# Coupling the Cell Method with the Boundary Element Method in Static and Quasi–Static Electromagnetic Problems

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## Abstract

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## 1. Introduction

## 2. Electromagnetic Field Problems

#### 2.1. Interior Problem

#### 2.2. Interface Conditions

#### 2.3. Exterior Problem

#### 2.4. Direct Approach

#### 2.5. Indirect Approach

## 3. Cell Method with Augmented Dual Grid

#### 3.1. Discrete Field Variables

#### 3.2. Topological Operators

#### 3.3. Discrete Constitutive Relations

## 4. Boundary Element Method

#### 4.1. Direct Approach

#### 4.2. Indirect Approach

## 5. Hydrid Formulations

#### 5.1. Unsymmetric Formulation

#### 5.2. Symmetric Formulation

#### 5.3. Multiply-Connected Domains

## 6. Numerical Results

^{®}software environment with a vectorized function in order to speed up the assembly of CM and BEM matrices. All simulations were run on a laptop with Intel

^{®}Core™i7-6920HQ processor (2.90 GHz clock) and 16 GB RAM. The direct formulation, based on the definition of Poincaré–Steklov operator (64), is presented here for the first time and compared to the indirect one, based on the alternative definition (70), by considering an axisymmetric model with highly-accurate third-order 2D FEM solution. Finally, the indirect hybrid formulation (which is shown to be equivalent to the direct formulation) is tested on a realistic problem with a more complex topology, i.e., the TEAM Workshop Problem 3 (the Bath Plate) proposed in [29].

#### 6.1. Axisymmetric Inductor

#### 6.2. Bath Plate

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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## Short Biography of Authors

Federico Moro received the Laurea degree in Electrical Engineering (2003), the Ph.D. degree in Bioelectromagnetic and Electromagnetic Compatibility (2007), and the B.S. degree in Mathematics (2012) from the University of Padova, Italy. He has been a Visiting Student at the Department of Physics, Swansea University, Wales, UK (2005) and a Visiting Professor at the G2ELab, Grenoble, France (2020). He was awarded the best oral presentation at UPEC 2006 and the best paper at ASME IDETC/CIE 2017 and Electrimacs 2019 conferences. He obtained the National Scientific Qualification as Full Professor (09/E1-Elettrotecnica) in 2021. From 2007 to 2010 he was a Research Associate at the Department of Electrical Engineering, University of Padova. From 2010 to 2020 he was an Assistant Professor of Electrical Engineering at the Department of Industrial Engineering of the same university. Since 2020 he has been working as an Associate Professor of Electrical Engineering at the same department. His research interests include numerical methods for computing electromagnetic problems and the numerical modeling of multiphysics and multiscale problems. He is author of more than 100 articles in peer-reviewed international journals and conference proceedings. | |

Lorenzo Codecasa received the Ph.D. degree in Electronic Engineering from Politecnico di Milano in 2001. From 2002 to 2010 he worked as an Assistant Professor of Electrical Engineering at the Department of Electronics, Information, and Bioengineering of Politecnico di Milano. Since 2010 he has worked as an Associate Professor of Electrical Engineering at the same department. His main research contributions are in the theoretical analysis and in the computational investigation of electric circuits and electromagnetic fields. In his research on heat transfer and thermal management of electronic components, he has introduced original industrial-strength approaches to the extraction of compact thermal models, currently available in market leading commercial software. For these activities, in 2016 he received the Harvey Rosten Award for Excellence. He has been serving as an Associate Editor for the IEEE Transactions of Components, Packaging and Manufacturing Technology. He has also been serving as a Chair of the conference THERMal INvestigation of Integrated Circuits (THERMINIC). In his research areas he has authored or coauthored over 200 papers in refereed international journals and conference proceedings. |

**Figure 1.**Computational domain for the eddy–current problem with open boundary: $\Omega $ is the interior region (possibly multiply-connected), ${\Omega}_{\mathrm{e}}$ is the exterior region (which is unbounded), and ${\Omega}_{0}$ is the source region (with given current density ${\mathbf{J}}_{\mathbf{0}}$). The interface $\Gamma $ separates $\Omega $ from ${\Omega}_{e}$.

**Figure 2.**Meshes for $\Omega ={[0,1]}^{2}$ used for CM discretization: (

**a**) primal grid ${\mathcal{G}}_{\Omega}$; (

**b**) barycentric subdivision ${\mathcal{G}}_{\Omega}^{\Delta}$; (

**c**) augmented dual grid ${\tilde{\mathcal{G}}}_{\Omega \partial \Omega}$. The latter is obtained from ${\mathcal{G}}_{\Omega}^{\Delta}$ by aggregating triangles in blue around any primal node in black. A one-to-one correspondence exists between primal nodes and dual cells (polygon in light red), and between dual nodes (red dots) and primal cells (triangle in light yellow). Boundary $\partial \Omega $ is split on its turn into barycentric cells (blue thick line),which are aggregated into 1D dual cells (red thick line).

**Figure 3.**Axisymmetric model used for 2D FEM validations ($\Omega $: conductive region; ${\Omega}_{e}$: exterior domain; ${\gamma}_{0}$: source coil; ${\gamma}_{1}$: virtual coil; line A-B is used for the current density evaluation in the shell region; line C-D is used for the magnetic flux density evaluation in the air region).

**Figure 4.**Real part of the azimuthal current density component along line A-B (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 5.**Imaginary part of the azimuthal current density component along line A-B (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 6.**Discrepancy (${L}^{2}$-norm) in $\Omega $ between the eddy–current density computed by 2D FEM (third order elements, taken as a reference) and 3D CM–BEM (direct/indirect approaches). The red line expresses theoretical first-order convergence typical of 3D FEM with linear elements.

**Figure 7.**Real part of the radial magnetic flux density component along line C-D (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 8.**Imaginary part of the radial magnetic flux density component along line C-D (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 9.**Real part of the axial magnetic flux density component along line C-D (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 10.**Imaginary part of the axial magnetic flux density component along line C-D (black line: direct formulation, blue line: indirect formulation, red line: 2D FEM, taken as a reference).

**Figure 11.**Tetrahedral mesh of the “Bath plate” model used for indirect CM–BEM simulations (field calculation line A-B is depicted in blue; virtual loops are depicted in red).

**Figure 13.**Real and imaginary parts of the y-axis magnetic flux density component at 50 Hz along the line A-B (indirect CM–BEM plot is in straight line, FEM plot is in dashed line).

**Figure 14.**Real and imaginary parts of the z-axis magnetic flux density component at 50 Hz along the line A-B (CM–BEM plot is in straight line, FEM plot is in dashed line).

**Figure 15.**Real and imaginary parts of the y-axis magnetic flux density component at 200 Hz along the line A-B (CM–BEM plot is in straight line, FEM plot is in dashed line).

**Figure 16.**Real and imaginary parts of the z-axis magnetic flux density component at 200 Hz along the line A-B (CM–BEM plot is in straight line, FEM plot is in dashed line).

**Table 1.**Real and imaginary parts of magnetic flux (μWb) computed through cut surfaces ${\Sigma}_{1},{\Sigma}_{2}$ by 3D CM–BEM and 3D FEM (2nd order) at 50 Hz and 200 Hz frequency.

$\mathit{R}\mathit{e}\left({\mathbf{\Phi}}_{{\mathbf{\Sigma}}_{1}}\right)$ | $\mathit{I}\mathit{m}\left({\mathbf{\Phi}}_{{\mathbf{\Sigma}}_{1}}\right)$ | $\mathit{R}\mathit{e}\left({\mathbf{\Phi}}_{{\mathbf{\Sigma}}_{2}}\right)$ | $\mathit{I}\mathit{m}\left({\mathbf{\Phi}}_{{\mathbf{\Sigma}}_{2}}\right)$ | ||
---|---|---|---|---|---|

50 Hz | 3D CM–BEM | $7.869$ | $-1.958$ | $7.872$ | $-1.958$ |

FEM (2nd ord.) | $7.862$ | $-1.974$ | $7.862$ | $-1.974$ | |

200 Hz | 3D CM–BEM | $4.435$ | $-3.126$ | $4.438$ | $-3.127$ |

FEM (2nd ord.) | $4.394$ | $-3.142$ | $4.394$ | $-3.142$ |

**Table 2.**Real and imaginary parts of eddy–current (A) computed through cut surfaces ${\Sigma}_{3},{\Sigma}_{4}$ by 3D CM–BEM and 3D FEM (2nd) at 50 Hz and 200 Hz frequency.

$\mathit{R}\mathit{e}\left({\mathit{I}}_{{\mathbf{\Sigma}}_{3}}\right)$ | $\mathit{I}\mathit{m}\left({\mathit{I}}_{{\mathbf{\Sigma}}_{3}}\right)$ | $\mathit{R}\mathit{e}\left({\mathit{I}}_{{\mathbf{\Sigma}}_{4}}\right)$ | $\mathit{I}\mathit{m}\left({\mathit{I}}_{{\mathbf{\Sigma}}_{4}}\right)$ | ||
---|---|---|---|---|---|

50 Hz | 3D CM–BEM | $21.811$ | $63.412$ | $-21.819$ | $-63.080$ |

FEM (2nd ord.) | $21.766$ | $63.258$ | $-21.765$ | $-63.259$ | |

200 Hz | 3D CM–BEM | $133.941$ | $98.886$ | $-133.687$ | $-97.393$ |

FEM (2nd ord.) | $133.550$ | $98.518$ | $-133.540$ | $-98.529$ |

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**MDPI and ACS Style**

Moro, F.; Codecasa, L.
Coupling the Cell Method with the Boundary Element Method in Static and Quasi–Static Electromagnetic Problems. *Mathematics* **2021**, *9*, 1426.
https://doi.org/10.3390/math9121426

**AMA Style**

Moro F, Codecasa L.
Coupling the Cell Method with the Boundary Element Method in Static and Quasi–Static Electromagnetic Problems. *Mathematics*. 2021; 9(12):1426.
https://doi.org/10.3390/math9121426

**Chicago/Turabian Style**

Moro, Federico, and Lorenzo Codecasa.
2021. "Coupling the Cell Method with the Boundary Element Method in Static and Quasi–Static Electromagnetic Problems" *Mathematics* 9, no. 12: 1426.
https://doi.org/10.3390/math9121426