Author: MingHsiu Hsu, YehLiang Hsu (20040902);
recommended: YehLiang Hsu (20050118).
Note: This paper is published in Computers and Structures, Vol.83,
Issues 45, January 2005, p. 327337.
Interpreting threedimensional
structural topology optimization results
Abstract
Topology
optimization result is usually a gray level image in discrete finite elements,
which is hard to interpret from a design point of view. It is especially
difficult to interpret threedimensional topology optimization result. Although
there are many techniques for interpreting topology optimization results, most
of them only deal with twodimensional problems, and some of the interpreting
techniques must be performed manually. This paper presents an automated process
for interpreting threedimensional topology optimization result into a smooth
CAD representation. A tuning process is employed before the interpretation
process to improve the quality of the topology optimization result. In the
tuning process, a continuity analysis is first used to transform the porous
topology formed by the elements with intermediate densities in the topology
optimization result into clear 01 elements. Next, trivial solids and trivial
voids, which are useless in mechanics but will complicate the implementation of
the interpretation process, are filtered out. After the tuning process, the
threedimensional structure is divided into representative crosssections. On
each crosssection, a density redistribution algorithm transfers the
blackandwhite finite element topology optimization result into a smooth
density contour represented by Bspline curves on each crosssection. A
threedimensional CAD model is obtained by sweeping through these cross
sections.
Keywords: topology optimization,
threedimensional structures, density distribution method, density contour,
topology interpretation.
1.
Introduction
Topology
optimization is the technique that finds the optimal layout of the structure
within a specified design domain. The possible constraints in the topology
optimization problems are the usable volume or material, and design restrictions
(or geometry constraints, such as holes or nondesign regions) in the design
domain, under certain applied loads and boundary conditions.
Bendsfe and Kikuchi
presented the homogenization method for topology optimization in 1988 [Bendsfe and Kikuchi,
1988], which is a milestone for topology optimization. The homogenization
method was based on the assumption of a microstructure in which the properties
are homogenized. Two other approaches were proposed in the early 90’s: the density distribution method
[Mlejnek, 1992, Yang and Chuang, 1994], in which the material density of each
element was selected as the design variable, and the evolutionary structural
optimization (ESO) [Xie and Steven, 1993], which gradually removed the lowstressed
elements to achieve the optimal design. Rozvany provided a complete survey on topology
optimization [Rozvany, 2001]. The density distribution method seems to be the
most convenient method and is used here to obtain topology optimization result.
After the
important advance by Bendsfe and Kikuchi in the field of topology optimization, many
researchers paid attention to integrating structural topology optimization and
shape optimization since early 1990s. Some researchers treat structural design
optimization as a threephase design process, and aim at developing an automated
design process [Bremicker et al., 1991, Chirehdast et al., 1994, Lin and Chao,
2000]. Phase I is the topology generation process, in which the optimal
topology of the structure is generated. In the Phase II topology interpretation
process, various approaches are used to interpret the topology optimization
result. Phase III is the detailed design phase, in which the shape and size
optimization are implemented.
The approaches
used in the Phase II interpretation process can be roughly divided into three
categories: image interpretation approach [Papalambros and Chirehdast, 1990,
Bendsfe and Rodrigues, 1991, Olhoff et al., 1991, Bremicker et al., 1991,
Chirehdast et al., 1994, Lin and Chao, 2000], density contour approach [Kumar
and Gossard, 1996, Youn and Park, 1997, Hsu et al., 2001], and geometric
reconstruction approach [Tang and Chang, 2001]. 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 the mathematical geometric
reconstruction technique.
Topology
optimization result is usually a gray level image in discrete finite elements,
which is hard to interpret from a design point of view. It is especially
difficult to interpret threedimensional topology optimization result. Although
there are many interpretation techniques available in the literature, most of
them only deal with the twodimensional problems. For example, the image
interpretation approach commonly seen in the literature can only represent the
boundaries of the twodimensional gray level topology result. Moreover, some of
the interpreting techniques must be performed manually. Thus, a technique that
can handle both two and threedimensional problems and can be performed
without human intervention is desired.
The concept of
using density contour to interpret the topology optimization results was first
introduced by Kumar and Gossard in 1996. In the density contour approach, the
boundaries of topology optimization result are represented by nodal density contours,
which are computed by material densities. Unlike the image interpretation
approach, the density contour approach has the potential of expanding to
threedimensional structural design.
This paper
presents an automated process for interpreting threedimensional topology
optimization results from the finite element representation into a smooth CAD
model that can be used in later design stages. As shown in Figure 1, the
density distribution method, or the socall Simple Isotropic Material with
Penalization (SIMP) method, is used in the topology optimization. A tuning
process is employed before the interpretation process to improve the quality of
the topology optimization result. In the tuning process, a continuity analysis
is first used to transform the porous topology formed by the elements with
intermediate densities in the topology optimization result into clear 01
elements. Referring to Figure 1, the normalized material densities (_{}) are obtained from topology optimization. In the continuity
analysis, a density value _{} is to be decided
so that a continuous topology can be obtained if only the elements whose normalized
material densities are equal to or greater than _{} 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_{} are set to be 0. Next, trivial solids and trivial voids,
which are useless in mechanics but will complicate the implementation of the
interpretation process, are filtered out.
After the tuning
process, the threedimensional structure is divided into representative crosssections.
A density redistribution algorithm then evaluates the nodal density values from
the element normalized material densities of elements on each cross section.
Spline curves are used to generate the boundary curves using the boundary nodes
as control points on each cross section. Till this step, the whole procedure is
fully automated and is applicable to both two and threedimensional structural
design. A threedimensional CAD model is obtained by sweeping through the cross
sections in any CAD software. In this paper, twodimensional examples are first
used to describe this process, and then the process is directly extended to
threedimensional structures.
Figure 1. The threedimensional interpretation
process
2.
The continuity analysis 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 the “porous
topology” [Rozvany et al., 1992], which is formed by elements with intermediate
normalized material densities between 0 and 1, often occurs practically.
Practically the
most convenient way is to force the intermediate normalized material densities
which are equal to or greater than a given threshold value of 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 of 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 discarded, 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 the porous topology B is filtered out.
Figure 2. The illustration of the porous topology
A structure with
continuous topology is stiffer than the discontinuous one, in other words, the
compliance of the structure with continuous topology is lower. Thus, a proper threshold
density value _{} has 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 of 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.
Figure 3. The compliances of the cantilever beam
example using different threshold densities
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


As shown in
Figure 3, there are 6 “jumps” in compliance when the threshold density value
increases. Compared with the figures in Table 1, there is more than 60%
increase (jump 2) in compliance upon increasing when the threshold density
value from 0.5 to 0.6, because a discontinuity of the topology occurs
(indicated by a circle). Other discontinuities result in “jumps” in the compliance
value. In this research, if the increase in compliance is more than 10% at
certain threshold density value, the threshold density value is set to be _{}. In the cantilever beam example, _{} is decided to be
0.2.
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 increases from 0.1 to 0.2, because a discontinuity of the topology
occurs. Therefore, _{} is decided to be
0.1.
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


3.
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 interpretation
process. Thus, the trivial solids should be filtered out before implementing
the interpretation process.
Figure 5. The trivial solids and void in
cantilever beam example
Trivial voids
are the voids containing few elements whose normalized material densities are
0. Figures 5 and 6 show the trivial voids in the examples of cantilever beam
and suspension arm. Similar to trivial solids, trivial voids are useless in
mechanics, as it complicates the interpretation of topology optimization
result, which should be filtered out.
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 do not connect with other groups, or connect with other groups in
only one node. Figure 7 is the result of cantilever beam example after applying
the trivial solid filter.
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 elements contained in a void is less than 1% of the total number of elements
multiplied by _{}, the ratio of the amount of usable material, it is defined
as a trivial void. Figure 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
4.
The density contour approach
We hope to
transfer the topology optimization result into a smooth CAD model that can be
used in later design stages. As discussed earlier, Kumar and Gossard [1996],
Youn and Park [1997] and Hsu et al. [2001] use the density contour approach to
interpret the topology optimization result to obtain a 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
[1997] evaluate the nodal density _{} by averaging the
normalized material densities of the neighboring elements as follows:
_{} (1)
where _{} is the density of
the kth node, W is the number of neighboring elements at this node, and
_{} 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), _{}, _{} and _{}. If the node is inside a void, the normalized material densities
of all neighboring elements are 0 and _{}. On the other hand, if the node is inside a solid, then _{}. 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, _{}, _{} and _{}.
Figure 9. The three types of nodal density value
Figure 10 shows
the nodal densities and nodal density contours of Case 1 in Figure 9. It is clear that the contour for _{} 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 _{} to represent the
structure. The nodes whose nodal density is 0.5 are the 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 _{}, using simple linear interpolation. For the cantilever beam
example, Figure 11 compares the nodal density contour for _{} and the
approximate nodal density contour using spline curves, which can be used in
later design stages. Figure 12 shows the resulting CAD model of the suspension
arm example using the approximate nodal density contour.
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. Representing the topology result of the
suspension arm example using Spline curves
5.
Interpreting the threedimensional
topology optimization result
The
interpretation process presented above can be directly extended to
threedimensional structures. In the interpretation process, first the same continuity
analysis, trivial solid and void filter are applied as in twodimensional
interpretation process to obtain a continuous and clear threedimensional
topology optimization result.
The trivial
solid for threedimensional topology optimization result is defined as the
solid containing a group of elements that does not connect with other groups,
or connects with other groups in less than two nodes. Figure 13 shows the
examples for solid elements that are connected and not connected. This
definition follows the criterion presented by Harzheim and Graf [2002]. The definition
of trivial void for the threedimensional topology optimization result is the
same as the twodimensional topology optimization result. If the number of
elements contained in a void is less than 1% of the total number of elements
multiplied by _{}, the ratio of the amount of usable material, it is defined
as a trivial void.
Figure 13. Examples for solid elements, connected
and not connected [Harzheim, L. and Graf, G., 2002]
How to present a
threedimensional structure is an important issue for interpreting
threedimensional topology optimization result. Hsu et al. [2001] used
crosssections to interpret the threedimensional topology optimization result.
Tang and Chang [2001] also used a series of preselected crosssections to
reconstruct the threedimensional topology optimization result. Representative
crosssections are used to interpret the threedimensional topology
optimization result in this research.
Referring to
Figure 1, in the “representative crosssection selection operation” developed
in this research, the selection of the crosssections can be along x, y,
or z direction, and the distance between crosssections can be
nonuniform. Table 3 shows the topology optimization result and boundaries of
selected crosssections of a cantilever beam example. There are 7
representative crosssections selected along the y direction in this
example. In each representative crosssection, the topology result becomes a
twodimensional problem. The density redistribution algorithm and density
contour approach presented above can be directly applied to obtain the boundary
of the section. Any CAD software can be used to construct the threedimensional
CAD model by sweeping through the boundaries of the representative
crosssections. Figure 14 is the reconstructed threedimensional CAD model of
the cantilever beam example.
Table 3. Topology results and boundaries of
selected sections of a cantilever beam example (selected along y
direction)


Topology result of sections

Topology boundaries of sections


Topology result of sections

Topology boundaries of sections

1



5



2



6



3



7



4






Figure 14. The reconstructed threedimensional CAD
model of Example 1
It should be
noted that the selected number of representative crosssection is related to
the resolution of threedimensional CAD model. More representative
crosssections are needed when the resolution of threedimensional model is
required. On the other hand, the direction of selected representative crosssection
will affect the efficiency in reconstructing the threedimensional CAD model.
Table 4 lists 6 threedimensional structures commonly seen in structural
topology optimization literature, their topology optimization results, and
their interpretation results using the process presented in this research.
Table 4. Topology optimization results and the
reconstructed CAD models
Examples

Topology optimization results

Reconstructed CAD models

[Olhoff et al., 1998;
Jacobsen et al., 1998; Beckers, M., 1999]



[Olhoff et al., 1998,
Jacobsen et al., 1998]



[Olhoff et al., 1998,
Jacobsen et al., 1998]



[Olhoff et al., 1998,
Jacobsen et al., 1998, Beckers, M., 1999]



[Díaz, A., Lipton, R., 1997]



[Díaz, A., Lipton, R., 1997]



6.
Conclusion
This paper
presents an automated process for interpreting threedimensional topology
optimization result from a gray level image in discrete finite elements into a
smooth CAD representation. The complexity of the interpretation process depends
heavily on the quality of the topology optimization result. Therefore in this
research, a tuning process is employed before the interpretation process.
However, there are still limitations to the process if the topology
optimization result is very poor:
(1)
The continuity analysis
discussed in this paper will not work well if there is a large region full of
elements with intermediate densities. Adding penalty function in the topology
optimization process is a common strategy in order to get a clear topology
optimization result.
(2)
The filters for trivial solids
and trivial voids will not work well if the socalled “checkerboard pattern”
appears in the topology optimization results. A common strategy to resolve the
checkerboard problem is to use higher order finite elements.
(3)
The topology optimization model
with finer finite element mesh will increase the complexity of the resulting
topology, the socalled “meshdependence problem” [Sigmund and Petersson,
1998]. Too many unnecessary details will complicate the interpretation process
but not necessarily reduces the compliance of the structure. Therefore, in the
topology optimization phase, we should first decide the proper number of
elements in the finite element mesh.
7.
Reference
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.
Bendsfe, M. P.,
Rodrigues, H. C., 1991, “Integrated Topology and Boundary Shape Optimization of
2D Solid,” Computer Methods in Applied Mechanics and Engineering, Vol.
87, pp.1534.
Beckers, M.,
1999, “Topology optimization using a dual method with discretevariables,” Structural
Optimization, Vol. 17, pp. 1424.
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.
Chirehdast, M.,
Gea, HC., Kikuchi, K., Papalambros, P. Y., 1994, “Structural Configuration
Examples of an Integrated Optimal Design Process,” Journal of Mechanical
Design, Vol. 116, pp. 9971004.
Díaz, A.,
Lipton, R., 1997, “Optimal material layout for 3D elastic structures,” Structural
Optimization, Vol. 13, pp. 6064.
Harzheim, L.,
Graf, G., 2002, “TopShape: An attempt to create design proposals including
manufacturing constraints,” International Journal of Vehicle Design,
Vol. 28, pp. 389409.
Hsu, Y. L., Hsu,
M. S., Chen, C. T., 2001, “Interpreting Results from Topology Optimization
Using Density Contours,” Computers & Structures, Vol. 79, pp.
10491058.
Kumar, A. V.,
Gossard, D. C., 1996, “Synthesis of Optimal Shape and Topology of Structures,” Journal
of Mechanical Design, Vol. 118, pp. 6874.
Lin, C.Y.,
Chao, L.S., 2000, “Automated Image Interpretation for Integrated Topology and
Shape Optimization,” Structural and Multidisciplinary Optimization, Vol.
20, pp.125137.
Mlejnek, H. P.,
1992, “Some Aspects of the Genesis of Structures,” Structural Optimization,
Vol. 5, pp. 6469.
Jacobsen, J. B.,
Olhoff, N., Rønholt, E., 1998, “Generalized shape optimization of
threedimensional structures using materials with optimum microstructures,” Mechanics
of Materials, Vol. 28, pp. 207225.
Olhoff, N., Bendsfe, M. P.,
Rasmussen, J., 1991, “On CADIntegrated Structural Topology and Design
Optimization,” Computer Methods in Applied Mechanics and Engineering,
Vol. 89, pp. 259279.
Olhoff, N., Rønholt,
E., Scheel, J., 1998, “Topology optimization of threedimensional structures
using optimum microstructures,” Structural Optimization, Vol. 16, pp.
118.
Papalambros, P.,
Chirehdast, M., 1990, “An Integrated Environment for Structural Configuration
Design,” Journal of Engineering Design, Vol. 1, pp. 7396.
Rozvany, G.I.N.,
Zhou, M., Birker, T., 1992, “Generalized Shape Optimization without
Homogenization,” Structural Optimization, Vol. 4, pp. 250252.
Sigmund, O.,
Petersson, J., 1998, “Numerical Instabilities in Topology Optimization: A
Survey on Procedures Dealing with Checkerboards, Meshdependencies and Local
Minima,” Structural Optimization, Vol. 16, pp. 6875.
Tang, P.S.,
Chang, K.H., 2001, “Integration of Topology and Shape Optimization for Design
of Structural Components,” Structural and Multidisciplinary Optimization,
Vol. 22, pp.6582.
Xie. Y. M.,
Steven, G. P., 1993, “A Simple Evolutionary Procedure for Structural
Optimization,” Computers & Structures, Vol. 49, pp. 885896.
Youn, S. K.,
Park, SH., 1997, “A Study on the Shape Extraction Process in the Structural
Topology Optimization Using Homogenized Material,” Computers &
Structures, Vol. 62, No. 3, pp. 527538.