# A Modification of the Monte Carlo Filtering Approach for Correcting Negative SEA Loss Factors

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## Abstract

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## 1. Introduction

- All elements off the main diagonal are negative.
- Elements on the main diagonal are positive and greater than the sum of the absolute values of the remaining elements for a given column.

- The present paper proposes a modification of the MCF method. The modification is called DESA (Diagonal Expansion of the Search Area), which is the main contribution to the experimental SEA field. DESA consists of applying a correction during the MCF, which causes a non-uniform expansion of the search area (Section 2.1). This novel approach expands the range of vibroacoustic systems that can be properly identified by MCF. It can be applied in the frequency bands for which correct results could not be obtained using the MCF method in the basic version (using a homogeneous expansion of the search area for the population $\left\{\left[{G}_{s}\right]\right\}$ with a normal distribution), as will be demonstrated by the example presented in Section 3.
- The effect of the expansion of the search area (parameter $\gamma $) on the errors introduced into the loss factors was investigated. The so-called shift error was observed and related to the asymmetry present in the generated population of the energy matrices. We pointed out that the asymmetry of the population increases with an increase in $\gamma $.
- We introduced a new parameter describing the degree of asymmetry of the energy matrix population, the asymmetry index $\alpha $, and proposed two methods (A and B) for eliminating the shift error. Method A involves detecting matrices that introduce asymmetry and rejecting them from the calculation, while method B involves using a log-normal distribution when generating the energy matrix population. A common feature of both methods is the need to perform $\gamma $ minimization.

## 2. Materials and Methods

#### 2.1. Expansion of the Search Area (UESA and DESA)

#### 2.2. Errors Associated with ESA

#### 2.2.1. Population Asymmetry and Shift Error

**Definition 1**.

^{5}times to obtain the populations $\left\{\left[{G}_{s}\right]\right\}$. Each energy matrix was then inverted using Equation (5) to obtain the resulting $\left\{\left[{L}_{s}\right]\right\}$ population. Next, the MCF procedure was performed, as described in “Introduction”, to obtain the set of correct matrices $\left\{\left[{L}_{s}^{P}\right]\right\}$. Elements of $\left\{\left[{L}_{s}^{P}\right]\right\}$ were averaged arithmetically to obtain $\left[{L}_{mean}\right]$. Then, based on Equation (2), the mean loss factors shown in the figure were extracted from $\left[{L}_{mean}\right]$ as follows. Coupling loss factor ${\eta}_{ij}$ is off-diagonal term ${L}_{ji}$ multiplied by −1, and damping loss factor ${\eta}_{ii}$ is obtained by summing the elements of column $i$.

#### 2.2.2. Scaling Factor Minimization

#### 2.2.3. Enforcing Symmetry of the Population

- Method A, which involves discarding from the calculation matrices that fall into the tail of the normal distribution.
- Method B, which involves generating a population with a log-normal distribution.
- The use of one of the presented SFM methods in combination with $\gamma $-minimization allows us to:
- Correct negative loss factors and replace them with factors that are free of offset error.
- Obtain results close to the original results in bands that do not require correction, which can be good in terms of the quality control of the applied methods.

^{5}samples). As expected, the population generated from Formulas (6) and (8) has a normal distribution. Method A is based on the rejection of elements introducing asymmetry, as seen in Figure 5.

- When $\mathrm{card}\left(\left\{\left[{G}_{s}^{P}\right]\right\}\right)>\mathrm{card}\left(\left\{\left[{G}_{s}^{A,P}\right]\right\}\right)$, or equivalently $\alpha <1$, the result obtained will be free of both shift error and negative LF. Then, the identification result obtained by Method A can be considered correct, and $\alpha =0$ occurs for the resulting population.
- When $\mathrm{card}\left(\left\{\left[{G}_{s}^{P}\right]\right\}\right)=\mathrm{card}\left(\left\{\left[{G}_{s}^{A,P}\right]\right\}\right)$, or equivalently $\alpha =1$ (which also means that $\left\{\left[{G}_{s}^{P}\right]\right\}=\left\{\left[{G}_{s}^{A,P}\right]\right\}$), all correct matrices will be discarded and the correction of negative LF coefficients will not take place. Method A is then ineffective. However, it is possible to take the LF coefficients determined for an asymmetric population of matrices as the final result. In such a situation, the result obtained will be affected by a shift error. However, this error will be minimized by using ${\gamma}_{min}$ during the calculation (the distance between the matrices $\left[{G}_{s}^{A}\right]$ and the original matrix $\left[G\right]$ will be relatively small).

## 3. Results

## 4. Discussion

^{5}, but trials continued (unsuccessfully) until the number of iterations reached 2 × 10

^{7}.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Subsets of the set $\left\{\left[{G}_{s}\right]\right\}$. ○: matrices that are correct and do not disturb symmetry, ●: correct matrices that disturb symmetry, ▲: incorrect matrices that disturb symmetry, △: incorrect matrices that do not disturb symmetry.

**Figure 2.**Effect of the scaling factor on shift errors. □—without MCF, ●: $\gamma =2.5,$▼: $\gamma =20$, ☆: $\gamma =100$. (

**a**) UESA, DLF; (

**b**) DESA, DLF; (

**c**) UESA, CLF; (

**d**) DESA, CLF. UESA: uniform expansion of the search area, DESA: diagonal expansion of the search area, DLF: damping loss factor, CLF: coupling loss factor.

**Figure 3.**The dependence of the error of loss factors (□) and the population asymmetry index (●) as a function of the scaling factor for a matrix population with a normal distribution. (

**a**) UESA, DLF; (

**b**) DESA, DLF; (

**c**) UESA, CLF; (

**d**) DESA, CLF.

**Figure 5.**Histogram of the selected item ${G}_{s,ij}$ (i = 1, j = 2) from a population with a normal distribution.

**Figure 6.**Dependence of the error of loss factors (□) and the population asymmetry index (●) on the scaling factor for a matrix population with a tail-free normal distribution. (

**a**) UESA, DLF; (

**b**) DESA, DLF; (

**c**) UESA, CLF; (

**d**) DESA, CLF.

**Figure 7.**Loss factors after applying Method A with arbitrary scaling factor values (influence of minimization omission). □—the original value, ▼—the value after expanding the search area, ●—the value after forcing the symmetry of the population using Method A. (

**a**) DLF for $\gamma =100$, (

**b**) DLF for $\gamma =1.5$, (

**c**) CLF for $\gamma =100$, (

**d**) CLF for $\gamma =1.5$.

**Figure 8.**Histogram of the selected item ${G}_{s,ij}$ (i = 1, j = 2) from a population with a log-normal distribution.

**Figure 9.**Dependence of the error of loss factors (□) and the population asymmetry index (●) on the scaling factor for a matrix population with log-normal distribution. (

**a**) DLF for $\gamma =100$, (

**b**) DLF for $\gamma =1.5$, (

**c**) CLF for $\gamma =100$, (

**d**) CLF for $\gamma =1.5$.

**Figure 10.**CLF coefficients between the flexural wave fields of the measured beams. ▼—MCF without ESA; ●—MCF+DESA with minimized γ = 6; □—MCF+UESA with forced population symmetry (Method B), γ = 1.5. Red circles indicate frequency bands where MCF without ESA was not successful.

**Figure 11.**Values of elements G_ (s,22) (▬) and G_ (s,23) (▬) for all the Monte Carlo iterations. Red dots indicate iterations with the correct loss matrix. (

**a**) no ESA; (

**b**) UESA with γ = 20; (

**c**) DESA with γ = 6; (

**d**) DESA with γ = 6 + Method A; (

**e**) DESA with γ = 1.5 + Method B; in this variant only one correct matrix (area indicated by the red circle) was found (

**f**) UESA with γ = 1.5 + Method B.

Geometry | |
---|---|

Thickness | 20 mm |

Length | 80 mm |

Width | 500 mm |

Mechanical Parameters | |

Material | Steel |

Density | 7827 kg/m^{3} |

Young’s modulus | 205 GPa |

Poisson number | 0.3 |

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**MDPI and ACS Style**

Nieradka, P.; Dobrucki, A.
A Modification of the Monte Carlo Filtering Approach for Correcting Negative SEA Loss Factors. *Acoustics* **2022**, *4*, 1028-1044.
https://doi.org/10.3390/acoustics4040063

**AMA Style**

Nieradka P, Dobrucki A.
A Modification of the Monte Carlo Filtering Approach for Correcting Negative SEA Loss Factors. *Acoustics*. 2022; 4(4):1028-1044.
https://doi.org/10.3390/acoustics4040063

**Chicago/Turabian Style**

Nieradka, Paweł, and Andrzej Dobrucki.
2022. "A Modification of the Monte Carlo Filtering Approach for Correcting Negative SEA Loss Factors" *Acoustics* 4, no. 4: 1028-1044.
https://doi.org/10.3390/acoustics4040063