Author: YehLiang Hsu, MingSho
Hsu,
ChuanTang Chen (20000408); last updated: YehLiang Hsu (20010216); recommended: YehLiang Hsu (20000418).
Note: This paper is published in Computers and
Structures,
Vol. 79, Issue 10, pp. 10491058, 2001.
Interpreting results
from topology optimization using density contours
Abstract
The topology
optimization result using material distribution method is a density
distribution of the finite elements in the design domain. Interpreting this
result has been a major difficulty for integrating topology optimization and
shape optimization into an automated structure design procedure. It is also one
of the factors that limit the extension of the optimization methods of
twodimensional structures into threedimensional structures.
This paper
presents a process for integrating topology optimization and shape
optimization. In this process, density contours are used to interpret the
topology optimization result, and then further integrate with shape
optimization. This is a fully automated procedure, since during this process no human
interpretation or intervening is required. This procedure can be extended to
the topology and shape optimization of threedimensional structures. The
optimizations of two and threedimensional structures are presented in design
examples.
Keywords: topology optimization, shape
optimization, density contours, material distribution method.
Introduction
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 using this initial topology is only the “optimal
design” under this nonoptimal topology. Therefore topology optimization, which
helps designer to find the optimal topology of the structure, has been a very
active research field.
Figure 1 shows
the structure design examples commonly used in topology optimization
literature. Three different methods for topology optimization can be found in
literature. One is the homogenization method, which is based on the assumption
of a microstructure in which the properties are homogenized. Another approach
is the material distribution method, in which the material density of each
element is selected as the design variable. During the optimization process,
intermediate density is penalized to force the design variables to approach 0
or 1. If the material density of an element is close to 0 at the end of
topology optimization, the element does not exist in the optimal topology. In
these two methods, the objective functions are usually to minimize compliance
of the structure, or to maximize stiffness of the structure. Maximizing the
lowest natural frequency of the structure is also used as the objective
function in some examples. In almost all examples, the only constraint is that
the amount of material that can be used in the design domain is limited. Stress
constraints are usually not included in topology optimization, though Yang [1]
considered stress constraints using global stress functions to approximate
local stresses.
Design Example

Authors

Objective

Optimization
Method


a. Suzuki
& Kikuhi [16]
b. Bendse
& Rodrigues [9]
c. Thomsen [17]
d. Ma et al [18]
e. Yang &
Chuang [6]
f. Xie &
Steven [2]
g. Chu et al.
[3]

a. compliance
b. stiffness
c. stiffness
d. compliance
e. compliance
f. compliance
g. weight

a.
homogenization
b. material
distribution
c.
homogenization
d.
homogenization
e. material
distribution
f.
evolutionary method
g.
evolutionary method


a. Olhoff et
al. [10]
b. Mlejnek [19]
c. Maute &
Ramm [7]

a. compliance
b. compliance
c. stiffness

a. material
distribution
b.
homogenization
c. material
distribution


a. Bendse
& Kikuchi [20]
b. Suzuki
& Kikuchi [16]
c. Bendse
& Rodrigues [9]

a. compliance
b. compliance
c. stiffness

a.
homogenization
b.
homogenization
c.
homogenization


a. Mlejnek [19]
b. Mlejnek
& Schirrmacher [21]
c. Youn &
Park [12]

a. compliance
b. compliance
c. compliance

a.
homogenization
b. material
distribution
c.
homogenization


a. Suzuki
& Kikuchi [16]
b. Mlejnek
& Schirrmacher [21]
c. Xie &
Steven [2]

a. compliance
b. compliance
c. compliance

a.
homogenization
b. material
distribution
c.
evolutionary method


a. Suzuki
& Kikuchi [16]
b. Yang &
Chuang [6]
c. Chu et al.
[3]

a. compliance
b. compliance
c. weight

a.
homogenization
b. material
distribution
c.
evolutionary method


a. Yang &
Chuang [6]
b. Ma et al. [22]

a. natural
frequency
b. natural
frequency

a.
homogenization
b. material
distribution


a. Yang &
Chuang [6]
b. Maute &
Ramm [7]
c. Xie &
Steven [2]
d. Youn &
Park [12]

a. compliance
b. stiffness
c. compliance
d. compliance

a. material
distribution
b. material
distribution
c.
evolutionary method
d.
homogenization


a. Chu et al.
[4, 5]

a.
displacement

a.
evolutionary method


a. Chu et al.
[4]

a. strain
energy

a.
evolutionary method


a. Chu et al.
[4]

a. strain
energy

a.
evolutionary method

Figure 1. Structure design examples commonly used
in literature
The third
approach is the evolutionary structural optimization (ESO) first proposed by
Xie and Steven [2]. The original idea of this method is to gradually remove
lowly stressed elements to achieve the optimal design. Chu et al. [3] extended
ESO to shape optimization problems that minimizes the weight of the structure
subject to stiffness constraints. Instead of completely removing the redundant
elements, Chu et al. [4, 5] further suggested an evolutionary method for
optimal design of plates with discrete variable thickness. Plates with
threedimensional loading are used as design examples.
The first
example in Figure 1 is shown in more details in Figure 2(a). Figure 2(b) shows
the optimal topology obtained by Yang and Chung [6] using the material
distribution method. As shown in the figure, the result from topology
optimization by any of the three methods is often a nonsmooth, or even
noncontinuous, skeleton type of structure, which may not be practical. Further
postprocessing, either by human interpretation using certain smoothing
algorithms [7], or integrating with shape optimization algorithms [810], is
required.
(a) The design domain and boundary conditions.
(b) The result from topology optimization (Yang and Chuang [6])
Figure 2. The Cantilever beam example.
When integrating
with shape optimization, cubic splines are often used to ensure the smoothness
of the boundary shape, and constraints on local stress are emphasized. The
constraint on the total mass of the structure is often treated as a “guideline”
instead of a rigid constraint, and is often relaxed in the post processing.
Another issue that has to be resolved is the formation of the socalled
“checkerboard” patterns [11, 12] during the topology optimization process.
Youn and Park [12] suggested a density redistribution algorithm to suppress the
checkerboard patterns of material densities obtained from the material
distribution method. Guedes and Taylor [13] presented a method in which the
optimal material properties are predicted along with the “highresolution”
definition of structural shape. Through this filtering process, the originally
opaque design is rendered into a distinct, “highresolution” design.
The topology
optimization result using material distribution method is a “density
distribution” of the finite elements in the design domain. Kumar and Gossard [14]
used contours of a “shape density function” to represent the boundaries of the
shape of structural components where both shape and topology are optimized.
Both shape and topology of the structure are modified simultaneously by the
optimization algorithm. In their work, the shape and topology of the structure
are optimized with the objective of minimizing the compliance subject to a
constraint on the total mass of the structure.
This paper
presents a process that uses density contours to interpret the result obtained
from topology optimization, and then integrated with shape optimization. In
this process, smooth boundary can be achieved, discontinuity in structure can
be identified, and both the stress constraint and the constraint on total mass
of the structure are considered and satisfied. The whole process is automated
and can be easily implemented for computer application, which is very crucial
for integrating topology optimization with shape optimization into a fully
automated structure design procedure. A twodimensional structure is used to
illustrate the ideas. However, this procedure can be easily extended to the topology and shape
optimization of threedimensional structures. Optimizations of two and
threedimensional structures are presented in design examples.
The Density Contours
As discussed in
the previous section, the material distribution method is commonly used in
topology optimization. The design variable in this method is the normalized
material density of each element, which is defined as _{}, where _{} is the assumed
material density of the element _{}; _{} is the true
material density and _{} is the normalized
material density of the element _{}.
The design
variable _{} may vary between
0 and 1. Intermediate density is usually penalized to force the design
variables to approach 0 or 1. If the normalized material density of an element
is close to 0 at the end of topology optimization, the element does not exist
in the optimal topology. On the other hand, if the normalized material density
of an element is close to 1 at the end of topology optimization, the element exists
in the optimal topology.
Figure 3 shows
the design domain and boundary conditions of a twodimensional structure, which
will be used to demonstrate the process using density contours to integrate
topology and shape optimization. Topology optimization using the material
distribution method is performed on this design domain. As in most topology
design cases in literature, the objective is to minimize compliance of the
structure, subject to the constraint on 25% mass usage. The design domain is
divided into _{} finite
elements, which is prescribed by the user. The initial density is set at _{} for all elements
in the design domain. The initial compliance of this structure is 2003.5_{}.
Figure 3. The design domain of a twodimensional
structure (F=500N).
Figure 4 shows
the result after topology optimization. In this Figure, a black element has
density value close to 1, and a blank element has density value close to 0. The
compliance of the structure decreases to 315.9_{} after 52
iterations.
Figure 4. The result after topology optimization.
The density
values of the elements are then extrapolated in a nonlinear fashion to obtain
the nodal density values. From the nodal density values, we can easily plot the
density contours for any value _{}, _{}, and compute the area covered by the contour. Figure 5 shows
the density contours for _{} and _{}. The total area of the design domain in this case is _{}. The area for the contour for _{} is _{}, which violates the constraint on 25% mass usage. The area
for the contour for _{} is _{}, which is less than the constraint on 25% mass usage. Using
simple secant method, we can find that the area for the contour _{} is exactly _{}, which satisfies the constraint on 25% mass usage with
strict equality.
Figure 5. Density contours for _{} and _{}.
Figure 6(a)
shows this design. Compared with the result shown in Figure 4, this design has
a smooth and continuous boundary, and can be used as the output of topology
optimization. The compliance of this design is 269.1_{}, which is lower than that of Figure 4. Using finite element
analysis on this design, stress concentration is found around the inside
corners of the structure (Figure 6(b)). The maximum stress of this structure is
_{}, which is higher than the yielding stress of the material _{}. Obviously further shape optimization is needed in order to
satisfy the stress constraint.
(a) Density contours for _{}.
(b)Stress analysis results.
Figure 6. The result of topology optimization.
Integrating with Shape Optimization
The design in
Figure 6 is used as the initial design for shape optimization. The objective is
still to minimize the compliance of the structure, subject to mass and stress
constraints. As shown in Figure 7, spline curves are used to represent the
initial shape. The movements along the normal vectors of the spline curves at
the control points are used as design variables. As shown in the Figure, 28
equally spaced control points are automatically selected to define the boundary
spline curves, so there are 28 design variables. The number of control points
used in shape optimization is also prescribed by the user. “Method of centers
[15]” is used as the optimization algorithm, and the sensitivities of the
design variables are calculated by finite difference.
Figure 7. Control points for the initial design
model for shape optimization.
Figure 8 shows the
shape optimization result after 40 iterations. The compliance of the structure
further drops to 231.8_{}, the area is _{}, while the maximum stress of the structure drops to _{}.
Figure 8. The result of shape optimization.
In summary, this structure design experiences four
different stages:
(1) Defining the initial design domain
(Figure 3).
(2) Topology optimization, to minimize
compliance under the constraint on mass usage (Figure 4).
(3) Using the density values obtained in
(2) to generate a smooth density contour of the structure, considering the
constraint on mass usage (Figure 6).
(4) Shape optimization, to minimize
compliance under the constraint on mass usage and stress, using the result in
(3) as the initial design (Figure 8).
This can be a fully integrated and fully automated
process, since during this process no human interpretation or intervening is
required. Comparing with the designs in Figure 3 and Figure 8, both topology
and shape of the structure are optimized. The compliance of the structure drops
from 2003.5_{} to 231.8_{}, while the constraints on mass usage and stress are both
satisfied.
Identifying Discontinuities of the Structure
The result from
topology optimization may have wide spread porous regions or checkerboard
patterns. Discontinuity may occur when generating the design contour in stage
(3) discussed above. To make the procedure described in the previous section a
fully automated process, discontinuities of the structure should also be
identified automatically in order to create a reasonable structure.
Figure 9 shows
the result of the topology optimization problem defined in Figure 2(a), using _{} grids. This
result is obtained by the material distribution method, but no penalty function
is introduced to push the density values of the elements into 0 and 1. As shown
in Figure 9, many elements have intermediate density values. Figure 10 shows
the density contours of _{}. There is a discontinuity within the design domain, which
has to be identified automatically before sending the density contour as the
initial design for shape optimization.
Figure 9. The result of topology optimization of a
cantelever beam.
Figure 10. Density contour forρ=0.60.
In the density
contour plot, a close region in the design domain is usually a “hole.” But as
shown in Figure 10, the close regions may also be discontinuous structure. Note
that the boundary of the design domain is also treated as part of the close
region. To identify those discontinuities for a certain density level _{}, first the close regions on the contour plot are
identified, and their areas are calculated. Then as shown in Figure 11, a
slightly lower density level (_{} in this case) is
used. The area of the close regions in this new density contour is calculated
again. The close region is a discontinuity if the area becomes larger;
otherwise, it is a hole.
Figure 11. Density contour for ρ=0.50.
Several things
can be done when a discontinuity is identified. If the area of the
discontinuity is smaller than a prescribed value, it can be ignored and
deleted. Or topology optimization should be resumed using higher penalty value
to push the density values to 0 or 1. If the discontinuities cannot be
improved, the designer is prompt with a message about this situation and
further shape optimization cannot proceed.
Design Examples
Finally three
examples that are commonly used in topology optimization literatures are shown
here to illustrate the practicality of this integrated procedure. For clarity,
all examples are presented in the format of the four sequential stages.
The Cantilever Beam Example
Stage (1): Defining the initial
design domain
The same
cantilever beam example in Figure 2 is shown again in Figure 12.
Figure 12. The design domain of a cantilever beam
(F=500N).
Stage (2): Topology
optimization
In
topology optimization, the objective is to minimize compliance of the
structure, subject to the constraint on 25% mass usage. The design domain is
divided into _{} finite
elements. The initial density is set at _{} for all elements
in the design domain. The initial compliance
is 453.5_{}.
Figure 13
shows the topology optimization result after 50 iterations. Its compliance
decreases to 123.7_{}.
Figure 13. Topology optimization result of the
cantilever beam.
Stage (3): Generating the density
contour
As shown
in Figure 14, the area covered by the density contour _{}, is found to be exactly _{} (25% of the area
of the design domain). It is used as the initial structure for shape
optimization. The maximum stress of this initial design is found to be _{}, which is higher than the yielding stress of the material _{}.
Figure 14. Density contour for _{}.
Stage (4): Shape optimization
In shape
optimization, 36 control points are automatically selected from Figure 14 as
design variables. The objective is still to minimize the compliance of the
structure, subject to mass and stress constraints. Figure 15 shows the shape
optimization result after 40 iterations. The compliance further decreases from
113.8_{} (Figure 14) to
93.8_{} (Figure 15). Its
area is _{}, and maximum stress is _{}, both are lower than the constrained values.
Figure 15. Shape optimization result of the
cantilever beam.
The Simply Supported Beam Example
Stage (1): Defining the initial
design domain
The
dimensions, boundary conditions and loads of a simply supported beam are shown
in Figure 16.
Figure 16. The design domain of a simply supported
beam (F=500N).
Stage (2): Topology
optimization
In
topology optimization, the objective is to minimize compliance of the
structure, subject to the constraint on 25% mass usage. The design domain is
divided into _{} finite
elements. The initial density is set at _{} for all elements
in the design domain. The initial compliance
is 1162.8_{}. Figure 17 shows the topology optimization result after 50
iterations. Its compliance decreases to 159.7_{}.
Figure 17. Topology optimization result of the
simply supported beam.
Stage (3): Generating the
density contour
As shown
in Figure 18, the area covered by the density contour _{}, is found to be exactly _{} (25% of the area
of the design domain). It is used as the initial structure for shape
optimization.
Figure 18. Density contour for ρ= 0.9
Stage (4): Shape optimization
For shape
optimization, 36 control points are automatically selected from Figure 18 as
design variables. The objective is still to minimize the compliance of the
structure, subject to mass and stress constraints. Figure 19 shows the shape
optimization result after 40 iterations. The compliance further decreases from
162.0_{} (Figure 18) to
160.8_{} (Figure 19). Its
area is _{}, and maximum stress is _{}, both are lower than the constrained values.
Figure 19. Shape optimization result of the simply
supported beam.
The threedimensional cantilever beam
The following
example is a threedimensional cantilever beam that is widely used in literature
discussing threedimensional structural topology and shape optimization [2,
2325]. It is presented here to illustrate that all four stages of the
procedure developed in this paper can be easily extended to threedimensional
structural optimization.
Stage (1): Defining the initial
design domain
The
threedimensional design domain, boundary conditions and load of a
threedimensional cantilever beam are shown in Figure 20.
Figure 20. The design domain of a
threedimensional cantilever beam (F=50000N).
Stage (2): Topology
optimization
In
topology optimization, the objective is to minimize compliance of the
structure, subject to the constraint on 25% mass usage. The threedimensional
design domain is divided into _{} finite
elements. The initial density is set at _{} for all elements
in the design domain. The initial compliance
is 52,485_{}. Figure 21 shows the topology optimization result after 75
iterations. Its compliance decreases to 11,516.5_{}.
Figure 21. Topology optimization result of a
threedimensional cantilever beam.
Stage (3): Generating the
density contour
As shown
in Figure 22, the threedimensional design domain is represented by 9 cross
sections. Each cross section is a twodimensional plane, and the same method
described in the previous sections can be used to find the density contours for
each cross section. 86 control points are automatically selected from Figure 22
as design variables. Spline curves are used to connect control points of each cross
section. The density contours of each cross section are then swept into a
threedimensional volume, as shown in Figure 23. The volume covered by the
density contour _{}, is found to be_{} (about 25% of the
volume of the design domain). It is used as the initial structure for shape
optimization. The maximum stress of this initial design is found to be _{}, which is higher than the yielding stress of the material _{}.
Figure 22. Density contour forρ= 0.75 of each cross section.
Figure 23. The volume of the initial design of the
threedimensional cantilever beam.
Stage (4): Shape optimization
For shape
optimization, the objective is still to minimize the compliance of the
structure, subject to mass and stress constraints. Figure 24 shows the shape
optimization result after 25 iterations. The compliance further decreases from
11,391.86 _{} (Figure 23) to 11,339_{} (Figure 24). Its volume
is _{}, and maximum stress is _{}, both are lower than the constrained values.
Figure 24. The shape optimization result of the
threedimensional cantilever beam.
Discussions and Conclusions
As described in
previous sections, the topology optimization result using material distribution
method is a density distribution of the finite elements in the design domain.
Interpreting this result has always been a major difficulty for integrating
topology optimization and shape optimization into an automated structure design
procedure.
This paper
presents a procedure for integrating topology optimization and shape
optimization. In this procedure, density contours are used to interpret the topology
optimization result, and then further integrate with shape optimization. As
illustrated in the design examples, this is a fully automated procedure, since during this process no human
interpretation or intervening is required. In this
process, smooth boundary can be achieved, discontinuity in structure can be
identified, and both constraints on stress and mass usage are satisfied.
The difficulty
of interpreting the result obtained from topology optimization has also limited
the extension of the methods of twodimensional structure optimal design into
threedimensional structures. All four stages in the procedure developed in
this paper can be easily extended to threedimensional structures. The topology
and shape optimization of a threedimensional structure using this procedure is
also presented.
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