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

Stand., (C), pp. 1–11.

[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.

[5] Rayegani, F., and Onwubolu, G. C., 2014, “Fused deposition modelling (fdm) process parameter

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