A New Parareal Algorithm for Problems with Discontinuous Sources I. - - PowerPoint PPT Presentation

A New Parareal Algorithm for Problems with Discontinuous Sources

• I. Kulchytska1, M. J. Gander2, S. Schöps1, I. Niyonzima3

1Institut Theorie Elektromagnetischer Felder and Graduate School CE, TU Darmstadt, 2Section de Mathématiques, University of Geneva, 3Department of Civil Engineering and Engineering Mechanics, Columbia University May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 1

Outline of the Talk

1

Introduction

Motivation The eddy current problem

2

Systems with highly-oscillatory excitations

Modified Parareal with reduced coarse dynamics Convergence results for nonsmooth sources Numerical example: induction machine

3

Acceleration of convergence to the steady state

Time-periodic eddy current problem Parareal for time-periodic problems Results for the induction machine

4

Conclusions and outlook

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 2

Outline of the Talk

1

Introduction

Motivation The eddy current problem

2

Systems with highly-oscillatory excitations

3

Acceleration of convergence to the steady state

4

Conclusions and outlook

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 3

Motivation

E-bike with a synchronous machine Robust geometry optimization Expensive time domain simulations May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 4

Motivation

E-bike with a synchronous machine Robust geometry optimization Expensive time domain simulations May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 4

The eddy current problem

Eddy current problem on domains Ω1 and Ω2

σ ∂ A ∂t ( x, t) = −∇ ×

• ν∇ ×

A( x, t)

• +

Js( x, t)

with magnetic vector potential

A( x, 0) = A0( x),

current density in coils and magnets

Js,

conductivity σ(

x) and reluctivity ν( x, A).

Spatial discretization yields initial value problem

Mdtu(t) = f

• t, u(t)
• , t ∈ (0, T],

u(0) = u0,

with unknown u(t), mass matrix M and right-hand-side f

• ...
• .

Γ

1

Γ

2

Ω2 Ω1 Ω2 Ω2 May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 5

The eddy current problem

Eddy current problem on domains Ω1 and Ω2

σ ∂ A ∂t ( x, t) = −∇ ×

• ν∇ ×

A( x, t)

• +

Js( x, t)

with magnetic vector potential

A( x, 0) = A0( x),

current density in coils and magnets

Js,

conductivity σ(

x) and reluctivity ν( x, A).

Spatial discretization yields initial value problem

Mdtu(t) = f

• t, u(t)
• , t ∈ (0, T],

u(0) = u0,

with unknown u(t), mass matrix M and right-hand-side f

• ...
• .

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 5

The eddy current problem

Eddy current problem on domains Ω1 and Ω2

σ ∂ A ∂t ( x, t) = −∇ ×

• ν∇ ×

A( x, t)

• +

Js( x, t)

with magnetic vector potential

A( x, 0) = A0( x),

current density in coils and magnets

Js,

conductivity σ(

x) and reluctivity ν( x, A).

Spatial discretization yields initial value problem

Mdtu(t) = f

• t, u(t)
• , t ∈ (0, T],

u(0) = u0,

with unknown u(t), mass matrix M and right-hand-side f

• ...
• .

Γ

1

Γ

2

Ω2 Ω1 Ω2 Ω2 May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 5

Challenges

Time / s Solution |u| / Wb

Machines operate most of their

life time in steady state

Long simulation time until

steady state is reached

Effects on several time scales,

e.g. due to pulsed excitations

Many time steps yield

time-consuming computation

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 6

Challenges

Time / s Solution |u| / Wb

Machines operate most of their

life time in steady state

Long simulation time until

steady state is reached

Effects on several time scales,

e.g. due to pulsed excitations

Many time steps yield

time-consuming computation

= ⇒

parallel-in-time method

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 6

Outline of the Talk

1

Introduction

2

Systems with highly-oscillatory excitations

Modified Parareal with reduced coarse dynamics Convergence results for nonsmooth sources Numerical example: induction machine

3

Acceleration of convergence to the steady state

4

Conclusions and outlook

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 7

Parareal for highly-oscillatory discontinuous excitation

Parareal

PWM (pulse width modulation): excitation

contains high-order frequency components

Propagators: fine F and coarse G Solver F resolves high-frequency pulses Solver G might not capture dynamics

10 20 −1 1

Time / ms Input function

PWM signal with a switching frequency of 500 Hz.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 8

Parareal for highly-oscillatory discontinuous excitation

Parareal

PWM (pulse width modulation): excitation

contains high-order frequency components

Propagators: fine F and coarse G Solver F resolves high-frequency pulses Solver G might not capture dynamics

Idea

Solve coarse problem for slowly-varying

smooth input

Low-frequency component: sinusoidal

waveform sin

2π

T t

• 10

20 −1 1

Time / ms Input function

PWM signal with a switching frequency of 500 Hz and a sine wave of 50 Hz.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 8

Parareal for highly-oscillatory discontinuous excitation

Parareal

PWM (pulse width modulation): excitation

contains high-order frequency components

Propagators: fine F and coarse G Solver F resolves high-frequency pulses Solver G might not capture dynamics

Idea

Solve coarse problem for slowly-varying

smooth input

Low-frequency component: sinusoidal

waveform sin

2π

T t

• Question

What about convergence?

10 20 −1 1

Time / ms Input function

PWM signal with a switching frequency of 500 Hz and a sine wave of 50 Hz.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 8

Modified Parareal with reduced coarse dynamics

Splitting of the nonsmooth excitation for t ∈ (0, T]

Mdtu(t) = f(t, u(t))

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9

Modified Parareal with reduced coarse dynamics

Splitting of the nonsmooth excitation for t ∈ (0, T]

Mdtu(t) = f(t, u(t)) = ¯ f(t, u(t))

• slow smooth

+ ˜ f(t)

• fast switching

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9

Modified Parareal with reduced coarse dynamics

Splitting of the nonsmooth excitation for t ∈ (0, T]

Mdtu(t) = f(t, u(t)) = ¯ f(t, u(t))

• slow smooth

+ ˜ f(t)

• fast switching

Reduced coarse propagator ¯

G Mdtu(t) = ¯ f(t, u(t)), u(0) = u0

Original fine propagator F

Mdtu(t) = f(t, u(t)), u(0) = u0

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9

Modified Parareal with reduced coarse dynamics

Splitting of the nonsmooth excitation for t ∈ (0, T]

Mdtu(t) = f(t, u(t)) = ¯ f(t, u(t))

• slow smooth

+ ˜ f(t)

• fast switching

Reduced coarse propagator ¯

G Mdtu(t) = ¯ f(t, u(t)), u(0) = u0

Original fine propagator F

Mdtu(t) = f(t, u(t)), u(0) = u0

Modified Parareal update formula

U(k+1) = u0, U(k+1)

n

= F

• Tn, Tn−1, U(k)

n−1

• + ¯

G

• Tn, Tn−1, U(k+1)

n−1

• − ¯

G

• Tn, Tn−1, U(k)

n−1

• May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9

Modified Parareal with reduced coarse dynamics

Splitting of the nonsmooth excitation for t ∈ (0, T]

Mdtu(t) = f(t, u(t)) = ¯ f(t, u(t))

• slow smooth

+ ˜ f(t)

• fast switching

Reduced coarse propagator ¯

G Mdtu(t) = ¯ f(t, u(t)), u(0) = u0

Original fine propagator F

Mdtu(t) = f(t, u(t)), u(0) = u0

Modified Parareal update formula

U(k+1) = u0, U(k+1)

n

= F

• Tn, Tn−1, U(k)

n−1

• + ¯

G

• Tn, Tn−1, U(k+1)

n−1

• − ¯

G

• Tn, Tn−1, U(k)

n−1

• Proof: based on perturbation results for ODEs with discontinuities

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9

Convergence of the modified approach

Theorem (Gander, K.-R., Schöps, Niyonzima, ’18)

• For I := [0, T] let ∆T = T/N denote window length and F
• Tn, Tn−1, U(k)

n−1

• be the

exact solution to the original problem at Tn, with the RHS f = ¯

f + ˜

• f. For p ≥ 1 we denote

Cp = ˜ fLp(I,Rn) and let q ≥ 1 be given by 1/p + 1/q = 1.

• Assume ¯

G

• Tn, Tn−1, U(k)

n−1

• is an approximation to the reduced problem with the

smooth RHS ¯

• f. The error is bounded by C3∆T l+1, and let ¯

G satisfy the Lipschitz

condition:

¯ G

• t + ∆T, t, U
• − ¯

G

• t + ∆T, t, Y
• ≤ (1 + C2∆T)U − Y .

Then at iteration k we have: ||u(Tn) − Uk

n|| ≤

¯ Ck

1

• ¯

C4Cp∆T (l+1)k+1/q + ¯ C3

• ∆T l+1k+1 (1 + C2∆T)n−k−1

(k + 1)!

k

• j=0

(n − j).

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 10

Convergence of the modified approach

Theorem (Gander, K.-R., Schöps, Niyonzima, ’18)

• For I := [0, T] let ∆T = T/N denote window length and F
• Tn, Tn−1, U(k)

n−1

• be the

exact solution to the original problem at Tn, with the RHS f = ¯

f + ˜

• f. For p ≥ 1 we denote

Cp = ˜ fLp(I,Rn) and let q ≥ 1 be given by 1/p + 1/q = 1.

• Assume ¯

G

• Tn, Tn−1, U(k)

n−1

• is an approximation to the reduced problem with the

smooth RHS ¯

• f. The error is bounded by C3∆T l+1, and let ¯

G satisfy the Lipschitz

condition:

¯ G

• t + ∆T, t, U
• − ¯

G

• t + ∆T, t, Y
• ≤ (1 + C2∆T)U − Y .

Then at iteration k we have: ||u(Tn) − Uk

n|| ≤

¯ Ck

1

• ¯

C4Cp∆T (l+1)k+1/q + ¯ C3

• ∆T l+1k+1 (1 + C2∆T)n−k−1

(k + 1)!

k

• j=0

(n − j).

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 10

Convergence of the modified approach

Theorem (Gander, K.-R., Schöps, Niyonzima, ’18)

• For I := [0, T] let ∆T = T/N denote window length and F
• Tn, Tn−1, U(k)

n−1

• be the

exact solution to the original problem at Tn, with the RHS f = ¯

f + ˜

• f. For p ≥ 1 we denote

Cp = ˜ fLp(I,Rn) and let q ≥ 1 be given by 1/p + 1/q = 1.

• Assume ¯

G

• Tn, Tn−1, U(k)

n−1

• is an approximation to the reduced problem with the

smooth RHS ¯

• f. The error is bounded by C3∆T l+1, and let ¯

G satisfy the Lipschitz

condition:

¯ G

• t + ∆T, t, U
• − ¯

G

• t + ∆T, t, Y
• ≤ (1 + C2∆T)U − Y .

Then at iteration k we have: ||u(Tn) − Uk

n|| ≤

¯ Ck

1

• ¯

C4Cp∆T (l+1)k+1/q + ¯ C3

• ∆T l+1k+1 (1 + C2∆T)n−k−1

(k + 1)!

k

• j=0

(n − j).

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 10

Convergence of the modified approach

Theorem (Gander, K.-R., Schöps, Niyonzima, ’18)

• For I := [0, T] let ∆T = T/N denote window length and F
• Tn, Tn−1, U(k)

n−1

• be the

exact solution to the original problem at Tn, with the RHS f = ¯

f + ˜

• f. For p ≥ 1 we denote

Cp = ˜ fLp(I,Rn) and let q ≥ 1 be given by 1/p + 1/q = 1.

• Assume ¯

G

• Tn, Tn−1, U(k)

n−1

• is an approximation to the reduced problem with the

smooth RHS ¯

• f. The error is bounded by C3∆T l+1, and let ¯

G satisfy the Lipschitz

condition:

¯ G

• t + ∆T, t, U
• − ¯

G

• t + ∆T, t, Y
• ≤ (1 + C2∆T)U − Y .

Then at iteration k we have: ||u(Tn) − Uk

n|| ≤

¯ Ck

1

• ¯

C4Cp∆T (l+1)k+1/q + ¯ C3∆T (l+1)k+l+1 (1 + C2∆T)n−k−1 (k + 1)!

k

• j=0

(n − j).

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 10

Numerical verification

RL-circuit model:

1 Rφ′(t) + 1 Lφ(t) = f (t) , t ∈ (0, T], φ(0) = 0,

with R = 0.01 Ω, L = 0.001 H, T = 0.02 s;

f− supplied PWM current source of 20 kHz. 10 20 −1 1

Time / ms PWM input

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 11

Numerical verification

RL-circuit model:

1 Rφ′(t) + 1 Lφ(t) = f (t) , t ∈ (0, T], φ(0) = 0,

with R = 0.01 Ω, L = 0.001 H, T = 0.02 s;

f− supplied PWM current source of 20 kHz. 10 20 −1 1

Time / ms PWM & step

Choice of the coarse reduced input:

¯ fstep(t) =

• 1,

t ∈ [0, T/2), −1, t ∈ [T/2, T)

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 11

Numerical verification

RL-circuit model:

1 Rφ′(t) + 1 Lφ(t) = f (t) , t ∈ (0, T], φ(0) = 0,

with R = 0.01 Ω, L = 0.001 H, T = 0.02 s;

f− supplied PWM current source of 20 kHz. 10 20 −1 1

Time / ms PWM & sine

Choice of the coarse reduced input:

¯ fstep(t) =

• 1,

t ∈ [0, T/2), −1, t ∈ [T/2, T) ¯ fsine(t) = sin 2π T t

• , t ∈ [0, T]

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 11

Numerical verification

RL-circuit model:

1 Rφ′(t) + 1 Lφ(t) = f (t) , t ∈ (0, T], φ(0) = 0,

with R = 0.01 Ω, L = 0.001 H, T = 0.02 s;

f− supplied PWM current source of 20 kHz. 10 20 −1 1

Time / ms PWM, step, sine

Choice of the coarse reduced input:

¯ fstep(t) =

• 1,

t ∈ [0, T/2), −1, t ∈ [T/2, T) ¯ fsine(t) = sin 2π T t

• , t ∈ [0, T]

= ⇒ ˜ f(t) := f(t) − ¯ f(t) ∈ L∞(0, T) ⇐ ⇒ 1/q = 1.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 11

Numerical verification

Step input: ∆T (l+1)k+1/q = ∆T 3; Sine wave:

• ∆T l+1k+1 = ∆T 4

101 102 103 104 10−15 10−12 10−9 10−6 10−3 N = T/∆T

• rel. error at t = T2

step input sine wave ∆T 3 ∆T 4

Convergence of the Parareal iteration k = 1 using the implicit Euler method (l = 1) and the reduced coarse step- and sine-input.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 12

Numerical verification

Step input: ∆T (l+1)k+1/q = ∆T 5; Sine wave:

• ∆T l+1k+1 = ∆T 6

101 102 10−15 10−12 10−9 10−6 10−3 N = T/∆T

• rel. error at t = T2

step input sine wave ∆T 5 ∆T 6

Convergence of the Parareal iteration k = 2 using the implicit Euler method (l = 1) and the reduced coarse step- and sine-input.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 13

Numerical verification

Step input: ∆T (l+1)k+1/q = ∆T 4; Sine wave:

• ∆T l+1k+1 = ∆T 6

101 102 103 10−15 10−12 10−9 10−6 10−3 N = T/∆T

• rel. error at t = T2

step input sine wave ∆T 6 ∆T 4

Convergence of the Parareal iteration k = 1 using the Crank-Nicolson scheme (l = 2) and the reduced coarse step- and sine-input.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 14

Application to an induction machine

Magnetic flux density 0 T 2 T Four-pole squirrel cage ’im_3kw’ model and its magnetic field at t = 20 ms if excited by a sinusoidal voltage source (author: Gyselinck).

1 2 ·10−2 −100 100

Time / s Voltage / V PWM voltage source of 5 kHz with a ramp-up and phase 1 of the corresponding sinusoidal waveform of 50 Hz.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 15

Numerical results

10 20 −20 20 Time / ms Current / A

Phase 1 Phase 2 Phase 3

Stator currents for the three-phase PWM voltage source of 20 kHz on [0, 20] ms. Software: implicit Euler within GetDP . 1 2 3 4 5 6 7 8 10−1 101 103 105 Iteration number

• Max. error at synchronization points

PWM coarse input Sine coarse input

Convergence of the standard Parareal and the modified Parareal algorithms to reach the prescribed tolerance 1.5 · 10−5.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 16

Outline of the Talk

1

Introduction

2

Systems with highly-oscillatory excitations

3

Acceleration of convergence to the steady state

Time-periodic eddy current problem Parareal for time-periodic problems Results for the induction machine

4

Conclusions and outlook

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 17

Time-periodic eddy current problem

Goal: obtain the steady-state solution

Time / s Solution |u| / Wb

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 18

Time-periodic eddy current problem

Goal: obtain the steady-state solution

Time / s Solution |u| / Wb

Solve periodic boundary value problem in time:

Mdtu(t) = f(t, u(t)), t ∈ (0, T)

with

u(0) = u(T).

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 18

Parareal for time-periodic problems

u(t) t T0 T1 T2 T3 T4 T5

PP-IC: periodic parareal algorithm with initial value coarse problem:

U(k+1) = U(k)

N ,

U(k+1)

n

= F(Tn, Tn−1, U(k)

n−1) + G(Tn, Tn−1, U(k+1) n−1 ) − G(Tn, Tn−1, U(k) n−1).

• M. J. Gander et al., Analysis of Two Parareal Algorithms for Time-Periodic Problems, SIAM Journal on

Scientific Computing 35 (5), 2013.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 19

Results for induction machine with PWM voltage source

8.92% 26.82% 100% Computational cost

Sequential Parareal PP-IC

Computational efforts to obtain the periodic (steady-state) solution:

Sequential: 9 periods until the steady

state =

⇒ 2 176 179 system solves

Parareal: calculation on [0, 9T], needs

effectively 583 707 linear solutions

PP-IC: applied on one period [0, T],

requires 194 038 linear solutions Period T = 0.02 s, available CPUs N = 20 Fine propagator F: three-phase PWM excitation of 20 kHz, δT = 10−6 s Coarse solver ¯

G: three-phase sinusolidal source of 50 Hz, ∆T = 10−3 s

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 20

Outline of the Talk

1

Introduction

2

Systems with highly-oscillatory excitations

3

Acceleration of convergence to the steady state

4

Conclusions and outlook

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 21

Conclusions and outlook

Conclusions

Introduced a new Parareal algorithm with reduced coarse dynamics Developed convergence theory for problems with (highly-oscillatory)

discontinuous excitation

Applied the modified Parareal method to the time-periodic eddy current

problem for an induction machine model

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 22

Conclusions and outlook

Conclusions

Introduced a new Parareal algorithm with reduced coarse dynamics Developed convergence theory for problems with (highly-oscillatory)

discontinuous excitation

Applied the modified Parareal method to the time-periodic eddy current

problem for an induction machine model Outlook

Prove convergence of the modified PP-IC algorithm Further development of parallel-in-time methods for the periodic eddy

current problem with PWM excitation

Combine the time-parallel techniques with spatial domain decomposition for

simulation of electric machines

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 22

Thank you!

ACKNOWLEDGEMENT

The work of I. Kulchytska-Ruchka has been supported by the Excellence Initiative of the German Federal and State Governments, the Graduate School of Computational Engineering at TU Darmstadt, and the BMBF (grant No. 05M2018RDA) in the framework

• f project PASIROM.
• M. J. Gander, I. Kulchytska-Ruchka, I. Niyonzima, and S. Schöps. A new Parareal algorithm for

problems with discontinuous sources, 2018. Submitted, https://arxiv.org/abs/1803.05503.

• S. Schöps, I. Niyonzima, and M. Clemens, Parallel-in-time simulation of eddy current problems

using Parareal, IEEE Trans. Magn. 54 (3), 2018.

• J. Gyselinck, L. Vandevelde, and J. Melkebeek. Multi-slice FE modeling of electrical machines

with skewed slots-the skew discretization error, IEEE Trans. Magn. 37(5), 2001.

May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 23