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Article published in
Computer Methods in Applied Mechanics and Engineering, Vol. 396 (2022) 115114
https://doi.org/10.1016/j.cma.2022.115114
Body-fitted bi-directional evolutionary structural optimization using nonlinear diffusion
regularization
Zicheng Zhuang 1, Yi Min Xie 1, Qing Li 2, and Shiwei Zhou 1,*
1 Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO
Box 2476, Melbourne 3001, Australia
2 School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney,
Sydney, NSW 2006, Australia
* Corresponding author. E-mail address: shiwei.zhou@rmit.edu.au.
Abstract
The bi-directional evolutionary structural optimization (BESO) method effectively uses basic
strategies of removing and adding material based on element sensitivity. However, challenges
remain in generating smooth boundaries to improve the finite element analysis accuracy and
achieve structural aesthetics. This work develops a body-fitted triangular/tetrahedral mesh
generation algorithm to yield smooth boundaries in the BESO method. The optimization problem is
regularized by adding a diffusion term in the objective function. We found that the first has the best
regularization effect of Lorentzian, Tikhonov, Perona–Malik, Huber, and Tukey functions. The void
elements are excluded from spatial optimization to save computation costs and computer memory.
Numerical examples show that the proposed method converges quickly, only taking dozens of
iterations to converge. Also, the smooth boundaries of the optimized structures in 2D/3D scenarios
are naturally obtained from the proposed method, not from smoothing post-processing. Compared
with the optimization toolbox in Abaqus, the example of the automotive control arm demonstrates
smoother boundaries and lower average mean compliance.
Keywords: Topology optimization, Body-fitted mesh, Bi-directional evolutionary structural
optimization, Nonlinear diffusion regularization.
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1 Introduction
Structural topology optimization seeks to find the optimal material distribution within a specified
design domain to minimize the costing functions under various constraints [1-3]. Over the past three
decades, researchers extensively investigated this problem. They developed remarkable techniques
such as the homogenization method [1], the solid isotropic microstructure with penalization (SIMP)
method [4], the evolutionary structural optimization (ESO) method [5, 6], and the level set method
[7-9]. Xie and Steven [5, 10, 11] originally proposed the ESO method to obtain the optimum
topology of continuum structures in the early 1990s. Yang et al. [12] and Querin et al. [13]
developed the BESO method based on the ESO method. The BESO method has been widely
implemented since it can provide a clear boundary in the optimized configuration with no gray
element. Thanks to the advances of these rapid-developed approaches and high-computing-power
computers, structural topology optimization has been widely applied in aerospace, automation,
aviation, buildings, and bioengineering to design lightweight, elegant, and high-performance
structures.
Conventional SIMP and ESO/BESO topology techniques employ rectangular and hexahedral
elements in geometric representation and finite element analysis for plane and spatial optimization
problems. The level set method presents smooth boundaries via the zero-level contour of a high-
level function. However, its finite element analysis still uses fixed elements with straight edges.
Although intuitive and straightforward, such a mesh essentially yields unwanted zig-zag boundaries
for non-horizontal/vertical interfaces in practical applications. Based on this scenario, we are
motivated to solve this problem in this paper by combining the body-fitted mesh generation method,
finite element analysis, nonlinear diffusions, and the BESO optimization techniques. The early
work of Kikuchi et al. [14] proposed the idea of adaptive grid design. Inspired by this pioneering
work, Persson [15] presented an innovative method to generate high-quality unstructured meshes.
By solving for a force equilibrium in the element edges, the quality of the body-fitted mesh is
improved iteratively. Then, Yaji et al. [16] constructed a convected level set optimization method
that employed the body-fitted mesh around the solid-void boundary. Salazar de Troya et al. [17]
proposed an efficient framework with adaptive mesh refinement and stress constraints. Xia et al. [18,
19] employed the body-fitted mesh in the stress-based optimization using a level set-based multiple-
type boundary method. Afterward, Baiges et al. [20] solved the large-scale stochastic topology
optimization using an adaptive mesh refinement method. Nana et al. [21] and Micheletti et al. [22]
designed lightweight structures with free-form features using the SIMP method and a mesh
adaption strategy. Christiansen et al. presented the deformable simplicial complex (DSC) method to
generate a well-formed mesh for 2D problems [23, 24] and 3D structures [23, 25, 26]. Micheletti et
al. [22] used a recovery-based a-posterior error estimator to eliminate the checkerboarding. On the
other hand, Allaire et al. [27-29] proposed a mesh evolution algorithm and applied it in the level set
topology optimization to generate smooth and elegant boundaries. Dapogny et al. [30] generated
high-quality tetrahedral mesh in the 3D structures using the mesh evolution algorithm. Very recently,
the utilization of body-fitted mesh in topology optimization algorithms has been in rapid
development. Li et al. [31, 32] adopted the mesh adaption technique and reaction-diffusion level set
method to generate excellent configurations. They employed FreeFEM, PETSc, and MMG re-
meshing tools for finite element analysis, distributed linear algebra, and mesh adaption. They also
applied this technique to solve the optimization problem of a fluid-structure system. Kuci et al. [33]
used the body-fitted mesh in the electro-mechanical optimization problems to accurately represent
the electromagnetic interface phenomena. Feppon et al. [34-36] adopted the body-fitted mesh in the
topology optimization problem of coupled thermal fluid-structure systems and 2D/3D fluid-to-fluid
heat exchangers.
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As introduced, many works have proposed to adopt the body-fitted mesh into the topology
optimization. However, efforts are still required to improve the performance and efficiency of
topology optimization using the body-fitted mesh. Overall, the main contribution of the present
paper is to find an efficient framework to fill the gaps. Most recent papers used various mesh
generation tools and commercial software to provide body-fitted mesh. However, the proposed
method utilizes a compact and straightforward MATLAB code to combine body-fitted mesh and
topology optimization. The non-rectangular elements, such as the polygons and the triangles, can
significantly enhance boundary smoothness in topology optimization [37-39]. Many researchers
have proposed to employ them to obtain flexibility in automated discretization [40] and make
optimized structures practical in the manufacture [41]. This paper employed the body-fitted
triangular/tetrahedral mesh for all numerical examples because of its stability and flexibility. The
body-fitted mesh is assumed as a truss structure [37] during the mesh generation procedure. The
joints of the initial mesh obtained from the standard Delaunay algorithm gradually move to make
the system balance under internal and external force. Based on solving a force equilibrium in
element edges, the generated body-fitted mesh smoothly matches the solid-void interfaces,
improving the algorithm accuracy and structural aesthetics. Compared with our previous paper [42],
which proposed a reaction diffusion-based level set method, the repeated nodes and the nodes too
close to each other are removed. The iteration to solve force equilibrium is also reduced by
adjusting parameters to increase mesh generation speed. Besides, the points outside the design
domain are pulled back to the closest point on the design domain boundary. The body-fitted mesh
generation procedure ensures the solid/void constraint, avoiding the gray-scale problem. The
numerical examples demonstrate that the body-fitted mesh composed of adaptive triangular or
tetrahedral elements can closely match smooth boundaries. Also, the previous works usually
consider all elements during the finite element analysis. Since the elastic property (Young’s
modulus) of the void element is considerably smaller than the solid element, only the degrees of
freedom for the nodes of the solid elements are calculated in this work. This method significantly
saves the computational cost of finite element analysis, particularly under the small volume
constraint.
Most previous works combined the SIMP and the level set method with the body-fitted mesh.
Although the sensitivity in the SIMP method is simple and easy to calculate, the generated gray
elements lead to the unsmooth solid-void interfaces [21]. The level set method [43-45] can
geometrically represent smooth boundaries according to the zero-level set. However, from the
perspective of computational cost, the implicit level set equation needs to be updated by solving the
PDE or reaction-diffusion equation. In contrast with the level set method, the BESO method uses
the basic strategies of removing and adding material according to element sensitivity. Compared to
the reaction diffusion-based level set method [42], the optimal bridge design displayed in Section 4
with the same volume constraint achieves less mean compliance by 4.42%. Since solving the
reaction-diffusion equation is unnecessary, the computational cost of the proposed BESO method is
about 23.0%~23.9% less than the reaction diffusion-based level set method. Combining this body-
fitted mesh and the BESO method, we found that the objective function value and convergence
speed can be even better. The proposed method uses the weighted-average method to interpolate the
design variable value to the vertices, generating high-quality boundaries. The BESO method [46, 47]
employs sensitivity analysis to find a gradient direction along which the costing function decreases
quickly to determine the removal or addition of an element [48]. The soft-kill BESO method [49, 50]
was proposed by introducing an artificial material interpolation scheme, where the zero-value used
to represent void domain density is replaced by a prescribed small number (denoted as xmin). Huang
and Xie [51], Huang [52], Zuo and Xie [53] published MATLAB codes and a Python code to
implement the BESO method for 2D/3D topology optimization problems, inspiring us to write the
BESO updating scheme in MATLAB. The body-fitted mesh is regenerated iteratively to capture the
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smooth boundaries between the solid/void domain and increase BESO accuracy without post-
processing.
Another main reason we selected the BESO method is that the nonlinear diffusion term can be
combined with it to regularize the optimization problem. The conventional BESO algorithm utilizes
the filter scheme to eliminate numerical instabilities such as checkerboarding and mesh-dependency
[49]. Then, the filtered sensitivity numbers can redistribute the solid and void material in each
iteration. However, it negatively affects the smoothness around the boundary of the material domain.
Also, the computational cost of the conventional filter in the body-fitted mesh is considerable since
it is necessary to find all neighboring points of each node iteratively. Thus, in this work, instead of
the conventional filter, a nonlinear diffusion term is added to the objective function to regularize the
optimization problem. The nonlinear diffusion technique is versatile and powerful in multiscale
image analysis. Weickert [54] reviewed the image enhancement techniques based on the parabolic
partial differential equations. Afterward, nonlinear diffusion regularization has been widely used in
topology optimization to improve image quality. Wang et al. [55] presented a nonlinear diffusion
technique and displayed the detailed derivation in the 2D rectangular mesh, formulating a well-
regularized problem. Keeling et al. [56] widened the range of sharpened edges using nonlinear
diffusion term. Platero et al. [57] introduced a nonlinear diffusion technique without control
parameters in their paper. They emphasized that the nonlinear diffusion term could simultaneously
generate smooth boundaries and avoid numerical instabilities. Due to this fact, the nonlinear
diffusion regularization is combined with the BESO method using body-fitted mesh in this paper.
The detailed derivation of the nonlinear diffusion term in the 2D and 3D scenarios is provided in
this paper. To the best of our knowledge, this is the first attempt to derive the nonlinear diffusion
matrix in the body-fitted mesh. As a result, the elegant structures and smooth boundaries in the
numerical examples demonstrate the effectiveness of nonlinear diffusion regularization. After
adding the nonlinear diffusion term, the material distribution is updated according to the relative
ranking of the element sensitivity number. The volume constraint is gradually attained by
multiplying the old value with an element removal ratio (ERR) in each iteration. For instance, the
volume constraint becomes Vnew=Vold(1-ERR) if Vold is the volume fraction ratio in the last iteration.
The ERR value in the conventional BESO method is usually prescribed to be smaller than 0.02. We
can define the ERR as large as 0.05 in this work to reduce iterations significantly.
The remainder of this paper is organized as follows. Section 2 reviews the topology optimization
problem and the diffusion process to regularize the problem. Then, the generation of the body-fitted
mesh and the design variable updating scheme is introduced in Section 3. The numerical examples
demonstrate the performance of our methodology in 2D/3D for compliance minimization problems
in Section 4. The proposed approach generates smoother boundaries and better objective function
values in fewer iterations than the conventional BESO and level set methods. Then, we summarized
our findings in Section 5.
2. Problem statement
2.1 Topology optimization
Topology optimization usually aims at searching for an optimal material distribution to obtain the
desired objective function under volume constraints in the design domain. Fig. 1 shows a typical
topology optimization problem in the design domain Ω⊂ℝ2 with boundary conditions. The
boundary of the design domain Γ includes the Dirichlet boundary ΓD and the Neumann boundary ΓN
and the other boundary ΓF. It is assumed that the Dirichlet boundary and the Neumann boundary are
fixed in this paper. The displacement in all directions is zero on the Dirichlet boundary. We used Fn
to represent the load applied on the Neumann boundary and assumed that the other boundary
sections are traction-free.
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Fig. 1. Schematic of the optimization problem with the design domain Ω, material domain Ωs, void
domain Ωv, and applied load Fn.
The design domain is divided into solid and void domains to meet the volume constraint. A
linear isotropic elastic material with constant Young’s modulus and Poisson’s ratio occupies the
solid domain Ωs. The void domain Ωv is occupied by a linear isotropic elastic material with a much
smaller Young’s modulus than the solid domain. The solid-void interface is the boundary between
the solid and void domains. The material distribution, also known as density field ρ(x), is defined as
the design variable in topology optimization problems. Thus, the optimization problem in Fig. 1
with an objective functional J under a volume constraint G can be generally expressed as:
max
min : ( ( ( ), ) ( ))
subject to ( ) 0
J C E j u dx
G dx V
Ω
ρ
Ω
µ ρ ρ τϕ ρ
ρρ
=+= +∇
= −≤
∫
∫
(1)
where j(u(ρ),ρ) denotes the objective function, depending on the optimization requirement, such as
compliance minimization. Vmax represents the upper limit value of the volume fraction ratio. An
unequal volume constraint G≤0 is included to restrict the set of admissible shapes. The inclusion of
the energy functional E works as a regularization method for the optimization problem. The
potential function φ is related to the diffusion process introduced in the next section. τ is a positive
diffusion coefficient. The diffusion regularization scheme is based on partial differential equations
(PDEs) to smooth the solution. It is a powerful tool in topology optimization problems that can
adjust the sensitivity around the solid-void interface, ensuring regularity. The displacement field u
across the domain is a state field that satisfies the linear elastic system
( ) 0 in
0 on
( ) on
( ) 0 on
s
D
nN
F
uf
u
un F
un
σΩ
Γ
σΓ
σΓ
∇⋅ − =
=
⋅=
⋅=
(2)
where n represents the outward normal to the Neumann boundary. f is the applied body force. The
second-order stress tensor σ at any points in the design domain can be calculated by the strain tensor
ε and the fourth-order stiffness tensor Es:
1
() () ( ( ))
2
T
ss
uEuE u u
σε
= = ∇ +∇
(3)
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Compliance minimization problem aims to find the stiffest structure with the least displacement
under the given boundary conditions. The strain energy is the objective function to measure the
displacement after prescribing the design domain and the boundary conditions. The elasticity
governing differential equations can be plugged in the general equation as follows:
max
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min : ( ( ) : ( ) ( )) ( ( ))
22
subject to ( ) 0
T
n
J u u dx dx F u dx
G dV
ΩΩ
ρ
Ω
σ ε τϕ ρ τϕ ρ
ρ ρΩ
= +∇ = +∇
= −≤
∫∫
∫
(4)
where J is the objective functional for the compliance minimization problem. ‘:’ denotes the
second-order tensor operator. In order to obtain the minimal value of the strain energy, we need to
separate the design domain Ω into many small elements. The detailed mesh generation method is
displayed in Section 3.
2.2 Diffusion processes
The existence and smoothness of the solution of topology optimization are usually questioned. The
previous literature [55] explained that the topology optimization problem introduced in the last
section is ill-posed. The conventional soft-kill BESO method employed the filter scheme [50] to
overcome the checkerboarding and mesh-dependency problems. This heuristic scheme defines the
nodal sensitivity numbers that are influenced by the neighboring points according to their distance.
Nevertheless, this method cannot ensure the smoothness of the solid-void boundary when the body-
fitted mesh is employed. Besides, it requires expensive computational costs because of the
dynamically changing mesh. Thus, this paper adds a diffusion term as the regularization technique,
effectively alleviating the numerical instabilities in topology optimization. This section will explain
how to employ the diffusion model to regularize the design variable. In a usual sense, diffusion is a
physical process that balances the concentration differences without mass creation or loss. Fick’s
law expresses the equilibration property as below:
R
ζρ
=− ⋅∇
(5)
where ∇ρ represents the gradient of mass concentration (material density), which causes a flow ζ to
compensate for this gradient. A positive definite diffusion tensor R describes the relation between
∇ρ and ζ. Since the diffusion process only transfers mass, mass cannot be created or destroyed. The
differential of ρ to the time t can be stated by the continuity equation:
div( )
t
ρζ
∂=−
(6)
The diffusion equation can be obtained by combining the two equations above as follows:
div( )
t
R
ρρ
∂ = ⋅∇
(7)
Inspired by the previous literature [54], we can replace the diffusion tensor R as a function of the
differential of the density field g(|∇ρ|2).
2
div( ( ) )
tg
ρ ρρ
∂= ∇ ∇
(8)
The nonlinear diffusion shown in the above equation can be employed to solve the energy
minimization problems. For example, the gradient of the potential function φ(│∇ρ│) can be defined
as ζ(∇ρ). Let us assume that:
2
() ( ) ( )g
ζρ ϕρ ρ ρ
∇=∇∇ = ∇ ∇
(9)
According to the classical variational method, the minimization of the energy functional E(ρ) is
related to the potential function φ, as shown in Eq. (1). Since the flow ζ is monotonously increasing
or decreasing, the potential function φ is convex. Thus, the energy functional E only has one
minimum. With the use of the descent gradient method, the minimization of the energy functional
leads to the nonlinear diffusion:
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2
div( ( )) div( ( )) div( ( ) )
t
g
ρ ϕρ ζρ ρ ρ
∂= ∇∇ = ∇ = ∇ ∇
(10)
Suppose η is used to denote ∇ρ for simplicity. In that case, the gradient of the potential function is
related to the diffusivity function as below:
2
() ( )g
ζη η η
=
(11)
2.3 Linear diffusions
If the flow ζ direction parallels the mass concentration gradient, the linear diffusions happen with
the diffusivity function g(│∇ρ│2) as a constant. The flow ζ is perpendicular to the tangent line of
the solid-void interface in the famous linear heat equation (g=1). The diffusion equation can be
expressed as:
= div ( ) =
t
ρ ρρ
∂ ∇∆
(12)
We can generally convolve the density field with a Gaussian kernel Gσ to obtain the solution of the
linear diffusions. The Gaussian kernel can be calculated by:
22
2
1exp( /2 )
2
w
G xw
w
π
= −
(13)
where the width w is related to the artificial time t. Then, the density field ρ(x) can be convolved as:
2
( , )= ( )
t
xt G x
ρρ
∗
(14)
The ‘*’ denotes the convolution operator. However, linear diffusion is an isotropic diffusion that
diffuses the image equally in all directions. The Gaussian smoothing blurs essential features around
the boundary, making them hard to identify. Thus, nonlinear diffusions are employed in the
following section to regularize the optimization problem.
2.4 Nonlinear diffusions
As displayed in Eq. (10), the diffusivity g is adopted to the gradient of the current configuration
ρ(x,t), which is called nonlinear diffusion. Since it is assumed that the potential function is convex,
the existence and uniqueness of the minimum of the energy functional E(ρ) can be ensured. Based
on the analysis of the nonlinear diffusions, a regularization model is introduced for the stiffness
optimization problems. Nonlinear diffusion terms can enhance the smoothness of the solid-void
interfaces if the nonlinear diffusivity function satisfies specific conditions. On the one hand, the
nonlinear diffusion should act like the linear diffusion inside the solid domain away from the
boundary. The │η│ value is usually small in these areas. On the other hand, the │η│ value is large
around the solid-void interfaces, where a regularization effect is required. In order to reduce noise
and enhance the contours, the nonlinear diffusivity function g(η2) should be regular and
continuously differentiable [9]. Besides, it requires to have the following properties [55]:
1. The nonlinear diffusivity function g(η2) is monotonously decreasing in the domain of function
η∈[0,∞) to make sure energy functional has a unique minimum.
2. As mentioned above, the regularization effect should be performed in the area where η→0. The
gradient of the potential function ζ(0)=0. Meanwhile, the second derivative of the potential function
φ(η) equals a positive constant γ as follows:
2
0
lim ( ) (0)g
η
ηϕ γ
→′′
= =
(15)
3. Besides, the smoothing effect should be strong around the boundary where η→∞. This condition
means that the potential function φ(η) is linear or sublinear when η→∞.
2
lim ( ) lim ( ) 0g
ηη
η ϕη
→∞ →∞ ′′
= =
(16)
Some typical potential functions φ(η) are widely used in nonlinear diffusion. Three of them are
plotted in Fig. 2 to illustrate their behaviors, including:
8
2
1
()
ϕη η
=
(17)
2
2
( ) log(1 )
ϕη η
= +
(18)
22
3
( ) / (1 )
ϕη η η
= +
(19)
It can be seen that all three potential functions satisfy the mentioned properties.
Fig. 2. Behaviors of three potential functions φ(η) with respect to η.
This paper employed Tikhonov, Lorentzian, Perona–Malik, Huber, and Tukey functions for the
nonlinear diffusion and compared their regularization effects in topology optimization [55]. Table 1
lists the potential functions φ and the diffusion function g of these functions with a positive
coefficient σ1. They will be used in Section 3.2 in the sensitivity analysis of the nonlinear diffusion
term in the body-fitted mesh.
Table 1 Nonlinear diffusion potentials and diffusivities
Diffusion function type Potential function φ(η) Diffusion function g(η2)
Tikhonov η2/2 1
Lorentzian σ12log[1+(η2/2σ12)] 1/(1+η2/2σ12)
Perona–Malik −σ12exp(−η2/2σ12) exp(−η2/2σ12)
Huber η2/2σ1+σ1/2, if η≤σ1
η, if η>σ1
1/σ1, if η≤σ1
sign(η)/η, if η>σ1
Tukey η2/σ12−η4/σ14+η6/3σ16, if η≤σ1
1/3, if η>σ1
(1−η2/σ12)2/2, if η≤σ1
0, if η>σ1
2.5 Sensitivity analysis of the Lagrangian functional
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The Euler–Lagrange equations are ordinary differential equations. Their solutions are stationary
points of the action functional, which is powerful in solving optimization problems. Let us consider
the mean compliance minimization problem. An unconstrained optimization problem is used to
replace the optimization problem in Eq. (4) as below:
max
1
min : ( ( )) ( )
2
T
n
L J G F u dx d V
ΩΩ
ρ
λ τϕ ρ λ ρ Ω
=+= +∇ + −
∫∫
(20)
where λ denotes the Lagrange multiplier. λ≥0, G≤0, λG=0. Considering the Karush–Kuhn–Tucker
(KKT) conditions [58] and the variational method, the solution of minimizing the modified
objective functional has to satisfy the Euler-Lagrange equation as follows:
1 d1
( ( )) / ( ( )) / =0
2 d2
TT
nn
Fu Fu
x
τϕ ρ ρ τϕ ρ ρ
′
∂ +∇ ∂−∂ +∇ ∂
(21)
We can simplify and plug the diffusivity function g into this Euler-Lagrange equation. Suppose
n+ is the outward normal to the boundary of the design domain Γ, the solution should satisfy the
following equations:
2
1/ div( ( ) )=0 for
2
=0 on
T
n
Fu g x
n
ρτ ρ ρ Ω
ρΓ
+
∂ ∂− ∇ ∇ ∈
∇⋅
(22)
As mentioned, body-fitted triangular/tetrahedral mesh is employed for 2D/3D models. Thus, the
element sensitivity number in the framework of the body-fitted mesh needs to be calculated using
the equations provided above. The following section introduces the generation of the body-fitted
mesh and sensitivity analysis in the body-fitted mesh. Besides, the detailed derivation of the
nonlinear diffusion term is displayed in Sections 3.2 and 3.3 for 2D/3D scenarios.
3. Numerical implementations
3.1 Body-fitted mesh generation
The most attractive advantage of the body-fitted mesh is the elegant and clear solid-void boundaries
[42]. The proposed BESO method adopts the adaptive triangular mesh as the body-fitted mesh in
the 2D conditions. The body-fitted mesh precisely captures the smooth boundaries to increase
accuracy. Besides, the elegant geometric representation can be obtained directly without post-
processing or material interpolation. Compared to our previous work, we found the solid-void
boundaries based on the relative density distribution instead of the level set function [45]. Let us
assume that Fig. 3(a) represents the density field over a rectangular design domain. The smooth
contour of the density field at specific heights (0.5 in this paper) can be found using the contour
function in MATLAB. Fig. 3(b) displays the isosurface (white) employed to capture the solid-void
interfaces.
The contour function can plot the smooth solid-void boundary and automatically generate a
point set P=[p1, p2, p3,…,ptotal] on the boundary. These nodes are fixed and defined as the element
vertices to describe the boundary in the proposed method. However, as illustrated in Fig. 4(a), too-
narrow point distribution on the contour may lead to the triangular elements with small angles,
resulting in the singularity problem. On the other hand, the generated adaptive mesh cannot
precisely express the smooth boundaries if the distance between neighboring points is too large.
Thus, we need to merge the too-close points and put additional points in the middle of the
neighboring points with large distances. The distance di between the neighbor points pi and pi+1 is
required to be adjusted to the appropriate values:
111
1
add ( , ) ( , ) ( , ) / 2
add ( , ) & remove ,
mid mid mid i i i i i i i max
mid mid mid i i i min
p x y p x y p x y if d d
p x y p p if d d
+++
+
=+>
<
(23)
10
where dmin and dmax denote the allowable minimal and maximal distance between adjacent points.
pmid is the midpoint of the neighbor points pi(xi,yi) and pi+1(xi+1,yi+1). The reorganization of point set
P can increase the body-fitted mesh quality by avoiding the singularity problem and the jagged
appearance, as illustrated in Fig. 4(b).
Fig. 3. (a) The density field across the design domain and (b) the generated body-fitted mesh
according to the specific isosurface.
Next, the body-fitted mesh can be generated based on the fixed node-set P and regularly
distributed unfixed points. The initial mesh is analogized to a truss structure where mesh vertices
are the nodes of the truss. Let us use a matrix p to represent the coordinates of all nodes in the
design domain as follows:
[ ]
=xyp
(24)
where x represents a vector including the x-coordinate of all nodes. y represents a vector including
the y-coordinate of all nodes. In the mesh generation algorithm, the unextended length l0 of each bar
should be defined manually to control the density or sparseness of the mesh. The force-
displacement relationship can be expressed as follows since we considered the repulsive forces in
the bars:
0 00 0
( , ) ( ), if f ll k l l l l=−<
(25)
where l is the current length of the bar and k0 is the modulus of elasticity. The internal forces come
from the difference between current and unextended bar length. Meanwhile, the reactions from the
boundaries, namely the external forces, are employed to avoid nodes moving outside the design
domain [37]. Considering both the internal force Fin and external force Fex, the force-displacement
function for the bars for equilibrium in the truss [37] can be expressed by:
() () () ( )
() [ () () () () ]
t in x ex x in y ex y
=+ +=0Fp F pF p F pF p
(26)
11
where the internal force is the force from bars, which depends on the length of the bars. The
position of the nodes can be found by solving this static force equilibrium.
Then, an artificial time t is introduced. The system of an ordinary differential equation can be
reviewed as:
= ()
t
d
dt
pFp
(27)
Fig. 4. (a) The point distribution on the contour before reorganization and (b) the point distribution
on the contour after reorganization.
Now, we need to find the stationary point that satisfies the resultant force Ft(p)=0. Let us
consider the initial node position as p0. With the help of the forward Euler method, the approximate
solution pnΔt at time Δt can be expressed as:
1
()
n n tn
t
+
= +∆p p Fp
(28)
where Δt is the step size. Then, the points outside the boundary should be moved back to the design
domain during this evolution process. The further away from the boundaries, the sparser the body-
fitted mesh as the unextended bar length l0 is proportional to the distance to the solid-void interfaces.
As shown in Fig. 5(a), we can gradually move the unfixed nodes to the best location when the
system gets into its balanced state and reset the topology using the Delaunay algorithm [37]. Using
this method, the elegant body-fitted mesh that accurately captures the boundaries can be obtained
for the solid (black) and void domain (blue), as illustrated in Fig. 5(b). The red curve represents the
smooth boundary between the solid and void domains. No element is passed through by the red
curve, avoiding the gray-scale problem in the proposed BESO method. Fig. 5(b) illustrates that
body-fitted mesh density varies throughout the design domain. The denser adaptive triangular mesh
generated around the solid-void boundaries ensures smoothness and accuracy. Meanwhile, the
sparse body-fitted mesh is produced in the areas away from the boundaries. The coarse mesh in the
low-sensitivity zones increases the finite element analysis efficiency by reducing the computational
cost. It should be emphasized that the mesh is regenerated according to the solid-void interfaces in
each iteration.
12
Fig. 5. (a) The node distribution in equilibrium across the design domain and (b) the generated
body-fitted mesh using the proposed method.
The mesh generated in this work is compared with the one created in MATLAB using the
generateMesh function. MATLAB can produce an adaptive triangular mesh using the typical
Delaunay triangulation shown in Fig. 6(a) for the unit circle with a quadrangle hole in its center.
The pdetriq function in MATLAB shows it has an average quality of 0.97 (the closer to 1, the better
the mesh) and a standard deviation of 0.031 (a value to measure the mesh uniformity). The body-
fitted mesh with a similar mesh amount generated by the proposed method is provided in Fig. 6(b).
Its average value of 0.98 and the standard deviation of 0.025 demonstrate that the proposed mesh
generation algorithm can also create a mesh with the same or even better quality. Even though
shape complexity and boundary node density will degrade mesh, the average quality of meshes in
all optimization problems is no less than 0.88, sufficient to avoid numerical instability and
guarantee smooth boundaries.
Fig. 6. (a) The adaptive triangular mesh generated using MATLAB polyshape function and (b) the
body-fitted mesh generated using the proposed method.
3.2 Discretized sensitivity in the triangular mesh
The optimization problem can be solved using the finite element method [38, 59] with the
assistance of high-computing-power computers. Suppose we discretized the design domain to T
individual triangular elements. In the finite element framework, the vector Fn=[Fn1, Fn2, Fn3,… FnT]
denotes the external force applied on the Neumann boundary. u=[u1, u2, u3,… uT] represents
13
displacement of each element under the external force Fn. The relationship between force and
displacement can be written as follows:
n=KuF
(29)
where K denotes the global stiffness matrix that is assembled by the element stiffness matrices.
ρ(x)=[ρ(x1), ρ(x2), ρ(x3),… ρ(xT)] and V(x)=[V1, V2, V3,…VT] represents the relative density and
volume of each element. ρ(xi) denotes the density of the ith element, whose value is either 1(solid) or
ρmin(void). The elastic property of the ith element is defined as:
[ ]
0 min
() () , () ,1
si i i
E x xE x
ρ ρρ
= ∈
(30)
where E0 is the elastic property of the solid material. Then, the mean compliance of the ith element
can be expressed as:
T TT
i ii ii ii i i ii
C dV
ρρ
= = ∫
u K u u B DB u
(31)
where Bi, D, Ki denote strain matrix, elasticity matrix, and element stiffness matrix. The element
stiffness matrix Ki keeps constant in the conventional BESO topology optimization using fixed
rectangular elements. However, the element stiffness matrix Ki varies in this work using adaptive
triangular elements, depending on the coordinates of the element vertices. Our previous paper [42]
provided the derivation of the element stiffness matrix Ki for the body-fitted mesh. Thus, the
sensitivity of the mean compliance for the ith element [52] can be expressed as:
( ) /( )
Ti
i i i ii i
CV
C
ρ
ρρ ρ
∂
∂∂
′=−=
∂∂ ∂
K
uu
(32)
On the other hand, we need to calculate the diffusion term in the triangular mesh. Firstly, let us
derive the discrete Laplacians considering the connectivity of nodes and edges in the body-fitted
mesh. Let us take point A0 displayed in Fig. 7 as an example. Suppose the point A1 to A7 are the
points directly adjacent to point A0. N(A0) is used to denote the list of all neighboring vertices to A0.
In that case, the Laplace-Beltrami operator [60] of the density field at point A0 can be
approximated as:
00
0
00 0
() ()
()
11
(cot cot )( ) (cot cot )( )
22
i ii i ii
iNA iNA
Ai
iTA
VT
ρ α βρ ρ α βρ ρ
∈∈
∈
∆= +−= +−
∑∑
∑
(33)
where αi and βi represent the two angles opposite of the edge A0Ai. VA denotes the vertex area of
point A0, which equals the sum of the area of the adjacent triangles T(A0). ρ0 and ρi are the density
values at vertex A0 and its adjacent points, respectively. The sum covers all adjacent points in set
N(A0).
14
Fig. 7. The Point A0 in the triangular mesh with seven neighboring vertices.
Subsequently, the matrices are used to describe the relationship between the density field and
the Laplacians. According to the previous literature [60-62], we can define ωAi and ωsumA as follows:
1
= (cot cot )
2
Ai i i
ω αβ
+
(34)
0
()
=
sumA i
iNA
ωω
∈
∑
(35)
Then, the per-vertex expression ∆ρ0 can be refactored as follows:
0
00
()
11
Ai i sumA
iNA
AA
VV
ρ ωρ ω ρ
∈
∆= −
∑
(36)
Suppose g(|∇ρ|2)Ai represents the function of the differential of the density field between point Ai
and A0. In that case, the nonlinear diffusion term value div(g(│∇ρ│2)∇ρ) at point A0 can be
expressed as:
0
22 2
0
() ()
11
div(())= (()) ( ())
A Ai Ai i Ai sumA
iNA iNA
AA
gg g
VV
ρ ρ ρ ωρ ρ ω ρ
∈∈
∇∇ ∇ − ∇
∑∑
(37)
According to the diffusion function g of Tikhonov, Lorentzian, Perona–Malik, Huber, and
Tukey functions listed in Table 1, the sensitivity analysis of the nonlinear diffusion term can be
calculated in the body-fitted mesh. As illustrated in Fig. 11, the Lorentzian diffusion function
proves its superiority over the other functions depending on the objective function value and
boundary smoothness.
Then, matrix calculation is employed to improve efficiency. Let us consider the discrete
Laplacians of the eight points in Fig. 7. Suppose the density values in every vertex A0, A1, A2, A3, A4,
A5, A6, A7 are included in an 8×1 vector Q=[ρ0, ρ1, ρ2,…, ρ7]. In that case, the matrix calculation is
employed to calculate the discrete Laplacians at all vertices.
-1 -1
= =( )
ij sum
ρ
∆−V LQ V L L Q
(38)
where V is an 8×8 diagonal matrix that stores the vertex area of each vertex. Lsum is an 8×8 diagonal
matrix that includes the sum of the cotangent weights in each vertex ωsumAi. Lij denotes an adjacency
8×8 matrix containing the individual cotangent weights ωij. The required matrices of an eight-node
system (Fig. 7) are displayed to illustrate their assembly considering the connectivity.
15
0
1
7
-1
1
00
0
0
00
A
A
A
V
V
V
−
V=
(39)
0
1
7
00
0
0
00
sumA
sumA
sum
sumA
ω
ω
ω
=
L
(40)
01 06 07
10 17
60 67
70 71 76
0
00
00
0
ij
ω ωω
ωω
ωω
ωω ω
=
L
(41)
where ωsumAi and ωij are defined as:
1
= (cot cot )
2
ij ij ij
ω αβ
+
(42)
()
= , {0,1, 2, 3,4, 5,6, 7}
i
sumAi ij
jNA
ij
ωω
∈
=
∑
(43)
Then, we can use V-1LQ to calculate the Laplacians by matrix calculation. The 8×8 Laplacian
matrix L can be expressed as:
0
1
6
7
01 06 07
10 17
60 67
70 71 76
0
==
0
sumA
sumA
Ai sum
sumA
sumA
ω ω ωω
ωω ω
ω ωω
ω ω ωω
−
−
−
−
−
LL L
(44)
Similarly, the discrete diffusion term matrix can be calculated using matrix calculation as G=V-1ZQ.
The 8×8 nonlinear diffusion matrix Ζ can be derived for the system in Fig. 7 as:
0
11
66
77
01 01 06 06 07 07
10 10 17 17
60 60 67 67
70 70 71 71 76 76
0
=
0
sumA sumA
sumA sumA
sumA sumA
sumA sumA
g g gg
gg g
g gg
g g gg
ωω ω ω
ωω ω
ω ωω
ωω ω ω
−
−
−
−
Z
(45)
where gsumAi and gij are defined as:
2
=( )
ij ij
gg
ρ
∇
(46)
()
= , {0,1,2, 3,4, 5,6, 7}
i
sumAi ij
jNA
g g ij
∈
=
∑
(47)
Using this method, G=V-1ZQ can calculate the diffusion term matrix G at all vertices over the
adaptive triangular design domain by matrix calculation.
3.3 Extension to the 3D tetrahedral mesh
16
The topology optimization method that cannot be readily extended to 3D is impractical in
manufacturing. In this section, the generation and the discretized sensitivity of the adaptive
tetrahedral mesh are introduced. 4-node linear tetrahedral elements are used as the body-fitted mesh
to extend the proposed method to the 3D conditions. The isosurface function in MATLAB can plot
the smooth solid-void surface and automatically generate a node-set on the boundary. Next, the
body-fitted mesh can be generated based on the fixed node-set P and regularly distributed unfixed
points. Like the 2D triangular mesh, the initial mesh can also be analogized to a 3D truss structure
where mesh vertices are the nodes of the truss. Let us use a matrix p to represent the coordinates of
all nodes in the design domain as follows:
[]
=x y zp
(48)
where x represents a vector including the x-coordinate of all nodes. y represents a vector including
the y-coordinate of all nodes. z represents a vector including the z-coordinate of all nodes.
Considering both the internal force and external force, we can also obtain the force-displacement
function for the bars for equilibrium in the truss can be obtained as follows:
()() ()()()()
() [ () () () () () ()]
t in x ex x in y ex y in z ex z
=+ + +=0
Fp F pF p F pF p F pF p
(49)
The equilibrium can be solved to obtain the approximate solution of node position using the
forward Euler method displayed in Eq. (27) and Eq. (28). Then, we pulled back the points outside
the boundary and used the Delaunay algorithm to generate the body-fitted tetrahedral mesh. As
displayed in Fig. 8(a), the nodes on the boundaries are fixed and defined as the tetrahedral
elements’ vertices. The regenerated body-fitted mesh using these nodes is shown in Fig. 8(b).
Fig. 8. (a) The nodes captured on the solid-void boundaries and (b) the body-fitted tetrahedral mesh
generated according to these nodes.
Subsequently, the iteratively generated mesh is plugged into the finite element analysis. In the
finite element analysis procedure, we only considered the solid element in the finite element
analysis to save time. The symbol calculation is also applied to assemble the stiffness matrix of the
tetrahedral mesh to improve efficiency further. The MATLAB code to calculate the element
stiffness matrix K using symbol calculation can be seen in Appendix. C of our previous paper [42].
The mean compliance value of the ith element can be calculated using Eq. (31).
17
Fig. 9. The Point B in the tetrahedral mesh and its neighboring points.
The sensitivity of the mean compliance for the ith element can be calculated using Eq. (32).
Then, the derivation to calculate the diffusion term in the tetrahedral mesh is displayed. Fig. 9
shows point B and its neighboring points in the tetrahedral mesh. Suppose the point B1 to BN are
the points directly adjacent to point B. N(B) denotes the list of all neighboring vertices to B. In that
case, according to the previous literature [63, 64], the Laplace-Beltrami operator of the density
function ρ at point B can be expressed as the function of density on point B and its neighboring
points.
() ()
()
B Bi i Bi B
iNB i NB
ρ ωρ ω ρ
∈∈
∆= −
∑∑
(50)
Suppose g(|∇ρ|2)Bi represents the function of the differential of the density field between points Bi
and B. In that case, the nonlinear diffusion term value div(g(│∇ρ│2)∇ρ) at point B can be
expressed as:
22 2
() ()
div(())= (())( ())
ii
B Bi Bi i Bi Bi B
B NB B NB
gg g
ρρ ρωρ ρωρ
∈∈
∇∇ ∇ − ∇
∑∑
(51)
Here i and j represent any two points in tetrahedron BB1B2B3. Then l and k denote the other two
points in tetrahedron BB1B2B3. Alexa et al. [64] derived the expression of ωij in the tetrahedron
BB1B2B3 accumulation formula as follows:
=2
ij ijlk jlki lkij kijl
ωω ω ω ω
+++
(52)
22
cot cot
= cot (2 cot cot )
4 cos
ij
ijlk
h
αβ
ω θ αβ
θ
−−
(53)
where hij is the height over the base. The dihedral angle θ at the edge BB1 and the angles α and β
inside the incident triangles are highlighted in Fig. 9. Then, we can calculate the ωBi for each
neighboring point of B and sum them in Eq. (51).
3.4 BESO updating scheme
The material distribution is updated by removing or adding solid materials according to the
sensitivity number relative ranking [50] in the soft-kill BESO method. Unlike the conventional
BESO method, a nonlinear diffusion term is added to the element sensitivity number. According to
a predefined element removal ratio (ERR), we removed the elements with the lowest sensitivity
numbers. Based on Eq. (22) and Eq. (32), the sensitivity number generally includes the nonlinear
diffusion term can be expressed as:
2
div( ( ) )
Ti
i ii i i
Jg
ρ τ ρρλ
ρ
∂
′= + ∇∇−
∂
K
uu
(54)
18
A Lagrange multiplier λ is added to control the volume fraction ratio. It should be mentioned that
the detailed derivation of the diffusion term div(g(│∇ρ│2)∇ρ) in the triangular and tetrahedral mesh
is shown in Sections 3.2 and 3.3.
The bi-section Lagrangian method can strictly control the volume fraction ratio of optimized
configurations to meet the volume constraint with the Lagrange multiplier λ, significantly reducing
the error. Meanwhile, the volume constraint in each iteration Vmax gradually reduces based on the
user-defined ERR value. Suppose λ1 and λ2 are the maximal and minimal values of element
sensitivity number. In that case, λ1 and λ2 are the bottom and top values of the Lagrange multiplier
defined in the bi-section method. Then, the average value λmid is subtracted from the element
sensitivity number to update the design variable of the ith element as follows:
'
'
if 0
1 if 0
i min i mid
i i mid
J
J
ρρ λ
ρλ
= −≤
= −>
(55)
Now, we compare the volume fraction ratio after obtaining the design variables of all elements. If it
is larger than the prescribed volume constraint, λ2 equals λmid. Otherwise, λ1 equals λmid.
1
2
if
if
mid i i max
mid i i max
VV
VV
λλ ρ
λλ ρ
= ≤
= >
∑
∑
(56)
12
0.5( )
mid
λ λλ
= +
(57)
Eq. (55), Eq. (56), and Eq. (57) are repeatedly solved until meeting the volume constraint to obtain
a precise λ and the updated material distribution. This bi-section updating method is essential
because it is flexible and precise in controlling the volume fraction ratio of the optimized
configuration.
4. Numerical examples
This section presents different numerical examples of structural optimization with the proposed
algorithm and implementation in 2D/3D scenarios. The optimization problem displayed below is
the mean compliance minimization problem that has been widely studied in the relevant literature [4,
42, 50]. The body force is assumed to be zero across the design domain. The objective function is
the strain energy value of the optimized structure under the prescribed material volume constraint.
The convergence check is performed iteratively after the minimal iteration number during the
optimization. The optimization terminates when the objective function values of the previous five
iterations differ by less than 0.5%. Meanwhile, the material volume of the optimized structure must
be strictly controlled to the prescribed value Vmax. The body-fitted triangular and tetrahedral mesh is
applied in Sections 4.1 and 4.2, respectively. The same body-fitted mesh is utilized in the finite
element analysis as in the geometrical representation for the numerical examples below. Solid initial
patterns with a 100% volume fraction ratio and porous structures are tested in this section to
demonstrate the robustness of the proposed method.
4.1 2D numerical examples
The optimized configurations of the 2D cantilever beam and MBB beam are displayed in this
section to demonstrate the effectiveness and feasibility of the proposed method. In this section, 2D
optimization designs start from porous structures. The mean compliance is conversely set as the
objective function to obtain a solid–void two-phase design with maximum stiffness under a given
volume constraint. We illustrated the design domain and boundary conditions of the 2D cantilever
beam optimization problem in Fig. 10. The 80 mm×50 mm×1 mm design domain in the red dotted
rectangular rectangle is fixed on the left edge. A 100 N downward load is applied to the center of
the right edge. Young’s modulus and Poisson’s ratio are assumed to be 100 GPa and 0.3. The
19
volume constraint of the optimized structure is prescribed as 50%. The element removal ratio (ERR)
for the 2D optimization problem is set as 0.05. A nonlinear diffusion coefficient of τ=2×10-8 is
applied to regularize 2D optimization designs. All numerical examples in this section are under a
plane stress condition.
We can see the optimization result of the 2D cantilever beam in Fig. 10(a) using the proposed
method with the body-fitted mesh. The boundary represented by red curves between the void and
solid domain is smooth and elegant with the proposed body-fitted mesh. The Lorentzian function is
utilized in nonlinear diffusion terms to regularize the optimization problem. Besides, the finite
element analysis can precisely follow this smooth boundary in the body-fitted mesh framework to
improve accuracy. The optimization result displayed in Fig. 10(a) converges in 23 iterations with
the mean compliance of 1.802. The volume fraction ratio of the solid material is strictly controlled
to 50.01%, 0.01% larger than the prescribed value. This result is compared to the conventional
BESO method under the same load and boundary condition with a similar amount of nodes using
fixed rectangular mesh [50]. As shown in Fig. 10(b), the mean compliance value of the optimized
structure becomes 1.870 when converged in 83 iterations. It can be found that the objective value
using the fixed rectangular mesh is 3.64% larger than the result obtained by the proposed method.
The difference includes the extra accuracy caused by the precise finite element analysis and the
flexibility of the body-fitted mesh. The proposed method replaces the conventional filter scheme
with the nonlinear diffusion term and uses a large ERR=0.05 to accelerate convergence. Thus, it
takes 152.1 sec to converge, faster than the conventional BESO method, which costs 188.2 sec. For
this example, the 2D mesh regeneration procedure takes up 21.0% of the total running time.
20
Fig. 10. The design domain, boundary conditions, and the optimized configuration of the 2D
cantilever beam using (a) proposed method and (b) conventional BESO method.
Table 1 compares the performance of Tikhonov, Perona–Malik, Huber, and Tukey functions in the
nonlinear diffusion regularization with the Lorentzian function. The optimized structures obtained
by various nonlinear diffusion functions are provided in Fig. 11. Since the nonlinear diffusion
regularization is not applied to the optimization problem, zig-zag boundaries appear around the
solid-void interfaces in Fig. 11(a). The Tikhonov and Tukey function can avoid the
checkerboarding and increase the smoothness of optimized structures, as shown in Fig. 11(b) and (f).
However, the objective function is 1.825 and 1.830, respectively, 1.26% and 1.53% larger than the
optimized result using the Lorentzian function. As displayed in Fig. 11(c) (d) (e), the Lorentzian,
Perona–Malik, and Huber functions can produce high-performance optimized results with smooth
interfaces. The objective function is 1.802, 1.803, and 1.803, respectively, indicating that the
Lorentzian function achieves the most satisfactory result. Thus, the Lorentzian function is employed
for the rest examples.
21
Fig. 11. The optimized configuration of the 2D cantilever beam using (a) zero nonlinear diffusion;
(b) Tikhonov function; (c) Lorentzian function; (d) Perona–Malik function; (e) Huber function and
(f) Tukey function.
Then we displayed another 2D optimization example called Messerschmitt–B¨olkow–Blohm
(MBB) beam. A pinned support and roller support are located on the corners of the bottom edge,
and a 100 N downward concentrated force is imposed on its bottom center. Because of symmetry,
only the symmetric right half of the design domain is considered, which is 120 mm in length, 40
mm in width, and 1 mm in thickness. Young’s modulus and Poisson’s ratio are assumed to be 200
GPa and 0.3. The volume constraint of the optimized structure is prescribed to be 50%. The
topology optimization of the MBB beam example using the proposed method converged after 20
iterations. The optimal structure displayed in Fig. 12 uses the BESO method with an adaptive
triangular mesh. The objective function value and the volume fraction ratio of the optimized
structure are 1.130 and 50.04%. It can be seen that the solid-void interface is smooth and precise as
the short cantilever beam example. The optimized result employing the fixed rectangular mesh with
the same load and boundary condition is solved by BESO for comparison. The iteration number to
convergence and mean compliance are 68 and 1.183, with a similar number of nodes and a
rectangular mesh. The computational time cost of the proposed method and conventional BESO
method is 174.0 and 176.9 sec, respectively. Besides, it should be noted that the 2D mesh
regeneration procedure takes up 33.6% of the total running time for this example.
22
Fig. 12. The design domain, boundary conditions, and the optimized configuration of the 2D MBB
beam using the (a) proposed method and (b) conventional BESO method.
For clarity, the convergence histories of 2D optimization problems using the proposed method
are plotted in Fig. 13(a) and Fig. 13(b). We can conclude from the two examples that the ideal mean
compliance values and smooth structures without zig-zag boundaries can be obtained using the
proposed method. Appropriate mesh density distribution improves the finite element analysis
efficiency. The high-quality structures indicate that a body-fitted triangular mesh and the BESO
method can be combined to obtain smoother boundaries and better stiffness performance. In
contrast to the fixed rectangular mesh, the objective function values of adaptive triangular mesh
results are 3.64% and 4.48% better. Also, the proposed method requires fewer iterations and less
time to convergence because of the larger ERR and the non-filter scheme. More importantly, the
void domains are excluded from finite element analysis. The volume fraction ratio of the optimized
configuration is strictly approaching the prescribed value because of the bi-section method. Overall,
the effectiveness and efficiency of the proposed BESO method can be guaranteed in the 2D mean
compliance minimization problems. The smooth and elegant boundaries can be achieved without
significantly increasing the mesh number or using a post-processing scheme.
Fig. 13. Convergence histories of the 2D optimization problems: (a) mean compliance value and (b)
volume fraction ratio.
23
4.2. 3D Numerical examples
This section displayed four 3D optimization designs to demonstrate the effectiveness, stability, and
extensibility of the proposed algorithm, including a 3D MBB beam, a 3D cantilever beam, an
optimal bridge, and a control arm example. The maximal volume constraint is prescribed as 20% of
the design domain for the 3D MBB beam and the optimal bridge examples. The volume constraint
for the 3D cantilever beam and the control arm examples are 30% and 40%, respectively. We
assumed that Young’s modulus (solid material) and Poisson’s ratio are 100 GPa and 0.3,
respectively. The numerical examples in this section are performed on the workstation with Intel(R)
Core(TM) i5-7500 CPU @ 3.40GHz. The GPU is an NVIDIA GeForce GTX 760. All 3D
optimization problems in this section start from a fully solid structure (100% volume fraction ratio
of the design domain). Large ERR values from 0.04 to 0.06 are used for 3D optimization designs to
boost convergence and improve efficiency.
Fig. 14 The design domain, boundary conditions, and optimal design of the 3D MBB beam
optimization using the body-fitted tetrahedral mesh.
Firstly, we presented a successful implementation of topology optimization with body-fitted
mesh adaption on the 3D MBB beam design. Fig. 14 illustrates the cuboid design domain of 200
mm in length, 20 mm in width, and 50 mm in thickness. The surface traction P=1 N/mm is imposed
along the z-direction to a narrow band located on the top surface center. Two corners of the bottom
surface are pinned to prevent rigid movement in the horizontal direction. The other two corners are
fixed along the z-direction. For clarity, the golden regions represent the solid domain, and the void
domain is transparent in the optimized configuration. The 3D MBB beam optimization result with
no edges is plotted in Fig. 14 to show the optimized structure with a 20.06% volume fraction ratio.
Also, we displayed two amplified parts and added the black edges on the surface. The surface of the
3D optimized configuration is clear and smooth using the proposed method with body-fitted
tetrahedral mesh. The tetrahedral element number of the optimized configuration is 1.32 million for
the whole design domain. The optimization is converged in 50 iterations with its objective function
value as 1.211×103. Besides generating the high resolution of boundaries, the computational cost
spent on the mesh generation procedure is relatively low compared to the optimization procedure.
For this example, the 3D mesh regeneration procedure takes up 29.6% of the total running time.
24
Fig. 15. The optimal designs of the 3D MBB beam optimization using: (a) ERR=0.04; (b)
ERR=0.05 and (c) ERR=0.06.
The ERR value and diffusion coefficient employed in Fig. 14 are 0.04 and 5×10-8, respectively.
To demonstrate the robustness of the proposed algorithm due to parameter choices, we changed the
ERR value to 0.05 and 0.06 without changing other parameters. The optimized structures of the 3D
MBB beam using different ERR values are displayed in Fig. 15. The solid-void interfaces in all the
results are smooth and elegant. However, it should be noted that the geometrical complexity of the
optimal design increases with a larger ERR value. Also, we found that more materials appear
around the fixed points in Fig. 15(c). The results indicate that it is better to use ERR<0.05 to achieve
structural aesthetics and avoid too many materials produced around the fixed points.
The second example we would like to display is a 3D cantilever beam example shown in Fig.16.
The design domain length, width, and height are 80 mm, 40 mm, and 40 mm. The surface traction P
is applied on the left edge of the bottom surface highlighted in red. Meanwhile, all the points on the
opposite surface are fixed on the wall. Like other 3D examples, the design domain is divided into
the body-fitted tetrahedral mesh to obtain a smooth and clear surface. To validate the efficiency of
the proposed method, the total run time of this example is compared to the cantilever beam example
in the previous paper [31].
25
Fig. 16. The design domain, boundary conditions, and optimal design of the 3D cantilever beam
optimization using the body-fitted tetrahedral mesh.
The isometric view of the 3D cantilever beam optimization result without edges is provided in
Fig. 16. After adding the black edge of the body-fitted mesh, we amplified two parts of the
optimized configuration. The total number of tetrahedral elements in the final design is 866,176 for
the whole design domain. The total run time of this example is 2.48 hours. In contrast with the level
set-based optimization using 874,835 body-fitted mesh [31], which costs 150 iterations and 3.26
hours to convergence, the proposed method saves 23.9% running time. It can be found that 28.3 %
of the total running time is spent on the mesh regeneration procedure for the 3D cantilever
optimization example. Besides, it can be observed that the proposed method generates elegant and
clear boundaries after adding the nonlinear diffusion term into the objective functional. The
optimization is converged in 31 iterations with the objective function value of 1.095×103 using the
proposed method. The volume fraction ratio of the optimized configuration is 30.20%. It can be
found that the 3D mesh generation algorithm slightly changed the volume fraction ratio by 0.20%.
Fig. 17. The optimal designs of the 3D cantilever beam optimization using different diffusion
coefficient τ values: (a) τ=5×10-8; (b) τ=5×10-9 and (c) τ=5×10-10.
26
The ERR value and diffusion coefficient employed in Fig. 16 are 0.05 and 5×10-8. To
demonstrate the robustness of the optimization due to various nonlinear diffusion coefficients τ, we
changed the τ value to 5×10-9 and 5×10-10 with other parameters constant. The optimized
configurations are displayed in Fig. 17 for comparison. Although the same ERR value is employed
in the optimization, the geometrical complexity of the optimal structures varies according to the
nonlinear diffusion coefficient τ. The holes are generated in different positions of the optimized
configurations. The results indicate that the users can obtain different optimal designs and choose
the most satisfactory result by changing the diffusion coefficient τ to an appropriate value.
Fig. 18. The design domain, boundary conditions, and optimal design of the 3D bridge optimization
using the body-fitted tetrahedral mesh.
Another typical application scenario of the proposed method is to design innovative
architectural structures. We presented an engineering case using the proposed method, an optimal
bridge optimization design. As illustrated in Fig. 18, the uniformly distributed vertical surface
traction P is applied on the top surface of the 200 mm×20 mm×60 mm design domain. The four
corners of the bottom surface are fixed as the bases. In the optimal bridge example, the top few
layers of the design domain are defined as the non-design domain (silver). The highlighted
tetrahedral elements located in the non-design domain must remain solid throughout the
optimization process. These elements are found before the optimization using our previous
algorithm [42]. The volume of elements in the non-design domain is also subtracted from the total
volume when calculating the volume fraction ratio of the final optimized design. Fig. 18 shows the
isometric view of the optimal bridge design (20.10% volume fraction ratio) without element edges
on the surface. Two zoomed parts of the optimized configuration are plotted with black surface
edges. The tetrahedral element number of the optimized configuration is 1.58 million for the body-
fitted mesh, including the solid and void domains. The optimization is smoothly converged in 30
iterations, with its objective function value as 1.146×106. The precise and smooth boundary can be
generated, demonstrating the effectiveness of the proposed method using the adaptive tetrahedral
mesh. It can be calculated that the 3D mesh regeneration procedure takes up 32.0% of the total
running time for the optimal bridge example. To validate the effectiveness of the proposed method,
we use the reaction diffusion-based level set method in our previous paper [42] to calculate the
same bridge design. Suppose the same amount of the body-fitted mesh is utilized for the level set
27
method. In that case, the optimization converged in 26 iterations with the objective function value
as 1.199×106. Compared with the reaction diffusion-based level set method, the optimization
example of a bridge under a 20% volume constraint attains 4.42% less mean compliance using 23.0%
less running time.
Fig. 19. The design domain, boundary conditions, and optimal design of another 3D bridge
optimization using the body-fitted tetrahedral mesh.
We can also apply the uniformly distributed vertical surface traction P on the middle layer of the
160 mm×40 mm×40 mm design domain. The volume fraction ratio of the optimized structure is
10.07%. As illustrated in Fig. 19, four points of the bottom surface are pinned as the supports, and
we defined the loaded layers as the non-design domain (silver). The optimization is smoothly
converged in 29 iterations, with its objective function value as 4.588×106. It can be seen that the
smooth branches generated in the design domain effectively increase the aesthetic and performance
of the optimized configuration.
Next, a practical optimization problem of an automotive control arm is presented to demonstrate
the effectiveness of the algorithm. The proposed method can generate smoother surfaces and lower
mean compliance than TOSCA, the optimization toolbox built-in Abaqus. As shown in Fig. 20(a),
the upper left and upper right end of the control arm are fixed and prescribed as the non-design
domain (red). They are fixed in all three translation degrees of freedom. However, they can be
rotated for practical applications. The lower bearing is also set as the non-design domain. The
center node of the lower bearing is connected to the lower bearing surface through a kinematic
coupling. Two concentrated loads, P1=100 N and P2=−100 N, are imposed at the center node of the
lower bearing in the x-axis and y-axis directions, respectively. The optimization aims to reduce
sixty percent of the material from the initial structure.
Since the load and boundary conditions are relatively complex, we employed Abaqus CAE to
perform the finite element analysis. Firstly, the node position and element vertices information are
written into the inp.-file in each iteration. Subsequently, the nodal displacement and element
volume can be read using the Python code from the obd.-file. After adding the nonlinear diffusion
term into the objective function, the material distribution can be updated using the BESO scheme
introduced in Section 3.4. Then, the node position and element vertices information is written into
the inp.-file for the next iteration. After 18 iterations, the optimization converges with the objective
function value as 2.222×103. The optimized configuration with 40.00% materials using the
proposed method is plotted in Fig. 20(a). We also used TOSCA to solve the same optimization
28
problem for comparison. The objective function value of the optimized structure (39.99% volume
fraction ratio) shown in Fig. 20(b) is 2.423×103, which is 8.30% larger than the proposed method
result. Also, the efficiency of the proposed method is compared with Abaqus based on the average
CPU time per degree of freedom (DOF). The computational cost of the proposed method is 0.021
sec per DOF, 8.70% less than TOSCA (0.023 sec per DOF). On the other hand, the proposed
method significantly eliminates the zig-zag interfaces between solid and void domains. The
amplified body-fitted tetrahedral mesh of the optimized configuration is provided in Fig. 20 for
clarity. It should be emphasized that the 3D examples in this section are plotted using MATLAB or
Abaqus without post-processing.
29
Fig. 20. The optimized configuration of the control arm optimization using (a) the proposed method
and (b) the Abaqus built-in function TOSCA.
The convergence histories of 3D optimization problems using the proposed method are
displayed in Fig. 21. It should be noted that all the optimization problems can reach convergence in
50 iterations. The reason is that we selected a larger ERR to meet volume constraints in fewer
iterations. Meanwhile, using the proposed method, the volume fraction ratio of the optimized
configuration is strictly approaching the prescribed value. Therefore, the stability and effectiveness
of the proposed method can be guaranteed in the 3D condition from the stiffness optimization
30
problems shown in this section. In order to verify the proposed method, all the numerical examples
are solved by the BESO updating scheme. In addition, the nonlinear diffusion term is added to the
objective functional to produce elegant boundaries and avoid numerical instability. The body-fitted
mesh can make use of these advantages to the maximum extent. In the 2D scenario, the
comparisons to the conventional BESO method well demonstrate the robustness of the proposed
BESO method. Setting ERR as 0.05 in the BESO updating scheme allows the 3D optimization
problems to converge in 30 iterations, significantly reducing the computational cost. It can be
observed that the smooth surface of the final 3D design has satisfactory aesthetic characteristics.
For the 3D models with complex load and boundary conditions, the node position, load, boundary
condition, non-design domain, and frozen areas can be written into the inp.-file. The control arm
example demonstrates that the proposed method can be combined with Abaqus to obtain
satisfactory optimized results with better stiffness performance. The coordinates of each node and
body-fitted element vertices information of the final design can also be exported to CAD for
fabrication.
Fig. 21. Convergence histories of the 3D optimization problems: (a) mean compliance value and (b)
volume fraction ratio.
One of the limitations of the proposed method is that the body-fitted mesh is required to be
regenerated according to the updated density field using the BESO updating scheme. To calculate
the computational power required for this procedure, we used MATLAB to profile the execution
time for the functions. After collecting the required data, it can be found that the execution time of
the body-fitted mesh regeneration procedure occupies 21.0%-33.6% of the total execution time in
2D and 3D scenarios. The 3D mesh generation algorithm slightly changes the volume fraction ratio
(error<0.20%). The computational cost can be further decreased by adjusting the parameter in the
mesh regeneration process. However, this will negatively affect the smoothness and precision of the
boundaries. Thus, we need to find a balance between accuracy and efficiency by selecting the
proper parameters to regenerate the mesh.
5. Conclusions
For the first time, this study employs the body-fitted mesh in the BESO method. The mesh
generation scheme originated from [37] and our previous work [42] is improved to match the
dynamically-updated boundaries exactly. The mesh density is controlled by the element distance to
the closest boundary node. Even though void elements are generated, they are excluded from finite
element analysis to save calculation time and computer memory. To complement the sensitivity in
the void region, they are set to zero value. A nonlinear diffusion item is minimized together with the
mean compliance to regularize the optimization problem for checkerboard-free structures.
Numerically, it is achieved by a simple and efficient matrix operation.
Numerical examples illustrate that the body-fitted mesh renders ultra-smooth solid-void
boundaries without post-processing in 2D/3D scenarios. Compared with the conventional BESO
31
and reaction diffusion-based level set methods, the proposed method enables lower mean
compliance for the same optimization problem, fewer iterations to convergence, and shorter
optimization time. The proposed method will present even better performances for fluid and optic
optimization problems, which are more sensitive to boundary smoothness.
Acknowledgments
This project was supported by the Australian Research Council (DP200102190, FL190100014). We
would like to thank the editor and reviewers for their valuable and constructive comments.
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