# Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Isogeometric Analysis

#### 2.2. Spatial Discretization of Nonlinear Magnetic Problems

**B**= ${B}_{x}{\mathbf{e}}_{x}+{B}_{y}{\mathbf{e}}_{y}+0{\mathbf{e}}_{z}$ and the magnetic vector potential $\mathbf{A}$ = ${A}_{z}{\mathbf{e}}_{z}$ is reduced to a scalar field. In 2D, ${A}_{z}$ is obviously independent of z, and therefore $\mathrm{div}\phantom{\rule{3.33333pt}{0ex}}\mathbf{A}$ = 0 falls naturally. Taking $\nu \left(\mathbf{x}\right)$ as the magnetic permeability, ${J}_{s}$ as the imposed electrical current density in the coil regions and ${\mathbf{B}}_{r}$ as the remanent magnetic flux density of the permanent magnets, the strong form in 2D is a Laplace problem that reads:

#### 2.3. Time-Stepping Technique

#### 2.4. Harmonic Balance Method

#### 2.5. Space-Time Galerkin Approach

#### 2.6. Benchmarks

## 3. Results

#### 3.1. Slotless Benchmark

#### 3.2. Slotted Benchmark

#### 3.3. Comparison with the Space-Time Approach

#### 3.4. Refinement Strategy

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1D, 2D, 3D, 4D | One-, Two-, Three-, Four-dimensional |

2.5D | Two and a half dimensional |

BE | Backward Euler |

B-spline | Basis spline |

CAD | Computer-Aided Design |

CG | Continuous Galerkin |

CK | Crank–Nicolson |

DC | Direct Current |

dofs | degrees of freedom |

DCT | Discrete Cosine Transform |

DFT | Discrete Fourier Transform |

DG | Discontinuous Galerkin |

FEM | Finite Element Method |

FE | Forward Euler |

FPM | Fixed-Point Method |

GAMG | Geometric Algebraic Multigrid |

GMG | Geometric Multigrid |

GMRES | Generalized Minimal Residual |

HB | Harmonic Balance |

IGA | Isogeometric Analysis |

LF | Leapfrog |

MEMS | Microelectromechanical Systems |

MG | Multigrid |

NRM | Newton–Raphson Method |

NURBS | Non-Uniform Rational B-splines |

PDEs | Partial Differential Equations |

QMR | Quasi-Minimal Residual |

RF | Radio Frequency |

rms | Root Mean Square |

ST | Space-Time |

THB-spline | Truncated Hierarchical B-spline |

THD | Total Harmonic Distortion |

TP | Time-Periodic |

TS | Time-Stepping |

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**Figure 1.**Schematic representations of (

**a**) the time-stepping technique, (

**b**) the space-time approach, and (

**c**) the harmonic balance method. The color code for the different regions is explained in Section 2.6.

**Figure 2.**B-spline basis of degree 0, 1, 2, 3 on an open uniform knot vector: (

**a**) B-spline basis of degree 0, (

**b**) B-spline basis of degree 1, (

**c**) B-spline basis of degree 2, and (

**d**) B-spline basis of degree 3. B-spline basis on an open nonuniform knot vector $\Xi =[3,\phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}3,\phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}3]$, (

**e**) B-spline basis of degree 2, and (

**f**) B-spline basis of degree 3.

**Figure 4.**Graphical representation of the numerical integration rule on the interval $({t}^{n},{t}^{n+1})$ for the $\theta $-schemes (28)–(31).

**Figure 5.**(

**a**) Slotless and (

**b**) slotted 2D benchmark geometries considered for validating the harmonic balance method.

**Figure 6.**Magnetic flux density modulus distribution B in [T] and flux lines for the slotless benchmark at speed $v\in \left(\right)open="\{"\; close="\}">0.1,1,10$ m/s, using the harmonic balance in (

**a**,

**c**,

**e**), and the time-stepping approach in (

**b**,

**d**,

**f**).

**Figure 7.**Absolute discrepancy in the magnetic flux density distribution $\Delta B$ in [T] between the harmonic balance and time-stepping solutions, (

**a**) at 1 m/s and (

**b**) at 10 m/s.

**Figure 8.**Attraction force ${F}_{y}$, damping force ${F}_{x}$, and eddy current losses P, for the slotless benchmark at speed $v=1$ m/s in (

**a**,

**c**,

**e**) and $v=10$ m/s in (

**b**,

**d**,

**f**), using the harmonic balance method with increasing number of harmonics and the time-stepping approach (TS).

**Figure 9.**(

**a**) Nonlinear convergence of the solution including up to the third harmonic, (

**b**) nonlinear convergence at 10 m/s, for different harmonic spectra H.

**Figure 10.**(

**a**) Computational effort for the time-stepping approach (TS) and harmonic balance method with increasing harmonic content N and different speeds, (

**b**) scaling of the computational effort by increasing the number of processors in parallel for the slotless benchmark.

**Figure 11.**Magnetic flux density modulus distribution B in [T] and flux lines for the slotted benchmark at speed $v\in \left(\right)open="\{"\; close="\}">0.1,1,10$ m/s, using the harmonic balance in (

**a**,

**c**,

**e**), and the time-stepping approach in (

**b**,

**d**,

**f**).

**Figure 12.**Absolute discrepancy in the magnetic flux density distribution $\Delta B$ in [T] between the harmonic balance and time-stepping solutions, (

**a**) at 1 m/s and (

**b**) at 10 m/s.

**Figure 13.**Attraction force ${F}_{y}$, damping force ${F}_{x}$, and eddy current losses P, for the slotted benchmark at speed $v=1$ m/s in (

**a**,

**c**,

**e**) and $v=10$ m/s in (

**b**,

**d**,

**f**), using the harmonic balance method with increasing number of harmonics and the transient approach.

**Figure 14.**(

**a**) Nonlinear convergence of the solution including up to the seventh harmonic, (

**b**) nlinear convergence at 10 m/s, for different harmonic spectra H.

**Figure 15.**(

**a**) Computational effort for the time-stepping approach (TS) and harmonic balance method with increasing harmonic content N and different speeds, (

**b**) scaling of the computational effort by increasing the number of processors in parallel for the slotted benchmark.

**Figure 16.**Space-time solutions for the slotted and slotless benchmarks of Figure 5, on the left and right, respectively.

**Figure 17.**(

**a**) Nonlinear convergence of the space-time approach and harmonic balance method with the third time harmonic for the slotless benchmark of Figure 5a, (

**b**) nonlinear convergence of the space-time approach and harmonic balance method with the seventh time harmonic for the slotted benchmark of Figure 5b.

**Figure 18.**Schematic representations of (

**a**) the progressive harmonic refinement strategy, (

**b**) the traditional approach.

**Figure 19.**(

**a**) Residual norm at successive Newton–Raphson iterations for the untrimmed problem, increasing the refinement levels without initial guess and (

**b**) with an initial guess obtained from the solution at the previous level; (

**c**) evolution of the number of degrees of freedom and corresponding computational time; (

**d**) convergence of the mean value of the eddy current losses for increasing refinement level.

**Figure 20.**Magnetic flux density distribution B in [T] and mesh at the last level for the slotless benchmark.

$\mathit{\theta}$ | Name | Scheme | Precision |
---|---|---|---|

0 | Forward Euler | explicit | $\mathcal{O}(\Delta t)$ |

0.5 | Crank–Nicolson | implicit | $\mathcal{O}{(\Delta t)}^{2}$ |

1 | Backward Euler | implicit | $\mathcal{O}(\Delta t)$ |

Parameter | Value [mm] | Parameter | Value [mm] |
---|---|---|---|

$2{\tau}_{p1}$ | 8.0 | $2{\tau}_{p2}$ | 10.5 |

${h}_{c}$ | 2.0 | ${\tau}_{i}$ | 2.0 |

g | 1.0 | ${\tau}_{a}$ | 1.5 |

${h}_{m}$ | 2.0 | ${h}_{i}$ | 1.5 |

${h}_{bi}$ | 2.0 | ${h}_{a}$ | 2.0 |

**Table 3.**Mean relative discrepancy in percentage, for the damping force $\Delta {F}_{x}$, attraction force $\Delta {F}_{y}$, and eddy current losses $\Delta P$, between the transient time-stepping solution and the ones obtained with harmonic balance of increasing harmonic content N and upper harmonic order H, for speed $v\in \left(\right)open="\{"\; close="\}">1,10$ m/s, for the slotless benchmark.

$\mathsf{\Delta}{\mathit{F}}_{\mathit{x}}$ [%] | $\mathsf{\Delta}{\mathit{F}}_{\mathit{y}}$ [%] | $\mathsf{\Delta}\mathit{P}$ [%] | |||||
---|---|---|---|---|---|---|---|

$\mathbf{N}$ | $\mathbf{H}$ | 1 m/s | 10 m/s | 1 m/s | 10 m/s | 1 m/s | 10 m/s |

1 | 1 | $1.88$ | $2.58$ | 4.41 × ${10}^{-2}$ | 2.36 × 10${}^{-1}$ | $1.84$ | $2.54$ |

2 | 3 | $1.68$ | $2.32$ | 9.63 × 10${}^{-3}$ | 9.20 × 10${}^{-2}$ | $1.64$ | $2.27$ |

3 | 5 | 1.54 × 10${}^{-1}$ | $2.10$ | 1.04 × 10${}^{-3}$ | 8.81 × 10${}^{-3}$ | $1.50$ | $2.05$ |

4 | 7 | 1.43 × 10${}^{-1}$ | $1.76$ | 2.84 × 10${}^{-3}$ | 1.73 × 10${}^{-2}$ | $1.39$ | $1.72$ |

**Table 4.**Mean and rms relative discrepancy in percentage, for the damping force $\Delta {F}_{x}$, attraction force $\Delta {F}_{y}$, and eddy current losses $\Delta P$, between the transient time-stepping solution and the ones obtained with harmonic balance of increasing harmonic content N and upper harmonic order H, for speed $v=10$ m/s, for the slotted benchmark.

$\mathsf{\Delta}{\mathit{F}}_{\mathit{x}}$ [%] | $\mathsf{\Delta}{\mathit{F}}_{\mathit{y}}$ [%] | $\mathsf{\Delta}\mathit{P}$ [%] | |||||
---|---|---|---|---|---|---|---|

$\mathbf{N}$ | $\mathbf{H}$ | mean | rms | mean | rms | mean | rms |

1 | 1 | $1.64$ | $1.40\times {10}^{1}$ | $6.25\times {10}^{-1}$ | $6.74\times {10}^{-1}$ | $1.71$ | $6.97$ |

2 | 3 | $5.40\times {10}^{-1}$ | $4.27$ | $5.22\times {10}^{-2}$ | $1.87\times {10}^{-1}$ | $4.86\times {10}^{-1}$ | $4.01$ |

3 | 5 | $7.85\times {10}^{-1}$ | $1.67$ | $3.77\times {10}^{-2}$ | $7.85\times {10}^{-2}$ | $7.29\times {10}^{-1}$ | $1.83$ |

4 | 7 | $8.60\times {10}^{-1}$ | $1.60$ | $6.46\times {10}^{-2}$ | $9.26\times {10}^{-2}$ | $8.05\times {10}^{-1}$ | $1.51$ |

5 | 9 | $8.90\times {10}^{-1}$ | $1.58$ | $6.85\times {10}^{-2}$ | $9.45\times {10}^{-2}$ | $8.36\times {10}^{-1}$ | $1.27$ |

**Table 5.**Computational effort comparison among the three nonlinear transient approaches for the 2D benchmarks, harmonic balance method with up to the third (HB(2)) or seventh (HB(4)) time harmonic, space-time (ST), and time-stepping (TS) methods, without adaptivity, and with a single core.

Slotless | Slotted | |||||
---|---|---|---|---|---|---|

HB(2) | ST | TS | HB(4) | ST | TS | |

Time [s] | 24 | 119 | 821 | 138 | 236 | 1470 |

Ratio [-] | - | 5.0 | 34.4 | - | 1.72 | 10.6 |

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**MDPI and ACS Style**

Friedrich, L.A.J.
Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems. *J* **2021**, *4*, 22-48.
https://doi.org/10.3390/j4010002

**AMA Style**

Friedrich LAJ.
Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems. *J*. 2021; 4(1):22-48.
https://doi.org/10.3390/j4010002

**Chicago/Turabian Style**

Friedrich, Léo A.J.
2021. "Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems" *J* 4, no. 1: 22-48.
https://doi.org/10.3390/j4010002