# Analysis and Optimization of Dimensional Accuracy and Porosity of High Impact Polystyrene Material Printed by FDM Process: PSO, JAYA, Rao, and Bald Eagle Search Algorithms

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## Abstract

**:**

^{2}equal to 94.56% for CE, and 99.10% for porosity, respectively. Four algorithms (bald eagle search optimization (BES), particle swarm optimization (PSO), RAO-3, and JAYA) were applied to determine optimal FDM conditions while examining six case studies (sets of weights assigned for porosity and CE) focused on minimizing both CE and porosity. BES and RAO-3 algorithms determined optimal conditions (layer thickness: 0.22 mm; shell thickness: 2 mm; infill density: 100%; print speed: 30 mm/s) at a reduced computation time equal to 0.007 s, differing from JAYA and PSO, which resulted in an experimental CE of 0.1215 mm and 2.5% of porosity in printed parts. Consequently, BES and RAO-3 algorithms are efficient tools for the optimization of FDM parts.

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussions

#### 3.1. Response: Cylindricity Error

#### 3.2. Response: Porosity

#### 3.3. Analysis of Variance of Responses: Cylindricity Error and Porosity

^{2}value examines both model accuracies and the goodness of fit of regression. It is important to note that both models showed an R

^{2}value close to 100% (i.e., 94.56% for cylindricity error and 99.1% for porosity). This strongly signifies the model is statistically significant for practical utility in industries for predictions and optimization.

#### 3.4. Multi-Objective Optimization Algorithms

#### 3.4.1. Rao Algorithm

^{th}variable corresponds to l

^{th}candidate at m iteration) is updated according to Equation (3),

_{1}and rand

_{2}are random numbers that operate in the range of 0 and 1. Equations (4) and (5), ${X}_{k,l,m}\mathrm{or}{X}_{K,l,m}$ represent the k

^{th}candidate solution compared with the random K

^{th}candidate solution and exchange information corresponding to fitness value. If fitness function f

_{k}produced a better function value than f

_{K}then ${X}_{k,l,m}\mathrm{or}{X}_{K,l,m}$ turns out to be ${X}_{k,l,m}$ and the term ${X}_{K,l,m}\mathrm{or}{X}_{k,l,m}$ turns out to be ${X}_{K,l,m}$. Conversely, if the fitness function value of K

^{th}candidate solutions produced a better solution, then the fitness function of the k

^{th}candidate solution ${X}_{k,l,m}\mathrm{or}{X}_{K,l,m}$ turns out to be ${X}_{K,l,m}$ and ${X}_{K,l,m}\mathrm{or}{X}_{k,l,m}$ turns out to ${X}_{k,l,m}$. To attain the optimal global solutions, Equation (3) is used for the RAO-1 algorithm, whereas Equation (4) is for the RAO-2 algorithm and Equation (5) is for the RAO-3 algorithm. The performance of globally optimal solutions of RAO algorithms is compared among themselves after comparing the fitness values, several function evaluations, and time. The RAO-3 algorithm was used to determine optimal conditions for the FDM process.

#### 3.4.2. BES Algorithm

_{best}is the previous best position of bald eagles in the search space. α is the parameter whose role is to control the changes in position and the corresponding value maintained between 1.5 to 2. The α value maintained is equal to 1.5. P

_{new}corresponds to the new position of bald eagles. P

_{mean}depicts the eagles using up all information from the previous points.

_{1}and C

_{2}is the eagle movement towards the best and centre point, and those values are maintained equal to 2. After ensuring the optimal search is concluded, the point corresponding to the minimum value of the objective function is chosen as the local best only when it produced a lower value than the previous best.

#### 3.4.3. JAYA Algorithm

^{th}decision variable corresponding to the k

^{th}candidate at i

^{th}iteration. The new solutions determined viz. $X{\prime}_{j,k,i}$ are compared with ${X}_{j,k,I}$ and the better solution of the two is updated. This procedure is carried out for pre-defined iterations and populations till it ensures optimal solutions are determined.

#### 3.4.4. PSO Algorithm

_{s}) and neighbor (social leader, Global best: P

_{g}) particles. In each iteration, the P

_{s}and P

_{g}of particle velocity and positions are determined and updated using Equation (10).

#### 3.5. Results of Optimization Models

#### 3.5.1. Mathematical Formulation for Multi-Objective Optimization

_{1}and w

_{2}were weights that corresponded to porosity and cylindricity error. Terms $porosit{y}_{min}$ and cylindricity $erro{r}_{min}$ were the minimum values that corresponded to porosity and cylindricity error. A single objective optimization task was carried out by all four algorithms to determine $porosit{y}_{min}$ and $Cylindricityerro{r}_{\mathrm{min}}$. All four algorithms (BES, PSO, JAYA, and RAO-3) were coded on Python (3.8.0) and executed on a computer (HP Intel (R) Core (TM) i3-7100U CPU at 2.40 GHz and RAM: 4G) to minimize f(z) and thereby minimize cylindricity error and porosity.

#### 3.5.2. Estimating Solution Accuracy and Determining Optimal Conditions

_{1}and w

_{2}are weight factors for porosity and cylindricity error), respectively. Six cases were considered, giving equal weight (case 1) importance to both outputs (w

_{1,}and w

_{2}= 0.5) and maximum importance (case 2–6) to one output minimal to the rest. Note that the summation of weight factors (w

_{1,}+ w

_{2}= 1) must be maintained equal to 1. The objective functions were evaluated to determine the fitness function value (solving Equation (12)) corresponding to different case studies (different sets of weights) by applying four algorithms. Note that all algorithms are capable of producing approximately similar results, and the obtained results are presented in Table 5. It was observed that the fitness function values differed from one another due to the different weight fractions (importance given to individual output) assigned to the individual output. The objective functions defined to minimize the fitness function value (goal to minimize both cylindricity error and porosity), and therefore, case 4 (porosity; w

_{1 =}0.4, and cylindricity error w

_{2}= 0.6) were recommended as optimal fused deposition modeling conditions due to their lower fitness function value equal to 2.494. Table 5 presents the results of optimal conditions corresponding to the FDM process subjected to different case studies.

#### 3.5.3. Estimate Computation Time and Solution Accuracy in Determining Optimal Conditions

#### 3.5.4. Confirmation Experiments

## 4. Conclusions

- All factors (except print speed for CE) were found statistically significant for both outputs. Shell thickness was the major contributing factor for cylindricity error, whereas least significant for the porosity of printed samples. Infill density was the most significant factor for porosity.
- The print speed relationship with cylindricity error and porosity was found to be linear, whereas shell thickness was found to have a non-linear relationship.
- All the interaction factor effects were significant, except the interactions among shell thickness and infill density (for CE and porosity) and layer thickness and infill density (for CE). Insignificant terms practically imply a lesser contribution to the outputs of a process. Both models produced better fit with a value of 99.1% for porosity and 94.56% for cylindricity error, respectively.
- Four algorithms (BES, RAO-3, PSO, and JAYA) were applied to determine the optimal fused deposition modeling conditions. Six case studies (set of weight fractions assigned to both outputs) were analyzed and the optimal conditions were determined. Case 4 (layer thickness 0.22 mm, shell thickness 2 mm, infill density 100%, print speed 30 mm/s) is recommended as the optimal condition, as they produced a minimum fitness value equal to 2.494. The recommended optimal conditions are experimentally evaluated and the resulting cylindricity error and porosity of printed parts were found equal to 0.1215 mm, and 2.5%.
- The computational time of all four algorithms (BES, RAO-3, PSO, and JAYA) were tested with common iterations and population size. BES and RAO algorithms were converged (population size: 20; iterations: 100) to optimize global solutions with a computation time equal to 0.007 s. JAYA and PSO algorithms converge on local solutions for population size: 20; iterations: 100, and require more population size and iteration to attain global solutions.
- BES and Rao algorithms are computationally efficient for attaining global solutions and efficient tools for optimizing FDM parts.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Porosity _{min} | Minimum values of porosity |

Cylindricity error _{min} | Minimum values of cylindricity error |

f(z) | Fitness function |

${w}_{1}$ | Weight importance of porosity |

${w}_{2}$ | Weight importance of cylindricity error |

$w$ | Inertia Weight |

R^{2} | Coefficient of correlation |

R^{2} Adj. | Adjusted Coefficient of Correlation |

A | Layer Thickness |

ABS | Acrylonitrile Butadiene Styrene |

AM | Additive Manufacturing |

ANN | Artificial Neural network |

ANFIS | Adaptive Network Fuzzy Interface System |

ASA | Amino-Salicylic Acid |

B | Shell Thickness |

BES | Bald Eagle Search Optimization |

BO | Build Orientation |

BT | Build Time |

C | Infill Density |

CAD | Computer Aided Design |

CCD | Central Composite Design |

CE | Cylindricity Error |

CT | Chamber Temperature |

D | Print Speed |

DA | Dimensional Accuracy |

DE | Differential Evolution |

DFA | Desirability Function Approach |

EC | Energy Consumption |

ET | Extruder Temperature |

FDM | Fused Deposition Modelling |

FP | Filling Pattern |

FS | Flexural Strength |

FT | Floor Thickness |

GA | Genetic Algorithm |

HIP | High Impact Polystyrene |

IDM | Inset Distance Multiplier |

IFD or ID | Infill Density |

IP | Infill Pattern |

IS | Infill Speed |

ISS | Inset Speed |

LT | Layer Thickness |

NC | Number Of Contours |

NS | Number Of Shells |

NT | Nozzle Temperature |

OS | Outline Speed |

OSS | Outer Shell Speed |

PC-ABS | Polycarbonate ABS |

PLA | Polylactic Acid |

PP | Printing Plane |

PS | Print Speed |

PT | Platform Temperature |

PSO | Particle Swarm Optimization |

RA | Raster Angle |

RSM | Response Surface Methodology |

RW | Road Width |

SHD | Sparse High Density |

SLD | Sparse Low Density |

SOS | Symbiotic Organism Search |

SR | Surface Roughness |

ST | Shell Thickness |

SST | Support Style |

TS | Tensile Strength |

UTS | Ultimate Tensile Strength |

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**Figure 4.**Surface plots of cylindricity error vs. (

**a**) LT and ST, (

**b**) LT and ID, (

**c**) LT and PS, (

**d**) ST and ID, (

**e**) ST and PS, and (

**f**) ID and PS.

Materials | Experimental Method | Process Variables | Analyzed Parameters | Remarks | Ref. |
---|---|---|---|---|---|

Optimization Method | |||||

ABS | Taguchi method | LT: 0.254–0.3302 mm; ID: SHD-SLD; SST: Sparse, smart | BT, SR | LT showed the highest contributions for both BT and SR. SLD and smart support style produce better results for both BT and SR. | [16] |

ABS | Taguchi method | LT: 0.16–0.24 mm; CT: 35–55 °C; ET: 207–230 °C; PT: 110–132 °C; NS: 1–3; IDM: 0.8–1.2; ISS: 0.56–0.84; FT: 0.64–0.94 mm; IP: H, L, D; ID: 25–75%; IS: 72–108 mm/s; OS: 24–40 mm/s; ISS: 54–90 mm/s. | DA | The set of high values of IS, IP, mid-values of CT, LT, PT, NS, IDM, FT-linear, ISS, and low values of OS, ID, ISSM, and ET resulted in better dimensional accuracy of parts. | [17] |

ABS | RSM method | LT: 0.12–0.4 mm; BO: 0–90°; ID: 0–100%; NC: 2–10 | DA | ANN-GA predictions and optimization results are better than RSM-GA. | [18] |

RSM-GA & ANN-GA | |||||

PLA | Taguchi method | LT: 0.1–0.3 mm; PS: 70–110 mm/s; NT: 220–240 °C; filling style: raster (short, long and offset); RW: 0.3–0.5 mm. | Distortion | Fast filling speed, low nozzle temperature, and layer thickness offset raster style ensures smaller distortion | [20] |

PLA | RSM method | ID: 20–100%; T: 190–210 °C; PS: 50–150 mm/s | TS | ↑ID and T, with mid-values of speed results in ↑TS. GA-ANN produced better results than other methods. | [37] |

GA-RSM, GA-ANN, GA-ANFIS | |||||

PLA | RSM method | LT: 0.18–0.3 mm; PS: 36–60 mm/s; PT: 185–205 °C; OSS: 29–40 mm/s | SR | PS and LT showed significant contributions to SR. PSO and SOS predicted identical optimal conditions | [21] |

PSO and SOS | |||||

ASA | Taguchi method | LT: 0.18–0.33 mm; FP: solid, sparse, and hexagonal; BO: 0–90°; PP: XY, XZ, YZ; TP: 1–9 | Processing time, EC, width, length, thickness | PP is the most significant factor for ↓process time and EC. FP influences the more on width. LT contributions are more for length thickness. PP influences more on part thickness. | [22] |

DFA | |||||

Nylon | Taguchi method | LT: 0.1–0.3 mm; IFD: 50–100%; PS: 60–70 mm/s | UTS, impact strength, hardness, FS | IFD showed the highest contribution on all outputs. ↓LT is better for all outputs except hardness. | [23] |

Property | Value |
---|---|

Density | 1.08 g/cm^{3} |

Surface Hardness | RM30 |

Tensile Strength | 42 MPa |

Input Variables | Output Variables | ||||
---|---|---|---|---|---|

Layer Thickness, (mm) | Shell Thickness, (mm) | Infill Density, (%) | Print Speed, (mm/s) | Porosity, (%) | Cylindricity Error, (mm) |

0.16 | 2 | 20 | 30 | 8.17 | 0.172 |

0.16 | 2 | 60 | 50 | 5.64 | 0.159 |

0.16 | 2 | 100 | 70 | 3.21 | 0.400 |

0.16 | 3 | 20 | 50 | 7.36 | 0.332 |

0.16 | 3 | 60 | 70 | 4.46 | 0.438 |

0.16 | 3 | 100 | 30 | 3.27 | 0.470 |

0.16 | 4 | 20 | 70 | 3.81 | 0.599 |

0.16 | 4 | 60 | 30 | 4.98 | 1.076 |

0.16 | 4 | 100 | 50 | 2.15 | 0.920 |

0.22 | 2 | 20 | 70 | 7.63 | 0.202 |

0.22 | 2 | 60 | 30 | 4.41 | 0.259 |

0.22 | 2 | 100 | 50 | 3.87 | 0.349 |

0.22 | 3 | 20 | 30 | 7.35 | 0.145 |

0.22 | 3 | 60 | 50 | 5.71 | 0.352 |

0.22 | 3 | 100 | 70 | 4.50 | 0.390 |

0.22 | 4 | 20 | 50 | 6.63 | 0.223 |

0.22 | 4 | 60 | 70 | 4.99 | 0.582 |

0.22 | 4 | 100 | 30 | 3.27 | 0.558 |

0.28 | 2 | 20 | 50 | 6.30 | 0.418 |

0.28 | 2 | 60 | 70 | 6.85 | 0.723 |

0.28 | 2 | 100 | 30 | 2.26 | 0.296 |

0.28 | 3 | 20 | 70 | 7.98 | 0.246 |

0.28 | 3 | 60 | 30 | 5.17 | 0.390 |

0.28 | 3 | 100 | 50 | 4.95 | 0.204 |

0.28 | 4 | 20 | 30 | 7.22 | 0.183 |

0.28 | 4 | 60 | 50 | 6.42 | 0.407 |

0.28 | 4 | 100 | 70 | 6.11 | 0.612 |

Response | Cylindricity Error | Porosity | |||||
---|---|---|---|---|---|---|---|

Source | DF | Adj. SS | p-Value | Significance | Adj. SS | p-Value | Significance |

Model | 14 | 1.2983 | 0.000 | S | 78.074 | 0.000 | S |

Linear | 4 | 0.5097 | 0.000 | S | 53.145 | 0.000 | S |

Layer thickness | 1 | 0.0656 | 0.007 | S | 5.7949 | 0.000 | S |

Shell thickness | 1 | 0.2645 | 0.000 | S | 0.4213 | 0.020 | S |

Infill density | 1 | 0.1566 | 0.000 | S | 46.271 | 0.000 | S |

Print speed | 1 | 0.0229 | 0.079 | IS | 0.6576 | 0.006 | S |

Square | 4 | 0.2556 | 0.001 | S | 1.1985 | 0.012 | S |

Layer thickness^{2} | 1 | 0.0686 | 0.006 | S | 0.0033 | 0.817 | IS |

Shell thickness^{2} | 1 | 0.0899 | 0.003 | S | 1.0608 | 0.001 | S |

Infill density^{2} | 1 | 0.0781 | 0.004 | S | 0.0269 | 0.512 | IS |

Print speed^{2} | 1 | 0.0190 | 0.106 | IS | 0.1075 | 0.201 | IS |

2-Term Interaction | 6 | 0.5329 | 0.000 | S | 23.730 | 0.000 | S |

Layer thickness × Shell thickness | 1 | 0.3863 | 0.000 | S | 04.338 | 0.000 | S |

Layer thickness × Infill density | 1 | 0.0014 | 0.647 | IS | 1.9970 | 0.000 | S |

Layer thickness × Print speed | 1 | 0.0370 | 0.031 | S | 6.9556 | 0.000 | S |

Shell thickness × Infill density | 1 | 0.0125 | 0.182 | IS | 0.1147 | 0.188 | IS |

Shell thickness × Print speed | 1 | 0.0188 | 0.108 | IS | 2.4309 | 0.000 | S |

Infill density × Print speed | 1 | 0.0374 | 0.030 | S | 1.0936 | 0.001 | S |

Error | 12 | 0.0746 | 0.7062 | ||||

Total | 26 | 1.3729 | 78.7802 | ||||

R^{2}: 94.56%; R^{2} adjusted: 88.22% | R^{2}: 99.10%; R^{2} adjusted: 98.06% |

^{2}: Coefficient of determination; p-value: preset confidence value.

Case Study (w1 and w2) | Layer Thickness (mm) | Shell Thickness (mm) | Infill Density (%) | Print Speed (mm/s) | Porosity (%) | Cylindricity Error (mm) | Min f(z) |
---|---|---|---|---|---|---|---|

Case 1 (w _{1}, w_{2} = 0.5) | 0.21 | 2 | 100 | 30 | 2.62 | 0.147 | 2.564 |

Case 2 (w _{1 =} 0.6, w_{2} = 0.4) | 0.207 | 2 | 100 | 30 | 2.65 | 0.145 | 2.639 |

Case 3 (w _{1 =} 0.7, w_{2} = 0.3) | 0.18 | 2.23 | 20 | 58.26 | 2.87 | 0.15 | 2.905 |

Case 4 (w _{1 =} 0.4, w_{2} = 0.6) | 0.216 | 2 | 100 | 30 | 2.55 | 0.15 | 2.494 |

Case 5 (w _{1 =} 0.3, w_{2} = 0.7) | 0.22 | 2 | 100 | 30 | 2.49 | 0.16 | 2.526 |

Case 6 (w _{1 =} 0.2, w_{2} = 0.8) | 0.24 | 2 | 100 | 30 | 2.31 | 0.20 | 2.939 |

Optimizing Algorithm | Trials (Iterations & Population Size) | Layer Thickness (mm) | Shell Thickness | Infill Density (%) | Print Speed (mm/s) | Computational Time (s) |
---|---|---|---|---|---|---|

PSO | Trial 1 (100 & 20) | 0.21 | 2 | 100 | 20 | 0.014 |

JAYA | 0.28 | 2.5 | 100 | 30 | 0.013 | |

RAO-3 | 0.21 | 2 | 100 | 30 | 0.007 | |

BES | 0.21 | 2 | 100 | 30 | 0.007 | |

PSO | Trial 2 (300 & 10) | 0.21 | 2 | 100 | 20 | 0.017 |

JAYA | 0.18 | 2 | 100 | 31 | 0.013 | |

RAO-3 | 0.21 | 2 | 100 | 30 | 0.011 | |

BES | 0.21 | 2 | 100 | 30 | 0.011 |

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## Share and Cite

**MDPI and ACS Style**

Chandrashekarappa, M.P.G.; Chate, G.R.; Parashivamurthy, V.; Kumar, B.S.; Bandukwala, M.A.N.; Kaisar, A.; Giasin, K.; Pimenov, D.Y.; Wojciechowski, S.
Analysis and Optimization of Dimensional Accuracy and Porosity of High Impact Polystyrene Material Printed by FDM Process: PSO, JAYA, Rao, and Bald Eagle Search Algorithms. *Materials* **2021**, *14*, 7479.
https://doi.org/10.3390/ma14237479

**AMA Style**

Chandrashekarappa MPG, Chate GR, Parashivamurthy V, Kumar BS, Bandukwala MAN, Kaisar A, Giasin K, Pimenov DY, Wojciechowski S.
Analysis and Optimization of Dimensional Accuracy and Porosity of High Impact Polystyrene Material Printed by FDM Process: PSO, JAYA, Rao, and Bald Eagle Search Algorithms. *Materials*. 2021; 14(23):7479.
https://doi.org/10.3390/ma14237479

**Chicago/Turabian Style**

Chandrashekarappa, Manjunath Patel Gowdru, Ganesh Ravi Chate, Vineeth Parashivamurthy, Balakrishnamurthy Sachin Kumar, Mohd Amaan Najeeb Bandukwala, Annan Kaisar, Khaled Giasin, Danil Yurievich Pimenov, and Szymon Wojciechowski.
2021. "Analysis and Optimization of Dimensional Accuracy and Porosity of High Impact Polystyrene Material Printed by FDM Process: PSO, JAYA, Rao, and Bald Eagle Search Algorithms" *Materials* 14, no. 23: 7479.
https://doi.org/10.3390/ma14237479