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 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 , (e) B-spline basis of degree 2, and (f) B-spline basis of degree 3.
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 , (e) B-spline basis of degree 2, and (f) B-spline basis of degree 3.
Figure 3.
Conforming multipatch isogeometric discretization schematic.
Figure 3.
Conforming multipatch isogeometric discretization schematic.
Figure 4.
Graphical representation of the numerical integration rule on the interval
for the
-schemes (
28)–(31).
Figure 4.
Graphical representation of the numerical integration rule on the interval
for the
-schemes (
28)–(31).
Figure 5.
(a) Slotless and (b) slotted 2D benchmark geometries considered for validating the harmonic balance method.
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 m/s, using the harmonic balance in (a,c,e), and the time-stepping approach in (b,d,f).
Figure 6.
Magnetic flux density modulus distribution B in [T] and flux lines for the slotless benchmark at speed 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 in [T] between the harmonic balance and time-stepping solutions, (a) at 1 m/s and (b) at 10 m/s.
Figure 7.
Absolute discrepancy in the magnetic flux density distribution 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 , damping force , and eddy current losses P, for the slotless benchmark at speed m/s in (a,c,e) and m/s in (b,d,f), using the harmonic balance method with increasing number of harmonics and the time-stepping approach (TS).
Figure 8.
Attraction force , damping force , and eddy current losses P, for the slotless benchmark at speed m/s in (a,c,e) and 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 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 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 m/s, using the harmonic balance in (a,c,e), and the time-stepping approach in (b,d,f).
Figure 11.
Magnetic flux density modulus distribution B in [T] and flux lines for the slotted benchmark at speed 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 in [T] between the harmonic balance and time-stepping solutions, (a) at 1 m/s and (b) at 10 m/s.
Figure 12.
Absolute discrepancy in the magnetic flux density distribution 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 , damping force , and eddy current losses P, for the slotted benchmark at speed m/s in (a,c,e) and m/s in (b,d,f), using the harmonic balance method with increasing number of harmonics and the transient approach.
Figure 13.
Attraction force , damping force , and eddy current losses P, for the slotted benchmark at speed m/s in (a,c,e) and 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 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 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 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 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 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 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.
Figure 20.
Magnetic flux density distribution B in [T] and mesh at the last level for the slotless benchmark.
Table 1.
-schemes for standard time-stepping approaches.
Table 1.
-schemes for standard time-stepping approaches.
| Name | Scheme | Precision |
---|
0 | Forward Euler | explicit | |
0.5 | Crank–Nicolson | implicit | |
1 | Backward Euler | implicit | |
Table 2.
Geometrical dimensions of the 2D benchmarks.
Table 2.
Geometrical dimensions of the 2D benchmarks.
Parameter | Value [mm] | Parameter | Value [mm] |
---|
| 8.0 | | 10.5 |
| 2.0 | | 2.0 |
g | 1.0 | | 1.5 |
| 2.0 | | 1.5 |
| 2.0 | | 2.0 |
Table 3.
Mean relative discrepancy in percentage, for the damping force , attraction force , and eddy current losses , 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 m/s, for the slotless benchmark.
Table 3.
Mean relative discrepancy in percentage, for the damping force , attraction force , and eddy current losses , 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 m/s, for the slotless benchmark.
| [%] | [%] | [%] |
---|
| | 1 m/s | 10 m/s | 1 m/s | 10 m/s | 1 m/s | 10 m/s |
1 | 1 | | | 4.41 × | 2.36 × 10 | | |
2 | 3 | | | 9.63 × 10 | 9.20 × 10 | | |
3 | 5 | 1.54 × 10 | | 1.04 × 10 | 8.81 × 10 | | |
4 | 7 | 1.43 × 10 | | 2.84 × 10 | 1.73 × 10 | | |
Table 4.
Mean and rms relative discrepancy in percentage, for the damping force , attraction force , and eddy current losses , 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 m/s, for the slotted benchmark.
Table 4.
Mean and rms relative discrepancy in percentage, for the damping force , attraction force , and eddy current losses , 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 m/s, for the slotted benchmark.
| [%] | [%] | [%] |
---|
| | mean | rms | mean | rms | mean | rms |
1 | 1 | | | | | | |
2 | 3 | | | | | | |
3 | 5 | | | | | | |
4 | 7 | | | | | | |
5 | 9 | | | | | | |
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.
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 |