Throughput and Strength Optimization for Fused Deposition Modeling of Ankle-Foot Orthotics Robert Chisena Dian-Ru Li

Transcription

1 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

2 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 timeconsuming and labor intensive manufacturing process with lead times of up to twoweeks. 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.

3 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

4 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 Subsystem 1 Symbols Definition Unit L Length of part mm W Width of part mm H Height of part mm T A Actual manufacturing time min T E Estimated manufacturing time min T I Ideal printing time min TP i Tool path of i layer mm α Part build orientation (medial axis) degree θ Part build orientation (longitudinal axis) degree N Number of layer - H L Layer height mm I Infill percentage %

5 Subsystem 2 I s Support infill percentage % W E Extruder width mm W C Contour width percentage % V Toolhead speed mm/min W P Weight of the part g W S Weight of the support material g A Infill angle (raster angle) degree O Outline overlap % N C Number of contours - 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 L ss Length of Support Span mm D LN 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.

6 Figure 5. System Level Design Optimization Problem for Optimizing Strength and Time of AM AFO. Red box defines the scope of this project 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. Figure 6. Example of a Warren Truss Bridge 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: T I = N i=1 TP i V

7 where V is the toolhead speed adjusted depending on users and TP i 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 datadriven 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]. Figure 7. Wavy Tool Path for Ankle-Foot Orthotic Cross-Section [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)

8 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. Figure 8.Wavy Infill Parameters 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.

9 Table 2. Design Variables and Parameters for the Wavy Infill Subsystem Notation Type Unit Value Description Variable Frequency, f continuous radians/mm Frequency of the internal infill sine wave. Thickness,T continuous mm 5-15 Thickness of the thin-walled structure. BeadWidth, BW continuous mm Thickness of the deposited filament bead. Interference, Intereference between the sine wave continuous % Int 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 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 Outer Thickness Frequency Interference [wave/mm]

10 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. L N L S Figure 9. Four-Point Bending Test Setup. L N is the nose span distance, which was 30mm. L S is the support span distance, which was 80 mm. Figure 10 shows the bending deflection versus stress-to-weight ratio. The maximum stress-toweight 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

11 Table 4. Strength-to-weight ratios from four-point bending tests. Experiment Weight [g] Strength-to-Weight Ratio [MPa/g] 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 Hidden Layer Output Layer Output Figure 11. Neural Network Setup for Wavy Infill Subsystem Data 1

12 Figure 12. Performance of the Neural Network Fit on Experimental 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

13 3.4 Summary Model The summary model of the wavy infill pattern is: Min SW(BW, OT, f, Int) (Neural network function) Subject to g 1 (BW) 1 BW 1.55 g 2 (OT) 5 OT 15 g 3 (f) 0.05 f 2 g 4 (Int) 60 Int Results 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 BW OT f [1/mm] Int SW [MPa/g] Optimal Result 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.

14 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. Perturbation Table 6. Strength-to-Weight Ratios from Perturbations from the Optimum Outer Beadwidth Frequency Interference Thickness [wave/mm] Strength-to-Weight Ratio [MPa/g] -30% % % 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 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,μ f, the upper frequency bound was relaxed and the function before and after relaxation was compared:

15 μ f = 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, μ f 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. H L continuous mm Layer height: This variable dominantly affect surface roughness, which will be evaluated in the future. I continuous % Infill percentage: The solid level of each layer. W C continuous % Contour width percentage: The percentage of extruder width. V continuous mm/min Tool head speed: Moving speed of extruder Parameter

16 L - mm 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 Width of part H - mm Height of part θ - degree 0 Part build orientation (longitudinal axis) I s - % 30 Support infill percentage W E - 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. N C - 2 Number of contours Figure 14. Printing Results with respect to The Changes of α. 4.2 Objective Function 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 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 H L 0.4, 20 I 80, 100 W C 300, 2000 V 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

17 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] H L I W C V [mm/min] Estimated Manufacturing Time (T E ) 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] H L I W C V [mm/min] T E [min]

18 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) 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

19 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 (T E) of 5 Variables in This Subsystem (Final Sets). Experiment α [deg] H L I W C V [mm/min] T E [min] 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 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.

20 Output ~= 1.2*Target Training: R= Data Fit Y = T Target Output ~= 0.81*Target Validation: R= Data Fit Y = T Target Output ~= 0.84*Target Test: R= Data Fit Y = T Target Output ~= 1*Target All: R= Data Fit Y = T Target Figure 17. Regression Plot of Data-driven Objective Function.

21 Error Histogram with 20 Bins 3 Training Validation Test Zero Error Instances Errors = Targets - Outputs 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 H L 0.4 Infill percentage: 28 I 80 Contour width percentage: 100 W C 281 Tool head speed: 2419 V 4000

22 4.4 Summary Model The summary model of subsystem 2: Min T E = Time(α, H L, I, W C, V) (Neural network function) Subject to g 1 (α) 0 α 19 g 2 (H L ) 0.31 H L 0.4 g 3 (I) 28 I 80 g 4 (W C ) 100 W C 281 g 5 (V) 2419 V Results 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 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 (T E) of 5 Variables and the Optimal Solution in This Subsystem. Experiment α [deg] H L I W C V [mm/min] T E [min]

23 Time (min) Optimal Results 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 (T E) of 5 Variables and the Optimal Solution in This Subsystem. Experiment α [deg] H L I W C V [mm/min] T E [min] T A [min] Optimal Results Optimal Results Experiment No. Actual manufacturing time Estimated manufacturing time Figure 19. The Actual Manufacturing Time and Estimated Manufacturing Time for Different Variable Sets.

24 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 H L 0.4 : 100 W C I 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 Table 14. The Function Minimum and Estimated Time in Real Model with Different Initial Points. Initial Points Optimal Points No. α [deg] H L [mm ] I W C V [mm/ min] α [deg] H L [mm ] I W C V [mm/ min] Function Minimum [min] Estimated Time [min] 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).

25 Table 15. The Function Minimum and Estimated Time in Real Model with Different Points around Optimum. V Function Output Estimated No. α [deg] H L I W C [mm/min] [min] Time [min] 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] H L I W C V [mm/min] Function Values Upper bound Lower bound Optimal solution Table 17. The Optimal Solution with the Relaxed Constraint on HL by 25% Increase of Upper Bound α [deg] H L I W C V [mm/min] Function Values Upper bound Lower bound Optimal solution Table 18. The Optimal Solution with the Relaxed Constraint on I by 25% Increase of Upper Bound α [deg] H L I W C V [mm/min] Function Values Upper bound Lower bound Optimal solution

26 Table 19. The Optimal Solution with the Relaxed Constraint on V by 25% Increase of Upper Bound α [deg] H L I W C V [mm/min] Function Values Upper bound Lower bound Optimal solution Also, the Lagrange multipliers are calculated for each constraint for 5 variables (α, HL, I, WC, V). Among them, μ α and μ WC = 0 since they are inactive constraints. The multipliers for active constraints was acquired from the following equations: μ HL = μ I = μ V = f g HL,active = ( ) 0.1 = f ( ) = = g I,active 20 f ( ) = = g V,active 1000 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.

27 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) + Strength to weight ratio(4 variables). Mean value of time (from the data sets) 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).

28 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 H L 0.4 Infill percentage: 28 I 80 Contour width percentage: 100 W C 281

29 Tool head speed: 2419 V 4000 AFO Thickness: 5 OT Cos (α) 0 OT Cos (α) 7 0 BW and CW Equivalency: BW + CW 4.1 = Summary Model Min: Time (5 variables) SW(4 variables) + Time mean Strength mean Subject to g 1 (α) 0 α 19 g 2 (H L ) 0.31 H L 0.4 g 3 (I) 28 I 80 g 4 (W C ) 100 W C 281 g 5 (V) 2419 V 4000 g 6 (BW) 1 BW 1.55 g 7 (OT) 5 OT 15 g 8 (f) 0.05 f 2 g 9 (Int) 60 Int 100 g 9 (Int) 5 OT Cos (α) 0 g 10 (Int) OT Cos (α) 7 0 h BW + CW 4.1 = 0

30 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. Figure 21. MDF System Optimization 5.6 Results 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. BW OT f [rad/mm] Table 20. The Optimal Solution in the System Level. Int α [deg] H L I W C V [mm/min] Function Output Time [min] Strength [MPa/g] 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.

31 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] Time [min] Strength [MPa/g] 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

32 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. Wavy Tool Path Extruder Figure 24. The Real Printing Process for the Prototype in Final System.

33 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 f 2 5 OT 15 0 α Int H L I V 4000 : 100 W C OT Cos (α) 0 OT Cos (α) 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 Table 23. Different Initial Points in the Feasible Domain of the Final System. Initial Point No. BW OT f [rad/mm] Int α [deg] HL I WC V [mm/min]

34 Table 24. The Optimal Solution, Function Minimum Output and the Results for Each Subsystem with Different Initial Points. No. BW OT f [rad/mm] Int α [deg] Optimal Solution HL I WC V [mm/min] Function Output Time [min] Strength [MPa/g] 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 OT f [rad/mm] Int α [deg] HL I WC V [mm/min] Function Output Time [min] Strength [MPa/g] 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.

35 Table 26. The Optimal Solution with the Original Bounded Constraints. BW OT f [rad/mm] Int α [deg] HL I WC V [mm/min] Function Output Upper bound Lower bound Optimal Solution Table 27. The Optimal Solution with the Relaxed Constraint on f by 25% Increase of Upper Bound BW OT f [rad/mm] Int α [deg] HL I WC V [mm/min] Function Output Upper bound Lower bound Optimal Solution Table 28. The Optimal Solution with the Relaxed Constraint on HL by 25% Increase of Upper Bound BW OT f [rad/mm] Int α [deg] HL I WC V [mm/min] Function Output Upper bound Lower bound Optimal Solution Table 29. The Optimal Solution with the Relaxed Constraint on V by 25% Increase of Upper Bound BW OT f [rad/mm] Int α [deg] HL I WC V [mm/min] Function Output Upper bound Lower bound Optimal Solution The Lagrange multipliers for active constraints was acquired from the following equations: μ f = μ HL = f ( ) = = g f,active 0.5 f g HL,active = ( ) 0.1 = 1.939

36 μ V = f ( ) = = g V,active 1000 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 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.