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Throughput and Strength Optimization for
Fused Deposition Modeling of Ankle-Foot Orthotics
By
Robert Chisena
Dian-Ru Li
ME555-Winter2016
Project Final Report
April 22, 2016
1 INTRODUCTION
1.1 Aim
Orthotics are medical devices used to help patients with various muscle deficiencies maintain
alignment during walking or sitting. The current process for producing custom orthotics is a time-
consuming and labor intensive manufacturing process with lead times of up to two-
weeks. Additive Manufacturing (AM) has made significant impacts on our ability to quickly and
accurately prototype custom parts with high complexity and detail. Fused-deposition modeling
(FDM), an inexpensive and reliable form of AM, has been suggested as a possible solution to
reducing custom orthotic manufacturing time [1]. One of the steps in creating a viable business
model around FDM orthotic manufacturing is minimizing the manufacturing time and maximizing
strength subjected to various manufacturing constraints.
In this project, we are optimizing the FDM manufacturing time and strength by manipulating
key variables, constraints, and parameters associated with the process. Variables such as
orientation angle, infill percentage and layer height can all affect the overall time for FDM printing.
Additionally, the type of infill plays a large role in the required AM time and strength. Jin proposed
using a wavy infill to reduce infill time [2]; however, the paper does not suggest the optimum
parameters of the wave. In this paper, we will attempt to answer the following two research
questions:
(1) What are the optimum wavy toolpath parameters that minimize layer time and maximize
strength (Robert Chisena)?
(2) What are the optimum set of build parameters that minimize the AM manufacturing time of
a solid ankle-foot orthotic (AFO) (Dian-Ru Li)?
1.2 Technical Features
Fused-deposition modeling (or 3D printing) is a process that builds three-dimensional parts by
stacking layers one-by-one. These layers are created by depositing lines, or roads, of molten plastic
through a toolhead, which is able to travel in the XY-plane (Figure 1). This material is usually in
the form of 1.75 mm round filament wrapped around a spool. This filament is then fed into the hot
end of the toolhead with a gear drive.
Figure 1. FDM Desktop TAZ 5 3D Printer by Lulzbot.
Additive manufacturing creates objects using many successive layers. To print, slicing
software called Simplify-3D was used to convert the computer-aided design (CAD) model into
printing instructions called G-Code. After importing the CAD model, the software slices the model
into multiple layers based on a given layer height. Upon slicing the model, the infill pattern is
assigned based on user settings. Figure 2 shows the process used in Simplify-3D to create the print
parameters for this project.
Figure 2. Simplify-3D Interface with Orthotic Toolpath
The path taken by the toolhead plays an important role in the overall time and strength of the
finished part. Common toolpaths include contours, rectilinear, and honeycomb fills (Figure 3).
However, these toolpaths are used without concern for the part being created. When using a
rectilinear infill pattern, for instance, the toolhead must accelerate to its normal operating velocity
and then decelerate to a stop at the end of each road. Because there are often thousands of stops
within a large part, the required accelerations result in large inefficiencies in time and energy.
Figure 3. (Left to Right) Contour Infill made by offsetting each road. Raster fill. Honeycomb fill.
Other build parameters that are important to the time of FDM additive manufacturing include
thickness of the layers, raster width (synonymous with beadwidth and contour width), maximum
toolhead speed, and part orientation (Figure 4). Each of these parameters can be altered within a
slicing software such as Simplify 3D; however, since so many tuning parameters exist, it is difficult
to determine which parameters affect the printing time the most.
Figure 4. (a) Longitudinal and Medial Angles and Location of the Part on the Print Bed are important
considerations. (b) Layer thickness plays an important role in manufacturing time because an increased layer height
reduces the total of overall layers. (3) Various parameters associated with a raster fill layer.
2 SYSTEM OVERVIEW
2.1 Notations
All symbols to be used in this project and units for each quantity are given in Table 1.
Table 1. Notations used for Optimization Project
Symbols Definition Unit
Su
bsy
stem
1
L Length of part mm
W Width of part mm
H Height of part mm
TA Actual manufacturing time min
TE Estimated manufacturing time min
TI Ideal printing time min
TPi Tool path of i layer mm
α Part build orientation (medial axis) degree
θ Part build orientation (longitudinal axis) degree
N Number of layer -
HL Layer height mm
I Infill percentage %
Is Support infill percentage %
WE Extruder width mm
WC Contour width percentage %
V Toolhead speed mm/min
WP Weight of the part g
WS Weight of the support material g
A Infill angle (raster angle) degree
O Outline overlap %
NC Number of contours -
Su
bsy
stem
2
σ Flexural stress MPa
SW Flexural Strength-to-weight ratio MPa/g
ext Extension mm
f Frequency 1/mm
Int Interference %
δ Interference Region mm
BW Beadwidth mm
OL Overall Length of Coupon mm
OT Overall Thickness of Coupon mm
t time per layer sec
x Distance along coupon mm
Lss Length of Support Span mm
DLN Distance between load-applying noses mm
W Weight of the coupon sample mm
T Thickness of AFO mm
2.2 System Description
The aim of this project is to minimize the additive manufacturing time and maximize the strength
of an AFO with a wavy infill. We approach the optimal solution by dividing an additive
manufactured AFO into two subsystem: (1) per layer parameters and (2) macro-system build
parameters. Each of these subsystems can be further divided into another two subsystems, strength
and time (Figure 5). In this project, we have limited our focus to optimizing the manufacturing
time for the macro-system and the strength for each layer.
Figure 5. System Level Design Optimization Problem for Optimizing Strength and Time of AM AFO.
Red box defines the scope of this project.
2.2.1 Subsystem 1 - Wavy Tool Path Strength Optimization
In this subsystem, the goal was to design an optimized infill pattern that maximizes strength while
minimizing the time required to complete the toolpath on each layer. According to Jin, a toolpath
that uses a sine wave between two outer contours will reduce the accelerations required to start
and stop the toolhead [2]. Reducing accelerations on the machine reduces print time, machine wear,
and energy consumption. Furthermore, the sine wave behaves similar to a truss bridge, a robust
structure that can dynamically and efficiently redistribute loads across its structure.
2.2.2 Subsystem 2 - AFO Build Parameters
In this subsystem, we aim to minimize the manufacturing time by adjusting the build parameters.
Those parameters include layer height, infill pattern, contour width, etc. Altering these parameters
will generate different tool paths per layer from Simplify3D based on the part geometry, and also
change the manufacturing time. The ideal objective time function can be described as follows:
𝑇𝐼 =∑ 𝑇𝑃𝑖
𝑁𝑖=1
𝑉
Figure 6. Example of a Warren Truss Bridge.
where V is the toolhead speed adjusted depending on users and 𝑇𝑃𝑖 is the tool path at i layer.
However, in the real practice, the manufacturing time actually depends on the velocity control
algorithm of the machine. More specifically, there exist lots of stopping points where the extruder
changes its printing direction to fill in the materials within the contours. The extruder will decrease
the speed down to zero on the stopping points and then accelerate to the given tool head speed (V).
Also, the algorithm implemented in Simplify3D will adjust the printing speed according to
different geometry and parameters sets for better printing quality of part, and thus the speed won’t
always stay on the maximum (the V we assigned). Therefore, the above equation fails to reflect
the real manufacturing time.
Due to the difficulty of describing the subsystem in a simple physical equation, we use data-
driven modeling technique to derive our objective function. Several experiments are performed to
capture system behavior for finding the minimal manufacturing time in our design space. The
methodology and results will be discussed in section 4.
3 SUBSYSTEM OPTIMIZATION – SUBSYSTEM 1 In the printing of thin-walled features such as orthotics, reducing time while maintaining overall
part strength is important. Jin proposed using a wavy structure that would use a sine wave to fill
in the areas between thin-walled features [2].
3.1 Design Variables and Parameters The wavy infill pattern is akin to a truss bridge that uses truss elements to dynamically support
compressive and tensile stresses. In a truss bridge, a number of design variables affect the strength
of the bridge: the number of links, width of each link, length of the links, and the strength of the
connections between the links and the outer structure. Likewise, in a wavy toolpath pattern, the
frequency of the wave, the beadwidth, the overall thickness, and the interference (or connection)
Figure 7. Wavy Tool Path for Ankle-Foot Orthotic Cross-Section [2]
between the wave and the outer structure are important design considerations for the strength of a
coupon (Figure 8).
The constraints on this subsystem were indicated by four bounded, continuous, inequality
constraints on each of the variables (Table 2). Frequency of the sine wave is defined by radians
per unit length and was constrained between 0.05 and 2 radians/mm. Beadwidth is the width of the
material extruded by the machine and is constrained between 1 and 1.55 mm. Outer thickness is
the distance between the outer contours of the coupon and was constrained between 5 and 15mm.
Interference is the overlap between the sine wave and the outer contour. In practice, interference
is conveyed as a percentage of the sine wave beadwidth within the contour beadwidth. For
example, 100% interference means that, during overlap, the sine wave passes through the center
of the outer contour beadwidth.
After a few experiments, it was determined that the minimum interference before the wave
separates from the contour is about 60%. Similarly, above 90% interference, the wave interfering
with the contour caused material “pushout.” However, because the part was still functional after
90% interference, the upper bound of the interference constraint was maintained at 100%. If a
designer decides that the material pushout is unacceptable for his or her design, the allowable
design region can be reduced.
The parameters of the design included overall length of the specimen, material-type used,
toolhead velocity, temperature and diameter of the nozzle, and orientation of the part in the print
bed. Furthermore, the four-point bending test experimental setup was kept constant throughout the
experiments.
Figure 8.Wavy Infill Parameters
Table 2. Design Variables and Parameters for the Wavy Infill Subsystem
Notation Type Unit Value Description
Variable
Frequency, f continuous radians/mm 0.05-2 Frequency of the internal infill sine wave.
Thickness,T continuous mm 5-15 Thickness of the thin-walled structure.
BeadWidth,
BW continuous mm 1-1.55 Thickness of the deposited filament bead.
Interference,
Int continuous % 60-100
Intereference between the sine wave
beadwidth and outer contour
Parameters
Length, OL - mm 120 Length of the thin-walled part
Layer Height - mm 0.5mm Layer height of Coupon Samples
Build
Velocity - mm/s 6000mm/s Number of contours
3.2 Objective Function
3.2.1 Data Collection and Experimental Setup
Our partner, Stratasys, a company that develops fused-deposition modeling machines,
manufactured coupon samples built with the wavy infill pattern. The following design variables
were altered: frequency, f, overall thickness, OT, beadwidth, BW, and interference, Int. To properly
sample the design space, a Latin-Hypercube sampling method was used. Ten wavy coupon sample
experiments were created between continuous upper and lower bound constraints on each of the
variables (Table 3).
Table 3. Experiments sampled from the design space.
Experiment Beadwidth
[mm] Outer Thickness
[mm] Frequency
[wave/mm] Interference
[%]
1 1.45 10.39 0.88 93
2 1.24 13.01 1.89 66
3 1.50 6.47 1.17 68
4 1.13 8.52 0.12 74
5 1.35 7.20 0.42 100
6 1.18 11.77 0.71 92
7 1.39 9.50 1.29 81
8 1.28 5.10 1.66 86
9 1.11 12.43 1.44 77
10 1.01 14.16 0.44 63
Each coupon was weighed, and a four point-bending test was performed to determine the
coupon’s resistance to bending force. The bending test was performed according to ASTM
Standard D7264 [3]. Nose length, LN , was fixed at 30 mm, and support span length, LS , was fixed
at 80 mm. Coupon sample strengths were compared to each other by dividing the stress by the
weight of each specimen.
Figure 10 shows the bending deflection versus stress-to-weight ratio. The maximum stress-to-
weight ratio was determined, and therefore, the strength-to-weight-ratio of each coupon was found.
Table 4 shows the weight and the strength-to-weight-ratio values for each experiment.
Figure 10. Results on Bending Test Performed on LHS Samples
LN
LS
Figure 9. Four-Point Bending Test Setup. LN is the nose span distance, which
was 30mm. LS is the support span distance, which was 80 mm.
Table 4. Strength-to-weight ratios from four-point bending tests.
Experiment Weight [g] Strength-to-Weight Ratio [MPa/g]
1 8.6 1.362
2 13.3 0.955
3 6.9 1.906
4 4.3 0.519
5 5.6 1.402
6 6.9 1.434
7 9.0 1.230
8 6.4 1.732
9 9.6 2.148
10 5.2 0.749
3.2.2 Model Construction
Using a Neural Network, the variable inputs were mapped to the variable outputs (Figure 11).
Figure 12 and Figure 13 show the performance of the Neural Network in fitting the data set and
the histogram of data points. We can see that there is a relatively good fit between the neural
network and the actual data points. More points need to be added to the sample set to increase the
goodness of fit, but due to time and material constraints, the number of experiments that could be
run was limited.
Input
4
Output
1
Hidden Layer Output Layer
10 4
Figure 11. Neural Network Setup for Wavy Infill Subsystem Data
Figure 13. Error Histogram chart for the Wavy Experiment Neural Network
3.3 Constraints The following constraints will be applied to wavy toolpath optimization problem:
Beadwidth (mm): 1 ≤ 𝐵𝑊 ≤ 1.55
Thickness (mm): 5 ≤ 𝑂𝑇 ≤ 15
Frequency (f): 0.05 ≤ 𝑓 ≤ 2
Interference (Int) %: 60 ≤ 𝐼𝑛𝑡 ≤ 100
Figure 12. Performance of the Neural Network Fit on
Experimental Data
3.4 Summary Model The summary model of the wavy infill pattern is:
Min −𝑆𝑊(𝐵𝑊, 𝑂𝑇, 𝑓, 𝐼𝑛𝑡)
(Neural network function)
Subject to 𝑔1(BW) ∶ 1 ≤ BW ≤ 1.55
𝑔2(OT) ∶ 5 ≤ OT ≤ 15
𝑔3(𝑓) ∶ 0.05 ≤ 𝑓 ≤ 2
𝑔4(Int) ∶ 60 ≤ Int ≤ 100
3.5 Results
3.5.1 Optimization Results
In this subsystem, a Matlab nonlinear constrained optimization programming solver package was
used to optimize the strength function. Sequential quadratic programming (SQP) method with
BFGS was used to find the optimal function value subject to given the constraints of the system.
The results of the optimization process for the wavy toolpath infill are shown in Table 5.
Table 5 Optimal Results for the Wavy Infill Toolpath
These results indicate that the optimum value will increase the strength-to-weight ratio of the
coupon sample by 30%. In regards to the variables, the only variable that is active is the frequency
because it achieves the upper limit at the optimum. A higher frequency will yield a stronger coupon
because the effective area of the coupon increases.
Counter to intuition, a higher interference value does not necessarily yield a higher part
strength. In fact, a higher interference value might negatively affect the strength of each part
because the high amount of interference might cause stress risers in the outer contour.
An interesting preliminary result can be seen when considering the time required to print each
part. Although higher frequencies create stronger parts, the time required to print these parts is
higher. Experiment 5 was an example of the trade-off between strength and time. The experiment
had one of the highest strength-to-weight ratios yet had one of the shortest print times making it
one of the likely final candidates for the design of the AFO.
BW [mm] OT [mm] f [1/mm] Int [%] SW [MPa/g]
Optimal Result 1.05 12.5 2 76 2.82
3.5.2 Model Evaluation
The optimization was run for multiple initial points in the feasible domain to determine
convergence. At each initial point, the same optimum value was found indicating that the data
driven objective function is convex in the design region. Additional experiments were run near the
optimum point to determine the convergence of the model. The results from perturbing the
optimum point are shown in Table 6. When the optimum is perturbed beyond the maximum
frequency, the strength-to-weight ratio of the coupon is expected to increase.
Table 6. Strength-to-Weight Ratios from Perturbations from the Optimum
The relatively few data-driven points in this experiment allowed the optimization problem to
be solved easily and without convergence issues. However, when more variables are added such
as performing four-point bending tests in multiple orientations, the function will become more
complex and convergence might become an issue.
Validating the optimum point with another coupon is the next step in the project. The CAD
data for the optimum coupon has been sent to Stratasys and will be received by the end of the week
for testing and validation of the model. Most likely, validating the model will be an iterative
process. According to one of the head engineers at Stratasys, the sharp radius of curvature caused
by the high frequency prevents proper interference between the wave and the outer contour. This
improper bonding may cause a weaker part than the model projects. If this is the case, the
additional point will be fed into the neural network and the optimization problem will be run again.
Until the model predicts experimental strength values with more accuracy, more data points will
be added.
Finally, because experimentation is costly and time-consuming, a computer FEA will be
generated to model the various variables in the design problem. The experiments that have been
performed thus far will be used to validate the computer model.
3.5.3 Post Analysis
Because only bounded constraints existed for this subsystem, the optimal design variables were
checked to determine constraint activity. The optimal frequency value was at the upper limit and
was the only active constraint in the problem. To determine the subsystem’s sensitivity to this
active constraint,𝜇𝑓, the upper frequency bound was relaxed and the function before and after
relaxation was compared:
Perturbation Beadwidth
[mm]
Outer
Thickness
[mm]
Frequency
[wave/mm]
Interference
[%]
Strength-to-Weight
Ratio [MPa/g]
-30% 0.735 8.75 1.4 53.2 1.81
-20% 0.84 10 1.6 60.8 1.90
-10% 0.945 11.25 1.8 68.4 2.30
Optimum 1.05 12.5 2 76 2.82
10% 1.155 13.75 2.2 83.6 2.85
20% 1.26 15 2.4 91.2 2.82
30% 1.365 16.25 2.6 98.8 2.78
𝜇𝑓 = Δ𝑆𝑊
Δ𝑓,
where Δ𝑆𝑊 is the change in the objective function and Δ𝑓 in the change in the frequency value.
With a 25% relaxation of the upper bound constraint, 𝜇𝑓 had a value of 0.70, which indicates the
sensitivity of this constraint.
Although it would have been interesting to alter some of the machine parameters such as layer
height, orientation angle, or the build temperature and speed, the printer settings could not be
changed unless agreed upon by our collaborator. Another interesting test would have been to differ
the material being used in the experiment.
4 SUBSYSTEM OPTIMIZATION – SUBSYSTEM 2 This subsystem aims to find the optimal macro-system build parameters of a 3D printing AFO.
Currently, we focus on the parameters that will affect the manufacturing time, but we also consider
some parameters that may cause differences in strength for future optimization. Neural network is
first used to construct the objective function, which is then optimized with a constrained
optimization procedure. Several experiments are performed to collect the data and then validate
the optimal results. The following sections provide the detailed descriptions of this subsystem.
4.1 Design Variables and Parameters Some studies has identified the crucial parameters will affect the AM manufacturing time,
including layer height, infill percentage, contour width and toolhead speed [4,5]. On the other hand,
the orientation angle will change the direction of layers (as shown in Figure 14), and we claim that
the different layer arrangement will affect the strength under the same loading from the same
direction. Therefore, we choose the 5 build parameters as our design variables that will affect the
manufacturing time and the performance of AFO, while the others maintain default. The type, unit,
value and description of variables and parameters are as follows:
Table 7. Design Variables and Parameters for AFO Build Parameter Subsystem
Notation Type Unit Value Description
Variable
α continuous degree 0-20 Part build orientation (medial axis): When
rotating α while extruder printing direction
is the same.
HL continuous mm 0.1-0.4 Layer height: This variable dominantly
affect surface roughness, which will be
evaluated in the future.
I continuous % 20-80 Infill percentage: The solid level of each
layer.
WC continuous % 100-300 Contour width percentage: The percentage
of extruder width.
V continuous mm/min 2000-4000 Tool head speed: Moving speed of extruder
Parameter
L - mm 59.07 Length of part: Due to the limitation that we
are unable to print the real size of orthotic in
the lab, we scale it to 40%
W - mm 121.18 Width of part
H - mm 163.43 Height of part
θ - degree 0 Part build orientation (longitudinal axis)
Is - % 30 Support infill percentage
WE - mm 0.4 Extruder width
A - degree 45/-45 Infill angle (raster angle): The angle will
change after printing one layer.
O - % 15 Outline overlap: The percentage of overlap
between infill and outer shells.
NC - 2 Number of contours
Figure 14. Printing Results with respect to The Changes of α.
4.2 Objective Function
4.2.1 Data Collection
In this subsystem, we had two steps to define the objective time function: 1) Design the
experiments for sampling 5 variables. 2) Acquire the estimated manufacturing time from
Simplify3D software. The 5 variables act as the inputs in our time function while the estimated
manufacturing time is the corresponding output. After collecting data points, a data-fitting method
is used to construct the model function.
4.2.1.1 Experiment Design Latin Hypercube Sampling (LHS) method is used to sample those 5 variables in the design space.
Each variable is sampled within the bounded constraints: 0 ≤ α ≤ 20, 0 ≤ H𝐿 ≤ 0.4, 20 ≤ I ≤
80, 100 ≤ W𝐶 ≤ 300, 2000 ≤ V ≤ 4000. These bounded constraints are assigned based on the
limitation of a 3D printer and used to construct the feasible domain in the design space. Besides
LHS, we also generate some additional experiment sets to get more data points around the possible
optimal region associated with these variables. Table 8 shows the experiment sets of 5 variables
that will be used to generate the AFO part, get the manufacturing time, and then derive the
objective time function in this subsystem. The complete table is shown in Appendix 8.2.
Table 8. Experiment Sets of 5 variables in This Subsystem
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min]
1 16 0.28 79 130 2325
2 13 0.20 65 221 3770
3 10 0.24 52 135 2073
4 5 0.39 71 206 3843
… … … … … …
38 1 0.36 32 169 3869
39 1 0.39 31 194 3914
40 0 0.37 38 152 3949
41 0 0.37 37 186 3822
42 1 0.37 35 179 3979
4.2.1.2 Estimated Manufacturing Time (𝑻𝑬)
We use Simplify3D to generate the tool path for AFO printing. By adjusting the part to the
appropriate printing position (around center of printing bed), the software will automatically
generate the tool path and support material for different variable sets (as shown in Figure 15).
Furthermore, the software will provide user the TE value. In this project, we use TE as our model
outputs since it is time-consuming to get sufficient data points on the actual printing time TA which
is often longer and could up to 9 hours. Although TE will always be less than TA, the trends between
each other should be consistent, and then we could use TE to represent the manufacturing time in
this subsystem. However, TA from the real printing procedure will be used to validate TE and the
final optimal results. The complete table is shown in Appendix 8.3.
Table 9. Experimental Results (TE) of 5 Variables in This Subsystem.
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min]
1 16 0.28 79 130 2325 279
2 13 0.20 65 221 3770 286
3 10 0.24 52 135 2073 327
… …. …. …. …. …. ….
40 0 0.37 38 152 3949 129
41 0 0.37 37 186 3822 137
42 1 0.37 35 179 3979 136
Figure 15. Estimated Printing Results with Different Printing Parameters in Simplify3D. The real tool head speed
will change according to the parameters and geometry per layer in a percentage of given tool head speed (V).
4.2.2 Model Construction
Neural network is a data-fitting technique to derive the model function within a given data set
where the exact relationship may not be apparent. Several neurons (transfer functions) are strung
together to form the final model function using a neural set to fit the data. In this subsystem, after
testing the function performance with different numbers of neuron, 5 neurons are used to model
the system.
Figure 16. The Schematic Diagram of Neural Network Function in This Subsystem.
As shown previously, we generate 42 experiments to sample 5 variables. However, we found
that although the derived model from these experiments perfectly fit the data points with the high
correlation coefficient, the function outputs are far from the results from the real model. For
example, we can get an extremely low time which is impossible in the reality, which means there
is a sharp valley curve between two data points with relatively low manufacturing time. Too few
data points around possible optimum caused data under fitting in the local region, while too many
data points that are not around the optimum lead to data overfitting in the final function. Therefore,
17 of 42 experiment sets (as shown in Table 10) with the manufacturing time lower than 200
minutes are chosen to construct the model within a smaller design space since we now only focus
on the feasible domain that may contain the optimal solution and remove those impossible designs
to avoid data overfitting. The complete table is in Appendix 8.4.
Table 10. Experimental Results (TE) of 5 Variables in This Subsystem (Final Sets).
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min]
1 5 0.39 71 206 3843 135
2 15 0.35 67 258 3464 176
3 14 0.31 61 181 3551 198
4 1 0.32 71 161 2889 174
5 19 0.37 76 208 3302 176
… … … … … …. …
14 1 0.39 31 194 3914 130
15 0 0.39 38 152 3949 129
16 0 0.37 37 186 3822 137
17 1 0.37 35 179 3979 136
After inputting the data points to neural network, the objective time function is derived with
these 5 variables (as shown in Appendix 8.5). Figure 17 and Figure 18 show the regression plot
and error histogram, respectively. With a high correlation coefficient up to 0.88, the derived
function is able to capture the system behavior. The model evaluation is performed in section 4.5.2
to validate the model (function outputs) with real model (TE). Finally, this time function is then
optimized the find the optimal build parameters with the minimal manufacturing time.
Figure 17. Regression Plot of Data-driven Objective Function.
140 160 180 200130
140
150
160
170
180
190
200
210
Target
Ou
tpu
t ~
= 1
.2*T
arg
et
+ -
21
Training: R=0.88235
Data
Fit
Y = T
140 160 180
130
140
150
160
170
180
190
Target
Ou
tpu
t ~
= 0
.81
*Ta
rge
t +
29
Validation: R=0.98251
Data
Fit
Y = T
130 140 150 160 170
130
135
140
145
150
155
160
165
170
Target
Ou
tpu
t ~
= 0
.84
*Ta
rge
t +
22
Test: R=0.99616
Data
Fit
Y = T
140 160 180 200
130
140
150
160
170
180
190
200
210
Target
Ou
tpu
t ~
= 1
*Ta
rge
t +
1.8
All: R=0.8859
Data
Fit
Y = T
Figure 18. Error Histogram of Data-driven Objective Function
4.3 Constraints Due to the limitation and recommendation of printer setting, there are some limitations on variables
in this subsystem, which have already been used to create the design space with LHS methods.
However, to avoid overfitting discussed previously, we defined a smaller feasible domain in the
design space according the following bounded constraints:
Orthotic orientation angle:
0 ≤ α ≤ 19
Layer height:
0.31 ≤ H𝐿 ≤ 0.4
Infill percentage:
28 ≤ 𝐼 ≤ 80
Contour width percentage:
100 ≤ W𝐶 ≤ 281
Tool head speed:
2419 ≤ V ≤ 4000
0
0.5
1
1.5
2
2.5
3
Error Histogram with 20 Bins
Ins
tan
ce
s
Errors = Targets - Outputs
-35.0
5
-32.7
-30.3
6
-28.0
2
-25.6
8
-23.3
4
-20.9
9
-18.6
5
-16.3
1
-13.9
7
-11.6
3
-9.2
83
-6.9
41
-4.5
99
-2.2
57
0.0
8554
2.4
28
4.7
7
7.1
12
9.4
54
Training
Validation
Test
Zero Error
4.4 Summary Model The summary model of subsystem 2:
Min 𝑇𝐸 = 𝑇𝑖𝑚𝑒(𝛼, 𝐻𝐿 , 𝐼, 𝑊𝐶 , 𝑉)
(Neural network function)
Subject to 𝑔1(α) ∶ 0 ≤ α ≤ 19
𝑔2(H𝐿) ∶ 0.31 ≤ H𝐿 ≤ 0.4
𝑔3(I) ∶ 28 ≤ 𝐼 ≤ 80
𝑔4(W𝐶) ∶ 100 ≤ W𝐶 ≤ 281
𝑔5(V) ∶ 2419 ≤ V ≤ 4000
4.5 Results
4.5.1 Optimization Results
In this subsystem, we use the nonlinear programming solver (fmincon) in MATLAB to optimize
the objective time function. The sequential quadratic programming (SQP) method implemented in
the solver is used to optimize a constrained problem with a multivariable function for improving
the global convergence. The function outputs will be further discussed in 4.5.2. Here we first
demonstrate that the minimal estimated manufacturing time is achieved with the optimal variable
sets within the bounded constraints (as shown in Table 11).
Table 11. Experimental Results (TE) of 5 Variables and the Optimal Solution in This Subsystem.
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min]
1 5 0.39 71 206 3843 135
2 15 0.35 67 258 3464 176
3 14 0.31 61 181 3551 198
4 1 0.32 71 161 2889 174
5 19 0.37 76 208 3302 176
6 6 0.34 60 190 3179 171
7 3 0.38 69 158 2622 160
8 8 0.39 73 173 2419 176
9 7 0.31 49 143 3077 191
10 13 0.36 28 281 3796 162
11 1 0.38 80 100 4000 134
12 0 0.40 33 157 4000 126
13 1 0.36 32 169 3869 141
14 1 0.39 31 194 3914 130
15 0 0.39 38 152 3949 129
16 0 0.37 37 186 3822 137
17 1 0.37 35 179 3979 136
Optimal Results 1 0.4 80 218 4000 124
Several variable sets are used to print a scaled AFO (40% in this study) to get the actual
manufacturing time (TA) for validating the optimal results. (The printer is LulzBot TAZ 5 and with
additive features built in our lab.) Table 12 shows that our optimal variables can really achieve the
optimum in the real printing. On the other hand, Figure 19 illustrates that the trends of TE and TA
are consistent, proving the feasibility on utilizing TE to derive the objective function and find the
optimal solution in real printing.
Table 12. Experimental Results (TE) of 5 Variables and the Optimal Solution in This Subsystem.
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min] TA [min]
1 13 0.13 53 121 3353 463 598
2 14 0.17 78 110 2253 466 567
3 12 0.30 56 287 3160 203 226
4 0 0.40 50 200 2500 146 158
5 1 0.38 80 100 4000 134 163
Optimal Results 1 0.4 80 218 4000 124 140
Figure 19. The Actual Manufacturing Time and Estimated Manufacturing Time for Different Variable Sets.
0
100
200
300
400
500
600
1 2 3 4 5 6
Tim
e (m
in)
Experiment No.
Actual manufacturing time Estimated manufacturing time
Optimal Results
4.5.2 Model Evaluation
In the optimization procedure, we identified three constraints are active as Table 13 shown. These
three constraints dominate the optimal results and we will perform the sensitivity analysis on them.
Table 13. Inactive and Active Constraints in Subsystem 2.
Inactive Active
0 ≤ α ≤ 19 0.31 ≤ H𝐿 ≤ 0.4
: 100 ≤ W𝐶 ≤ 281 28 ≤ 𝐼 ≤ 80
2419 ≤ V ≤ 4000
Some model evaluations were performed to validate the results between the outputs from
neural network function and the time (estimated time) from the real model. Firstly, the exploration
in the design space was performed with different initial points. As Table 14 shows, the values from
neural network function was closed to the estimated time and have the same trend when changing
the initial points. Also, two local minimums have been identified. The main difference in these
two optimums was the combination with I and WC, which is reasonable since the larger contour
width can achieve the higher infill percentage with the same manufacturing time. The optimal
solution is the variable set with the function minimum 116.7753.
Table 14. The Function Minimum and Estimated Time in Real Model with Different Initial Points.
No.
Initial Points Optimal Points
Function
Minimum [min]
Estimated
Time [min] α
[deg]
HL
[mm
]
I
[%]
WC
[%]
V
[mm/
min]
α
[deg]
HL
[mm
]
I
[%]
WC
[%]
V
[mm/
min]
1 2 0.35 40 200 3500 2 0.4 42 100 4000 119.2249 128
2 2 0.35 60 200 3500 2 0.4 42 100 4000 119.2249 128
3 2 0.35 70 200 3500 1 0.4 80 218 4000 116.7753 124
4 2 0.35 75 200 3500 1 0.4 80 218 4000 116.7753 124
5 10 0.35 75 200 3500 1 0.4 80 218 4000 116.7753 124
6 10 0.32 75 200 3500 1 0.4 80 218 4000 116.7753 124
7 10 0.32 75 110 3500 2 0.4 42 100 4000 119.2249 128
Secondly, the exploitation around the region of optimum was performed to test the converging
trend around the local minimum, the optimal solution in this subsystem. We changed the variables
with active bounded constraints, input the values to the function, and the results showed the
optimal point was the minimum around the points nearby (as shown in Table 15).
Table 15. The Function Minimum and Estimated Time in Real Model with Different Points around Optimum.
No. α [deg] HL [mm] I [%] WC [%] V
[mm/min]
Function Output
[min]
Estimated
Time [min]
1 0.7996 0.4 80 218 4000 116.7753 124
2 0 0.39 79 217 3950 122.0687 127
3 0 0.38 78 216 3900 129.8019 130
4 0 0.37 77 215 3850 139.6122 134
5 1 0.39 76 214 3800 126.9197 128
6 1 0.38 75 213 3750 136.4206 132
7 1 0.37 74 212 3700 147.3746 136
4.5.3 Post Analysis
The sensitivity analysis was performed on the active constraints in this subsystem. There were
three active constraints on layer height HL, infill percentage I and toolhead speed V, reaching their
upper bounds. Therefore, we increase the upper bound by 25% for each variable to observe the
relative changes in function output. Table 16 to Table 19 show the results of sensitivity analysis.
Table 16. The Optimal Solution with the Original Bounded Constraints
α [deg] HL [mm] I [%] WC [%] V [mm/min] Function Values
Upper bound 19 0.4 80 281 4000
116.7753 Lower bound 0 0.31 28 100 2419
Optimal solution 0.7997 0.4 80 218 4000
Table 17. The Optimal Solution with the Relaxed Constraint on HL by 25% Increase of Upper Bound
α [deg] HL [mm] I [%] WC [%] V [mm/min] Function Values
Upper bound 19 0.5 80 281 4000
91.4404 Lower bound 0 0.31 28 100 2419
Optimal solution 4.1041 0.5 28 100 4000
Table 18. The Optimal Solution with the Relaxed Constraint on I by 25% Increase of Upper Bound
α [deg] HL [mm] I [%] WC [%] V [mm/min] Function Values
Upper bound 19 0.4 100 281 4000
114.2811 Lower bound 0 0.31 28 100 2419
Optimal solution 0 0.4 100 281 4000
Table 19. The Optimal Solution with the Relaxed Constraint on V by 25% Increase of Upper Bound
α [deg] HL [mm] I [%] WC [%] V [mm/min] Function Values
Upper bound 19 0.4 80 281 5000
110.634 Lower bound 0 0.31 28 100 2419
Optimal solution 2.406 0.4 80 246.1595 5000
Also, the Lagrange multipliers are calculated for each constraint for 5 variables (α, HL, I, WC,
V). Among them, 𝜇α and 𝜇𝑊𝐶 = 0 since they are inactive constraints. The multipliers for active
constraints was acquired from the following equations:
𝜇𝐻𝐿=
−∆𝑓
∆𝑔𝐻𝐿,𝑎𝑐𝑡𝑖𝑣𝑒=
−(91.44 − 116.78)
0.1= 253.4
𝜇𝐼 =−∆𝑓
∆𝑔𝐼,𝑎𝑐𝑡𝑖𝑣𝑒=
−(114.28 − 116.78)
20= 0.125
𝜇𝑉 =−∆𝑓
∆𝑔𝑉,𝑎𝑐𝑡𝑖𝑣𝑒=
−(110.63 − 116.78)
1000= 0.00615
From the above results, we identified HL as the most sensitive variable in the subsystem. To
achieve the minimal time, layer height would be the priority that designer should focus on.
However, in the real practice, layer height will greatly affect the material strength of printing part
due to the connection between two layers. As a result, there would be a trade-off within this
subsystem when considering the strength. In future studies, the strength of printing part will bring
the new constraints to find an optimal solution with the minimal manufacturing time without
sacrificing the mechanical property.
5 SYSTEM OPTIMIZATION As mentioned previously, the system level objective contains four individual subsystems, two of
which were in the scope of this project. Because only two of the subsystems were studied, we have
assumed that our constraints are within the allowable design regions for the other two subsystems
not studied. Although this assumption may be incorrect, performing a system-level analysis with
results from the two subsystems will provide us with useful insight into the experimental design
of the remaining subsystems.
5.1 Design Variables and Parameters The design variables in the system level optimization problem have not changed from the original
subsystems (Table 2 and Table 7). However, because the wavy structure is being built into the
final orthotic, clinicians were consulted to verify each of the design parameters for the orthotic.
The clinicians required that the thickness of the AFO be between 4-6 mm so that bulkiness is
reduced and patient comfort is maintained.
5.2 Model Construction In order to find the minimum function value in the system-level optimization, the time and strength
functions were both normalized using mean time and strength values, respectively. The normalized
function values were added together, and the resulting function was then minimized:
System Function =Time (5 variables)
Mean value of time (from the data sets)+
−Strength−to−weight ratio(4 variables)
Mean value of strength (from the data sets).
Ultimately, designers will use the results from this optimization problem in determining the
3D print setup. Therefore, it is best that a weight is assigned to each of the subsystem functions
such that the designer can decide which design aspect is more important. For instance, if a stronger
part is needed, the strength function can be weighted more than the time function. This model is
given by:
System Function =w1 x Time (5 variables)
Mean value of time (from the data sets)+
−w2 x Strength−to−weight ratio(4 variables)
Mean value of strength (from the data sets),
where w1 and w2 are the weights for each system.
5.3 Constraints In addition to the bounded constraints from subsystems 1 and 2, two constraints are added to link
system variables. First, the AFO thickness specified by the orthotists was added as a constraint to
the problem. However, the thickness depends on the overall thickness per layer and the orientation
of the part during printing according to the following function (Figure 20):
𝑇 = 𝑂𝑇 × 𝑆𝑖𝑛 (𝛼).
Figure 20. Orientation, thickness, and overall thickness constraint relationship.
Additionally, beadwidth (subsystem 1) and contour width (subsystem 2) are linking variables
between the subsystems. Since contour width was indicated as a percentage, the upper bound for
contour width was assumed to be equivalent to the upper bound of the beadwidth. From this
relationship, an equality constraint was added to the system.
The following are the constraints for the system-level optimization problem:
Beadwidth (mm): 1 ≤ 𝐵𝑊 ≤ 1.55
Thickness (mm): 5 ≤ 𝑂𝑇 ≤ 15
Interference (Int) %: 60 ≤ 𝐼𝑛𝑡 ≤ 100
Orthotic orientation angle: 0 ≤ α ≤ 19
Layer height: 0.31 ≤ H𝐿 ≤ 0.4
Infill percentage: 28 ≤ 𝐼 ≤ 80
Contour width percentage: 100 ≤ W𝐶 ≤ 281
Tool head speed: 2419 ≤ V ≤ 4000
AFO Thickness: 5 − 𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) ≤ 0
𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) − 7 ≤ 0
BW and CW Equivalency: −190.9 𝐵𝑊 + 𝐶𝑊 − 4.1 = 0
5.4 Summary Model
Min: Time (5 variables)
Time𝑚𝑒𝑎𝑛
+−SW(4 variables)
Strength𝑚𝑒𝑎𝑛
Subject to 𝑔1(α) ∶ 0 ≤ α ≤ 19
𝑔2(H𝐿) ∶ 0.31 ≤ H𝐿 ≤ 0.4
𝑔3(I) ∶ 28 ≤ 𝐼 ≤ 80
𝑔4(W𝐶) ∶ 100 ≤ W𝐶 ≤ 281
𝑔5(V) ∶ 2419 ≤ V ≤ 4000
𝑔6(BW) ∶ 1 ≤ BW ≤ 1.55
𝑔7(OT) ∶ 5 ≤ OT ≤ 15
𝑔8(𝑓) ∶ 0.05 ≤ 𝑓 ≤ 2
𝑔9(Int) ∶ 60 ≤ Int ≤ 100
𝑔9(Int) ∶ 5 − 𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) ≤ 0
𝑔10(Int) ∶ 𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) − 7 ≤ 0
ℎ10 ∶ −190.9 𝐵𝑊 + 𝐶𝑊 − 4.1 = 0
5.5 Optimization Method Using SQP and multidisciplinary feasible design (MDF), the system-level results were found. The
MDF technique was used because there was little coupling between the systems and no coupling
variables. Figure 21 shows the problem setup for the system-level MDF.
5.6 Results
5.6.1 Optimization Results
In the final system, we use the nonlinear programming solver (fmincon) in MATLAB to optimize
the objective time function subject to 9 bounded constraints, 1 linear equality constraint and 1
nonlinear inequality constraint as shown previously. The local minimum was achieved with the
optimal variable set as shown in Table 20. Due to the new constraints and the changes in some
bounded constraints, the optimal values for some variables were different from the results in
subsystem. However, some variables like f, HL and V still remained the same for both system and
subsystem level.
Table 20. The Optimal Solution in the System Level.
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Time
[min]
Strength
[MPa/g]
1.35 6 2 78 0 0.4 39 261 4000 -1.08 125 2.52
To acquire high strength of AFO, the manufacturing time may be longer for producing a part
strong enough. Due to the different requirement from users on strength and manufacturing time of
AFO, the weights can be applied on these two subsystems to perform a tradeoff in the system level.
Figure 21. MDF System Optimization
Table 21. The Function Minimum in The System Level and The Optimal Results for Each Subsystem with Different
Weights Applied.
Weight on Time Weight on Strength Function Minimum Time [min] Strength [MPa/g]
0.7 0.3 -0.001934472 125.05561 2.51509706
0.6 0.4 -0.269812106 125.07438 2.5154058
0.5 0.5 -0.537713978 125.08579 2.5155283
0.4 0.6 -0.80562805 125.0934 2.51558281
0.3 0.7 -1.073549087 125.09883 2.51560833
Figure 22. Pareto Curve between Time and Strength.
Figure 22 shows the Pareto curve between strength and time in the system level. As strength
was increased, the time increased accordingly. Although there exists a tradeoff between these two
subsystems, the differences are not apparent. This is because we have not established sufficient
relationships between each variables and also the constraints in the system. Therefore, a further
study on the real performance in the final system is needed to confirm the tradeoff phenomenon.
The experiments on strength test and actual printing time are necessary for validating the
optimal results we have in the system level. We currently work closely with Stratasys (our partner
in this project) to produce the specimens with optimal variables for further experiments, and aim
2.515 2.5151 2.5152 2.5153 2.5154 2.5155 2.5156 2.5157 2.5158125.055
125.06
125.065
125.07
125.075
125.08
125.085
125.09
125.095
125.1
125.105
Strength [MPa/g]
Tim
e [
min
]
to print the part of AFO with the optimal build parameters with wavy tool path. For now, we have
successfully made the prototype of calf part of AFO. The procedure to make the prototype in the
system lever is shown in Figure 23. The wavy tool path is generated first and then the calf part
would be printed with the optimal build parameters. Figure 24 further illustrates the real printing
process of our prototype.
Figure 23. The Procedure to Make a Prototype of the Calf Part of a Real AFO.
Figure 24. The Real Printing Process for the Prototype in Final System.
Wavy Tool Path
Extruder
5.6.2 Model Evaluation
In the optimization procedure on the final system, we identified seven constraints are active as
Table 22 shown. These active constraints dominate the optimal results and we will perform the
sensitivity analysis on them.
Table 22. The Inactive Constraints and Active Constraints in the System Level.
Inactive Active
1 ≤ 𝐵𝑊 ≤ 1.55 0.05 ≤ 𝑓 ≤ 2
5 ≤ 𝑂𝑇 ≤ 15 0 ≤ α ≤ 19
60 ≤ 𝐼𝑛𝑡 ≤ 100 0.31 ≤ H𝐿 ≤ 0.4
28 ≤ 𝐼 ≤ 80 2419 ≤ V ≤ 4000
: 100 ≤ W𝐶 ≤ 281 4 − 𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) ≤ 0
𝑂𝑇 × 𝐶𝑜𝑠 (𝛼) − 6 ≤ 0
−190.9 𝐵𝑊 + 𝐶𝑊 − 4.1 = 0
In model evaluation on the system level, we first performed the exploration with the different
initial points in the feasible domain. As Table 23 shows, with the different initial points, there are
two region (with/without marked in orange) containing the local optimum. The different
combinations of infill percentage I and toolhead speed V can lead to different function minimum
as shown in Table 24 However, the data points marked in orange provide a more stable optimal
solution in this system; that is, the different initial points in that region will converge to the same
optimal result. On the other hand, the data points without marked in orange can change a lot in
function minimum when changing the initial point. Without enough constraints in the system level,
the neural network function in subsystem 2 may lose global convergence. Future studies are needed
to improve this problem. Therefore, the suitable optimal solution in the system level will be the
variable set with the function minimum -1.08.
Table 23. Different Initial Points in the Feasible Domain of the Final System.
Initial Point
No. BW [mm] OT [mm] f [rad/mm] Int [%] α [deg] HL [mm] I [%] WC [%] V [mm/min]
1 1.3 6 1.5 80 0 0.35 50 200 4000
2 1.3 10 1 70 0 0.35 50 200 4000
3 1.3 10 1 70 0 0.35 30 280 3800
4 1.3 10 1 70 0 0.35 30 200 4000
5 1.3 10 1 70 0 0.35 70 200 3000
6 1.3 10 1 70 0 0.35 75 200 3500
7 1.3 10 1 70 0 0.35 80 200 4000
8 1.3 10 1 70 0 0.35 80 200 3900
Table 24. The Optimal Solution, Function Minimum Output and the Results for Each Subsystem with Different Initial Points.
Optimal Solution
No. BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Time
[min]
Strength
[MPa/g]
1 1.35 6.00 2.00 77.90 0.00 0.40 38.65 261.37 4000.00 -1.08 125.09 2.52
2 1.35 6.00 2.00 77.90 0.00 0.40 38.70 261.36 4000.00 -1.08 125.09 2.52
3 1.34 6.00 2.00 77.13 0.00 0.40 39.26 259.04 3801.17 -1.06 127.22 2.51
4 1.35 6.00 2.00 77.97 0.00 0.40 36.16 261.54 4000.00 -1.08 125.13 2.52
5 1.35 6.00 2.00 77.81 0.00 0.40 60.49 261.09 3003.16 -0.97 142.18 2.52
6 1.33 6.00 2.00 76.58 0.00 0.40 80.00 257.28 3501.72 -1.05 129.58 2.51
7 1.33 6.00 2.00 77.10 1.88 0.40 80.00 258.87 4000.00 -1.12 118.09 2.51
8 1.33 6.00 2.00 76.92 1.59 0.40 80.00 258.33 3901.09 -1.11 119.87 2.51
Finally, the exploitation around the region of optimum was performed to test the converging
trend around the local minimum. We changed the variables with active constraints, input the values
to the function, and the results showed the optimal point was the minimum around the points
nearby (as shown in Table 25)
Table 25. The Function Minimum and Estimated Time in Real Model with Different Points around Optimum.
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Time
[min]
Strength
[MPa/g]
1.35 6 2 77.90 0 0.4 38.72 261.37 4000 -1.0754 125 2.52
1.35 6 2 77.90 0 0.35 35 270 3900 -0.9504 144.59 2.52
1.35 7 1.8 79 0 0.4 38.72 261.37 4000 -0.9431 125.09 2.34
1.35 7 1.8 79 0 0.35 35 270 3900 -0.8181 144.593 2.34
5.6.3 Post Analysis
The sensitivity analysis was performed on the active constraints in the system level. There were
seven active constraints including the bounded constraints including frequency f, orientation α,
layer height HL and toolhead speed V, the linear equality constraint on bead width BW and contour
width WC and two nonlinear inequality constraints on thickness and orientation. Three equality
and inequality constraints assigned in this study are based on the current AFO printing process and
thus have little space to relax the constraints. We mainly performed the sensitivity analysis on the
variables reaching their upper bounds (it is not allowable to change the constraint on lower bound
in the printing setting). We increase the upper bound by 25% for each variable to observe the
relative changes in function output. Table 26 to Table 29 shows the results of sensitivity analysis.
Table 26. The Optimal Solution with the Original Bounded Constraints.
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Upper bound 1.55 15 2 100 19 0.4 80 300 4000
-1.0754 Lower bound 1 5 0.05 60 0 0.31 28 195 2914
Optimal Solution 1.35 6 2 77.90 0 0.4 38.71 261.38 4000
Table 27. The Optimal Solution with the Relaxed Constraint on f by 25% Increase of Upper Bound
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm] I [%]
WC
[%]
V
[mm/min]
Function
Output
Upper bound 1.55 15 2.5 100 19 0.4 80 300 4000
-1.2693 Lower bound 1 5 0.05 60 0 0.31 28 195 2914
Optimal Solution 1.43 6 2.5 78.09 0 0.4 39.05 277.33 4000
Table 28. The Optimal Solution with the Relaxed Constraint on HL by 25% Increase of Upper Bound
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Upper bound 1.55 15 2 100 19 0.5 80 300 4000
-1.284 Lower bound 1 5 0.05 60 0 0.31 28 195 2914
Optimal Solution 1.35 6.022 2 78.28 4.88 0.5 46.14 262.43 3999.97
Table 29. The Optimal Solution with the Relaxed Constraint on V by 25% Increase of Upper Bound
BW
[mm]
OT
[mm]
f
[rad/mm]
Int
[%]
α
[deg]
HL
[mm]
I
[%]
WC
[%]
V
[mm/min]
Function
Output
Upper bound 1.55 15 2 100 19 0.4 80 300 5000
-1.0755 Lower bound 1 5 0.05 60 0 0.31 28 195 2914
Optimal Solution 1.35 6 2 77.90 0 0.4 38.61 261.37 4000.48
The Lagrange multipliers for active constraints was acquired from the following equations:
𝜇𝑓 =−∆𝑓
∆𝑔𝑓,𝑎𝑐𝑡𝑖𝑣𝑒=
−(−1.2693 + 1.0754)
0.5= 0.3878
𝜇𝐻𝐿=
−∆𝑓
∆𝑔𝐻𝐿 ,𝑎𝑐𝑡𝑖𝑣𝑒=
−(−1.284 + 1.0754)
0.1= 1.939
𝜇𝑉 =−∆𝑓
∆𝑔𝑉,𝑎𝑐𝑡𝑖𝑣𝑒=
−(−1.0755 + 1.0754)
1000= 0.0000001
From the above results, we identified HL as the most sensitive variable in the final system.
Based on this finding which consistent with the results in subsystem 2, layer height is the dominant
factor that will affect the performance of 3D printing AFO. This indicates we should investigate
more on how layer height interacts with other variables. Future studies are needed to find out the
real behavior of these relatively sensitive variables in the system level.
5.6.4 Discussion
The neural network in subsystem 1 was produced from a small number of experiments and was
therefore stable. However, given the difficulty in characterizing polymeric materials, creating a
model that captures the subsystem response to changes in variables will be difficult. Multiple
coupons will need to be tested using the four-point bending test to find a range of strengths per
coupon.
There are some variables remain the same optimal values between subsystem and system level,
including frequency, layer height and toolhead speed. Those would be the priority to be adjusted
to achieve our goal for optimal printing processes in the future studies.
The neural network in subsystem 2 did not provide a stable optimized result. We believe that
the issue lies in the large number of potential variables that have not been included in our
subsystem. Additionally, our subsystem constraints are not specific enough to properly link each
variable. Increasing the number of data points around the optimum would better capture the system
behavior.
Another important conclusion from the subsystem analysis is that the designer can choose the
relative importance between the time and strength of the printed AFO. After adding the additional
two subsystems not considered in this report, there will exist more coupling between the systems,
which will yield a reliable result with a greater tradeoff.
6 CONCLUSION AND FUTURE WORK In this project, we optimized the FDM manufacturing time and strength by manipulating key
variables, constraints, and parameters associated with the process. Variables such as orientation
angle, infill percentage and layer height can all affect the overall time for FDM printing.
Additionally, the type of infill plays a large role in the required AM time and strength. In this
paper, experiments were performed and a neural network was fitted to the data and output.
Optimum values were determined for each subsystem and system.
Future work will include ensure good fit between actual values and neural network output,
validating the model with additional experiments, conducting a parametric study, increasing the
number of variables, and including the additional subsystems into our system level design. For the
future parametric study, we will change parameters such as temperature, thickness, and material
to test the performance of our models.
7 REFERENCE [1] Chen, R., Chen, L., Tai, B., Wang, Y., Shih, A., and Wensman, J., 2014, “Additive manufacturing
of personalized ankle-foot orthosis,” Proc. NAMRI/SME, 42.
[2] Jin, Y., He, Y., and Shih, A., 2016, “Process Planning for the Fuse Deposition Modeling of Ankle-
Foot-Othoses,” 00.
[3] Specimens, P., and Materials, E. I., 2011, “Standard Test Methods for Flexural Properties of
Unreinforced and Reinforced Plastics and Electrical Insulating Materials 1,” Annu. B. ASTM
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[4] Panda, S. K., 2009, “Optimization of Fused Deposition Modelling (FDM) Process Parameters
Using Bacterial Foraging Technique,” Intell. Inf. Manag., 01(02), pp. 89–97.
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prediction and optimization using group method for data handling (gmdh) and differential
evolution (de),” Int. J. Adv. Manuf. Technol., 73(1-4), pp. 509–519.
8 APPENDIX
8.1 Neural Network Function of Subsystem 1
function [y1] = myNeuralNetworkFunction(x1) %MYNEURALNETWORKFUNCTION neural network simulation function. % % Generated by Neural Network Toolbox function genFunction, 31-Mar-2016
12:33:35. % % [y1] = myNeuralNetworkFunction(x1) takes these arguments: % x = Qx4 matrix, input #1 % and returns: % y = Qx1 matrix, output #1 % where Q is the number of samples.
%#ok<*RPMT0>
% ===== NEURAL NETWORK CONSTANTS =====
% Input 1 x1_step1_xoffset =
[1.00740087749428;5.099147511468;0.115048837622704;63.1623796639163]; x1_step1_gain =
[4.06941912343373;0.220855999561719;1.12950721286545;0.0543926281969955]; x1_step1_ymin = -1;
% Layer 1 b1 = [-2.5318651503251894;-1.9615075774660269;1.4078629175457122;-
1.1252428109866988;0.13493944212014625;-0.28015702613874821;-
0.8785118823807978;1.4378638473184249;1.9721292398409371;2.4836207160677]; IW1_1 = [1.4133596613200874 -1.5497368965025347 0.73254633712660289
1.0150292262404521;1.066957833338136 1.3312037773494667 -1.2973742264166119 -
1.2392843104582965;-1.041254824288099 1.3341200650053244 1.2139626291959091 -
1.0202940264458165;0.76684677307598004 0.21780398736025941 1.4007989305824822
-1.6025288630971821;0.90239281668040416 1.375890853919814 0.88341225963777792
-1.6470436948185967;-1.4594300989243003 -1.3227174899292462
0.93897267975215115 1.1898702526201121;-1.3305372082821787 -
0.55994305043535264 1.5214668507905715 1.2650626722790301;0.34265215385366216
2.1036914552214183 -0.58103030793038524 -
1.0564241769956548;1.5969676091287928 1.0432493259223132 0.63410552214530758
1.5342281251751959;1.3232471977149691 1.3423984437065508 -0.95407985999964429
-1.3399999853663482];
% Layer 2 b2 = -0.40260210834282323; LW2_1 = [0.11459937688430968 -0.21718827702732141 -0.081533060333279783
0.29910330198952378 0.18528709972373869 0.30565718596745994
0.56199156212330892 0.20840547435317311 0.55299547413167804
0.57031173499777699];
% Output 1 y1_step1_ymin = -1; y1_step1_gain = 1.22755241251639; y1_step1_xoffset = 0.518576632109835;
% ===== SIMULATION ========
% Dimensions Q = size(x1,1); % samples
% Input 1 x1 = x1'; xp1 = mapminmax_apply(x1,x1_step1_gain,x1_step1_xoffset,x1_step1_ymin);
% Layer 1 a1 = tansig_apply(repmat(b1,1,Q) + IW1_1*xp1);
% Layer 2 a2 = repmat(b2,1,Q) + LW2_1*a1;
% Output 1 y1 = mapminmax_reverse(a2,y1_step1_gain,y1_step1_xoffset,y1_step1_ymin); y1 = -y1'; end
% ===== MODULE FUNCTIONS ========
% Map Minimum and Maximum Input Processing Function function y = mapminmax_apply(x,settings_gain,settings_xoffset,settings_ymin) y = bsxfun(@minus,x,settings_xoffset); y = bsxfun(@times,y,settings_gain); y = bsxfun(@plus,y,settings_ymin); end
% Sigmoid Symmetric Transfer Function function a = tansig_apply(n) a = 2 ./ (1 + exp(-2*n)) - 1; end
% Map Minimum and Maximum Output Reverse-Processing Function function x =
mapminmax_reverse(y,settings_gain,settings_xoffset,settings_ymin) x = bsxfun(@minus,y,settings_ymin); x = bsxfun(@rdivide,x,settings_gain); x = bsxfun(@plus,x,settings_xoffset); end
8.2 Experiment Sets of 5 Variables in Subsystem 2
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min]
1 16 0.28 79 130 2325
2 13 0.20 65 221 3770
3 10 0.24 52 135 2073
4 5 0.39 71 206 3843
5 14 0.12 57 271 3925
6 8 0.16 65 124 3229
7 12 0.19 75 288 2342
8 18 0.27 69 104 2510
9 15 0.35 67 258 3464
10 18 0.23 64 263 3724
11 19 0.18 58 117 2195
12 14 0.31 61 181 3551
13 0 0.22 52 280 3970
14 1 0.32 71 161 2889
15 2 0.21 80 242 3353
16 4 0.27 55 195 2962
17 1 0.26 54 227 2558
18 9 0.14 78 240 3608
19 7 0.13 59 111 3489
20 11 0.11 51 153 2675
21 19 0.37 76 208 3302
22 4 0.15 56 177 3124
23 6 0.34 76 298 2262
24 17 0.30 55 218 2019
25 6 0.34 60 190 3179
26 3 0.38 69 158 2622
27 10 0.14 67 146 2823
28 17 0.18 73 279 2765
29 8 0.39 73 173 2419
30 13 0.29 63 251 3033
31 7 0.31 49 143 3077
32 1 0.20 36 229 3397
33 13 0.36 28 281 3796
34 17 0.27 39 203 2073
35 10 0.12 26 122 2482
36 1 0.38 80 100 4000
37 0 0.40 33 157 4000
38 1 0.36 32 169 3869
39 1 0.39 31 194 3914
40 0 0.37 38 152 3949
41 0 0.37 37 186 3822
42 1 0.37 35 179 3979
8.3 Experimental Results (TE) of 5 Variables in Subsystem 2
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min]
1 16 0.28 79 130 2325 279
2 13 0.20 65 221 3770 286
3 10 0.24 52 135 2073 327
4 5 0.39 71 206 3843 135
5 14 0.12 57 271 3925 462
6 8 0.16 65 124 3229 360
7 12 0.19 75 288 2342 376
8 18 0.27 69 104 2510 293
9 15 0.35 67 258 3464 176
10 18 0.23 64 263 3724 262
11 19 0.18 58 117 2195 477
12 14 0.31 61 181 3551 198
13 0 0.22 52 280 3970 224
14 1 0.32 71 161 2889 174
15 2 0.21 80 242 3353 246
16 4 0.27 55 195 2962 212
17 1 0.26 54 227 2558 222
18 9 0.14 78 240 3608 395
19 7 0.13 59 111 3489 432
20 11 0.11 51 153 2675 601
21 19 0.37 76 208 3302 176
22 4 0.15 56 177 3124 363
23 6 0.34 76 298 2262 203
24 17 0.30 55 218 2019 298
25 6 0.34 60 190 3179 171
26 3 0.38 69 158 2622 160
27 10 0.14 67 146 2823 444
28 17 0.18 73 279 2765 383
29 8 0.39 73 173 2419 176
30 13 0.29 63 251 3033 224
31 7 0.31 49 143 3077 191
32 1 0.20 36 229 3397 254
33 13 0.36 28 281 3796 162
34 17 0.27 39 203 2073 321
35 10 0.12 26 122 2482 561
36 1 0.38 80 100 4000 134
37 0 0.40 33 157 4000 126
38 1 0.36 32 169 3869 141
39 1 0.39 31 194 3914 130
40 0 0.37 38 152 3949 129
41 0 0.37 37 186 3822 137
42 1 0.37 35 179 3979 136
8.4 Experimental Results (TE) of 5 Variables in Subsystem 2 (Final Sets)
Experiment α [deg] HL [mm] I [%] WC [%] V [mm/min] TE [min]
1 5 0.39 71 206 3843 135
2 15 0.35 67 258 3464 176
3 14 0.31 61 181 3551 198
4 1 0.32 71 161 2889 174
5 19 0.37 76 208 3302 176
6 6 0.34 60 190 3179 171
7 3 0.38 69 158 2622 160
8 8 0.39 73 173 2419 176
9 7 0.31 49 143 3077 191
10 13 0.36 28 281 3796 162
11 1 0.38 80 100 4000 134
12 0 0.40 33 157 4000 126
13 1 0.36 32 169 3869 141
14 1 0.39 31 194 3914 130
15 0 0.39 38 152 3949 129
16 0 0.37 37 186 3822 137
17 1 0.37 35 179 3979 136
8.5 Neural Network Function of Subsystem 2
function [y1] = myNeuralNetworkFunction(x1) %MYNEURALNETWORKFUNCTION neural network simulation function. % % Generated by Neural Network Toolbox function genFunction, 16-Apr-2016
23:10:00. % % [y1] = myNeuralNetworkFunction(x1) takes these arguments: % x = 5xQ matrix, input #1 % and returns: % y = 1xQ matrix, output #1 % where Q is the number of samples.
%#ok<*RPMT0>
% ===== NEURAL NETWORK CONSTANTS =====
% Input 1 x1_step1_xoffset =
[0;0.307179466462819;28.2462886356219;100;2418.99286454918]; x1_step1_gain =
[0.106278233354178;21.5469565168882;0.0386445715152439;0.0110785933038194;0.0
0126501642854996]; x1_step1_ymin = -1;
% Layer 1 b1 = [1.6814784892953174;-0.98740610451424649;0.11104783086249836;-
0.3690567020114679;1.6702416680606311]; IW1_1 = [-0.69004876269499937 -1.2169666620110782 -1.3695719657099525
0.72239447414460778 -0.95746722925021466;0.48079335256102312
1.0141769318918776 0.013881006661929547 0.78978676919689128
1.4541036474493136;-0.30822024707907458 0.37609664323067049 -
1.8927785291903785 0.45946958356878947 -
1.0535577117649275;0.0062701134923072166 -1.0630607379694237
1.2079224593847768 -1.3667109966772102 -
0.50172169464525407;1.1503096637341084 -0.91335861156908971 -
0.28031799283264019 0.17877256371422207 -1.0106166369120366];
% Layer 2 b2 = -0.41835758649001942; LW2_1 = [0.41961267740956965 -0.0046557555199523981 -0.26922015246682252
0.29618445874830562 0.58254784087792744];
% Output 1 y1_step1_ymin = -1; y1_step1_gain = 0.0277777777777778; y1_step1_xoffset = 126;
% ===== SIMULATION ========
% Dimensions Q = size(x1,2); % samples
% Input 1 xp1 = mapminmax_apply(x1,x1_step1_gain,x1_step1_xoffset,x1_step1_ymin);
% Layer 1 a1 = tansig_apply(repmat(b1,1,Q) + IW1_1*xp1);
% Layer 2 a2 = repmat(b2,1,Q) + LW2_1*a1;
% Output 1 y1 = mapminmax_reverse(a2,y1_step1_gain,y1_step1_xoffset,y1_step1_ymin); end
% ===== MODULE FUNCTIONS ========
% Map Minimum and Maximum Input Processing Function function y = mapminmax_apply(x,settings_gain,settings_xoffset,settings_ymin) y = bsxfun(@minus,x,settings_xoffset); y = bsxfun(@times,y,settings_gain); y = bsxfun(@plus,y,settings_ymin); end
% Sigmoid Symmetric Transfer Function function a = tansig_apply(n) a = 2 ./ (1 + exp(-2*n)) - 1; end
% Map Minimum and Maximum Output Reverse-Processing Function function x =
mapminmax_reverse(y,settings_gain,settings_xoffset,settings_ymin) x = bsxfun(@minus,y,settings_ymin); x = bsxfun(@rdivide,x,settings_gain); x = bsxfun(@plus,x,settings_xoffset); end
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