# Rapid Electromagnetic Modeling and Simulation of Eddy Current NDE by MLKD-ACA Algorithm with Integral Kernel Truncations

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## Abstract

**:**

## 1. Introduction

**Z**with dimension m × n, ACA algorithm generates the U and V matrices with dimensions m × r and r × n, which resulting in the computational cost to be ${r}^{2}\left(m+n\right)$, here r is the rank to approximate

**Z**necessarily with a predefined tolerance. FFT projects the current basis functions to regular grids by interpolation to produce the same fields as the original currents [12]. $\mathscr{H}$ matrix algorithm provides an efficient mathematical framework to categorize the blocks into admissible and inadmissible ones. The inadmissible blocks are computed by full matrices and the admissible blocks are approximated by the low rank matrix compression techniques [13,14,15].

## 2. Description of MLKD-ACA with Kernel Truncations

#### 2.1. Multilevel Partition

#### 2.2. MLKD Algorithm

**K**,

**L**and

**R**operators shown in Equations (1) and (2) relate to the Green function [10]

**K**,

**L**and

**R**operators for the far block interactions in the submatrices ${\mathbf{R}}_{1}^{\times}$, $-{\mathbf{R}}_{2}^{n}$, $-{\mathbf{K}}_{1}^{\times}$, ${\mathbf{K}}_{2}^{\times}$, ${\mathbf{K}}_{2}^{n}{\mu}_{2}/{\mu}_{1}$, ${\mathbf{L}}_{2}^{\times}{\mu}_{2}/{\mu}_{1}$, and $-{k}_{2}^{2}{\mathbf{L}}_{2}^{n}$ can be degenerated accordingly [10].

**c**and d at coarser levels can be accessed by those at leaf level and the transfer matrices

#### 2.3. MLACA Algorithm with Kernel Truncations

**Z**of far block pair t and s is with dimension T by S. Application of the ACA algorithm yields the following approximate factorization [9,10,11,20,21,22,23,24,25]

**Z**are calculated in ACA. The dimensions of

**U**and

**V**matrices are T by rank and rank by S, respectively. Since the rank is much smaller than T or S, only rank × (T + S) elements need to be computed and stored instead of T × S.

## 3. Numerical Experiments

#### 3.1. Coil with Finite Cross Section Placing above a Conductive Plate

#### 3.2. Single Turn Coil Placing above a Conductive Sphere

#### 3.3. Coil with Finite Cross Section Placing above a Conductive Plate with Surface Slot

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The ratios between block 6′s diagonal, near and far interactions, and its diagonal interactions of seven submatrices.

**Figure 4.**A coil with finite cross section. ${r}_{i}$ and ${r}_{o}$ represent the inner and outer radii of the coil, respectively. S is the lift-off distance between the coil and the conducting plate. l is the thickness of the coil. ${a}_{s}$ and ${h}_{s}$ are the continuous variables in the radial and vertical directions, respectively [2].

**Figure 5.**Case of placing coil C5 above the conductive plate. Memory requirements for total and far block interactions of MLKD-ACA algorithm with threshold values of kernel truncations.

**Figure 6.**Case of placing coil C27 above the conductive plate. Memory requirements for total and far block interactions of MLKD-ACA algorithm with threshold values of kernel truncations.

**Figure 7.**Problem description of placing single turn coil above a conducting sphere [29]. R

_{0}and R

_{1}represent the free space and conducting sphere. The radius of the single turn coil is ${r}_{c}$ with lift-off distance h between the coil and the sphere with radius ${\rho}_{1}$. ${\rho}_{c}$ is the distance between the origin of the sphere and the edge of the coil.

**Figure 9.**Impedance variations of scanning the coil above the conductive plate with surface slot predicted and measured by the MLKD-ACA algorithm with kernel truncations solver and measurements: (

**a**) Resistance variations; (

**b**) reactance variations.

**Table 1.**Impedance variations achieved by different solvers for placing coil C5, operating at 850 Hz, above a conductive plate.

Solver or Method | $\mathbf{Impedance}\text{}\mathbf{Variation}\text{}\left(\mathsf{\Omega}\right)$ |
---|---|

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-2}$ | $22.054-69.312j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-3}$ | $22.237-70.437j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-4}$ | $22.141-70.431j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-5}$ | $22.143-70.428j$ |

MoM [10] | $22.153-70.409j$ |

Experiment [27] | $20.00-70.5j$ |

Theodoulidis and Bowler [27] | $22.25-70.45j$ |

Dodd and Deeds [28] | $22.20-70.49j$ |

**Table 2.**Impedance variations achieved by different solvers for placing coil C27, operating at 20 kHz, above a conductive plate.

Solver or Method | $\mathbf{Impedance}\text{}\mathbf{Variation}\text{}\left(\mathsf{\Omega}\right)$ |
---|---|

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-1}$ | $10.583-127.438j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-2}$ | $12.825-124.837j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-3}$ | $12.721-124.992j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-4}$ | $12.721-124.990j$ |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-5}$ | $12.722-124.989j$ |

MoM [10] | $12.734-124.98j$ |

Experiment [27] | $12.650-125.1j$ |

Theodoulidis and Bowler [27] | $12.801-125.329j$ |

Dodd and Deeds [28] | $12.801-125.388j$ |

**Table 3.**Impedance variations predicted, and memory of far block pairs required, by different solvers for placing single turn coil, operating at 500 Hz, above a conductive sphere.

Solver or Method | $\mathbf{Impedance}\text{}\mathbf{Variation}\text{}\left(\mathsf{\mu}\mathsf{\Omega}\right)$ | Memory Requirement for Far Block Pairs (MB) |
---|---|---|

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-2}$ | $0.2293-1.497j$ | 24.8 |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-3}$ | $0.2431-1.478j$ | 104.3 |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-4}$ | $0.2422-1.478j$ | 151.3 |

$\mathrm{MLKD}\text{-}\mathrm{ACA}\text{}\mathrm{with}\text{}{\Delta}_{1}={10}^{-5}$ | $0.2421-1.480j$ | 190.3 |

MoM [30,31] | $0.2416-1.484j$ | NA |

Analytical method [29] | $0.2425-1.489j$ | NA |

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**MDPI and ACS Style**

Bao, Y.; Liu, Z.; Song, J.
Rapid Electromagnetic Modeling and Simulation of Eddy Current NDE by MLKD-ACA Algorithm with Integral Kernel Truncations. *Symmetry* **2022**, *14*, 712.
https://doi.org/10.3390/sym14040712

**AMA Style**

Bao Y, Liu Z, Song J.
Rapid Electromagnetic Modeling and Simulation of Eddy Current NDE by MLKD-ACA Algorithm with Integral Kernel Truncations. *Symmetry*. 2022; 14(4):712.
https://doi.org/10.3390/sym14040712

**Chicago/Turabian Style**

Bao, Yang, Zhiwei Liu, and Jiming Song.
2022. "Rapid Electromagnetic Modeling and Simulation of Eddy Current NDE by MLKD-ACA Algorithm with Integral Kernel Truncations" *Symmetry* 14, no. 4: 712.
https://doi.org/10.3390/sym14040712