Suitability of SIMP and BESO topology optimization algorithms for additive manufacture

Abstract

Additive manufacturing (AM) is expanding the range of designable geometries, but to exploit this evolving design space new methods are required to find optimum solutions. Finite element based topology optimisation (TO) is a powerful method of structural optimization, however the results obtained tend to be dependent on the algorithm used, the algorithm parameters and the finite element mesh. This paper will discuss these issues as it relates to the SIMP and BESO algorithms. An example of the application of topological optimization to the design of improved structures is given.

Full text

Suitability of SIMP and BESO Topology
Optimization Algorithms for Additive Manufacture
Aremu A., Ashcroft I., Hague R., Wildman R., Tuck C.
Wolfson School of Mechanical and Manufacturing Engineering,
Loughborough University, Loughborough, LE11 3TU, UK
Abstract
Additive manufacturing (AM) is expanding the range of designable geometries,
but to exploit this evolving design space new methods are required to find
optimum solutions. Finite element based topology optimisation (TO) is a
powerful method of structural optimization, however the results obtained tend to
be dependent on the algorithm used, the algorithm parameters and the finite
element mesh. This paper will discuss these issues as it relates to the SIMP and
BESO algorithms. An example of the application of topological optimization to
the design of improved structures is given.
Nomenclature
FR = Filter radius
ER = Evolution rate
VF = Volume Fraction
b
η
= Sensitivity of node b
a
V = Volume of element a
a
λ
= Sensitivity of element a
s = Number of elements connected to node
1+m
V = Iterative new target volume
m
V = Current volume
ab
d = Distance between centre of an element a and node b
th
del
λ
= Element sensitivity threshold value (deleting)
th
add
λ
= Element sensitivity threshold value (adding)
tol = Convergence tolerance, 1e-5
m = Current iteration number
T = Number of iterations over which convergence is measured, 5
s
E = Young’s Modulus Solid
v
E = Young’s Modulus Void
TO = Topology Optimization
V* = Volume fraction constraint
SE = Strain Energy
SE
∆
= change in strain energy
y = Distance between the centre and a node of same element.
e
u
= Elemental displacement vector
e
k
= Elemental stiffness matrix
P = Penalization factor
e
x
= Elemental density distribution
Reviewed, accepted September 23, 2010
679

m
µ
= Displacement field at iteration m
min
ρ
= Parameter used to prevent singularity
m
ρ
= Density at the previous cycle
ζ
= Move limit
η
= Tuning parameter
m
Λ
= Lagrange multiplier at cycle m
f
H
)
= Convolution operator
),( yxL = Distance between centres of element x and y.
I Introduction
Additive manufacturing (AM) is a relatively recent approach to manufacturing
whereby a component is built up [1-3] layer by layer, usually from sliced 3D CAD
data. It is a contrasting approach to traditional manufacturing techniques such as
subtractive (e.g. machining) or formative (e.g. casting). This layer by layer approach
requires less manufacturing constraints. Its ability to build components with intricate
complexities opens up the design domain significantly, enabling the production of
optimal parts with improved structural performance. Established topology
optimization algorithms (TO) could be adopted for AM by relaxing constraints within
these algorithms originally meant for traditional manufacturing routes.
TO is a type of structural optimization that seeks the optimum layout of a design
by determining the number of members required and the manner in which these
members are connected. Unlike shape and size optimization, TO achieve designs that
are not greatly constrained by the nature of the initial design. Hence, TO is a better
route to take towards optimum parts. Several algorithms have been developed for TO.
These include homogenization [4, 5], solid isotropic microstructure with penalization
(SIMP) [6-8], and bi-directional evolutionary structural optimization (BESO) [9-11].
Stochastic algorithms used in the broader field of optimization have also been adopted
for TO, among which are genetic algorithms [12-14] and ant colony optimization
[15]. Optimum topologies depend on which of these algorithm is used, starting
design, finite element mesh, parametric settings. A detailed study of these is necessary
if AM’s design flexibility is to be completely exploited. In this paper, we investigate
the effects of these factors on optimum achieved by the SIMP and BESO algorithm,
since they have been widely implemented in literature to achieve practical designs.
II Method
SIMP and BESO are the most widely used TO algorithms, owing to their
efficiency and simplicity. The section describes the main features of these two
algorithms.
A BESO
The BESO algorithm is a combination of additive evolutionary structural
optimization (AESO) [9] and evolutionary structural optimization (ESO) [16]. Querin
et al [9, 11] originally proposed and implemented BESO to improve results and
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convergence time of both AESO and ESO. Huang and Xie [10] presented a different
version to BESO to solve compliance problems. BESO is a finite element based TO
method, where inefficient material is iteratively removed from a structure as efficient
material is simultaneously added to the structure. Figure 1 show a BESO flow chart
for the minimization of strain energy (SE) for a given volume fraction constraint (V*).
An evolution rate (ER), filter radius (FR), V* and design domain are supplied to the
algorithm. The ER is the rate at which the volume is allowed to change per iteration.
FR is a distance limit. Sensitivity values of nodes within FR from the centre of an
element are used to recalculate elemental sensitivity values of the same element when
filtering sensitivities. This is done to eliminate the occurrence of undesired
checkerboard patterns in optima.
Figure 1: BESO Flow chart
The design domain is discretized and a finite element analysis (FEA) is
performed. An initially fully (fig 2a) or partially (fig 2b) solid design domain has
often been used in past works. Where a partially solid design domain has been used,
the solid elements have been concentrated in a particular region of the design domain.
This sort of starting design is intuitive in nature and might have constrained the
Start
Input
ER, V*,
FR
Finite
Element
Analysis
Filter
Sensitivities
Solid
Element
s
th
dela
λλ
≤
th
adda
λλ
>
Set Property
to void
Set Property
to Solid
Yes
Yes
No
Yes
tolSE
<
∆
Stop
No
No
No
Yes
Target
V*
New Tar
get
Volume
Yes
No
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topology to a local minimum
. This is illustrated later by stochastically distributing
solid elements in the design domain.
(a)
(b)
Fig 2: Starting designs (a) fully solid start design
(b) Partially solid start design
Elemental sensitivities are then filtered by first distributing them into nodes, to
which they are connected using equation 1,
∑
∑
=
=
=
s
aa
s
aaa
b
V
V
1
1
λ
η
(1)
The sensitivity of node b,
b
η
is computed by finding all elements a connected to this
node, and averaging their sensitivities values
a
λ
according to equation 1. Elemental
sensitivities are then recomputed by finding nodes whose distance to the centre of an
element a, is less than or equal to FR. Sensitivity values of these nodes are averaged
according to equation 2 to obtain elemental sensitivity values.
∑
∑
=
=
=
s
bab
s
b
c
bab
a
dv
dv
1
1
)(
)(
η
λ
(2)
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Where
abab
dFRdv
−
=
)( .
Structural detail might be lost while eliminating
checkerboards in this manner
. These filtered sensitivities are averaged with values
they assumed in the previous optimization iteration to further improve these
sensitivities. The volume fraction (VF) of the design domain is checked against the
target volume fraction V*, if they were not equal a new iterative target volume is
computed equation 3,
mm
VV
=
+1
(1±ER) (3)
Elemental sensitivity values can then be ranked. Solid elements having sensitivity
values below
th
del
λ
are reclassified as having void property. Void elements have been
modelled in this work by multiplying the elemental stiffness matrix of element
concerned by 1e-12. This is done to reduce the stiffness contribution of these elements
before the global stiffness matrix is assembled. Reducing the structural stiffness this
way is a soft kill approach to TO. This has been adopted in this work to avoid
connectivity problems associated with hard kill [10] optimization procedures. The
numbers of void elements reclassified as solid brings the current volume of solid
elements to
1+
m
V at iteration m. The TO is repeated until SE
∆
is less than a specified
tolerance (tol) and the specified volume fraction is reached. SE
∆
is computed using
equation 4,
( )
∑
∑
=+−
=+−−+−
−
=∆
T
im
T
iiTmim
SE
SESE
SE
111
111
(4)
The mesh does not change from the start to the end of the TO. BESO algorithm might
be constrained by the starting design, ER, FR and the finite element mesh. In the next
sections we investigate these effects.
B SIMP
Rozvany et al [7] developed the SIMP algorithm to achieve practical designs for
generalized shape optimization (optimization involving higher volume fraction).
Figure 3 shows flow chart for the SIMP algorithm to minimize SE for a volume
fraction constraint. Sigmund [17] implementation of SIMP is described in this paper.
The compliance problem can be expressed mathematically as,
∑
=
=
N
eee
T
e
p
e
x
ukuxSE
1
)(:min ,
subject to *
)( V
VxV
o
= 10
min
≤
≤
<
e
x
ρ
(5)
The penalty factor P is a main feature of the SIMP algorithm. This factor
suppresses the occurrence of fractional densities in the optimum design. Its inclusion
has been justified by assuming a high expense of making fractional densities.
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Figure 3: SIMP flow chart
According to Zhou and Rozvany [6],
“the extra manufacturing cost of cavities
would increase with the size of the cavities if we consider a casting process
requiring some sort of formwork for the cavities”.
This is not true for AM, since
manufacturing cost of AM is independent [2] of part complexity. Most commercial
TO software have implemented SIMP for TO, hence raising questions about their
suitability for AM.
An initial distribution of density in the design domain is used as the starting
design. Sigmund suggests [17] an initial even distribution of these densities, Bendsoe
[4] terms it an initial guess.
There isn’t any rigorous mathematical proof for
choosing an even density distribution
.
It is unclear the sort of initial density
distribution implemented in commercial TO software.
In later sections we
investigate the effect of random density distribution. Using these densities, an FEA is
performed; the displacement vector from the FEA is used to calculate the SE
(equation 6) and sensitivities. These sensitivities are calculated by differentiating
equation 6 to get
ee
T
e
p
e
e
ukuxp
x
SE
1
)(
)(
−
−=
∂
∂
(6)
Initial
Design
Finite Element Analysis
Sensitivity Analysis
Low pass filtering
Update Elemental
Densities
Converge
Stop
Yes
No
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Sensitivities are filtered to eliminate the existence of checker board patterns in
the optima using equation 7,
f
f
N
ff
N
ffe
e
x
SE
xH
Hx
x
SE ∂
∂
=
∂
∂
∑
∑
=
=
)(1)(
1
1
)
)
)
(7)
where
),( yxLFRH
f
−
=
Other methods to eliminate checkerboard patterns [4, 18, 19] include perimeter
control, filtering the densities, patch smoothing, image processing, higher order
elements and monotonic scale length based control. Each of these techniques bring
new challenges into the TO. Filtered sensitivities are then used to update densities
using the optimality criteria method as expressed in equation 8. The process is
continued until convergence is reached.
{
}
},)1max{(
min1
ρ
ρ
ζ
ρ
mm
−
=
+
If
{
}
},)1max{(
min
ρ
ρ
ζ
ρ
mmm
B
−
=
{
}
}1,)1max{(
1mm
ρ
ζ
ρ
+
=
+
If
{
}
min1
,1,)1min(
ρ
ρ
ρ
ζ
ρ
mmm
≤
+
=
+
Otherwise
mmm
B
ρρ
η
=
+1
,
)()()(
011
mklmijijkl
p
mm
uExpB
µεερ
−−
Λ=
(8)
Again as with BESO, most applications of SIMP in the literature has been with a
constant mesh. The continuous change in topology during a TO suggest results might
by improved by an iterative mesh improvement.
III 2D Example
A cantilever plate problem (figure 4) is solved to show the effect of the different
aspects of the BESO and SIMP algorithm on the optimum topologies.
Figure 4: Cantilever plate
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The objective is to minimize
SE
for a
VF
constraint of 0.5. The cantilever plate
was meshed with 16000 quadrilateral elements. The left side of the plate is fixed
while a 100N force is imposed at the middle of the right side of the plate.
A BESO
Figure 5 shows results of BESO algorithm at different parametric settings and starting
point. FR was set at 3mm for ER=10% to achieve figure 5a. Reducing both FR and
ER to 1mm and 0.5% results in topology shown in figure 5b. Figure 5c is for a
random starting point where solid elements are randomly distributed in the design
domain. It can be seen from this figure that the parametric settings does affect the
nature of an optimal topology since figure 5a and 5c and different.
(a) (b) (c)
Figure 5: BESO topologies
(a) FR=3mm, ER=10%, solid start, SE=1.87Nmm
(b) FR=1mm, ER=0.5%, solid start, SE=1.84Nmm
(c) FR=3mm, ER=1%, random start, SE=1.82Nmm
Also, the random start converged to a different topology with lower SE. Hence the
starting point also affects the nature of the optima. Starting with a fully solid domain
moved the TO towards a local minimum. Figure 5c might not be the best design
possible for this problem as it differ from known Mitchell’s [7] analytical solutions to
this problem. A detail study of the parameters and starting point might improve the
BESO algorithm.
B SIMP
Figure 6 shows the SIMP topologies for different starting points and parametric
values. As shown in Figure 6a, a low FR of 1mm does not totally eliminate
checkerboard pattern. If both the FR and P are kept at 1 (Figure 6b), checkerboards
disappear but there is a large grey region owing to unpenalized intermediate densities.
While the main feature of SIMP is to steer TO towards feasible topologies without
grey region, there might be an interpretation for intermediate densities in AM as this
has the lowest SE.
If P=3 and FR=1.5, the grey regions disappear (Figure 6c).
Unlike figure 6(a-c), figure 6d was carried out with a random starting point. As
before symmetrical constraint were imposed to attain figure 6d by ensuring the
density distribution of the upper half of the plate was a mirror image of the lower half.
This topology (figure 6d) is characterized by an SE of 3.8Nmm which is lower than
SE
for figure 6a and 6c where and even density distribution was used at the start of
the TO.
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(a)
(b)
(c) (d)
Figure 6: SIMP topologies
(a) P=3, FR=1mm,
SE
=4.0Nmm, 0.5 even density start
(b) P=1, FR=1mm,
SE
=3.3Nmm, 0.5 even density start
(c) P=3, FR=1.5mm,
SE
=3.9Nmm, 0.5 even density start
(d) P=3, FR=1.5mm,
SE
=3.8Nmm, Random density start
IV 3D BESO application
The BESO algorithm is used to solve a practical 3D problem of an aerospace
arm shown in figure 7. All degrees of freedom are constrained in surface A, while
pressure of 594KPa is imposed on surface B. This arm is meshed with approximately
300,000 tetrahedral elements to investigate the effects of changing FR and ER on the
optima. The red part is set to non-design while the green is the design domain.
Figure 7: 3D Aerospace metallic arm
Two sets of experiments are conducted, in the first set the ER is set at 5%, while
the FR is varied by setting it as a multiple (FRF-filter radius factor) of the distance
between the centroid of a tetrahedral element and a node on that element. This factor
is set at 1.5, 2.0, 2.5, 3.0. The FRF is fixed at 2.0 while the ER is set at 1%, 3%, 7%
and 10% in the second set experiments.
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A FR Results
Figure 8 shows the optimum topologies for the different FRF. A plot of
SE
against iteration is shown in figure 9.
(a)
(b)
(c) (d)
(e)
Figure 8: Optimum topologies for FR Experiment (ER=5%)
(a) FRF=1.5, (b) FRF=2.0, (c) FRF=2.5, (d) FRF=3.0 (e) Unfiltered
Figure 9:
SE
against iteration (FR)
The Structures appear truss-like but are significant different since the location and
number of trusses are significantly different. The number of trusses decrease as the
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FRF is increased. This orients the BESO algorithm towards a less optimal part as
shown in Figure 9 and 10.
Figure 10:
SE
and Number of members against FRF
B ER Results
Fig 11 shows the results for the ER experiments. Optimum topologies appear
truss-like, similar to results for FR experiments. The locations of these trusses are
again different, significantly dependent on the ER. Though the graph of
SE
against
ER suggest a quadratic relationship (figure 12), the TO is less sensitive to increasing
ER as compared to FR.
(a)
(b)
(b)
(d)
Fig 11: Optimum topologies for ER Experiments (FRF=2):
(a)
ER=1%, (b) ER=3%, (c) ER=7%, (d) ER=10%
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Figure 12:
SE
and Number of members against ER
IV Conclusion
Though BESO and SIMP attains optimal topologies efficiently, these topologies
are often local since the TO is constrained by starting design, finite element mesh and
parametric values. These less complex optima are suitable for traditional
manufacturing; AM’s ability to make complex parts allows the production of truly
optimal parts. Hence this algorithms needs to be improved for AM.
In the BESO algorithm an increase in FR decrease the complexity in the part
and orients the TO towards less optimum topologies. In the three dimensional case,
the FR has a greater influence on the TO than the ER. The optimum ER occurs at 6%,
though more data point is needed to confirm this.
The basis for inclusion of a penalty factor in the SIMP might be inappropriate
for AM, since its manufacturing cost is independent of complexity. A different
penalization approach might resolve this problem.
The stagnant mesh often used for the iterative FEA in both BESO and SIMP
needs to be improved. Numerical errors might have constrained the TO, since
topologies achieved while solving the cantilever problem differ significantly from
Michell’s analytically optima [7]. Also, the issue of different optima topology for
different starting point might have been caused by these errors. While mesh
refinement might solve this, computational cost limits resolution attainable. An
adaptive mesh improvement strategy could be incorporated in these algorithms to
focus refined elements at new boundaries and coarsen mesh away from boundaries.
Old void elements could be purge from the design domain to allow the introduction of
smaller new elements at the boundaries while reducing computational cost. Further
work would investigate these proposed amendments.
These improvement might cause the TO to efficiently attain highly complex
topologies with improved performance. These topologies can be made via AM.
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