7th World Congress on Structural and Multidisciplinary Optimization
COEX Seoul, 21 May - 25 May 2007, Korea
A Density Filtering Approach for Topology Optimization
M.Y. Wang1 , S.Y. Wang2 , K.M. Lim2,3
1
Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, NT,
Hong Kong, [email protected]
2
Singapore-MIT Alliance, E4-04-10, 4 Engineering Drive 3, Singapore 117576, [email protected]
3
Department of Mechanical Engineering, National University of Singapore, Singapore 119260,
[email protected]
1. Abstract
Traditional filtering approaches have been widely used for topology optimization using the SIMP (Simple Isotropic Material with Penalization) method to overcome the well-recognized numerical instabilities.
These approaches can be generally effective to obtain a well-posed problem, however, they usually generate final designs with blurry boundaries, which may become an undesirable solution to the original
problem. In this work, a heuristic density filtering approach is proposed to resolving the topology optimization problem more properly. The standard form of the original SIMP method is retained since
the filtering operation is not directly involved. The filtered densities are only introduced in a heuristic
way in a density updating scheme of the optimality criteria method. The present approach can thus be
physically meaningful and mathematically simple. In the present implementation, a simple linear edge
smoothing Gaussian filter in the spatial domain is used to filter the natural density directly. Checkerboard patterns can be eliminated and mesh-dependent designs can be avoided due to the direct edge
smoothing. The 0-1 convergence of a final topology is further improved by using a continuation method
based on a nonlinear bilateral filter at a post-processing step. The two bilateral filter design parameters
are gradually decreased to generate a practical black-and-white design without creating new small holes
to destroy a minimum length scale and a hill-climbing process is allowed to avoid getting stuck at a local
minimum. The proposed approach is illustrated with examples in minimum compliance design and its
efficiency and accuracy are discussed. It is suggested that the present filtering approach be an appealing
alternative to the traditional approaches for topology optimization.
2. Keywords: Topology optimization, filtering approach, checkerboard patterns, black and white design, continuation method.
3. Introduction
Topology optimization is an attractive design tool for obtaining more efficient and/or lighter structures.
An optimum topology can be generally obtained by optimal modifications of holes and connectivities of
the structural design domain, which is actually implemented by redistributing material in an iterative
and systematic manner [1–3]. Generally, topology optimization is one of the most important structural
optimization methods because of its ability in achieving greatest savings [2,4,5]. The topology optimization as a conceptual design tool has the highest importance in the developing process of all structural
optimization methods.
The finite element (FE) based continuum topology optimization as a generalized shape optimization
problem [4] has experienced tremendous progress since the seminal work of Bendsøe and Kikuchi [2]
in 1988. The power-law model or the SIMP method [4, 6–8], in which the material properties can be
expressed in terms of the design variable material density using a simple “power-law” interpolation as
an explicit means to suppress intermediate values of the bulk density, has been generally accepted in
topology optimization [4] because of its computational efficiency and conceptual simplicity. As shown
by Bendsøe and Sigmund [7], the power-law interpolation is physically permissible as long as some
simple conditions on the power are satisfied. However, like most of the other topology optimization
methods, the SIMP method does not directly resolve the problem of non-existence of solutions (illposedness) [9] and thus numerical instabilites may occur. One of the most serious numerical instabilities
is the occurrence of checkerboard patterns in the final solutions. Other numerical instabilities may include
mesh-dependency and local minima. Various approaches have been proposed to relieve these problems,
including adding slope constraints [10,11] or move limit constraints [12] or perimeter controls [13], using
higher order or non-conforming finite elements [14], and employing filters for suppressing the chattering
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solutions [4, 7–9, 15–17].
Among all these approaches to resolving the problem of ill-posedness of the SIMP method, a filtering
technique is most widely used [9, 17] since filtering techniques have been quite popular and successful in
various domains of applications as a numerical method to ensure regularity or existence of solutions to
an ill-posed engineering problem [17, 18]. Sigmund [19] first introduced a filtering approach for topology
optimization using the SIMP method. In his approach, a heuristic sensitivity filter was used to modify the
design sensitivity of a specific element, making it dependent on a weighted average over its neighboring
elements. It should be noted that not only the sensitivities but also the the material densities are included
in the weighted average. Because of the involvement of the densities, this sensitivity filter can distinct
itself from others and usually results in checkerboard pattern-free and mesh-independent optimization
results with moderate computational cost. It has been extensively used in the SIMP method [8, 9].
However, this is a heuristic method since the sensitivities are not consistent with the primal SIMP
model and the optimization problem may not be well-posed mathematically. Filtering the material
densities was first developed by Bruns and Tortorelli [20] and analyzed in detail by Bourdin [18], who
addressed the issue of existence of solutions by applying a more general filtering theory with a relationship
between the density and the material properties. It was shown that filtering the density can also lead to
checkerboard-free and mesh-independent designs [18, 20]. Nevertheless, this approach may be physically
less meaningful than the original SIMP method since the problem definition was modified due to the
introduction of density measure [9] and thus the optimization may be limited within a subspace of the
solutions [9]. It should be noted that grey regions containing intermediate densities along the solid-void
boundary of the final design can be quite apparent using either the sensitivity or the density filtering
approaches. This widely observed phenomenon may cause difficulties in boundary identification and
design realization in a post-processing step which is necessary for shape recovery from the optimization
solution [9]. Furthermore, the topology optimization problem may be inappropriately resolved if the
original objective is to generate distinct solids and voids. In order to obtain a discrete black-andwhite optimal design, many approaches have been proposed in the literature [21–26]. However, these
approaches may be inappropriate if both the computational cost and the mathematical rigorousness are
taken into account. More recently, Wang et al. [27] introduced the anisotropic diffusion [28], a popular
edge-preserving nonlinear filtering technique, into topology optimization using the SIMP method to
obtain a good quality black-and-white design. Although anisotropic diffusion is a popular tool for
edge-preserving filtering [29], its discrete diffusion nature makes it slow and computationally expensive
[27]. Wang and Wang [17] presented a more efficient nonlinear bilateral filtering approach for topology
optimization using the SIMP method. Practical black-and-white final designs without checkerboard
patterns were illustrated. However, an efficient and robust method to reach the 0-1 convergence and
ensure a minimum length scale was not developed.
The objective of the present study is to present an alternative heuristic density filtering approach for
discrete topology optimization using the SIMP method. In the present implementation, natural densities
are used in evaluating the objective and constraint functions to preserve the physical meaning of the
standard SIMP method, while filtered densities are only introduced in a density updating scheme in the
optimality criteria. A linear Gaussian filter is used to achieve checkerboard-free and mesh-independent
designs. A continuation method based on the bilateral filter [17] is developed to obtain practical blackand-white designs in a post-processing step. Numerical examples in minimum compliance design are
used to show the efficiency and accuracy of the present method.
4. Topology Optimization Using the SIMP Method
The topology optimization problem as a generalized shape optimization problem of finding the optimal
material distribution [4] is considered. It is confined in a fixed design reference domain, or design domain,
Ω ∈ Rd (d = 2 or 3) to allow for the applied loads and boundary conditions [9]. The FE discretized design
domain Ω can be taken as a digital image and each element a pixel or voxel whose color is represented by
its density ρ (in a gray scale, white is void and black is solid material). The geometric representation of
a structure corresponds to a black-and-white raster representation of the geometry with pixels or voxels
given by the FE discretization and the material properties are modeled as a function of the material
density ρ, ρ ∈ {0, 1}, in which 0 represents void and 1 solid material. Hence, the original topology
optimization problem is a distributed, discrete valued design problem (a 0-1 problem) [9].
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In the present study, to simplify the analysis procedure, we consider the classical minimum compliance
topology optimization problem in linear elasticity subject to the applied body force f in Ω and the surface
traction forces t on the traction part Γt of the smooth boundary ∂ Ω, which can be defined as
min ` (ρ, u)
s.t. : aρ (u, v) = ` (ρ, v)
ρ ∈ {0, 1}
Z
∀ v ∈ U (equilibrium)
(1)
ρ d Ω = V 6 γV0
(volume constraint)
Ω
where
Z
Z
fT u d Ω +
` (ρ, u) =
Ω
tT u d s
(2)
Γt
Z
aρ (u, v) =
ρ
Ω
∂ui
∂vk
Eijkl
dΩ
∂xj
∂xl
(3)
and u is the displacement field that defines the equilibrium of the elastic body, v the kinematically admissible virtual displacement field, U the set of kinematically admissible displacements, V the allowable
volume of solid material (V > 0), V0 the total volume of the fixed reference domain, and γ the volume
fraction.
In the design domain Ω, according to the SIMP method in the power-law interpolation of material
properties [6], the material properties are modeled as a function of material density ρ, which is a continuous rather than discrete design variable, raised to some power p (p > 1) times the material properties
of solid material, i.e., E(ρ) = ρp E0 , where E0 is the Young’s modulus of a given solid material and
E(ρ) the effective Young’s modulus. The standard SIMP version of the minimum compliance topology
optimization problem in Eq. (1) can be expressed as follows:
min ` (ρ, u)
s.t. : aρ (u, v) = ` (ρ, v), ∀ v ∈ U
0<ρ6ρ6ρ
Z
(4)
ρ d Ω 6 γV0
Ω
where
Z
Z
fT u d Ω +
` (ρ, u) =
tT u d s
(5)
∂ui
∂vk
Eijkl
dΩ
∂xj
∂xl
(6)
Ω
Γt
Z
ρp
aρ (u, v) =
Ω
in which ρ and ρ are the lower and upper limits on the material density ρ. In this standard SIMP method,
the original discrete 0-1 topology optimization problem is converted into a continuous optimization
problem with intermediate material densities.
Using the finite element method, the discrete form of Eq. (4) can be written as
min
s.t. :
J(ρ) = UT KU =
N
X
N
X
(ρe )p uTe Ke ue
e=1
(ρe )p Ke ue = F
(7)
e=1
N
X
ρe ve = γV0
e=1
0 < ρ 6 ρe 6 ρ
e = 1, . . . , N
where ρ is the design variable vector, J(ρ) the objective function (compliance), U the global displacement
vector, K the global stiffness matrix, ue and Ke the element displacement vector and stiffness matrix,
respectively, ρe the unfiltered element density, N the total number of elements used to discretize the
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design domain Ω, ve the element volume, and F the global force vector. Hence, the corresponding
sensitivity of the objective function J(ρ) can be written [9] as
∂J
∂K
= −UT
U = −p (ρi )p−1 uTi Ki ui
∂ρi
∂ρi
(8)
In order to overcome the problems of numerical instabilities, a filtering technique can be introduced
in the SIMP method. In the present study, a heuristic density filtering approached is presented as an
alternative approach to the popular ones.
4.1. Present Filtering Approach
The topology optimization problem described in Eqs. (7) and (8) can be solved by many approaches
[9]. In the present study, the efficient OC (Optimality Criteria) method proposed by Bendsøe [8, 9, 30]
is adopted, but a density filtering operation is introduced to overcome the numerical instabilities. As a
result, the present heuristic updating scheme for the design variable ρ can be formulated as follows:

 max(ρ, ρee − m) if ρe (Be )η 6 max(ρ, ρee − m)
new
ρee (Be )η
if max(ρ, ρee − m) < ρe (Be )η < max(ρ, ρee + m)
ρe =
(9)

min(ρ, ρee + m) if ρe (Be )η > max(ρ, ρee + m)
is the updated element density, ρee the filtered element density, m a move limit, η a numerical
where ρnew
e
damping coefficient, and Be can be found from the optimality condition [8, 30], in which the sensitivity
∂J
∂ρe and a Lagrangian multiplier λ, which can be obtained by a bi-sectioning algorithm [8], are involved
as
Be = −
∂J
∂ρe
(10)
λve
As shown in Eq. (9), in the present implementation, filtered element density ρee , rather than the
natural element density ρe , is used in the heuristic density updating scheme of the OC method. Since
the standard SIMP method using the power-law material interpolation model keeps untouched, the
simplicity, efficiency and physical meaning of the standard SIMP method can be retained. In this
sense, the present heuristic density filtering approach can be consistent with a number of techniques to
regularize the topology optimization problem using the SIMP method, such as the mesh-independent
sensitivity filtering method [19], the density slope control methods [10, 11], and the move-limit filtering
method [12]. Furthermore, the present heuristic density filtering approach is basically different from the
those approaches filtering the densities [18, 20], which introduce a density measure to modify the SIMP
method.
The design of a filter is of crucial importance to overcome the numerical instabilities and to obtain black-and-white topologies. In the present study, a linear Gaussian filter is adopted to generate
checkerboard-free and mesh-independent optimal topologies, while a continuation method based on a
nonlinear bilateral filter [17] is used to drive 0-1 convergence of the optimal topologies. In the Gaussian
filter, the relationship between the filtered density and natural density can be written as
X
ρe(x) = S(ρ(x)) = k −1 (x)
wd (kq − xk) ρ (q)
(11)
q∈N (x)
where
k(x) =
X
wd (kq − xk)
(12)
q∈N (x)
wd (z) = e−(1/2)(z/σd )
2
(13)
in which N (x) is the neighborhood of the current element x, kq − xk the Euclidean distance between
element q and element x, and σd the geometric spread of the Gaussian kernel function wd . In the
present implementation, the neighborhood N (x), defined by a filter window size rmin , is related with the
geometric spread σd as follows:
rmin = 2.1σd
(14)
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such that only the relatively near elements are involved in the filtered densities and the computational
cost can be reduced. Furthermore, a bilateral filter can be defined as
X
ρe(x) = S(ρ(x)) = k −1 (x)
wd (kq − xk) wr ( | ρ(q) − ρ(x)| ) ρ (q)
(15)
q∈N (x)
where
k(x) =
X
wd (kq − xk) wr ( | ρ(q) − ρ(x)| )
(16)
q∈N (x)
wr (z) = e−(1/2)(z/σr )
2
(17)
in which σr is the photometric spread [31]. Bilateral filtering was developed by Tomasi and Manduchi in
1998 [31] as an alternative nonlinear filter to the popular anisotropic diffusion [28] in image processing.
It is a simple and fast nonlinear filtering technique to perform good-quality edge-preserving smoothing
in a single pass and produce PDE (Partial Differential Equation)-like (similar or possibly better [31,32])
results without an iterative solver or instability risks [29, 31–33]. In this study, bilateral filtering is
adopted in the continuation method as a post-processing step for the present heuristic density filtering
approach to achieve practical black-and-white designs.
4.3. A Continuation Method
Continuation methods have been well adopted in the optimization field [9]. Unlike the monotone
line search methods, continuation methods allows uphill steps to be taken and can generate good initial
guesses not only for the optimization variables but also for the design parameters. More importantly,
continuation methods may globalize trivially without incorporating any other globalization strategy. In
this study, the continuation method is further developed for the present density filtering approach as a
post-processing step to generate practical black-and-white designs without changing the topology. The
present continuation method is based on the bilateral filter defined in Eq. (15) and starts with a large
value σr of the photometric spread σr and gradually decreases σr by a fixed small amount of ∆σr until
convergence or a given lower bound σr is reached or the topology is changed. For each σr value, the
present continuation method is to start with a large value σd of the geometric spread σd to perform a separate topology optimization and to gradually decrease σd by a fixed small amount of ∆σd to repeat the
same optimization procedure using the updated design parameters σd and σr and the optimal solution at
the previous step as an initial guess until convergence or a given lower bound σd is reached or the topology is changed. Using this continuation method, the 0-1 convergence of the optimal topologies can be
greatly improved due to the excellent edge preserving smoothing effect of a nonlinear bilateral filter [31].
5. Examples
Numerical examples are used to illustrate the efficiency and accuracy of the present filtering approach
for topology optimization. Unless otherwise stated, all the units are consistent and the following parameters are assumed as: ρ = 0.001, ρ = 1, E0 = 1, ν = 0.3, and p = 3. The optimization process is
terminated when the relative difference between two successive objective function values is less than a
small value ² = 10−6 or a given maximum number of iterations has been reached. The FE analysis is
based on the bilinear rectangular elements and all the comparisons on the objective function values are
based on the final results when the iteration is terminated.
The minimum compliance design problem of a MBB beam is shown in Fig. 1, which is loaded with a
concentrated vertical force of P at the centre of the top edge and is supported on rollers at the bottomright corner and on fixed supports at the bottom-left corner. The basic parameters are assumed to be
E0 = 1, ν = 0.3, L = 3, H = 1, thickness t = 1.0, load P = 1 and a volume fraction of 0.5.
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P
L
L
∂Ω
Ω
Designdomain
H
Figure 1: Definition of a MBB beam topology optimization problem
The capacity of the present density filtering approach in handling the checkerboard patterns is demonstrated in Fig. 2. The initial design with significant checkerboard patterns, which is obtained from the
SIMP method without filtering, is effectively eliminated due to the strong edge smoothing effect of the
present Gaussian filter. Hence, the final design can be checkerboard-free.
(a) Initial design with checkerboard patterns
(b) Final design
Figure 2: Checkerboard-free optimal topology using the present filtering approach (mesh 300 × 50)
The existence of solutions is demonstrated in Fig. 3. It can be seen that the final designs can be
mesh-independent so that the existence of solutions can be guaranteed and the optimization problem
can become well-posed. Because of the relatively large filter window size caused by the geometric spread,
the topology can be quite simple but the boundary can be too blurry. The 0-1 convergence of the final
topology can be improved by applying the present continuation method. Figure 4 shows that the final
topology becomes a practical black-and-white design after using the present continuation method due
to the edge preserving smoothing of a nonlinear bilateral filter.
The existence of solutions is further demonstrated in Fig. 5, in which the filter window size is reduced.
The topology becomes more complicated due to the fact that the minimum length scale on the size of
holes is also reduced. Again, the final designs can be mesh-independent provided the mesh is fine enough.
The mesh 180 × 30 becomes too coarse to generate the same topology as the finer meshes. As expected,
mesh independency may require that the mesh be fine enough. Because of the relatively small filter
window size, the final topology can be with much distinct boundary. The 0-1 convergence of the final
topology is further improved by applying the present continuation method. Figure 6 shows that the final
topology may become a practical black-and-white design.
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(a) Mesh 180 × 30
(b) Mesh 240 × 40
(c) Mesh 300 × 50
(d) Mesh 360 × 60
Figure 3: Mesh-independent design using the present filtering approach (σd = 0.0833)
(a) Mesh 180 × 30
(b) Mesh 240 × 40
(c) Mesh 300 × 50
(d) Mesh 360 × 60
Figure 4: Practical black-and-white design using the present continuation method
6. Conclusions
A heuristic density filtering approach is proposed to overcoming the numerical instabilities and achieving
better 0-1 convergence of the final designs. The filtered densities are introduced in a density updating
scheme of the optimality criteria method. The standard form of the original SIMP method remains unchanged so that the present approach may be physically meaningful and mathematically simple. A linear
edge smoothing Gaussian filter is used in the filtering operation. Checkerboard patterns can be eliminated and mesh-dependent designs can be avoided and better 0-1 convergence can be obtained because
of the direct edge smoothing on the material densities. The 0-1 convergence of a final topology is further
improved by using a continuation method based on a nonlinear bilateral filter at a post-processing step.
A practical black-and-white design can be generated due to the edge preserving smoothing of the bilateral filter. The probability of converging to a local minimum can be avoided by allowing hill-climbing
steps. The proposed approach is illustrated with examples in minimum compliance design and its efficiency and accuracy are discussed. The present filtering approach can be an effective alternative to the
traditional approaches for topology optimization.
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Figure 5: Mesh-independent design using the present filtering approach (σd = 0.0267)
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