# Finite Element Method for the Estimation of Insertion Loss of Noise Barriers: Comparison with Various Formulae (2D)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mathematical Formulation, Exact Solutions and Approximate Analytical Solutions

#### 1.2. Formulae of Insertion Loss

#### 1.3. FEM, Other Numerical Methods and Noise Barriers

#### 1.4. Aims and Novelties

- Validate of the accuracy of FEM for calculation of insertion loss of noise barriers.
- Present a simple and applicable methodology for the accurate calculation of insertion loss utilizing commercial software which can be extended to various cases (e.g., predict the behavior of noise barriers with various shapes, with a profile which absorbs or disperses sound, in 3D space, etc.).
- Lay the groundwork for application of FEM for urban acoustic microscale modeling.

- To the best of our knowledge this is the first study that extensively compares insertion loss results of FEM with Kurze–Anderson, ISO 9316-2/Tatge and Menounou formulae results.
- This is the first study to present the accuracy of FEM for the calculation of insertion loss of noise barriers especially in the cases where the receiver is near the barrier or in the shadow border and when both source and receiver are near the barrier.

## 2. Methods

#### 2.1. Elements Regarding Insertion Loss

#### 2.2. Formulae for the Calculation of Insertion Loss

#### 2.2.1. Kurze–Anderson Formula

#### 2.2.2. ISO 9613-2/Tatge Formulae

#### 2.2.3. Menounou Formula

_{1}, which is a measure of the relative position of the receiver to the source. The second term is a function of N

_{2}/N

_{1}, which is a measure of the proximity of either the source or the receiver to the barrier surface. The third term is appreciable and needs to be computed only when N

_{1}is very small, which in turn is a measure of the proximity of the receiver to the shadow border. Finally, the fourth term, ILsp, is a term that depends on the type of incident radiation and is a function of the ratio (A + B)/d. The Menunu formula is:

#### 2.3. FEM Setup and Methodology for the Calculation of Insertion Loss

#### 2.3.1. FEM Models

#### 2.3.2. FEM Formulation

## 3. Results

#### 3.1. Sound Pressure Levels and Acoustic Pressures of the Domain via FEM

#### 3.2. Calculation of Insertion Loss via FEM and Various Formulae

- S
_{1}: source in medium distance from the barrier (4 m) - S
_{2}: source in long distance from the barrier (16 m) - S
_{3}: source in short distance from the barrier (0.2 m) - S
_{4}: source above the barrier (6 m)

- R
_{SZ}: receiver in the shadow zone - R
_{SB}: receiver near the shadow border - R
_{NB}: receiver near the barrier

#### 3.2.1. Receiver in the Shadow Zone

#### 3.2.2. Receiver near Barrier (0.1 m)

#### 3.2.3. Receiver in Increasing Distance from the Barrier (0.1 m–5 m)

#### 3.2.4. Receiver near the Shadow Border (0.1 m)

#### 3.2.5. Receiver in Increasing Distance from the Shadow Border (0.1 m–5 m)

## 4. Discussion

- FEM and various formulae insertion loss calculations.
- Future work and further applications of FEM for noise barriers and microscale urban acoustic modeling.

#### 4.1. FEM and Various Formulae Insertion Loss Calculations

^{−5}(Source: S3, Receiver: R

_{SB31}, Frequency: 20 Hz), while the maximum was 18.374 (Source: S2, Receiver: R

_{SZ2}, Frequency: 500 Hz). Previous research has indeed shown that the Menounou’s formula is very close in each case to the analytical solution. There are differences with the other formulae when the receiver is near the barrier and near the shadow border. The differences between the FEM/Menounou and the other formulae, have arisen probably because these formulae have been formed from semi-empirical data (e.g., Maekawa chart and thereafter Kurze–Anderson, Tatge/ISO 9613-2) in which the cases close to the barrier and near the shadow border have not been properly taken into account.

#### 4.2. Future Work and Further Applications of FEM for Noise Barriers and Microscale Urban Acoustic Modeling

- Accuracy
- Availability
- Low cost

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ABC | Absorbing Boundary Condition |

BEM | Boundary Element Method |

CAD | Computer-Aided Design |

FDTD | Finite-Difference Time-Domain |

FEM | Finite Element Method |

GPU | Graphics Processing Unit |

IL | Insertion Loss |

PML | Perfectly Matched Layer |

PSTD | Pseudo-Spectral Time-Domain |

R_{NB} | Receiver Near the Barrier |

R_{SB} | Receiver in the Shadow Border (near) |

R_{SZ} | Receiver in the Shadow Zone |

SPL | Sound Pressure Levels |

WHO | World Health Organization |

## Appendix A

**Table A1.**Source and receiver positions for calculation of insertion loss via FEM and common formulae (correspond to positions shown in Figure 5).

Receiver Area | Distance d(m) | Source Positions (x,y) | |||
---|---|---|---|---|---|

S_{1}(−4.00, −4.00) | S_{2}(−16.00, −10.00) | S_{3}(−0.20, −5.00) | S_{4}(−6.00, 6.00) | ||

Receiver Positions (x,y) | |||||

Shadow zone | R_{SZ1}(4.00, −4.00) | R_{SZ2}(10.00, −9.00) | R_{SZ3}(5.00, −4.00) | R_{SZ4}(3.00, −8.00) | |

Shadow border (0.10 m in the shadow zone) | R_{SB11}(4.07, 3.93) | R_{SB21}(12.05, 7.42) | R_{SB31}(0.50, 10.00) | R_{SB41}(8.93, −9.07) | |

Near barrier | R_{NB1}(0.10, −3.00) | R_{NB2}(0.10, −8.00) | R_{NB3}(0.10, −5.00) | R_{NB4}(0.10, −7.00) | |

Varying distance d from shadow border (in the shadow zone) | 0.10 | R_{SB11}(4.07, 3.93) | R_{SB21}(12.05, 7.42) | R_{SB31}(0.50, 10.00) | R_{SB41}(8.93, −9.07) |

0.20 | R_{SB12}(4.14, 3.86) | R_{SB22}(12.11, 7.33) | R_{SB32}(0.60, 9.99) | R_{SB42}(8.86, −9.14) | |

0.30 | R_{SB13}(4.21, 3.79) | R_{SB23}(12.16, 7.25) | R_{SB33}(0.70, 9.99) | R_{SB43}(8.79, −9.21) | |

0.40 | R_{SB14}(4.28, 3.71) | R_{SB24}(12.21, 7.16) | R_{SB34}(0.80, 9.98) | R_{SB44}(8.72, −9.28) | |

0.50 | R_{SB15}(4.35, 3.65) | R_{SB25}(12.27, 7.08) | R_{SB35}(0.90, 9.98) | R_{SB45}(8.65, −9.35) | |

0.75 | R_{SB16}(4.53, 3.47) | R_{SB26}(12.40, 6.86) | R_{SB36}(1.15, 9.97) | R_{SB46}(8.47, −9.53) | |

1.00 | R_{SB17}(4.71, 3.29) | R_{SB27}(12.53, 6.65) | R_{SB37}(1.40, 9.96) | R_{SB47}(8.29, −9.71) | |

1.50 | R_{SB18}(5.06, 2.94) | R_{SB28}(12.80, 6.23) | R_{SB38}(1.90, 9.94) | R_{SB48}(7.94, −10.06) | |

2.00 | R_{SB19}(5.41, 2.59) | R_{SB29}(13.06, 5.80) | R_{SB39}(2.40, 9.92) | R_{SB49}(7.59, −10.41) | |

3.00 | R_{SB110}(6.12, 1.88) | R_{SB210}(13.59, 4.96) | R_{SB310}(3.40, 9.88) | R_{SB410}(6.88, −11.12) | |

4.00 | R_{SB111}(6.83, 1.17) | R_{SB211}(14.12, 4.11) | R_{SB311}(4.40, 9.84) | R_{SB411}(6.17, −11.83) | |

5.00 | R_{SB112}(7.54, 0.46) | R_{SB212}(14.65, 3.26) | R_{SB312}(5.40, 9.80) | R_{SB412}(5.46, −12.54) | |

Varying distance d near barrier (in the shadow zone) | 0.10 | R_{NB11}(0.10, −5.00) | R_{NB21}(0.10, −7.00) | R_{NB31}(0.10, −7.00) | R_{NB41}(0.10, −12.00) |

0.20 | R_{NB12}(0.20, −5.00) | R_{NB22}(0.20, −7.00) | R_{NB32}(0.20, −7.00) | R_{NB42}(0.20, −12.00) | |

0.30 | R_{NB13}(0.30, −5.00) | R_{NB23}(0.30, −7.00) | R_{NB33}(0.30, −7.00) | R_{NB43}(0.30, −12.00) | |

0.40 | R_{NB14}(0.40, −5.00) | R_{NB24}(0.40, −7.00) | R_{NB34}(0.40, −7.00) | R_{NB44}(0.40, −12.00) | |

0.50 | R_{NB15}(0.50, −5.00) | R_{NB25}(0.50, −7.00) | R_{NB35}(0.50, −7.00) | R_{NB45}(0.50, −12.00) | |

0.75 | R_{NB16}(0.75, −5.00) | R_{NB26}(0.75, −7.00) | R_{NB36}(0.75, −7.00) | R_{NB46}(0.75, −12.00) | |

1.00 | R_{NB17}(1.00, −5.00) | R_{NB27}(1.00, −7.00) | R_{NB37}(1.00, −7.00) | R_{NB47}(1.00, −12.00) | |

1.50 | R_{NB18}(1.50, −5.00) | R_{NB28}(1.50, −7.00) | R_{NB38}(1.50, −7.00) | R_{NB48}(1.50, −12.00) | |

2.00 | R_{NB19}(2.00, −5.00) | R_{NB29}(2.00, −7.00) | R_{NB39}(2.00, −7.00) | R_{NB49}(2.00, −12.00) | |

3.00 | R_{NB110}(3.00, −5.00) | R_{NB210}(3.00, −7.00) | R_{NB310}(3.00, −7.00) | R_{NB410}(3.00, −12.00) | |

4.00 | R_{NB111}(4.00, −5.00) | R_{NB211}(4.00, −7.00) | R_{NB311}(4.00, −7.00) | R_{NB411}(4.00, −12.00) | |

5.00 | R_{NB112}(5.00, −5.00) | R_{NB212}(5.00, −7.00) | R_{NB312}(5.00, −7.00) | R_{NB412}(5.00, −12.00) | |

Shadow border line equations | y = x | y = 0.625x | y = 25x | y = −x | |

Perpendicular line equations (for varying distance from the shadow border points) | y = 8 − x | y = 26.7 − 8x/5 | y = 1252/125 –x/25 | y = −18 + x | |

Points of intersection of the above lines (x,y) | (4.00, 4.00) | (12.00, 7.50) | (0.40, 10.00) | (9.00, −9.00) |

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**Figure 2.**Finite Element Method (FEM) model with sound barrier and Perfectly Matched Layer (Violet color). Axes are in meters (m).

**Figure 3.**Acoustic pressures of the domain calculated via FEM for different frequencies. Colorbar units are in Pascal (Pa). Axis units are in meters (m).

**Figure 4.**Sound pressure levels of the domain calculated via FEM for different frequencies. Colorbar units are in SPL (dB). Axis units are in meters (m).

**Figure 5.**Source and receiver positions for calculation of insertion loss via FEM and various formulae (source (S): red, receiver in the shadow zone (R

_{SZ}): orange, receiver near shadow border (R

_{SB}): green, receiver near barrier (R

_{NB}): blue). The exact positions for each source and receiver are presented in the Appendix A (Table A1).

**Figure 6.**Comparison of insertion loss results between FEM and various formulae for various source and receiver positions in the shadow zone.

**Figure 7.**Comparison of calculation of insertion loss between FEM and various formulae for various source and receiver positions near the barrier inside the shadow zone.

**Figure 8.**Comparison of calculation of insertion loss between FEM and various formulae for various source and receiver positions near the barrier inside the shadow zone (300 Hz).

**Figure 9.**Comparison of calculation of insertion loss between FEM and various formulae for various source and receiver positions in the shadow border.

**Figure 10.**Comparison of calculation of insertion loss between FEM and various formulae for various source and receiver positions near the barrier inside the shadow zone (300 Hz).

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**MDPI and ACS Style**

Papadakis, N.M.; Stavroulakis, G.E.
Finite Element Method for the Estimation of Insertion Loss of Noise Barriers: Comparison with Various Formulae (2D). *Urban Sci.* **2020**, *4*, 77.
https://doi.org/10.3390/urbansci4040077

**AMA Style**

Papadakis NM, Stavroulakis GE.
Finite Element Method for the Estimation of Insertion Loss of Noise Barriers: Comparison with Various Formulae (2D). *Urban Science*. 2020; 4(4):77.
https://doi.org/10.3390/urbansci4040077

**Chicago/Turabian Style**

Papadakis, Nikolaos M., and Georgios E. Stavroulakis.
2020. "Finite Element Method for the Estimation of Insertion Loss of Noise Barriers: Comparison with Various Formulae (2D)" *Urban Science* 4, no. 4: 77.
https://doi.org/10.3390/urbansci4040077