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

A New Parareal Algorithm for Problems with Discontinuous Sources I. Kulchytska 1 , M. J. Gander 2 , S. Schöps 1 , I. Niyonzima 3 1 Institut Theorie Elektromagnetischer Felder and Graduate School CE, TU Darmstadt, 2 Section de Mathématiques, University of Geneva, 3 Department 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 Introduction 1 � Motivation � The eddy current problem Systems with highly-oscillatory excitations 2 � 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 Conclusions and outlook 4 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 Systems with highly-oscillatory excitations 2 Acceleration of convergence to the steady state 3 Conclusions and outlook 4 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 Ω 1 and Ω 2 σ ∂ � A � � Ω 1 ν ∇ × � + � ∂t ( � x, t ) = −∇ × A ( � x, t ) J s ( � x, t ) Ω 2 Γ with magnetic vector potential � x, 0) = � 2 A ( � A 0 ( � x ) , Ω 2 Ω 2 current density in coils and magnets � J s , x, � conductivity σ ( � x ) and reluctivity ν ( � A ) . Γ 1 � Spatial discretization yields initial value problem � � M d t u ( t ) = f t, u ( t ) , t ∈ (0 , T ] , u (0) = u 0 , � � 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 Ω 1 and Ω 2 σ ∂ � A � � ν ∇ × � + � ∂t ( � x, t ) = −∇ × A ( � x, t ) J s ( � x, t ) with magnetic vector potential � x, 0) = � A ( � A 0 ( � x ) , current density in coils and magnets � J s , x, � conductivity σ ( � x ) and reluctivity ν ( � A ) . � Spatial discretization yields initial value problem � � M d t u ( t ) = f t, u ( t ) , t ∈ (0 , T ] , u (0) = u 0 , � � 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 Ω 1 and Ω 2 σ ∂ � A � � Ω 1 ν ∇ × � + � ∂t ( � x, t ) = −∇ × A ( � x, t ) J s ( � x, t ) Ω 2 Γ with magnetic vector potential � x, 0) = � 2 A ( � A 0 ( � x ) , Ω 2 Ω 2 current density in coils and magnets � J s , x, � conductivity σ ( � x ) and reluctivity ν ( � A ) . Γ 1 � Spatial discretization yields initial value problem � � M d t u ( t ) = f t, u ( t ) , t ∈ (0 , T ] , u (0) = u 0 , � � 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

Challenges � Machines operate most of their life time in steady state � Long simulation time until Solution | u | / Wb steady state is reached � Effects on several time scales, e.g. due to pulsed excitations � Many time steps yield time-consuming computation Time / s May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 6

Challenges � Machines operate most of their life time in steady state � Long simulation time until Solution | u | / Wb 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 Time / s 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 Acceleration of convergence to the steady state 3 Conclusions and outlook 4 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 1 Input function � Propagators: fine F and coarse G � Solver F resolves high-frequency pulses 0 � Solver G might not capture dynamics − 1 0 10 20 Time / ms 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 1 Input function � Propagators: fine F and coarse G � Solver F resolves high-frequency pulses 0 � Solver G might not capture dynamics Idea − 1 � Solve coarse problem for slowly-varying 0 10 20 smooth input Time / ms � Low-frequency component: sinusoidal PWM signal with a switching � 2 π � waveform sin T t 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 1 Input function � Propagators: fine F and coarse G � Solver F resolves high-frequency pulses 0 � Solver G might not capture dynamics Idea − 1 � Solve coarse problem for slowly-varying 0 10 20 smooth input Time / ms � Low-frequency component: sinusoidal PWM signal with a switching � 2 π � waveform sin T t frequency of 500 Hz and a sine wave of 50 Hz. Question � What about convergence? 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 ] M d t u ( 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 ] M d t u ( t ) = f ( t, u ( t )) = ¯ ˜ f ( t, u ( t )) + f ( t ) � �� � ���� slow smooth 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 ] M d t u ( t ) = f ( t, u ( t )) = ¯ ˜ f ( t, u ( t )) + f ( t ) � �� � ���� slow smooth fast switching � Reduced coarse propagator ¯ G � Original fine propagator F M d t u ( t ) = ¯ f ( t, u ( t )) , M d t u ( t ) = f ( t, u ( t )) , u (0) = u 0 u (0) = u 0 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 ] M d t u ( t ) = f ( t, u ( t )) = ¯ ˜ f ( t, u ( t )) + f ( t ) � �� � ���� slow smooth fast switching � Reduced coarse propagator ¯ G � Original fine propagator F M d t u ( t ) = ¯ f ( t, u ( t )) , M d t u ( t ) = f ( t, u ( t )) , u (0) = u 0 u (0) = u 0 � Modified Parareal update formula U ( k +1) = u 0 , 0 � � � � � � T n , T n − 1 , U ( k ) T n , T n − 1 , U ( k +1) T n , T n − 1 , U ( k ) U ( k +1) + ¯ − ¯ = F G G n n − 1 n − 1 n − 1 May 4, 2018 | TU Darmstadt | Institut Theorie Elektromagnetischer Felder and Graduate School CE | Iryna Kulchytska-Ruchka | 9