under tension. The two legs in the BESO-
FPS solution are thicker and supported by two slim bars that
are under tension. The structural compliance of the BESO-
FPS solution is 11% higher, but the maximum FPS is only
~ 39% (1.059/2.728) of the BESO solution.
It can be concluded that boundary conditions affect the
solutions, and a larger fixed area on the structure leads to a
lower stress level.
Case II: Different locations
When n= 1, the two solutions in Fig. 7are the same as
those in Fig. 6because the structures have the same boundary
conditions.
When n= 2, in the BESO solution, more material is under
low tensile stress. The maximum FPS is at the fixed ends. In
the BESO-FPS solution, the material is mainly under com-
pression. The maximum FPS is about half of that in the
Fig. 6 Structural topologies with
the distribution of FPS in Case I.
BESO solutions (left column)
indicate the optimal topologies
without stress constraints in the
optimization. BESO-FPS
solutions (middle column) are
obtained by the present approach.
Color bars (right column) are
given for five types of boundary
conditions. σmax
min is the minimum
allowable value of the max FPSin
afeasibledesign.
<. Chen et al.

BESO solution. However, the compliance of the structure in-
creases by 29% compared with the BESO solution.
When n= 3, material in the final designs are mainly under
compression. The BESO-FPS design is simpler and has a very
low-stress level, i.e., 1927 MPa, which is much smaller than the
BESO solution (5.115 MPa). However, the structural compli-
ance of the BESO-FPS design is higher than the BESO design.
When n= 4, the two designs are slightly different but have
significantly different stress levels. The maximum FPS in the
BESO-FPS solution is about half of that in the BESO solution.
When n= 5, some parts (in yellow) of the BESO solution
are under tension, which can reduce the structural compliance.
In the BESO-FPS solution, there only exist two thick compo-
nents, and the maximum FPS is only 0.4 MPa.
The conclusion is that the material is mainly under com-
pression or tension depends on the location of the fixed areas
in this example.
3.3 Effect of stress constraint on the optimal design
Case I: Rectangle structure
In this example of Fig. 8, we set V
f
= 45%, p= 20, and
ER = 0.5%. To illustrate the effect of the critical value of
FPS, i.e., σ*
1, five different cases are considered in the
BESO-FPS calculations, σ*
1= 4.0 MPa, 3.9 MPa, 3.7 MPa,
3.5 MPa, and 3.2 MPa.
Figure 9shows optimal topologies of the structure with
different FPS constraints. For the BESO solution, the material
layout in the structure is nearly symmetric about the non-
design domain. Because the force is applied on the upper edge
of the non-design domain. The maximum FPS is 5.286 MPa.
When the maximum FPS is confined to be 4 MPa, a strong
enclosed frame and several slim bars support the non-design
domain. The structural compliance increases by 4% compar-
ing the BESO solution.
By decreasing the maximum FPS to 3.9 MPa, the two
trunks below the non-design domain have a higher slope and
connect the loading area via a triangle with thick sides. The
structural compliance is 4% higher than that of the BESO
solution.
As we reduce the maximum FPS to approximately 3 MPa,
the optimal topologies become simple. The part below the
non-design domain has less material and mainly under ten-
sion, and its bottom becomes narrower. Therefore, the mate-
rial layout is sensitive to the allowable stress in this example.
It can be found that constraining the stress level leads to
material layouts, which are greatly different from the optimal
design without stress constraints.
Case II: L-bracket design
Figure 10 shows an L-bracket example, where V
f
=40%,
ER = 2%, and σ*
1= 1.5, 1.6, or 1.7 MPa. The design domain
contains 25,600 unit square elements, and the norm parameter
Fig. 7 Structural topologies with
the distribution of FPS in Case II.
Unit of stress: megapascal.
Fig. 8 A 480-m × 160-m design domain having a non-design area with a
thickness of 8 m at the central part. The two edges of the non-design area
are fixed. The upper and lower design domains are symmetric about the
non-design area. A uniformly distributed force of 400 kN/m is applied
near the center of the non-design area.
Controlling the maximum first principal stress in topology optimization

pis set as 10. To compare the BESO-FPS method with the
BESO method and the BESO-VM method (Fan et al. 2019),
the optimal topologies attaching FPS fields are shown in
Fig. 11.
The BESO method produces the stiffest structure, where
the maximum FPS is 2.934 MPa. The maximum FPS appears
at the reentrant corner. In the BESO-VM solution, the reen-
trant corner is rounded, the maximum FPS is 1.669 MPa, and
the compliance is increased by 8% compared with the BESO
solution.
For the BESO-FPS method, the maximum FPS is reduced
to 1.7 MPa, which is almost the same as that of the BESO-VM
method. However, the increment in compliance is only 3%
compared with the BESO solution. The reentrant corner is
rounded, which is quite similar to the BESO-VM design.
When the maximum FPS is reduced to 1.6 MPa or
1.5 MPa, the bar connecting the sharp reentrant corner and
the part under loading is removed. The compliance does not
change too much; i.e., both are 1.11, but the maximum FPS is
reduced by 49% compared with the BESO method. Figure 11
shows that the BESO-FPS solution with σ*
1¼1:5MPawill
generate more elements with maximum FPS than other solu-
tions, and the stress distribution is smoother as well.
Figure 12 shows the evolution histories of the mean com-
pliance and the maximum FPS. The mean compliance in-
creases as the material is gradually removed from the design
domain. Oscillations in the mean compliance are caused by
the removal of bars. Convergent solutions are obtained using
the three methods. In the BESO-FPS method, the FPS has
fewer oscillations. It is seen that controlling the stress level
in topology optimization leads to distinctly different layouts.
4 Conclusion
In this work, we have developed an effective algorithm, the
BESO-FPS to minimize the structural compliance of continu-
um structures with constraints on both the maximum first
principal stress (FPS) and the structural volume fraction. The
local stress constraint is converted into a global constraint by
using the p-norm function. The effectiveness of the present
approach is demonstrated by four plane-stress examples. The
following conclusions can be drawn from the results.
Fig. 9 Optimal topologies of the
structure with different FPS
constraints. One BESO solution
and five BESO-FPS solutions are
included and have different
maximum FPS.
Fig. 10 Illustration of the L-bracket benchmark problem (Xia et al.
2018b; Fan et al. 2019). The structure is clamped at the top end, and a
uniform load of 400 kN/m is applied to the 10-m length nearby the right
end of the top edge.
<. Chen et al.

Firstly, compared with the BESO solution without stress
constraint, the BESO-FPS approach can generate a design
with slightly higher compliance but significantly lower stress
level.
Secondly, there exists a lower limit of the maximum FPS
for design; i.e., a BESO-FPS solution might not exist when the
allowable FPS is too low.
Thirdly, the sharp reentrant corners in an initial design can
be well smoothed by the present method with a higher value of
p.
Finally, in some cases, where the solution becomes poor
for a given high value of p, a perturbation of pcould yield a
reasonable solution.
In this study, a linear elastic material model is used. In our
future work, nonlinear elastic or elastic-plastic materials will
be considered in the stiffness optimization with stress con-
straint(s). As a typical nonlinear elastic model, bi-modulus
materials layout optimization (Cai et al. 2016)willbeconsid-
ered as well.
Funding information The authors received financial support from the
National Natural Science Foundation of China (51778283 and
51678082) and the Australian Research Council (FL190100014,
DE200100887).
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of
interest.
Replication of results The results of the optimized designs and the basic
code of this work are available from the corresponding author on reason-
able request.
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