# Design and Additive Manufacturing of Porous Sound Absorbers—A Machine-Learning Approach

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## Abstract

**:**

## 1. Introduction

#### 1.1. Description of Porous Media by Mechanical Models

#### 1.2. Additive Manufacturing of Porous Materials for Sound Absorption

#### 1.3. Aims and Scope of the Presented Work

#### 1.4. Outline of the Paper

## 2. Materials and Methods

#### 2.1. General Description of the Applied Process

#### 2.2. Specimen Design (Step 1)

#### 2.3. Additive Manufacturing of the Specimens (Step 2)

^{®}(4.1.2, Simplify3D, LLC, Cincinnati, OH, USA, 2020) and manufactured on the X400, a pro-consumer additive manufacturing machine from German RepRap GmbH (Feldkirchen, Germany), whose original extruders were replaced by E3D Hemera (Direct Extruder, 24 V). A $0.20$ mm Micro-Swiss nozzle (Ramsey, MN, USA) was used as the extrusion nozzle. The process parameters were used as listed in Table 2.

#### 2.4. Acoustical Investigation of Specimen Population (Step 3)

#### 2.4.1. Measurement of the Flow Resistivity

#### 2.4.2. Measurement of the Absorption Coefficient

#### 2.5. Inverse Parameter Identification Using the JCAL-Model (Step 4)

^{−1}. Therefore, it is assumed that indeed some differences occur for each run but since the deviations are rather small, the general behavior is kept. Thus, it is assumed that the data augmentation yields meaningful results to enrich the database and the choice of input parameters is reasonable.

#### 2.6. Machine-Learning for the Geometry-Biot Parameter Relations (Step 5)

#### 2.6.1. Scaling of the Input and Output Data

^{−1}, the magnitude of the plane angle is in the order of 1 × 10

^{1}. The input data is scaled using mean and standard deviation of the data with:

^{3}–1 × 10

^{5}, the magnitude of the static thermal permeability is approx. 1 × 10

^{−8}. It could be found that the training process becomes rather unsuccessful when the input and output data is used directly. This is expected to be a result of the large range of values within the data. Therefore, variable scaling is introduced and applied to the Biot parameters in order to result in training data comprising a smaller value range. The resulting scaled Biot parameters are shown in Table 3:

#### 2.6.2. K-Nearest Neighbors

#### 2.6.3. Artificial Neural Network

## 3. Results and Discussion

#### 3.1. Inspection of the Specimen Population

#### 3.1.1. Results of the Optical Investigation of the Additively Manufactured Specimens

#### 3.1.2. Results of Acoustic Measurements of the Specimen Population

#### 3.1.3. Results of the Inverse Parameter Identification

#### 3.2. Setup of Machine-Learning Models for Predicting Biot Parameters from the Specimen Geometry

#### 3.2.1. Performance of the KNN Model for Predicting Biot Parameters

#### 3.2.2. Performance of the ANN Model for Predicting Biot Parameters

^{−8}–1.00 × 10

^{−8}m

^{2}can be seen that are predicted by the ANN model with rather low values of 0.01 × 10

^{−8}–0.45 × 10

^{−8}m

^{2}. Here as well, it is assumed that the geometry parameter combination has only a low effect on this acoustic parameter which leads to rather inaccurate predictions. To summarizing, the training of the ANN model can be assumed to be successful here as well.

#### 3.3. Application of the Machine-Learning Models for Absorber Design (Step 6)

#### 3.3.1. Specimen Design with K-Nearest Neighbor Approach

#### 3.3.2. Specimen Design with Artificial Neural Network Approach

#### 3.4. Design of a Material with Different Height and Prediction of Another Frequency Range

## 4. Conclusions and Outlook

#### 4.1. Conclusions

#### 4.2. Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | specimen cross sectional area (flow resistivity measurement) |

d | bar width |

h | bar height |

k | number of neighbors in KNN model |

${k}_{0}^{\prime}$ | static thermal permeability |

l | specimen height |

q | volume flow through the specimen (flow resistivity measurement) |

R | airflow resistance |

R | coefficient of determination (performance measure of ML model training) |

s | bar spacing |

$\alpha $ | absorption coefficient |

${\alpha}_{\infty}$ | tortuosity (high frequency limit) |

$\Delta p$ | pressure drop over the specimen (flow resistivity measurement) |

$\mathsf{\Lambda}$ | viscous characteristic length |

${\mathsf{\Lambda}}^{\prime}$ | thermal characteristic length |

$\mathsf{\Xi}$ | flow resistivity |

$\rho $ | Pearson’s Correlation Coefficient |

$\phi $ | plane angle |

$\varphi $ | porosity |

AM | additive manufacturing |

ANN | artificial neural network |

DOE | design of experiment |

JCAL | Johnson–Champoux–Allard–Lafarge |

KNN | k-nearest neighbor |

LHS | latin hypercube sampling |

MEX | material extrusion |

ML | machine learning |

PBF-P | powder bed fusion of polymers |

PET-G | glycol modified polyethylene terephthalate |

PLA | polylactide |

VAT | vat photopolymerization |

## Appendix A. Overview of the Design Parameter Space LHS Sampling

**Table A1.**Overview of the generated test Spec.s. The design parameters (d, s, h, $\phi $) were varied using a Latin hypercube sampling. The process parameters (layer height, the extrusion width) are used during the manufacturing process and are given here for information only.

Variation Parameters (Spec.) | Process Parameters (AM) | |||||
---|---|---|---|---|---|---|

Bar Width (d) | Bar Spacing (s) | Bar Height (h) | Angle ($\mathit{\phi}$) | Layer Height | Extrusion Width | |

Spec. 0 | $0.20$ mm | $0.70$ mm | $0.16$ mm | $40.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 1 | $0.50$ mm | $0.40$ mm | $0.11$ mm | $70.00$${}^{\circ}$ | $0.11$ mm | $0.25$ mm |

Spec. 2 | $0.40$ mm | $0.40$ mm | $0.08$ mm | $60.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 3 | $0.40$ mm | $0.80$ mm | $0.06$ mm | $70.00$${}^{\circ}$ | $0.06$ mm | $0.20$ mm |

Spec. 4 | $0.50$ mm | $0.20$ mm | $0.12$ mm | $80.00$${}^{\circ}$ | $0.06$ mm | $0.25$ mm |

Spec. 5 | $0.30$ mm | $0.90$ mm | $0.15$ mm | $50.00$${}^{\circ}$ | $0.075$ mm | $0.15$ mm |

Spec. 6 | $0.30$ mm | $0.70$ mm | $0.13$ mm | $50.00$${}^{\circ}$ | $0.065$ mm | $0.15$ mm |

Spec. 7 | $0.20$ mm | $0.30$ mm | $0.17$ mm | $40.00$${}^{\circ}$ | $0.085$ mm | $0.20$ mm |

Spec. 8 | $0.40$ mm | $0.50$ mm | $0.11$ mm | $80.00$${}^{\circ}$ | $0.11$ mm | $0.20$ mm |

Spec. 9 | $0.30$ mm | $0.90$ mm | $0.10$ mm | $40.00$${}^{\circ}$ | $0.10$ mm | $0.15$ mm |

Spec. 10 | $0.20$ mm | $0.90$ mm | $0.12$ mm | $50.00$${}^{\circ}$ | $0.06$ mm | $0.20$ mm |

Spec. 11 | $0.30$ mm | $0.80$ mm | $0.07$ mm | $80.00$${}^{\circ}$ | $0.07$ mm | $0.15$ mm |

Spec. 12 | $0.30$ mm | $0.50$ mm | $0.16$ mm | $80.00$${}^{\circ}$ | $0.08$ mm | $0.15$ mm |

Spec. 13 | $0.50$ mm | $0.30$ mm | $0.14$ mm | $40.00$${}^{\circ}$ | $0.07$ mm | $0.25$ mm |

Spec. 14 | $0.20$ mm | $0.90$ mm | $0.07$ mm | $30.00$${}^{\circ}$ | $0.07$ mm | $0.20$ mm |

Spec. 15 | $0.30$ mm | $0.20$ mm | $0.06$ mm | $70.00$${}^{\circ}$ | $0.06$ mm | $0.15$ mm |

Spec. 16 | $0.50$ mm | $0.80$ mm | $0.15$ mm | $80.00$${}^{\circ}$ | $0.075$ mm | $0.25$ mm |

Spec. 17 | $0.20$ mm | $0.30$ mm | $0.08$ mm | $80.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 18 | $0.40$ mm | $0.20$ mm | $0.08$ mm | $60.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 19 | $0.20$ mm | $0.70$ mm | $0.07$ mm | $50.00$${}^{\circ}$ | $0.07$ mm | $0.20$ mm |

Spec. 20 | $0.20$ mm | $0.70$ mm | $0.08$ mm | $70.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 21 | $0.20$ mm | $0.50$ mm | $0.11$ mm | $50.00$${}^{\circ}$ | $0.11$ mm | $0.20$ mm |

Spec. 22 | $0.40$ mm | $0.70$ mm | $0.09$ mm | $50.00$${}^{\circ}$ | $0.09$ mm | $0.20$ mm |

Spec. 23 | $0.30$ mm | $0.60$ mm | $0.18$ mm | $90.00$${}^{\circ}$ | $0.09$ mm | $0.15$ mm |

Spec. 24 | $0.50$ mm | $0.60$ mm | $0.17$ mm | $70.00$${}^{\circ}$ | $0.085$ mm | $0.25$ mm |

Spec. 25 | $0.50$ mm | $0.60$ mm | $0.06$ mm | $40.00$${}^{\circ}$ | $0.06$ mm | $0.25$ mm |

Spec. 26 | $0.20$ mm | $0.40$ mm | $0.20$ mm | $80.00$${}^{\circ}$ | $0.10$ mm | $0.20$ mm |

Spec. 27 | $0.20$ mm | $0.50$ mm | $0.16$ mm | $90.00$${}^{\circ}$ | $0.08$ mm | $0.20$ mm |

Spec. 28 | $0.40$ mm | $0.50$ mm | $0.17$ mm | $40.00$${}^{\circ}$ | $0.085$ mm | $0.20$ mm |

Spec. 29 | $0.30$ mm | $0.90$ mm | $0.13$ mm | $60.00$${}^{\circ}$ | $0.065$ mm | $0.15$ mm |

Spec. 30 | $0.30$ mm | $0.30$ mm | $0.10$ mm | $40.00$${}^{\circ}$ | $0.10$ mm | $0.15$ mm |

Spec. 31 | $0.40$ mm | $0.40$ mm | $0.09$ mm | $60.00$${}^{\circ}$ | $0.09$ mm | $0.20$ mm |

Spec. 32 | $0.50$ mm | $0.60$ mm | $0.14$ mm | $60.00$${}^{\circ}$ | $0.07$ mm | $0.25$ mm |

Spec. 33 | $0.30$ mm | $0.60$ mm | $0.14$ mm | $50.00$${}^{\circ}$ | $0.07$ mm | $0.15$ mm |

Spec. 34 | $0.20$ mm | $1.00$ mm | $0.19$ mm | $80.00$${}^{\circ}$ | $0.095$ mm | $0.20$ mm |

Spec. 35 | $0.40$ mm | $0.90$ mm | $0.10$ mm | $30.00$${}^{\circ}$ | $0.10$ mm | $0.20$ mm |

Spec. 36 | $0.30$ mm | $0.70$ mm | $0.05$ mm | $70.00$${}^{\circ}$ | $0.05$ mm | $0.15$ mm |

Spec. 37 | $0.40$ mm | $0.80$ mm | $0.13$ mm | $60.00$${}^{\circ}$ | $0.065$ mm | $0.20$ mm |

Spec. 38 | $0.30$ mm | $0.90$ mm | $0.19$ mm | $90.00$${}^{\circ}$ | $0.095$ mm | $0.15$ mm |

Spec. 39 | $0.40$ mm | $0.80$ mm | $0.15$ mm | $70.00$${}^{\circ}$ | $0.075$ mm | $0.20$ mm |

Spec. 40 | $0.20$ mm | $1.00$ mm | $0.18$ mm | $90.00$${}^{\circ}$ | $0.09$ mm | $0.20$ mm |

Spec. 41 | $0.30$ mm | $0.50$ mm | $0.09$ mm | $30.00$${}^{\circ}$ | $0.09$ mm | $0.15$ mm |

Spec. 42 | $0.40$ mm | $0.40$ mm | $0.17$ mm | $50.00$${}^{\circ}$ | $0.085$ mm | $0.20$ mm |

Spec. 43 | $0.40$ mm | $0.30$ mm | $0.20$ mm | $30.00$${}^{\circ}$ | $0.10$ mm | $0.20$ mm |

Spec. 44 | $0.30$ mm | $0.60$ mm | $0.19$ mm | $70.00$${}^{\circ}$ | $0.095$ mm | $0.15$ mm |

Spec. 45 | $0.20$ mm | $0.80$ mm | $0.18$ mm | $30.00$${}^{\circ}$ | $0.09$ mm | $0.20$ mm |

Spec. 46 | $0.40$ mm | $0.40$ mm | $0.14$ mm | $80.00$${}^{\circ}$ | $0.07$ mm | $0.20$ mm |

Spec. 47 | $0.20$ mm | $0.30$ mm | $0.06$ mm | $60.00$${}^{\circ}$ | $0.06$ mm | $0.20$ mm |

Spec. 48 | $0.40$ mm | $0.40$ mm | $0.12$ mm | $40.00$${}^{\circ}$ | $0.06$ mm | $0.20$ mm |

Spec. 49 | $0.30$ mm | $1.00$ mm | $0.12$ mm | $60.00$${}^{\circ}$ | $0.06$ mm | $0.15$ mm |

## Appendix B. Photographs of the Used Measurement Setups

## Appendix C. Inverse Parameter Identification for the Biot Parameters

Property/Parameter | Setting/Value |
---|---|

objective function | takes the Biot parameters, applies Equations (1)–(A2) to compute the absorption coefficient, computes the error using Equation (5) with regard to the measured results and returns the error |

bounds | $\varphi \in \left[0.20,0.80\right]$ ${\alpha}_{\infty}\in \left[0,10\right]$ $\mathsf{\Lambda}\in \left[1\times {\mathrm{10}}^{-7},1\times {\mathrm{10}}^{-3}\right]$ ${\mathsf{\Lambda}}^{\prime}\in \left[1e-7,1e-3\right]$ ${k}_{0}^{\prime}\in \left[1e-12,1e-8\right]$ |

convergence tolerance (rel.) | 1 × 10^{−2} |

mutation parameter | 0.50 |

recombination parameter | 0.70 |

max. iterations | 1000 |

population size | 75 |

**Figure A2.**A correlation plot for the inversely estimated porosity of the specimen (vertical axis) and an analytical porosity estimate, computed using $s/(s+d)$. A general correlation (correlation coefficient $\rho =0.63$) can be found. Therefore, it is assumed that the general procedure can be trusted.

**Figure A3.**Relative standard deviation for the data augmentation procedure. The standard deviation is computed for all ten runs of the inverse parameter estimates for each specimen and normalized by the mean of the ten runs. It can be seen, that the data varies about approx. 10% for most Biot parameters.

## Appendix D. Mathematical Formulation o the JCAL Model Used Here

Symbol | Quantity | Symbol | Quantity |
---|---|---|---|

$\mathsf{\Xi}$ | flow resistivity | $\eta $ | dynamic viscosity of the fluid |

${\alpha}_{\infty}$ | tortuosity | ${\rho}_{0}$ | ambient density of the fluid |

$\varphi $ | porosity | $\omega $ | angular frequency |

$\mathsf{\Lambda}$ | viscous characteristic length | ${C}_{p}$ | heat capacity at constant pressure |

${\mathsf{\Lambda}}^{\prime}$ | thermal characteristic length | $\gamma $ | heat capacity ratio |

${k}_{0}^{\prime}$ | static thermal permeability | $\kappa $ | thermal conductivity |

## Appendix E. Settings of the Machine Learning Models

**Table A4.**Settings of the KNN and ANN model (implementation see python-library scikit-learn [71]).

(a) Settings of the KNN model | |

parameter | setting |

n neighbors | 5 |

weights | distance (weighting by inverse of the distance) |

algorithm | auto (default, chooses according input data) |

leaf size | 30 (default) |

p | 2 (default) |

metric | minkowski (default) |

metric params | None (default) |

n jobs | Nonde (default) |

parameter | setting |

hidden layers | 4000-100 |

activation | ReLU |

solver | Adam |

alpha | 1 × 10^{−4} (default) |

batch size | auto (default) |

learning rate | constant (default) |

learning rate init | 1 × 10^{−3} (default) |

power t | 0.50 (default) |

max iter | 50.00 |

shuffle | True (default) |

random state | None (default) |

tol | 1 × 10^{−9} |

verbose | False (default) |

warm start | False (default) |

momentum | 0.90 (default) |

nesterovs momentum | True (default) |

early stopping | False (default) |

validation fraction | 0.10 (default) |

beta 1 | 0.90 (default) |

beta 2 | 1.00 (default) |

epsilon | 1 × 10^{−8} (default) |

n iter no change | 100 |

max fun | 15,000 (default) |

## Appendix F. Results of the Hyperparameter-Tuning of the ML Models

**Figure A4.**Hyperparameter tuning of the KNN and ANN model. The tuning is done on the validation data during the 3-fold cross-validation. The error bars indicate the 95% confidence interval of the variation during the cross-validation procedure.

## Appendix G. Settings of the Evolutional Algorithm for the Inverse Absorber Design and Results

Property/Parameter | Setting/Value |
---|---|

objective function | takes the design variables, inputs into ML model, computes Biot parameters, applies Equations (1)–(A2) to compute the absorption coefficient, computes and returns the error using Equation (8) with regard to target curve |

bounds | see Table 6 |

convergence tolerance (rel.) | 1 × 10^{−2} |

mutation parameter | 0.50 |

recombination parameter | 0.70 |

max. iterations | 1000 |

population size | 60 |

**Figure A5.**Overview over all designed Spec.s using the proposed ML approach. The colors refer to the ML model and the Spec. start population: Blue: basic population, Orange: KNN model, Green: ANN model. The symbols indicate the absorption coefficient target curve: ♦: target curve “low”, ■: target curve: “medium”, ▲: target curve: “high”.

## References

- Labia, L.; Shtrepi, L.; Astolfi, A. Improved Room Acoustics Quality in Meeting Rooms: Investigation on the Optimal Configurations of Sound-Absorptive and Sound-Diffusive Panels. Acoustics
**2020**, 2, 451–473. [Google Scholar] [CrossRef] - Nayfeh, A.H.; Kaiser, J.E.; Telionis, D.P. Acoustics of Aircraft Engine-Duct Systems. AIAA J.
**1975**, 13, 130–153. [Google Scholar] [CrossRef] - Sutliff, D.L.; Elliott, D.; Jones, M.; Hartley, T.C. Attenuation of FJ44 Turbofan Engine Noise With a Foam-Metal Liner Installed Over-the-Rotor. In Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), Miami, FL, USA, 11–13 May 2009. [Google Scholar]
- Wilby, J.F.; Scharton, T. Acoustic Transmission through a Fuselage Sidewall; National Aeronatics and Space Administration: Cambridge, MA, USA, 1973.
- Blech, C.; Appel, C.K.; Ewert, R.; Delfs, J.W.; Langer, S.C. Numerical prediction of passenger cabin noise due to jet noise by an ultra–high–bypass ratio engine. J. Sound Vib.
**2020**, 464, 114960. [Google Scholar] [CrossRef] - Antonio, J.; Tadeu, A.; Godinho, L. Analytical evaluation of the acoustic insulation provided by double infinite walls. J. Sound Vib.
**2003**, 263, 113–129. [Google Scholar] [CrossRef][Green Version] - Van den Wyngaert, J.C.; Schevenels, M.; Reynders, E.P. Predicting the sound insulation of finite double-leaf walls with a flexible frame. Appl. Acoust.
**2018**, 141, 93–105. [Google Scholar] [CrossRef] - Beck, S.; Langer, S. Modeling of flow-induced sound in porous materials. Int. J. Numer. Methods Eng.
**2014**, 98, 44–58. [Google Scholar] [CrossRef] - Geyer, T.; Sarradj, E.; Fritzsche, C. Measurement of the noise generation at the trailing edge of porous airfoils. Exp. Fluids
**2010**, 48, 291–308. [Google Scholar] [CrossRef] - Geyer, T.F.; Sarradj, E. Trailing Edge Noise of Partially Porous Airfoils. In Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 16–20 June 2014. [Google Scholar] [CrossRef][Green Version]
- Ewert, R.; Appel, C.; Dierke, J.; Herr, M. RANS/CAA Based Prediction of NACA 0012 Broadband Trailing Edge Noise and Experimental Validation. In Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), Miami, FL, USA, 11–13 May 2009. [Google Scholar] [CrossRef]
- Herr, M.; Dobrzynski, W. Experimental Investigations in Low-Noise Trailing Edge Design. AIAA J.
**2005**, 43, 1167–1175. [Google Scholar] [CrossRef] - Yang, M.; Sheng, P. Sound absorption structures: From porous media to acoustic metamaterials. Annu. Rev. Mater. Res.
**2017**, 47, 83–114. [Google Scholar] [CrossRef] - Koponen, A.; Kataja, M.; Timonen, J.v. Tortuous flow in porous media. Phys. Rev. E
**1996**, 54, 406. [Google Scholar] [CrossRef] - Herr, M.; Rossignol, K.S.; Delfs, J.; Lippitz, N.; Mößner, M. Specification of porous materials for low-noise trailing-edge applications. In Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, USA, 16–20 June 2014; p. 3041. [Google Scholar]
- Biot, M.A. Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range. J. Acoust. Soc. Am.
**1956**, 28, 168–178. [Google Scholar] [CrossRef] - Biot, M.A. Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. II. Higher Frequency Range. J. Acoust. Soc. Am.
**1956**, 28, 179–191. [Google Scholar] [CrossRef] - Dazel, O.; Brouard, B.; Dauchez, N.; Geslain, A. Enhanced Biot’s finite element displacement formulation for porous materials and original resolution methods based on normal modes. Acta Acust. United Acust.
**2009**, 95, 527–538. [Google Scholar] [CrossRef] - Johnson, D.L.; Koplik, J.; Dashen, R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J. Fluid Mech.
**1987**, 176, 379–402. [Google Scholar] [CrossRef] - Champoux, Y.; Allard, J. Dynamic tortuosity and bulk modulus in air-saturated porous media. J. Appl. Phys.
**1991**, 70, 1975–1979. [Google Scholar] [CrossRef] - Lafarge, D.; Lemarinier, P.; Allard, J.F.; Tarnow, V. Dynamic compressibility of air in porous structures at audible frequencies. J. Acoust. Soc. Am.
**1997**, 102, 1995–2006. [Google Scholar] [CrossRef][Green Version] - Ogam, E.; Fellah, Z.E.A.; Sebaa, N.; Groby, J.P. Non-ambiguous recovery of Biot poroelastic parameters of cellular panels using ultrasonicwaves. J. Sound Vib.
**2011**, 330, 1074–1090. [Google Scholar] [CrossRef] - Atalla, Y.; Panneton, R. Inverse acoustical characterization of open cell porous media using impedance tube measurements. Can. Acoust.
**2005**, 33, 11–24. [Google Scholar] - Kutscher, K.; Geier, M.; Krafczyk, M. Multiscale simulation of turbulent flow interacting with porous media based on a massively parallel implementation of the cumulant lattice Boltzmann method. Comput. Fluids
**2019**, 193, 103733. [Google Scholar] [CrossRef] - Rosen, D.W. Research supporting principles for design for additive manufacturing. Virtual Phys. Prototyp.
**2014**, 9. [Google Scholar] [CrossRef] - Ring, T.P.; Langer, S.C. Design, Experimental and Numerical Characterization of 3D-Printed Porous Absorbers. Materials
**2019**, 12, 3397. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gebhardt, A. Additive Fertigungsverfahren: Additive Manufacturing und 3D-Drucken für Prototyping—Tooling—Produktion; 5. neu bearbeitete und erweiterte auflage ed.; Carl Hanser: München, Germany, 2016. [Google Scholar]
- Kumke, M.; Watschke, H.; Hartogh, P.; Bavendiek, A.K.; Vietor, T. Methods and tools for identifying and leveraging additive manufacturing design potentials. Int. J. Interact. Des. Manuf. (IJIDeM)
**2018**, 12, 481–493. [Google Scholar] [CrossRef] - Cai, X.; Guo, Q.; Hu, G.; Yang, J. Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar Helmholtz resonators. Appl. Phys. Lett.
**2014**, 105, 121901. [Google Scholar] [CrossRef][Green Version] - Jiang, C.; Moreau, D.; Doolan, C. Acoustic Absorption of Porous Materials Produced by Additive Manufacturing with Varying Geometries. In Proceedings of the ACOUSTICS 2017, Perth, Australia, 19–22 November 2017. [Google Scholar]
- Liu, Z.; Zhan, J.; Fard, M.; Davy, J.L. Acoustic properties of a porous polycarbonate material produced by additive manufacturing. Mater. Lett.
**2016**, 181, 296–299. [Google Scholar] [CrossRef] - Guild, M.D.; García-Chocano, V.M.; Kan, W.; Sanchez-Dehesa, J. Acoustic metamaterial absorbers based on multi-scale sonic crystals. J. Acoust. Soc. Am.
**2014**, 136, 2076. [Google Scholar] [CrossRef] - Liu, Z.; Zhan, J.; Fard, M.; Davy, J.L. Acoustic properties of multilayer sound absorbers with a 3D printed micro-perforated panel. Appl. Acoust.
**2017**, 121, 25–32. [Google Scholar] [CrossRef] - Fotsing, E.R.; Dubourg, A.; Ross, A.; Mardjono, J. Acoustic properties of periodic micro-structures obtained by additive manufacturing. Appl. Acoust.
**2019**, 148, 322–331. [Google Scholar] [CrossRef] - Cai, X.; Yang, J.; Hu, G.; Lu, T. Sound absorption by acoustic microlattice with optimized pore configuration. J. Acoust. Soc. Am.
**2018**, 144, EL138–EL143. [Google Scholar] [CrossRef] - Guild, M.D.; Rothko, M.; Sieck, C.F.; Rohde, C.; Orris, G. 3D printed sound absorbers using functionally-graded sonic crystals. J. Acoust. Soc. Am.
**2018**, 143, 1714. [Google Scholar] [CrossRef] - Zieliński, T.G.; Opiela, K.C.; Pawłowski, P.; Dauchez, N.; Boutin, T.; Kennedy, J.; Trimble, D.; Rice, H.; Van Damme, B.; Hannema, G.; et al. Reproducibility of sound-absorbing periodic porous materials using additive manufacturing technologies: Round robin study. Addit. Manuf.
**2020**, 36, 101564. [Google Scholar] [CrossRef] - Boulvert, J.; Costa-Baptista, J.; Cavalieri, T.; Perna, M.; Fotsing, E.R.; Romero-García, V.; Gabard, G.; Ross, A.; Mardjono, J.; Groby, J.P. Acoustic modeling of micro-lattices obtained by additive manufacturing. Appl. Acoust.
**2020**, 164, 107244. [Google Scholar] [CrossRef][Green Version] - Setaki, F.; Tenpierik, M.; Turrin, M.; van Timmeren, A. Acoustic absorbers by additive manufacturing. Build. Environ.
**2014**, 72, 188–200. [Google Scholar] [CrossRef] - Boulvert, J.; Cavalieri, T.; Costa-Baptista, J.; Schwan, L.; Romero-García, V.; Gabard, G.; Fotsing, E.R.; Ross, A.; Mardjono, J.; Groby, J.P. Optimally graded porous material for broadband perfect absorption of sound. J. Appl. Phys.
**2019**, 126, 175101. [Google Scholar] [CrossRef] - Ring, T.; Kuschmitz, S.; Watschke, H.; Vietor, T.; Langer, S. Additive Fertigung und Charakterisierung akustisch wirksamer Materialien. In Proceedings of the Tagungsband DAGA 2018-44. Jahrestagung für Akustik, Munich, Germany, 19–22 March 2018; pp. 451–455. [Google Scholar]
- Comiti, J.; Renaud, M. A new model for determining mean structure parameters of fixed beds from pressure drop measurements: Application to beds packed with parallelepipedal particles. Chem. Eng. Sci.
**1989**, 44, 1539–1545. [Google Scholar] [CrossRef] - Yun, M.; Yu, B.; Xu, P.; Wu, J. Geometrical models for tortuosity of streamlines in three-dimensional porous media. Can. J. Chem. Eng.
**2006**, 84, 301–309. [Google Scholar] [CrossRef] - Bo-Ming, Y.; Jian-Hua, L. A geometry model for tortuosity of flow path in porous media. Chin. Phys. Lett.
**2004**, 21, 1569. [Google Scholar] [CrossRef] - Iannace, G.; Ciaburro, G.; Trematerra, A. Modelling sound absorption properties of broom fibers using artificial neural networks. Appl. Acoust.
**2020**, 163, 107239. [Google Scholar] [CrossRef] - Ciaburro, G.; Iannace, G.; Ali, M.; Alabdulkarem, A.; Nuhait, A. An artificial neural network approach to modelling absorbent asphalts acoustic properties. J. King Saud Univ.-Eng. Sci.
**2020**. [Google Scholar] [CrossRef] - Gardner, G.C.; O’Leary, M.E.; Hansen, S.; Sun, J. Neural networks for prediction of acoustical properties of polyurethane foams. Appl. Acoust.
**2003**, 64, 229–242. [Google Scholar] [CrossRef] - Liu, J.; Bao, W.; Shi, L.; Zuo, B.; Gao, W. General regression neural network for prediction of sound absorption coefficients of sandwich structure nonwoven absorbers. Appl. Acoust.
**2014**, 76, 128–137. [Google Scholar] [CrossRef] - Watschke, H.; Kuschmitz, S.; Heubach, J.; Lehne, G.; Vietor, T. A Methodical Approach to Support Conceptual Design for Multi-Material Additive Manufacturing. Proc. Des. Soc. Int. Conf. Eng. Des.
**2019**, 1, 659–668. [Google Scholar] [CrossRef][Green Version] - Blösch-Paidosh, A.; Shea, K. Design Heuristics for Additive Manufacturing Validated Through a User Study1. J. Mech. Des.
**2019**, 141. [Google Scholar] [CrossRef] - Pradel, P.; Zhu, Z.; Bibb, R.; Moultrie, J. Investigation of design for additive manufacturing in professional design practice. J. Eng. Des.
**2018**, 29, 165–200. [Google Scholar] [CrossRef][Green Version] - Iman, R.L. Latin Hypercube Sampling. In Encyclopedia of Quantitative Risk Analysis and Assessment; Melnick, E.L., Everitt, B.S., Eds.; John Wiley & Sons, Ltd.: Chichester, UK, 2008. [Google Scholar] [CrossRef]
- Stocki, R. A method to improve design reliability using optimal Latin hypercube sampling. Comput. Assist. Mech. Eng. Sci.
**2005**, 12, 393–411. [Google Scholar] - Helton, J.C.; Davis, F.J. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab. Eng. Syst. Saf.
**2003**, 81, 23–69. [Google Scholar] [CrossRef][Green Version] - Sheikholeslami, R.; Razavi, S. Progressive Latin Hypercube Sampling: An efficient approach for robust sampling-based analysis of environmental models. Environ. Model. Softw.
**2017**, 93, 109–126. [Google Scholar] [CrossRef] - Florian, A. An efficient sampling scheme: Updated Latin Hypercube Sampling. Probabilistic Eng. Mech.
**1992**, 7, 123–130. [Google Scholar] [CrossRef] - Iman, R.L.; Conover, W.J. A distribution-free approach to inducing rank correlation among input variables. Commun. Stat.-Simul. Comput.
**1982**, 11, 311–334. [Google Scholar] [CrossRef] - McKay, M.D.; Beckman, R.J.; Conover, W.J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics
**1979**, 21, 239. [Google Scholar] [CrossRef] - OpenSCAD. Available online: https://github.com/openscad/openscad (accessed on 23 February 2021).
- Ngo, T.D.; Kashani, A.; Imbalzano, G.; Nguyen, K.T.; Hui, D. Additive manufacturing (3D printing): A review of materials, methods, applications and challenges. Compos. Part B Eng.
**2018**, 143, 172–196. [Google Scholar] [CrossRef] - Acoustics—Materials for Acoustical Applications—Determination of Airflow Resistance; Technical Report ISO 9053-2:2020; International Standards Organization: Geneva, Switzerland, 2020.
- Acoustics—Determination of Sound Absorption Coefficient and Impedance in Impedance Tubes—Part 2: Transfer-Function Method; Technical Report ISO 10534-2:1998; International Standards Organization: Geneva, Switzerland, 1998.
- APMR—Acoustical Porous Material Recipes. Available online: https://apmr.matelys.com/ (accessed on 22 February 2021).
- Beck, S.C.; Müller, L.; Langer, S.C. Numerical assessment of the vibration control effects of porous liners on an over-the-wing propeller configuration. CEAS Aeronaut. J.
**2016**, 7, 275–286. [Google Scholar] [CrossRef] - Allard, J.; Atalla, N. Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials 2e; John Wiley & Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- Storn, R.; Price, K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mitchell, R.; Michalski, J.; Carbonell, T. An Artificial Intelligence Approach; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Kubat, M. An Introduction to Machine Learning; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Ring, T.P. Effiziente Unsicherheitsquantifizierung in der Akustik mittels eines Multi-Modell-Verfahrens. Ph.D. Thesis, TU Braunschweig, Braunschweig, Germany, 2020. [Google Scholar]
- Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; et al. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res.
**2011**, 12, 2825–2830. [Google Scholar] - Ho, T.K. Nearest neighbors in random subspaces. In Joint IAPR International Workshops on Statistical Techniques in Pattern Recognition (SPR) and Structural and Syntactic Pattern Recognition (SSPR); Springer: Berlin/Heidelberg, Germany, 1998; pp. 640–648. [Google Scholar]
- Kramer, O. K-nearest neighbors. In Dimensionality Reduction with Unsupervised Nearest Neighbors; Springer: Berlin/Heidelberg, Germany, 2013; pp. 13–23. [Google Scholar]
- Laaksonen, J.; Oja, E. Classification with learning k-nearest neighbors. In Proceedings of the International Conference on Neural Networks (ICNN’96), Washington, DC, USA, 3–6 June 1996; Volume 3, pp. 1480–1483. [Google Scholar]
- Rosenblatt, F. The perceptron: A probabilistic model for information storage and organization in the brain. Psychol. Rev.
**1958**, 65, 386–408. [Google Scholar] [CrossRef][Green Version] - Rosenblatt, F. Perceptron simulation experiments. Proc. IRE
**1960**, 48, 301–309. [Google Scholar] [CrossRef] - LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] - LeNail, A. NN-SVG: Publication-Ready Neural Network Architecture Schematics. J. Open Source Softw.
**2019**, 4, 747. [Google Scholar] [CrossRef] - Barrett, J.P. The Coefficient of Determination—Some Limitations. Am. Stat.
**1974**, 28, 19–20. [Google Scholar] [CrossRef] - Nagelkerke, N.J. A note on a general definition of the coefficient of determination. Biometrika
**1991**, 78, 691–692. [Google Scholar] [CrossRef]

**Figure 1.**General procedure of the work to obtain ML models that predict Biot parameters from specimen geometry and their application to absorber design.

**Figure 2.**Overview of investigated specimen. (

**a**) shows the test specimen including the design variables (from [26]) which are used to vary the test specimen. (

**b**) shows the additive manufactured specimen 2.

**Figure 3.**Additive manufacturing of three test specimens using material extrusion. Used 3D printer: X400, ppro-consumer additive manufacturing machine from German RepRap GmbH.

**Figure 4.**Flowchart of the inverse parameter identification process. Starting with an initial guess for the Biot parameters, the absorption coefficient is computed using the JCAL model and the results is compared to the measured data. Based on the error measure, the Biot parameters are updated until convergence is reached.

**Figure 5.**Classical model building process (from [70]).

**Figure 6.**Sketch of the ANN used within this work, drawn using [78]. The input layer has four neurons and takes the geometry parameters w, s, h, $\phi $ of the specimen. The output layer has six neurons for the Biot parameters $\mathsf{\Xi}$, $\varphi $, ${\alpha}_{\infty}$, $\mathsf{\Lambda}$, ${\mathsf{\Lambda}}^{\prime}$ and ${k}_{0}^{\prime}$. The two hidden layers have 4000 and 100 neurons (not to scale), respectively. The activation function for all neurons is the ReLU function.

**Figure 7.**Measured data (flow resistivity and absorption coefficient) of all 50 investigated specimens. The specimens 2 and 39 are marked in orange and blue, respectively, and represent those specimens with the highest and lowest mean absorption coefficient in the frequency range.

**Figure 8.**Actual-target comparison of specimens 2 and 39 using Keyence VR5200 3D surface profilometer. (

**a**) shows specimen 2 with only slight deviations in the actual-target comparison. However, some small tapers can be seen in the filament strands. (

**b**) shows specimen 39, which also shows only minimal deviations in the actual-target comparison.

**Figure 9.**Correlation of the flow resistivity and the design variables of all manufactured specimens. It can be seen that only the bar width and the bar spacing show a relevant correlation with the flow resistivity, whereas the flow resistivity does not (linearly) depend on the bar height and plane angle.

**Figure 10.**Comparison of the measured absorption coefficient, the absorption coefficient computed from the inversely identified Biot parameters (’Fit’) and absorption coefficient computed using the Biot parameters outputted by the KNN model.

**Figure 11.**Correlation graphs for inversely identified Biot parameters and their corresponding prediction by the KNN model. It can be seen that all quantities are predicted with reasonable accuracy, thus the model is qualified for further application.

**Figure 12.**Comparison of the measured absorption coefficient, the absorption coefficient computed from the inversely identified Biot parameters (’Fit’) and absorption coefficient computed using the Biot parameters outputted by the ANN model.

**Figure 13.**Correlation graphs for inversely identified Biot parameters and their corresponding prediction by the ANN model. It can be seen that all quantities are predicted with reasonable accuracy, thus the model is qualified for further application.

**Figure 14.**Target curves for the porous absorber design. Red and green areas mark known/unknown regions that are/are not covered by the existing specimen yet. The target curve “low” lies completely within the known data range, it is assumed that here only interpolation within the available data is required. The target curves “medium” and “high” leave the known range above approx. 5000 Hz, here the ability to extrapolate from the known data is required.

**Figure 15.**Flow chart of the inverse absorber design process (Step 6 from the flowchart in Figure 1). Based on an initial guess of the design parameters, the corresponding Biot parameters and the resulting absorption coefficient is computed. Based on the given error measure with respect to the target, the design parameters are updated until convergence is reached.

**Figure 16.**Design of porous media using an KNN machine-learning model for three different target curves. It can be seen that, especially for the targets “low” and “high” the predictions are met with high accuracy. However, the KNN model is not able to extrapolate from the know data range, as the prediction does not follow the target curves for “medium” and “high” above approx. 5000 $\mathrm{Hz}$.

**Figure 17.**Design of porous media using an ANN machine-learning model for three different target curves. It can be seen that, especially for the targets “low” and “high” the predictions are met with reasonable accuracy. The ANN model is able to extrapolate from the know data range, as the prediction follows the target curves for “medium” and “high” above approx. 5000 Hz. The large deviations of the measurement for the target “medium” are expected to result from manufacturing inaccuracies.

**Figure 18.**Computed and measured absorption coefficient of a specimen with a height $l=30$ mm and enlarged frequency range 150–6600 Hz. The computation employs the Biot parameters obtained from the KNN model for traget curve “high” and computes the absorption coefficient using the Equations (1)–(4), (A1) and (A2) for the new specimen height.

**Figure 19.**Actual-target comparison of specimen 26 with the Keyence VR5200 3D surface profilometer. (

**a**) shows specimen 26 without measurement lines at 80× magnification. (

**b**) shows specimen 26 with measurement lines and only slight deviations in the actual-target comparison

**Table 1.**Overview of the design variables (see Figure 2a) and their variation ranges.

Bar Width (d) | Bar Spacing (s) | Bar Height (h) | Plane Angle ($\mathit{\phi}$) |
---|---|---|---|

0.10–0.50 mm | 0.10–1.00 mm | 0.05–0.20 mm | 0${}^{\circ}$–90${}^{\circ}$, |

Nozzle Diameter | |
---|---|

0.20 mm | |

Nozzle temperature (${}^{\circ}$C) | 210 |

Bed temperature (${}^{\circ}$C) | 60 |

Layer height ${}^{1}$ (mm) | 0.05–0.11 |

Flow (%) | 85 |

Extrusion Speed (mm/s) | 36 |

Cooling (%) | 40 |

Outline direction | Inside-Out |

Extrusion width (mm) | 0.15; 0.20; 0.25 |

^{1}Increment of 0.005 mm.

Biot Parameter | Scaling | Biot Parameter | Scaling |
---|---|---|---|

${\mathsf{\Xi}}^{1}$ | ${\mathsf{\Xi}}^{\u2605}=\mathsf{\Xi}$ | ${\varphi}^{2}$ | ${\varphi}^{\u2605}=1\times {\mathrm{10}}^{3}\phantom{\rule{4pt}{0ex}}\varphi $ |

${{\alpha}_{\infty}}^{3}$ | ${\alpha}_{\infty}^{\u2605}=1\times {\mathrm{10}}^{3}{\alpha}_{\infty}$ | ${\mathsf{\Lambda}}^{4}$ | ${\mathsf{\Lambda}}^{\u2605}=1\times {\mathrm{10}}^{6}\phantom{\rule{4pt}{0ex}}\mathsf{\Lambda}$ |

${{\mathsf{\Lambda}}^{\prime}}^{5}$ | ${\mathsf{\Lambda}}^{{\prime}^{\u2605}}=1\times {\mathrm{10}}^{6}\phantom{\rule{4pt}{0ex}}{\mathsf{\Lambda}}^{\prime}$ | ${{k}_{0}^{\prime}}^{6}$ | ${{k}_{0}^{\prime}}^{\u2605}=1\times {\mathrm{10}}^{12}\phantom{\rule{4pt}{0ex}}{k}_{0}^{\prime}$ |

^{1}flow resistivity,

^{2}porosity,

^{3}tortuosity,

^{4}thermal characteristic length,

^{5}viscous characteristic length,

^{6}static thermal permeability.

**Table 4.**Training and test scores of the KNN and ANN model; uncertainty measure is two times the standard deviation of the cross-validation process (equiv. to approx. 95% confidence interval).

${\mathit{R}}^{2}$-Score | KNN ${}^{1}$ | ANN ${}^{2}$ |
---|---|---|

cross-validation | 0.80 ± 0.10 | 0.73 ± 0.13 |

test on unseen data | 0.76 | 0.80 |

^{1}K-Nearest Neighbor,

^{2}Artificial Neural Network.

Bar Width (d) | Bar Spacing (s) | Bar Height (h) | Angle ($\mathit{\varphi}$) | |
---|---|---|---|---|

Specimen 2 | $0.40$ mm | $0.40$ mm | $0.08$ mm | $60.00$${}^{\circ}$ |

Specimen 39 | $0.40$ mm | $0.80$ mm | $0.15$ mm | $70.00$${}^{\circ}$ |

Quantity | Lower Bound | Upper Bound |
---|---|---|

bar width (mm) | 0.15 | 0.50 |

bar height (mm) | 0.10 | 0.30 |

bar spacing (mm) | 0.10 | 1.00 |

plane angle (${}^{\circ}$) | 30 | 90 |

**Table 7.**Design parameters and measured values of the specimens designs using the KNN model. The printed specimens are measured using a 3D surface profilometer, quantities marked with a dash (’-’) could not be measured due to the measurement principle.

Target “low” | Target “medium” | Target “high” | ||||
---|---|---|---|---|---|---|

Quantity | Design | Measured | Design | Measured | Design | Measured |

bar width (mm) | 0.17 | 0.21 | 0.26 | 0.24 | 0.20 | 0.19 |

bar height (mm) | 0.12 | – | 0.11 | – | 0.15 | – |

bar spacing (mm) | 0.93 | 0.89 | 0.44 | 0.41 | 0.33 | 0.325 |

plane angle (${}^{\circ}$) | 63.40 | 63.399 | 63.000 | 63.256 | 33.40 | 33.466 |

**Table 8.**Design parameters and measured values of the specimens designs using the ANN model. The printed specimens are measured using 3D surface profilometer, quantities marked with a dash (‘-’) could not be measured due to the measurement principle.

Target “low” | Target “medium” | Target “high” | ||||
---|---|---|---|---|---|---|

Quantity | Design | Measured | Design | Measured | Design | Measured |

bar width (mm) | 0.15 | 0.16 | 0.17 | 0.162 | 0.15 | 0.16 |

bar height (mm) | 0.12 | - | 0.20 | - | 0.30 | - |

bar spacing (mm) | 0.89 | 0.843 | 0.66 | 0.665 | 0.10 | 0.11 |

plane angle (${}^{\circ}$) | 78.00 | 78.107 | 30.00 | 30.296 | 62.40 | 61.927 |

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**MDPI and ACS Style**

Kuschmitz, S.; Ring, T.P.; Watschke, H.; Langer, S.C.; Vietor, T. Design and Additive Manufacturing of Porous Sound Absorbers—A Machine-Learning Approach. *Materials* **2021**, *14*, 1747.
https://doi.org/10.3390/ma14071747

**AMA Style**

Kuschmitz S, Ring TP, Watschke H, Langer SC, Vietor T. Design and Additive Manufacturing of Porous Sound Absorbers—A Machine-Learning Approach. *Materials*. 2021; 14(7):1747.
https://doi.org/10.3390/ma14071747

**Chicago/Turabian Style**

Kuschmitz, Sebastian, Tobias P. Ring, Hagen Watschke, Sabine C. Langer, and Thomas Vietor. 2021. "Design and Additive Manufacturing of Porous Sound Absorbers—A Machine-Learning Approach" *Materials* 14, no. 7: 1747.
https://doi.org/10.3390/ma14071747