Time-parallel solution of the eddy current problem Iryna Kulchytska-Ruchka 1,2 , Sebastian Schöps 1,2 , Herbert De Gersem 1,2 1 Graduate School of Computational Engineering, 2 Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt 4th STEAM Collaboration Meeting, Darmstadt www.graduate-school-ce.de September 21, 2017
Outline Introduction � Motivation � The eddy current problem Parallel-in-time solution � The Parareal method for IVPs � Numerical example: coaxial cable model Fourier basis for time-periodic systems � Coarse solution by spectral collocation � Numerical results: coaxial cable model � Systems with nonsmooth excitations Conclusions and outlook TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 2/33
Overview Introduction � Motivation � The eddy current problem Parallel-in-time solution � The Parareal method for IVPs � Numerical example: coaxial cable model Fourier basis for time-periodic systems � Coarse solution by spectral collocation � Numerical results: coaxial cable model � Systems with nonsmooth excitations Conclusions and outlook TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 3/33
Motivation → Transient FEM simulation Fig.: Cross-section of an induction machine (J. Gyselinck). TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 4/33
Motivation → Transient FEM simulation Fig.: Cross-section of an induction machine (J. Gyselinck). → System evolution in time TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 4/33
Motivation → Transient FEM simulation Fig.: Cross-section of an induction machine (J. Gyselinck). → System evolution in time TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 4/33
Motivation → Transient FEM simulation • Long settling time till the steady state Fig.: Cross-section of an induction machine (J. Gyselinck). → System evolution in time TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 4/33
Motivation → Transient FEM simulation • Long settling time till the steady state Fig.: Cross-section of an induction machine (J. Gyselinck). • Many time steps = ⇒ time-consuming computation! → System evolution in time TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 4/33
Overview Introduction � Motivation � The eddy current problem Parallel-in-time solution � The Parareal method for IVPs � Numerical example: coaxial cable model Fourier basis for time-periodic systems � Coarse solution by spectral collocation � Numerical results: coaxial cable model � Systems with nonsmooth excitations Conclusions and outlook TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 5/33
The eddy current problem Fundamentals of electromagnetism: Maxwell’s equations. Assumptions: � � ∂ D � � � Quasi-static regime: | J | ≫ � ; � � ∂ t � � Neglect hysteresis. The eddy current equation: σ∂ A ∂ t + curl ( ν ( | curl A | ) curl A ) = J s , A − unknown magnetic vector potential; J s − impressed current density; σ, ν − electric conductivity and magnetic reluctivity. TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 6/33
Semi-discrete problem Solve the IVP in time: M d t u ( t ) = f ( t , u ) , t ∈ I := ( 0 , T ) , u ( 0 ) = u 0 , where u : I �→ R N dof denotes the space-discretization of A . TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 7/33
Semi-discrete problem Solve the IVP in time: M d t u ( t ) = f ( t , u ) , t ∈ I := ( 0 , T ) , u ( 0 ) = u 0 , where u : I �→ R N dof denotes the space-discretization of A . • f ( t , u ) = − Ku ( t ) + g ( t ); M , K ∈ R N dof × N dof − mass and stiffness matrices; • • g ( t ) − excitation (e.g., impressed current). TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 7/33
Overview Introduction � Motivation � The eddy current problem Parallel-in-time solution � The Parareal method for IVPs � Numerical example: coaxial cable model Fourier basis for time-periodic systems � Coarse solution by spectral collocation � Numerical results: coaxial cable model � Systems with nonsmooth excitations Conclusions and outlook TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 8/33
The Parareal method: splitting of the time interval Partitioning the time interval into N windows (e.g., one per core) yields M d t u 0 = f ( t , u 0 ) , u 0 ( T 0 ) = U 0 , t ∈ ( T 0 , T 1 ] , M d t u 1 = f ( t , u 1 ) , u 1 ( T 1 ) = U 1 , t ∈ ( T 1 , T 2 ] , . . . M d t u N − 1 = f ( t , u N − 1 ) , u N − 1 ( T N − 1 ) = U N − 1 , t ∈ ( T N − 1 , T N ] , TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 9/33
The Parareal method: splitting of the time interval Partitioning the time interval into N windows (e.g., one per core) yields M d t u 0 = f ( t , u 0 ) , u 0 ( T 0 ) = U 0 , t ∈ ( T 0 , T 1 ] , M d t u 1 = f ( t , u 1 ) , u 1 ( T 1 ) = U 1 , t ∈ ( T 1 , T 2 ] , . . . M d t u N − 1 = f ( t , u N − 1 ) , u N − 1 ( T N − 1 ) = U N − 1 , t ∈ ( T N − 1 , T N ] , u ( t ) t T 0 T 1 T 2 T 3 T 4 T 5 TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 9/33
The Parareal method: splitting of the time interval Partitioning the time interval into N windows (e.g., one per core) yields M d t u 0 = f ( t , u 0 ) , u 0 ( T 0 ) = U 0 , t ∈ ( T 0 , T 1 ] , M d t u 1 = f ( t , u 1 ) , u 1 ( T 1 ) = U 1 , t ∈ ( T 1 , T 2 ] , . . . M d t u N − 1 = f ( t , u N − 1 ) , u N − 1 ( T N − 1 ) = U N − 1 , t ∈ ( T N − 1 , T N ] , u ( t ) t T 0 T 1 T 2 T 3 T 4 T 5 TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 9/33
Parareal as the Newton-Raphson method (I) Let F ( t , T i , U ) be the solution operator of the IVP on ( T i , T i + 1 ] , for i = 0 , . . . , N − 1 , which propagates the initial value U in time. TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 10/33
Parareal as the Newton-Raphson method (I) Let F ( t , T i , U ) be the solution operator of the IVP on ( T i , T i + 1 ] , for i = 0 , . . . , N − 1 , which propagates the initial value U in time. Matching conditions can be satisfied by root finding U 1 − F ( T 1 , T 0 , U 0 ) = 0 , . . . U N − 1 − F ( T N − 1 , T N − 2 , U N − 2 ) = 0 TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 10/33
Parareal as the Newton-Raphson method (I) Let F ( t , T i , U ) be the solution operator of the IVP on ( T i , T i + 1 ] , for i = 0 , . . . , N − 1 , which propagates the initial value U in time. Matching conditions can be satisfied by root finding U 1 − F ( T 1 , T 0 , U 0 ) = 0 , . . . U N − 1 − F ( T N − 1 , T N − 2 , U N − 2 ) = 0 or, equivalently, � � U T = U T 0 , U T 1 , ..., U T i , ..., U T F ( U ) = 0 , with . N − 1 M. J. Gander and E. Hairer, Nonlinear convergence analysis for the parareal algorithm , Domain Decomposition Methods in Science and Engineering XVII, Springer Berlin Heidelberg, 2008. TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 10/33
Parareal as the Newton-Raphson method (II) The Newton-Raphson iteration: for k = 0 , 1 , . . . U ( k + 1 ) = u 0 , 0 � � + ∂ F ( T n , T n − 1 , U ) � � U ( k + 1 ) T n , T n − 1 , U ( k ) U ( k + 1 ) − U ( k ) = F , n n − 1 n − 1 n − 1 ∂ U where n = 1 , . . . , N − 1 . TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 11/33
Parareal as the Newton-Raphson method (II) The Newton-Raphson iteration: for k = 0 , 1 , . . . U ( k + 1 ) = u 0 , 0 � � + ∂ F ( T n , T n − 1 , U ) � � U ( k + 1 ) T n , T n − 1 , U ( k ) U ( k + 1 ) − U ( k ) = F , n n − 1 n − 1 n − 1 ∂ U where n = 1 , . . . , N − 1 . How to calculate the derivative? TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 11/33
Parareal as the Newton-Raphson method (II) The Newton-Raphson iteration: for k = 0 , 1 , . . . U ( k + 1 ) = u 0 , 0 � � + ∂ F ( T n , T n − 1 , U ) � � U ( k + 1 ) T n , T n − 1 , U ( k ) U ( k + 1 ) − U ( k ) = F , n n − 1 n − 1 n − 1 ∂ U where n = 1 , . . . , N − 1 . How to calculate the derivative? Cheap approximation by a coarse propagator G : � � ∂ F ( T n , T n − 1 , U ) U ( k + 1 ) − U ( k ) ≈ n − 1 n − 1 ∂ U � � � � T n , T n − 1 , U ( k + 1 ) T n , T n − 1 , U ( k ) ≈ G − G . n − 1 n − 1 TU Darmstadt | GSC CE | Iryna Kulchytska | Time-parallel solution of the eddy current problem | 11/33
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