作者： 許銘修(20021017)；推薦：徐業良(20021018)。
附註：本文發表於第二十六屆全國力學會議。
Integrating 2D topology and shape optimization using the
density contour approach
Abstract
In this research,
a process for integrating 2D structural topology optimization and shape
optimization using the density contour approach is presented. The critical
issue is how to interpret topology optimization result into a representation
that can be used in shape optimization. In the interpretation process presented
in this research, a continuity analysis is first used to filter out the
socalled “porous topology” formed by the elements with intermediate densities
in the topology optimization result. Next, trivial solids and trivial voids are
also filtered out. Finally the blackandwhite finite element topology
optimization result is transferred into a smooth density contour represented by
Bspline curves, which will be used as the initial design of shape
optimization. 14 twodimensional structural topology optimization examples
commonly seen in literature are used to demonstrate the integration process.
Keywords: Topology optimization, shape
optimization, density contour approach
The topology optimization process
In structural
shape optimization, as in most design optimization problems, an initial design
model has to be defined first. During the optimization process, the size and
shape of the structure may be altered, but the topology of the structure is not
changed. If the topology of the initial design model is not optimal, the result
after shape optimization is only the optimal design under this nonoptimal
topology. Therefore, using the topology optimization result as the initial
design of the shape optimization is a natural concept.
After the
important advance by Bendsfe and Kikuchi [1] in the field of topology optimization, many
researchers paid attention to integrating structural topology optimization and
shape optimization since early 1990. Some researchers treat structural design
optimization as a threephase design process, and aim at developing an automated
design process [2, 3, 4]. Phase I is the topology generation process. The
optimal topology of the structure is generated in this phase. Phase II is the
topology interpretation process. In this process various approaches are used to
interpret the topology optimization result. Phase III is the detail design
phase. The shape and size optimization are implemented in phase III.
The approaches
used in the Phase II interpretation process can be roughly divided into three
categories: image interpretation approach [2, 3, 4, 5, 6, 7], density contour
approach [8, 9, 10], and geometric reconstruction approach [11]. The image
interpretation approach uses graphic facilities or computer vision technologies
to represent the boundary of the blackandwhite finite element topology
optimization result. The density contour approach generates the boundaries of
the structure by redistributing densities from the topology optimization
result. In the geometric reconstruction approach, the boundaries are
represented by mathematical geometric reconstruction technique.
In this
research, the concept of threephase design process is adopted. Density contour
approach is used. Figure 1 shows the phase II interpretation process that
connects topology optimization and shape optimization. The normalized material
densities r_{i,final} are obtained from topology optimization. The continuity analysis is
implemented in order to filter out the porous topologies. In the analysis, a density
value r_{continuous} is to be decided so that a continuous topology can be obtained if
only elements whose normalized material densities are equal or greater than r_{continuous} are considered. The normalized material densities of these elements are set to
be 1. On the other hand, the normalized material densities which are less
than r_{continuous} are set to be 0.
Figure 1 The interpretation process
After the
continuity analysis, two “filters” are implemented to filter out the trivial
solids and trivial voids in the topology result. A density redistribution
algorithm then evaluates the nodal density values from the element normalized
material densities. Spline curves are used to generate the boundary curves
using the boundary nodes as control points. Thus, the blackandwhite finite
element topology result is represented by smooth Bspline curves, which will be
used as the initial design of shape optimization.
The continuity of the topology optimization result
Topology
optimization with penalty function attempts to generate the result with
elements whose normalized material densities are either 0 or 1. But practically
the “porous topology” [12], which is formed by elements with intermediate
normalized material densities between 0 and 1, often occurs. From engineering
point of view, the porous topology is meaningless because materials with
intermediate densities do not exist.
Practically the
most convenient way is to force the intermediate normalized material densities
which are equal to or grater than a given threshold value to 1 directly. On the
other hand, the densities that are less than the given threshold value are
forced to be 0. The continuity of the structure should be the most important
consideration in deciding this threshold value. As shown in Figure 2, if the threshold
value is high and a portion of porous topology A is filtered out, a
discontinuous structure will be generated, which may greatly affect the
compliance of the structure. On the other hand, the continuity of the structure
is not affected if porous topology B is filtered out.
Figure 2 The illustration of the porous topology
A structure with
continuous topology is stiffer than discontinuous one, in other words, the
compliance of the structure with continuous topology is lower. Thus, a proper threshold
density value r_{continuous} is to be decided by comparing the compliances of the structure
using different threshold density values.
Figure 3 shows
the compliances of the cantilever beam example using different threshold
density values. The xaxis is the threshold density value starting from
0.1 to 1.0, with increment 0.1. Using the compliance when the threshold density
is 0.1 as the base, the increasing percentages of the compliances are plotted.
The respective topology results are shown in Table 1.
As shown in
Figure 3, there are 6 “jumps” in compliance when the threshold density value
increases. Compare with the figures in Table 1, there is a more than 60%
increase (jump 2) in compliance when the threshold density value increase from
0.5 to 0.6, because a discontinuity of the topology occurs. Other
discontinuities occur at jump 4, 5 and 6. Although there is no discontinuity
occurs in jump 1 and jump 3, but the material in critical connection of the
topology decreases. In this research, if the increase in compliance is more
than 10% at certain threshold density value, this threshold density value is
set to be r_{continuous}. In the cantilever beam example, r_{continuous} is decided to be 0.2.
Figure 3 The compliances of the cantilever beam
example using different threshold densities
Figure 4 shows
the compliances of the suspension arm example using different threshold density
values. The respective topology results are shown in Table 2. As shown in
Figure 4, there is a 24% increase in compliance when the threshold density
value increase from 0.1 to 0.2, because a discontinuity of the topology occurs.
Therefore, r_{continuous} is decided to be 0.1.
Table 1 The topology of different threshold density
_{}

Topology

_{}

Topology

0.1


0.6


0.2


0.7


0.3


0.8


0.4


0.9


0.5


1.0


Figure 4 The compliances of the suspension arm example
using different threshold densities
Table 2 The topology of different threshold
density of suspension arm example
_{}

Topology

_{}

Topology

0.1


0.6


0.2


0.7


0.3


0.8


0.4


0.9


0.5


1.0


Table 2 shows
the r_{continuous} values and the topology results after continuity analysis for the
14 examples discussed in the previous chapter. Note that in example 5, 6, 11,
12, 13, some structural connections in topology optimization results are
deleted after the continuity analysis because they have very small influence on
the compliance of the structure.
Table 3 The r_{continuous} values and topology
results after continuity analysis

The topology optimization result

Compliance analysis

r_{continuous}

The topology results after continuity analysis

1



1.0


2



0.9


3



1.0


4



1.0


5



1.0


6



1.0


7



0.1


8



1.0


9



1.0


10



1.0


11



1.0


12



0.9


13



1.0


14



0.1


The trivial solid and void filter
After the
continuity analysis, it is observed that trivial solids and voids appear in the
topology result. Figure 5 shows the typical trivial solids, such as islands or
small salient, in the cantilever beam example. These trivial solids are useless
in mechanics and will complicate the implementation of the shape optimization.
Thus, the trivial solids should be filter out before implementing shape
optimization.
Trivial voids
are the voids containing few elements whose normalized material densities are
0. Figure 5 and 6 show the trivial voids in the cantilever beam and suspension
arm examples. Similar to trivial solids, trivial voids are useless in
mechanics, will complicate the implementation of the shape optimization, and
should also be filtered out.
Figure 5 The trivial solids and void in cantilever
beam example
Figure 6 The trivial voids in the suspension arm
example
In this
research, a trivial solid and void filter is developed to filter those trivial
topologies. The trivial solid is defined as the solid containing a group of
elements that does not connect with other groups, or connects with other groups
in only one node. Figure 7 is the result of the cantilever beam example after
the trivial solid filter is applied.
Figure 7 The result of the cantilever beam example
after the trivial solid filter
There can be
many voids in the topology optimization result. In this research, if the number
of element contained in a void is less than 1% of the total number of elements
multiplied by r_{,0}, the ratio of the amount of usable material, it is defined as a
trivial void. Figure 4.8 shows the topology result of the suspension arm example
after the trivial void filter. The checkerboard like trivial voids shown in
Figure 6 are filtered out.
Figure 8 The result of suspension arm example
after the trivial void filter
The density contour approach
Finally, we hope
to transfer the topology optimization result into a smooth CAD model that can
be used in shape optimization. As discussed earlier, Kumar and Gossard [8],
Youn and Park [9] and Hsu et al. [10] use the density contour approach to
interpret the topology optimization result to obtain smooth boundary. In their
works, the element densities of topology optimization result are redistributed
into nodal densities, and the nodal density contour is generated at a specified
density value. This contour is then transferred into a smooth CAD representation
(such as Bsplines) of the structure. This research also adopts the density
contour approach, as discussed below.
Youn and Park
[9] evaluate the nodal density (x) by
averaging the normalized material densities of the neighboring elements as
follows:
_{} (1)
where x_{k} is the density of the kth node, W is the number of
neighboring elements at this node, and r_{k,e} is the normalized material density of the
eth neighboring element of the kth node. In two dimensional
topology optimization, almost all nodes have four neighboring elements, and
thus W=4.
Figure 9 shows
the three possibilities of nodal density values using Equation (1), x=0, x=1 and 0<x<1. If the node is inside a void, the normalized material densities
of all neighboring elements are 0, and x=0. On the other hand, if the node is inside a solid, then x=1. For the nodes on the boundary between solid and void, the nodal
density value is between 0 and 1. Figure 9 also shows three possible cases, x=0.25, x=0.5
and x=0.75.
Figure 9 The three types of nodal density value
Figure 10 shows
the nodal densities and nodal density contours of the Case 1 in Figure 9. It is clear that the contour
for x=0.5 properly describes the
boundary of the solid. The nodal density contours are linearly interpolated
from nodal densities. Spline curves, which are commonly used in CAD models, are
generated to approximate the nodal density contour for x=0.5 to represent the structure. The nodes whose nodal density is
0.5 are control points of the spline curves directly. As shown in Figure 10,
for the nodes whose nodal densities are 0.25 or 0.75, the positions of the
corresponding control points are moved inward or outward from the node to the
nodal density contour for x=0.5,
using simple linear interpolation. For the cantilever beam example, figure 11
compares the nodal density contour for x=0.5 and the approximate nodal density contour using spline curves,
which can be used in shape optimization.
Figure 10 The nodal densities and nodal density
contour of Case 1
(a) The nodal density contour
(b) The approximate nodal density contour
Figure 11 Representing the topology result of the
cantilever beam example using Spline curves
Figure 12 shows
the resulting CAD model of the suspension arm example using the approximate
nodal density contour. Table 4 shows the resulting CAD models of the 14
examples.
Figure 12 Representing the topology result of the
suspension arm example using Spline curves
Table 4 Representing the topology result of the 14 examples
using Spline curve

The topology result after continuity analysis

The topology result after filtering trivial voids
out

Representing the topology result using Spline
curve

1




2




3




4




5




6




7




8




9




10




11




12




13




14




Conclusion
A process for
integrating 2D structural topology optimization and shape optimization using
the density contour approach is presented in this research. There are four
steps to interpret the topology optimization result in the interpretation
process. The continuity analysis is first used to filter the porous topology
out. Second, trivial solids and trivial voids are also filtered out. Next, the
nodal density value is evaluated from the normalized material density of
elements by density redistribution algorithm. In the last step of the process,
the blackandwhite finite element topology optimization result is transferred
into a smooth density contour represented by Bspline curves, which will be
used as the initial design of shape optimization. In this research, 14
twodimensional structural topology optimization examples are demonstrated to
show the generality of this integration process. This integration process can
be easily expanded to deal with 3D structures.
References
[1]
Bendsfe, M. P., Kikuchi, N.,
1988, “Generating Optimal Topologies in Structural Design Using Homogenization
Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 71,
pp. 197244.
[2]
Bremicker, M., Chirehdast, M.,
Kikuchi, N., Papalambros, P. Y., 1991, “Integrated Topology and Shape
Optimization in Structural Design,” Mechanics of Structures and Machines,
Vol. 19, pp.551587.
[3]
Chirehdast, M., Gea, HC.,
Kikuchi, K., Papalambros, P. Y., 1994, “Structural Configuration Examples of an
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116, pp. 9971004.
[4]
Lin, C.Y., Chao, L.S., 2000, “Automated
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[5]
Papalambros, P., Chirehdast,
M., 1990, “An Integrated Environment for Structural Configuration Design,” Journal
of Engineering Design, Vol. 1, pp. 7396.
[6]
Bendsfe, M. P., Rodrigues, H. C.,
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[7]
Olhoff, N., Bendsfe, M. P.,
Rasmussen, J., 1991, “On CADIntegrated Structural Topology and Design
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[8]
Kumar, A. V., Gossard, D. C.,
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[9]
Youn, S. K., Park, SH., 1997, “A
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[10]Hsu, Y. L., Hsu, M. S., Chen, C. T., 2001, “Interpreting Results
from Topology Optimization Using Density Contours,” Computers &
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[11]Tang, P.S., Chang, K.H., 2001, “Integration of Topology and Shape
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[12]
Rozvany, G.I.N., Zhou, M.,
Birker, T., 1992, “Generalized Shape Optimization without Homogenization,”
Structural Optimization, Vol. 4, pp. 250252.
利用密度等高線法整合二維結構形式幾何最佳化與型態最佳化
摘要
本研究發展了一套利用密度等高線法整合二維結構形式幾何最佳化與型態最佳化的程序，研究的重點在於如何闡述形式幾何最佳化的結果，使其能為型態最佳化所使用。於闡述形式幾何最佳化的過程中，首先對其進行了連續性分析，用以找到一適合的門檻值，並強制所有元素密度值為0或1。接著，於形式幾何最佳化結果中，對整體結構剛性沒有貢獻之結構以及孔洞將予以去除；最後，將元素密度值完全為0或1的形式幾何最佳化結果，以Bspline曲線描繪出其密度等高線，此Bspline曲線所表示的結構即可作為型態最佳化的初始設計。本研究對14個文獻中常出現的二維結構最佳化範例進行了整合設計，用以展示此整合程序的可行性。
關鍵詞: 形式幾何最佳化, 型態最佳化, 密度等高線法