Author: MingHsiu Hsu, YehLiang Hsu(20040319);
recommended: YehLiang Hsu (20041027).
Note: This paper is published in Engineering Optimization, Vol.
37, No. 1, January 2005, p. 83–102.
Generalization of two and
threedimensional structural topology
optimization
Abstract
This paper
explores the fundamental issues on the quality of structural topology
optimization results, and presents a generalized topology optimization process.
In the generalized process, nonrectangular design domains with geometrical
constraints can be accepted, and the use of an automatic mesh generator to mesh
the design domain is allowed. The proper number of elements in the design
domains to avoid the meshdependence problem is also suggested. Higher order
elements are used to deal with the checkerboard problem, and a twostage
penalty function method is proposed for the topology optimization. Finally a
continuity analysis is used to deal with the porous topology, and two filters
are implemented to filter out the trivial solids and voids. The whole process
is generalized to two and three dimensional structures and can be fully
automated. Fourteen twodimensional and six threedimensional examples are used
to demonstrate the effectiveness of this process.
Keywords: topology optimization, structure optimization
1.
Introduction
Topology
optimization is a welldeveloped field. The purpose is to obtain the optimal
layout of structural components to achieve a predetermined performance goal.
The objective function and constraints commonly used in topology optimization
research are to minimize the compliance of the structure under a given loading
condition, subject to the amount of usable material. The same objective
function and constraints are also used here.
Bendsfe and Kikuchi
developed the homogenization method for topology optimization in 1988 [1],
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 for topology optimization in
the early 90’s: the
density distribution method [2, 3], in which the material density of each
element was selected as the design variable, and the evolutionary structural
optimization technique (ESO) [4], which gradually removed the lowstressed
elements to achieve the optimal design. Rozvany provided a complete survey on topology
optimization [5]. All three methods have their own advantages and have many
followers. The density distribution method, the socalled Simple Isotropic
Material with Penalization (SIMP) method, seems to be the most convenient
method and is used here to obtain topology optimization result.
Figure 1 shows a
general topology optimization process. The process begins with a userdefined
design domain _{}. Figure 2 shows a cantilever beam example that has a
rectangular design domain. The design domain may also have geometrical
constraints, namely, void regions and fixed regions as shown in the suspension
arm example in Figure 3. A general design processor, for example,
Pro/Engineering or AutoCAD, can be used to construct the design domain and to
create a corresponding CAD model for topology optimization.
Figure 1. A general topology optimization process
Figure 2. The design domain and boundary
conditions of the cantilever beam example
Figure 3. The suspension arm example with hole and
support regions in design domain
An analysis
processor, for example, ANSYS or NASTRAN, generates the finite element model
from the CAD model. The boundary conditions, for example, displacement
constraints on boundaries _{}, loads and body force _{}, and surface force _{}, are applied to the finite element model.
The user defines
the amount of usable material _{}, and selects true material properties including Young’s
modulus _{}, material density _{}, and Poisson’s ratio _{}. The design variables in the material density distribution
method are the “normalized material densities” r_{i,j} (the normalized material density of the ith element at the jth
iteration). Note that 0_{}r_{i,j}_{}1, and r_{i,j} = 0 denotes that the area of
this element contains no material, while r_{i,j} = 1 denotes that the area of this element contains material. The
initial value of the normalized material density r_{i,}_{0} is the ratio of the amount of usable
material M_{0} to the total material if the whole design domain is
occupied.
The density that
is actually used in the finite element model can be calculated by
_{}= r_{i,j}
r_{0} (1)
The Young’s modulus of the ith element at the jth iteration
actually used in the finite element model is assumed to be:
_{} (2)
where _{} is an exponent usually between 2 and 4 to penalize the small _{} [3, 6, 7]. In this research, the value of
_{} is taken as 2.
The finite
element analysis solver, for example, ANSYS or NASTRAN, is used to analyze the
finite element model under given boundary conditions and loads. The
displacement of nodes and the strain energy of elements _{} are evaluated by the solver. At the jth
iteration, the compliance of the structure _{} and its sensitivities
with respect to the normalized density of each element can then be evaluated by
the following equations [7]:
_{} (3)
_{} (4)
This objective function value and its sensitivities are required
when solving the optimization model.
Userdefined
termination criterion is used to check if the topology optimization process
should be stopped. If the result is false, a topology optimization model that
minimizes compliance of the structure and subjects it to the usable material
constraint will be generated:
_{} (5)
An optimization solver is used to find a new set of normalized
material densities for the next iteration. The normalized material density will
then be used to compute material properties _{} and _{} used in the
finite element model in the next iteration using Equations (3) and (4).
If the termination
criterion result is true, the topology optimization process will stop and
output the set of final normalized material densities of each element, _{} in Figure 1. This
topology optimization result can be plotted using different gray levels in the
elements to represent the corresponding _{}, as shown in Figure 4.
Figure 4. A cantilever beam example with poor
topology optimization result
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 1990s. Some researchers treat structural design
optimization as a threephase design process, and aim at developing an automated
design process [8, 9, 10]. Phase I is the topology generation process, in which
the optimal topology of the structure is generated. Phase II is the 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 [813], density contour approach [1416],
and geometric reconstruction approach [17]. 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.
The quality of
the topology optimization result generated in Phase I directly affects the
implementation of the interpretation process in Phase II. For example, it will
be very difficult to interpret the poor topology optimization result shown in
Figure 4 using any interpretation technique.
Thus, several
fundamental issues with respect to the quality of topology optimization result
must be considered in the topology optimization phase:
(1)
Could the design domains be
shapes other than rectangular ones?
(2)
Could the automatic mesh
generation be used to create the finite element model?
(3)
What is the proper element
size?
(4)
How can the checkerboard
problem be dealt with?
(5)
How can a strictly
blackandwhite topology optimization result be obtained?
(6)
How can the result of topology
optimization be interpreted to ensure that a continuous structure is obtained?
In the following
sections, these issues are explored and suggestions are made for the
generalization of the topology optimization process to general two or
threedimensional structural design problems.
2.
The mesh generation and
meshdependence problem
The structural design
examples in the topology optimization literature often have rectangular design
domains and are manually meshed using square elements, such as the cantilever
beam example in Figure 4. Realistic structural design examples may have complicated
design domains with geometry constraints, such as the suspension arm example
shown in Figure 3. It is not easy to mesh the complicated design domain
manually using square elements. Bendsfe [18] suggested that the use of an automatic
mesh generator would, of course, simplify the treatment of problems with
complicated design domains.
As the CAD model
is developed or generated into finite element model, the design domain _{} is divided into _{} nodes and _{} finite elements.
The topology optimization model in Equation (5) can be rewritten in a discretized
form as:
_{} (6)
where _{} and are the body
and surface forces applied to node k; _{} is the volume or
area of element i.
The quality of
topology optimization result is dependent on the discretization of the finite
element model, the socalled “meshdependence problem.” Figure 5(a) shows a simply
supported beam example to illustrate the meshdependence problem using the
density distribution approach [19]. Figure 5(b) shows the topology optimization
result for the discretization using 600 elements, and Figure 5(c) shows the
result using 5,400 elements. The result in Figure 5(c) is much more detailed
than that in 5(b), and more importantly, the two topologies are different in
nature.
Figure 5. The simply supported beam example for
meshdependence problem [19]
It has been
shown that the meshdependence problem often relates to the problem of
nonexistence of solutions. Several restriction methods were proposed in the
literature to avoid this type of problems. In the “perimeter control method,”
an upperbound constraint on the perimeter is used to ensure a wellposed
design problem [20, 21]. In “mesh independent filtering,” the filter modifies
the design sensitivity of the elements within a specific zone to guarantee the
radius of the members [22, 23]. In “global gradient constraint” method, the density
function and its gradient variation are bounded to ensure the existence of a
solution [18]. In “local gradient constraint” method, a local gradient
constraint on the density variation is introduced to guarantee the existence of
solution [24, 25]. Further details of these methods can be found in the survey
by Sigmund and Petersson [19].
The model with a
finer finite element mesh not only results in a better description of
boundaries but also increases the complexity of the resulting topology. Zhou et
al. [26] consider the manufacturability of the design from the engineering
viewpoint, instead of worrying about the fact that different mesh densities may
result in different final solutions. Therefore, in their work, the algorithm of
minimum member size control is developed to improve the manufacturability of
the design. Haber et al. [21] reported that the
reduction in compliance obtained by increasing the complexity of the design
topology is the modest. Effective designs can be obtained with relatively simpler
topologies. Following this finding, a “convergence test” is used to find a
proper topology optimization result. Table 1 shows the topology optimization
results using different number of elements, and Figure 6 plots the number of
elements vs. final compliance. It is observed that as the number of elements
increases the resulting compliance decreases but converges after the number of
elements exceeds 2,000. Table 1 also shows that, as the number of elements
increases the topology of the structure becomes more complicated, which is
impractical.
Table 1. The convergence test of the cantilever
beam example
No. of Elements

Final compliance

Topology optimization results

No. of Elements

Final compliance

Topology optimization results

112

23286


2432

12702


135

20760


2680

12435


198

20289


3053

12524


240

18928


3375

12686


360

17174


3840

12690


540

16688


4472

12173


960

13973


5208

12434


2160

12461


6000

12604


Figure 6. Final compliance vs. number of elements
The same
convergence test is also applied in the example of automaticmeshed suspension
arm. The convergence test results given in Figure 7 are similar to the result
of cantilever beam example. The compliance tends to converge when the number of
elements exceeds 2,000. Although this element number should be
problemdependent, for the 14 twodimensional topology optimization examples
and the 6 three dimensional topology optimization examples presented later in
Table 4, the element number around 2,000 has been a good reference number.
Figure 7. Final compliance vs. number of elements
3.
The penalty function
Ideally, the
normalized material densities of the elements should be either 0 or 1 after
topology optimization. But there can still be intermediate normalized material
densities at the end of the topology optimization, which results in an unreasonable
structure. Using a penalty function to penalize the intermediate normalized
material densities is the most common strategy to obtain a “blackandwhite”
topology.
Bensfe [18] embedded
a penalization in the material property that is actually used in the finite
element model by the following power law formulation:
_{} (7)
where _{} is the Young’s
modulus of the ith element at the jth iteration and _{} is the Young’s
modulus of a given isotropic material. The penalization factor that is larger
than 1 is represented by p. A similar penalization method was also used
by Sigmund [23] and Zhou et al. [26]
Kumar and Gossard
[14] added a penalty function to their objective function to penalize the
intermediate densities:
_{} (8)
The last term of Equation (8) is the penalty function, in which c_{p}
is a penalty constant and is increased in steps. It can be verified easily that
the penalty function has the maximum value when _{} and has the
minimum value when _{} and _{}.
Chen and Wu [7]
presented a similar penalty function to minimize the number of elements with
normalized densities not equal to either 0 or 1. The penalty function is
included in the objective function, and is defined as
_{} (9)
where M is the total number of finite elements and R
is a given penalty parameter which can vary iteration by iteration. This
penalty function has several advantages. The function is continuous and
differentiable and is symmetric about the most unwanted density value of 0.5. Secondly,
as a result of the nonlinear nature of this penalty function, heavier penalties
are imposed on those densities that are close to 0.5. The sum of the penalties
is an indicator of the clearness of the topology and is small if most densities
are close to 0 or 1.
Borrvall and
Petersson [27] treat the penalty function as an explicit constraint:
g(r)=_{}(_{} r)(r _{}) _{} (10)
where _{} for pure
materialvoid problem, in which _{} is either _{} or _{} at almost each
element. The value e_{p} can be small upon allowing some
intermediate density. In order to guarantee the existence of solutions, a
compact linear operator _{} is used to
regularize the constraint. Thus, the constraint _{} is
replaced by _{}, and is called “regularized intermediate density
control”. The _{} is called “regularized
penalty function,” and is defined by a composite mapping _{}.
In most
approaches discussed above, there are penalty parameters that control the
amount of penalty, for example, p in Equation (7), _{} in
Equation (8), and R in Equation (9). The value of the penalty parameter
has been shown to regularly increase with iteration times. If the value of the penalty
parameter is small, the effect of penalizing the intermediate densities is not apparent.
On the other hand, if the value is large, it will affect the objective function
value, which could lead to nonoptimal topology results. The initial value and
the amount of increment of the penalty parameter are often decided by experience
of the numerical process and are dependent on individual cases.
In order to avoid
numerical problems, such as premature convergence and the difficulty of the densities
crossing 0.5 arising from the addition of the penalty function in the early
stage, Chen and Wu [7] presented a twotiming approach to add the penalty
function. The first timing is at the fifth iteration, when the topologies have
been formed roughly. The second timing is at the iteration when the number of
densities whose values are greater than 0.95 and less than 0.05 is over 50% of
the total number of the density values.
In the research
presented here, the topology optimization process is divided into two stages.
The purpose of the first stage is to reduce the objective function value, and
the purpose of the second stage is to drive the normalized material densities
to either 0 or 1. The amount of penalty added in each stage is planned ahead
according to the different purposes of each stage.
The user defines
the total number of iterations _{} as one of the
termination criteria. The user also defines the ratio of the number of
iterations of the first stage to the total number of the iterations _{}. In order to have an adequate time to reduce the objective
function, the ratio _{} is suggested to
be 75%. Therefore, the first stage starts at the initial iteration _{} to 75% of the
total number of iterations (_{}, rounded to an integer).
The penalty
function at the jth iteration is defined as follows:
_{} (11)
where _{} is a penalty
parameter and is increased iteratively. The function _{} is the base of
penalty function defined as
_{} (12)
where M is the total number of finite elements. As shown in Equation
(12), _{} is a reward
function to reward the normalized material densities close to either 0 or 1. Equation
(12) is also a continuous and differentiable function, and is symmetric at
about 0.5. The function receives maximum reward when the normalized material
density is 0 or 1, and receives no reward when it is 0.5._{}
In Equation
(11), the penalty parameter _{} is defined by the
amount of “reward” intended for each stage, instead of numerical experience. At
the end of the first stage, the magnitude of _{} is set to be
equal to initial compliance. At the end of the second stage, the magnitude of _{} is set to be
equal to 10 times of initial compliance. Thus, _{} can be derived
from the following equations:
_{} for the first
stage,
_{} for the second
stage (13)
The multipliers in Equation (13), 1 in the first stage and 10 in the second stage, can be adjusted if a heavier
penalty is desired.
Figures 8 and 9
compare the topology optimization results with and without the penalty
function. The effect of using a penalty function is more apparent in the
suspension arm example shown in Figure 9. Figure 9(a) is the topology optimization
result without using a penalty function. It is observed that the gray element
covers a significant part of the design domain, and the members are not
distinct in this region. In Figure 9(b), the penalty function is used. Although
there are still gray elements, the members are much clearer than the result in
Figure 9(a).
(a) The result without penalty function
(b) The result with penalty function
Figure 8. The comparison of the topology
optimization results with and without penalty function in the cantilever beam
example
(a) The result without penalty function
(b) The result with penalty function
Figure 9. The comparison of the topology
optimization results with and without penalty function in the suspension arm example
4.
The checkerboard problem
The “optimal
topology” often contains a checkerboard pattern, which is the alternately solid
and void elements as shown in Figure 4. This is a typical result in topology
optimization using finite elements. It was believed that some sort of
microstructure exists in these regions. Bendsfe [18] demonstrated that the
checkerboard problem is related to the features of finite element approximation
as a numerical phenomenon. This interpretation is now widely accepted.
At least four
types of methods are proposed to prevent the checkerboard problem. In the “smoothing
method,” image processing is used to smooth the output picture of topology
optimization within the checkerboards. The use of higherorder finite elements
is also suggested to avoid the checkerboard problem [28]. In the “patches method,”
a “superelement” is introduced to the finite element formulation to damp the
appearance of checkerboards [18]. Finally in the “filter method,” the modified
design sensitivities of each iteration are used to prevent checkerboard patterns
[22]. A survey of the checkerboard pattern problem was given by Sigmund and
Petersson [19]. Of the four methods, the higherorder finite elements is
probably the most convenient one. No external techniques are needed other than
altering the element that is used to discretize the design domain to a
higherorder finite element. Rodrigues and Fernandes [29] showed that the use
of the ninenode element could prevent the occurrence of checkerboard pattern
that occurred when using the fournode element. Díaz and Sigmund [28] and Jog
and Haber [30] also showed that checkerboard pattern could be prevented efficiently
when using eight or ninenode elements.
Figure 4 shows
the topology optimization result of the cantilever beam example using a fournode
element. The finite element analysis solver used here is ANSYS, and the twodimensional
fournode element “Plane 42” provided
by ANSYS [31] was used. It is observed that the
checkerboard pattern appears in many parts of the “optimal structure” of this
example. The same example is then solved again using the twodimensional
eightnode element “Plane 82” provided
by ANSYS. The topology optimization result is shown in Figure 10. Comparing
Figure 4 with Figure 10, the checkerboard patterns disappear resulting in a
clear topology of the structure.
Figure 10. The topology optimization result of the
cantilever beam example using eightnode element
Figure 11 shows
the topology optimization result using fournode elements for the suspension
arm example. The element distribution shown in Figure 11 are the elements in
which the normalized material density is greater than 0.5. The checkerboard
patterns are also distinct in this example. The topology optimization result
when using eightnode elements is shown in Figure 9(b). The distributed
elements shown in Figure 9(b) are also the elements in which the normalized
material density is greater than 0.5. The checkerboard patterns disappear.
Figure 11. The topology optimization result of the
suspension arm example using fournode element
These examples
demonstrate that the use of higherorder finite elements can effectively prevent
checkerboard patterns, even for the example with complicated design domain.
Thus, the higherorder finite element is suggested and is actually used in the
topology optimization process presented in this research.
5.
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” [32], which is formed by elements with intermediate normalized
material densities between 0 and 1, often occurs. In this paper, it is assumed
that the material is isotropic, and materials with intermediate densities are
not allowed.
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 12, if the threshold
value is high and a portion of porous topology A is discarded, a
discontinuous structure (a structure with dangling ribs) 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 12. 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 _{} is to be decided
by comparing the compliances of the structure using different threshold density
values.
Table 2 shows
the compliance increase and the respective topology results of the cantilever
beam example using different threshold density values. As shown in Table 2,
when a discontinuity of the topology occurs (indicated by a circle) as the
threshold value increases, there will be a corresponding “jump” in compliance.
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, it was decided to set _{} at 0.2. In this
way a proper threshold density value _{} can be selected
to filter the porous topology without generating a discontinuous structure.
Table 2. The topology of different threshold
density
_{}

Topology

_{}

Topology

0.1


0.6


Compliance increase: 0%

Compliance increase: 98.6%

0.2


0.8


Compliance increase: 34.9%

Compliance increase: 1937.4%

After the
continuity analysis, it is observed that trivial solids (such as islands or
small salient) and trivial voids (the voids containing few elements whose
normalized material densities are 0) appear in the topology result. Trivial
solids and trivial voids are useless in mechanics and will complicate the
implementation of the shape optimization. Thus, the trivial solids should be
filtered out before implementing shape optimization. In this research two “filters”
are implemented to filter out the trivial solids and trivial voids in the
topology result after the continuity analysis.
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 at only one node.
Figure 13 is the result of cantilever beam example after applying 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 14 shows the topology result of the suspension arm
example after the trivial void filter. The checkerboard like trivial voids
shown in Figure 14 are filtered out.
Figure 13. The result of the cantilever beam
example after the trivial solid filter
Figure 14. The result of suspension arm example
after the trivial void filter
6.
Two and threedimensional
topology optimization examples
After a
highquality topology optimization result is obtained in the Phase I (topology
generation process), it becomes much easier to interpret the result in the
Phase II (topology interpretation process), then complete the shape and size
optimization in the Phase III (detailed design phase). Table 3 shows 14
twodimensional structural topology optimization examples commonly seen in
literature. The topology optimization procedure developed in this research
worked well for all 14 examples. The results are also presented in Table 3.
Table 3. Twodimensional topology optimization
examples

Examples

Topology optimization results in literatures

Topology optimization results

1


Bendsfe & Rodrigues [12]


2


Bendsfe & Rodrigues [12]


3


Bendsfe & Rodrigues [12]


4


Olhoff et al. [13]


5


Bremicker et al. [8]


6


Bremicker et al. [8]


7


Bremicker et al. [8]


8


Chirehdast et al. [9]


9


Chirehdast et al. [9]


10


Chirehdast et al. [9]


11


Tang & Chang, [17]


12


Youn & Park, [15]


13


Youn & Park, [15]


14


Youn & Park, [15]


Topology
optimization has been extended to threedimensional continuum structures in
recent years. Cherkaev and Palais [33] considered the topology optimization of threedimensional
axisymmetric elastic structures. In their work, the method of finding optimal
bounds upon the effective properties of a composite is used. Díaz and Lipton
[34] presented a full relaxation strategy to find the optimal topology of minimum
compliance for threedimensional elastic structures subjected to the amount of
available material. Olhoff et al. [35] and Jacobsen et al. [36] used
threedimensional microstructures developed by Gibianski and Cherkaev [37] to
get the optimum topology for threedimensional structures.
Beckers [38]
developed a dual method in which the design variables can only be 1 (present)
or 0 (absent) to deal with the topology optimization of continuous structure.
In his work, two and threedimensional problems are solved. The examples of
threedimensional structural topology optimization using the density
distribution method can be found in the papers by Yang and Chen [39] and Hsu et
al. [16].
For threedimensional
structures, it is very difficult to interpret the topology optimization result
manually. Therefore a generalized and automated process is even more important.
The generalized topology optimization process shown in Figure 1 and the
discretized optimization model Equation (6) can be applied to the topology optimization
of threedimensional structures without any modifications. Table 4 shows 6
threedimensional structural topology optimization examples that are commonly
seen in the literature. The same twostage penalty function presented in Section
3 is used to penalize the intermediate normalized material density, and the
higher order 20nodes cubic element is used in order to avoid the checkerboard
problem. The same continuity analysis and trivial solids and voids filters
discussed in Section 5 are used to finetune the final topology. Both the
topology optimization results reported in the literature and in this research
are listed in Table 4 for comparison.
Table 4. Threedimensional topology optimization
examples

Examples

Topology
optimization results in literatures

Topology
optimization results

1


Olhoff et al. [35],
Jacobsen et al. [36], Beckers, [38]


2


Olhoff et al. [35],
Jacobsen et al. [36]


3


Olhoff et al. [35],
Jacobsen et al. [36]


4


Olhoff et al. [35],
Jacobsen et al. [36], Beckers, [38]


5


Díaz, & Lipton,
[34]


6


Díaz, & Lipton,
[34]


7.
Conclusion
A generalized
topology optimization process is presented in this paper. Several fundamental
issues on the quality of the topology optimization result are considered in
order to achieve a clear topology optimization result. They are
meshdependence, elements with intermediate densities, the checkerboard
problem, and porous topology.
A convergence
analysis is implemented to find the minimum number of elements that does not
significantly affect the compliance. A twostage penalty function strategy is
developed in this research to determine a proper topology optimization result
in the first stage using a light penalty, then heavily penalize the elements
with intermediate densities in the second stage. Higher order finite element is
used, which is convenient to the users, and the checkerboard pattern is avoided
effectively. Finally a continuity analysis is used to deal with the porous
topology, and two filters are implemented to filter out the trivial solids and
voids.
The whole
process is generalized to two and three dimensional structure and can be
fully automated. Fourteen twodimensional and six threedimensional examples
are used to demonstrate the effectiveness of this process.
References
[1] Bendsfe, M. P., Kikuchi, N. (1988) Generating Optimal Topologies in
Structural Design Using Homogenization Method. Computer Methods in Applied
Mechanics and Engineering, 71, 197244.
[2] Mlejnek, H. P., 1992, Some Aspects of the Genesis of Structures. Structural
Optimization, 5, 6469.
[3] Yang, R. J., Chuang, C. H. (1994) Optimal Topology Design Using
Linear Programming. Computers & Structures, 52, 265275
[4] Xie. Y. M., Steven, G. P. (1993) A Simple Evolutionary Procedure for
Structural Optimization. Computers & Structures, 49, 885896.
[5] Rozvany, G. I. N. (2001) Aim, Scope, Methods, History and Unified
Terminology of ComputerAided Topology Optimization in Structural Mechanics. Structural
and Multidisciplinary Optimization, 21, 250252.
[6] Yang, R. J., Chahande, A. I. (1995) Automotive Applications of
Topology Optimization. Structural Optimization, 9, 245249.
[7] Chen, T. Y., Wu, S. –C. (1998) Multiobjective Optimal Topology
Design of Structures. Computational Mechanics, 21, 483492.
[8] Bremicker, M., Chirehdast, M., Kikuchi, N., Papalambros, P. Y.
(1991) Integrated Topology and Shape Optimization in Structural Design. Mechanics
of Structures and Machines, 19, 551587.
[9] Chirehdast, M., Gea, HC., Kikuchi, K., Papalambros, P. Y. (1994)
Structural Configuration Examples of an Integrated Optimal Design Process. Journal
of Mechanical Design, 116, 9971004.
[10] Lin, C. Y. Chao, L. S. (2000) Automated Image Interpretation for
Integrated Topology and Shape Optimization. Structural and Multidisciplinary
Optimization, 20, 125137.
[11] Papalambros, P., Chirehdast, M., (1990) An Integrated Environment
for Structural Configuration Design. International Journal for Numerical
Methods in Engineering, 41, 14171434.
[12] Bendsfe, M. P., Rodrigues, H. C. (1991) Integrated Topology and Boundary
Shape Optimization of 2D Solid. Computer Methods in Applied Mechanics and
Engineering, 87, 1534.
[13] Olhoff, N., Bendsfe, M. P., Rasmussen, J. (1991) On CADIntegrated Structural Topology
and Design Optimization. Computer Methods in Applied Mechanics and
Engineering, 89, 259279.
[14] Kumar, A. V., Gossard, D. C. (1996) Synthesis of Optimal Shape and
Topology of Structures. Journal of Mechanical Design, 118, 6874.
[15] Youn, S. K., Park, SH. (1997) A Study on the Shape Extraction
Process in the Structural Topology Optimization Using Homogenized Material. Computers
& Structures, 62, 527538.
[16] Hsu, Y. L., Hsu, M. S., Chen, C. T. (2001) Interpreting Results from
Topology Optimization Using Density Contours. Computers & Structures,
79, 10491058.
[17] Tang, P. S., Chang, K. H. (2001) Integration of Topology and Shape
Optimization for Design of Structural Components. Structural and
Multidisciplinary Optimization, 22, pp.6582.
[18] Bendsfe, M. P. (1995) Optimization of Structural Topology, Shape, and
Material, SpringerVerlag Berlin Heidelberg.
[19] Sigmund, O., Petersson, J. (1998) Numerical Instabilities in
Topology Optimization: A Survey on Procedures Dealing with Checkerboards,
Meshdependencies and Local Minima. Structural Optimization, 16,
6875.
[20] Ambrosio, L., Buttazzo, G.. (1993) An Optimal Design Problem with
Perimeter Penalization. Calculus of Variations and Partial Differential
Equations, 1, 5569.
[21] Haber, R. B., Jog, C. S., Bendsfe, M. P. (1996) A New
Approach to VariableTopology Shape Design Using a Constraint on Perimeter. Structural
Optimization, 11, 112.
[22] Sigmund, O. (1994) Design of material structures using topology
optimization, Ph.D thesis, Department of Solid Mechanics, Technical
University of Denmark.
[23] Sigmund, O. (1997) On the Design of Compliant Mechanisms Using
Topology Optimization. Mechanics of Structures and Machines, 4,
493524.
[24] Niordson, F. I. (1983) Optimal Design of Plates with a Constraint on
the Slope of the Thickness Function. International Journal of Solids and
Structures, 19, 141151.
[25] Petersson, J., Sigmund, O. (1998) Slope Constrained Topology
Optimization. International Journal for Numerical Methods in Engineering,
41, 14171434.
[26] Zhou, M., Shyy, Y. K., Thomas, H. L. (2001) Checkerboard and Minimum
Member Size Control in Topology Optimization. Structural and
Multidisciplinary Optimization, 21, 152158.
[27] Borrvall, T., Petersson, J. (2001) Topology Optimization Using
Regularized Intermediate Density Control. Computer Methods in Applied
Mechanics and Engineering, 190, 49114928.
[28] Díaz, A. R., Sigmund, O. (1995) Checkerboard patterns in layout
optimization. Structural Optimization, 10, 4045.
[29] Rodrigues, H., Fernandes, P. (1995) A Material Based Model for
Topology optimization of Thermoelastic Structures. International Journal for
Numerical Methods in Engineering, 38, 19511965.
[30] Jog, C. S., Haber, R. B. (1996) Stability of Finite Element Models
for Distributedparameter Optimization and Topology Design. Computer Methods
in Applied Mechanics and Engineering, 130, 203226.
[31] ANSYS Inc. (1999) PLANE42 2D Structural Solid. ANSYS Element
Reference, 11 edition, ANSYS Inc.
[32] Rozvany, G. I. N., Zhou, M., Birker, T., (1992) Generalized Shape
Optimization without Homogenization, Structural Optimization, 4,
250252.
[33] Cherkaev, A., Palais R. (1996) Optimal design of threedimensional
axisymmetric elastic structures. Structural Optimization, 12,
3345.
[34] Díaz, A., Lipton, R. (1997) Optimal material layout for 3D elastic
structures, Structural Optimization, 13, 6064.
[35] Olhoff, N., Rønholt, E., Scheel, J. (1998) Topology optimization of
threedimensional structures using optimum microstructures. Structural
Optimization, 16, 118.
[36] Jacobsen, J. B., Olhoff, N., Rønholt, E. (1998) Generalized shape
optimization of threedimensional structures using materials with optimum
microstructures. Mechanics of Materials, 28, 207225.
[37] Gibiansky, L. V., Cherkaev, A. V. (1987) Microstructures of
composites of extremal rigidity and exact estimates of provided energy density.
A. F. Ioffe PhysicoTechnical Institute, 1115. English translation to
appear in: Kohn, R. (1994) Topics in the mathematical modeling of composite
materials, Birkhauser.
[38] Beckers, M. (1999) Topology optimization using a dual method with
discrete variables. Structural Optimization, 17, 1424.
[39] Yang R. J., Chen, C. J., 1996 Stressbased Topology Optimization, Strucutral
Optimization, 12, 98105.