# Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Continuous Formulation

#### 2.1. Eddy-Current Model

**B**and

**H**are the magnitudes of the $\mathit{B}$- and $\mathit{H}$-fields, respectively.

#### 2.2. Weak Form

**Remark**

**1.**

- 1.
- Perfect electric conductor (PEC) boundary conditions imply ${\gamma}_{D}^{-}\mathit{A}=\mathit{0}$ (and therefore ${\mathit{B}}^{-}\xb7\mathit{n}=0$). In this case, one usually makes use of the function space ${\mathit{H}}_{0}(\mathbf{curl},\mathsf{\Omega})$, that is the subspace of (12) with vanishing ${\gamma}_{D}$ on the boundary. Application of the weighted-residual method in this subspace does not yield any boundary term.
- 2.
- Perfect magnetic conductor (PMC) boundary conditions state ${\mathit{H}}_{\times}^{-}=\mathit{0}$, which means that the boundary term can be dropped from Equation (19).
- 3.
- Dirichlet-to-Neumann maps allow for a (global) functional relation between ${\mathit{H}}_{\times}^{-}$ and ${\gamma}_{D}^{-}\mathit{A}$. In this case, the boundary term is either approximated by absorbing or asymptotic boundary conditions or represented accurately by means of boundary integral equations as in the following. For an overview of these techniques, see [9].

#### 2.3. Boundary Integral Equations

**Remark**

**2.**

## 3. Discretisation

#### 3.1. Cohomologies

**Example**

**1.**

**λ**by means of ${\mathbf{curl}}_{\mathsf{\Gamma}}\psi $, and the fact that ${\gamma}^{\prime}=\partial \mathsf{\Sigma}$ is a closed circle. Clearly, for the given situation, there is a contradiction in that I cannot be zero for the considered non-zero current density. We thus need to augment the trial (41) by adding extra functions.

#### 3.2. Spatial Discretisation

**Remark**

**3.**

#### 3.3. Time Integration

**Remark**

**4.**

#### 3.4. Newton Method

## 4. Solution of Linear System of Equations

- The matrix ${\underset{\xaf}{J}}_{n+1}^{(k)}$, that is, the linearisation of $\underset{\xaf}{h}(\underset{\xaf}{a})$ at ${\underset{\xaf}{a}}_{n+1}^{(k)}$, has a large kernel due to the fact that $\mathbf{curl}\mathbf{grad}=\mathit{0}$. In case there are non-conducting regions, or in other words $supp(\sigma )\u228a\mathsf{\Omega}$, this deficiency carries over to the entire top-left block of the system matrix. In case of a direct solver, one would need to classify the degrees of freedom on the finite element mesh and eliminate the redundant ones, see [13]. For iterative solvers, on the other hand, careful preconditioning is needed as discussed below.
- The matrices $\underset{\xaf}{W}$, $\underset{\xaf}{K}$, and $\underset{\xaf}{V}$ are fully populated. This implies a quadratic complexity for storage and cost of matrix-vector products. Since a direct solution is complicated from the onset—see the previous point—it can be ruled out completely when considering its cubic complexity due to the boundary element matrices. In case of an iterative solution, only the actions of these matrices on a given vector need to be calculated.

#### 4.1. Fast Multipole Method

#### 4.2. Preconditioning

## 5. Force Computation

## 6. Results

#### 6.1. Force between Permanent Magnets

#### 6.2. TEAM 24

#### 6.3. Eddy-Current Brake

#### 6.4. Electromagnetic Launcher

**Remark**

**5.**

#### 6.5. Magnetic Metal Forming

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Embedding domain $\mathsf{\Omega}$ with conducting part ${\mathsf{\Omega}}_{c}$ and source current region ${\mathsf{\Omega}}_{s}$.

**Figure 3.**Geometric clusters of the same partition level (

**left**) and up and downward passes in the multilevel scheme (

**right**).

**Figure 4.**Two parallelepiped permanent magnets. Position of the magnets where the red is shifted through various positions in x-direction (

**left**). Analytic and simulated force components (

**right**).

**Figure 5.**Two axis-aligned cylindrical permanent magnets. Position of the magnets where the red is shifted through various positions in z-direction (

**left**). Analytic and simulated force component ${F}_{z}$ (

**right**).

**Figure 6.**TEAM 24 setup (

**left**) with rotor (blue), stator (green), and coils; material behaviour (

**right**) with provided data and employed curve representation.

**Figure 7.**TEAM 24 result snapshots for induced currents at $t=0.03\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ (

**left**) and the B-field at $t=0.3\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ (

**right**).

**Figure 11.**EM launcher input data: $B(H)$ relation (

**left**, logarithmic H-axis), current in the wire vs. time (

**right**).

**Figure 12.**Different snapshots of the electromagnetic launcher, projectile is coloured by the magnitude of the B-field (max = 6.8 T).

**Figure 13.**Magnetic force on the projectile (

**left**) and its velocity (

**right**). Dashed lines show results carried out with FEMM.

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**MDPI and ACS Style**

Rüberg, T.; Kielhorn, L.; Zechner, J. Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling. *Mathematics* **2021**, *9*, 1804.
https://doi.org/10.3390/math9151804

**AMA Style**

Rüberg T, Kielhorn L, Zechner J. Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling. *Mathematics*. 2021; 9(15):1804.
https://doi.org/10.3390/math9151804

**Chicago/Turabian Style**

Rüberg, Thomas, Lars Kielhorn, and Jürgen Zechner. 2021. "Electromagnetic Devices with Moving Parts—Simulation with FEM/BEM Coupling" *Mathematics* 9, no. 15: 1804.
https://doi.org/10.3390/math9151804