Introduction to Simulation - Lecture 16. Methods for Computing Periodic Steady-State - Part II Jacob White

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1 Inroducion o Simuaion - ecure 6 Mehods for Compuing Periodic Seady-Sae - Par II Jacob Whie hanks o Deepak Ramaswamy, Micha Rewienski, and Karen Veroy

2 Ouine hree Mehods so far ime inegraion uni seady-sae achieved Finie difference mehods Shooing Mehods Shooing Mehods Sae ransiion funcion Sensiiviy marix Marix-Free Approach Specra Mehods Gaerkin and Coocaion Mehods

3 Periodic Seady-Sae Basics Basic Definiion () dx d = F x() + u () sae inpu Suppose he sysem has a periodic inpu 2 3 Many Sysems evenuay respond periodicay SMA-HPC 2003 MI x( + ) = x( ) for >> 0

4 Periodic Seady-Sae Basics Compuing Seady Sae ime Inegraion Mehod ime-inegrae Uni Seady-Sae Achieved dx d ( ()) ˆ ˆ ( ˆ ) = F x + u() x = x + F x + u( ) Need many imepoins for ighy damped case! SMA-HPC 2003 MI

5 Boundary-Vaue Probem Basic Formuaion Periodiciy Consrain Differenia Equaion Souion d N Differenia Equaions: x i = F i x d () () N Periodiciy Cons rains: x = x 0 i i SMA-HPC 2003 MI

6 Boundary-Vaue Probem Finie Difference Mehods Noninear Probem dx d SMA-HPC 2003 MI ( ()) () [ 0, ] = F x + u x = x inpu periodiciy consrain Discreize wih Backward-Euer xˆ xˆ ( F( xˆ ) + u( ) ) xˆ xˆ 2 xˆ ( F( xˆ 2 ) + u( 2 ) ) 2 xˆ H FD = ˆ x xˆ xˆ F xˆ + u = 0 ( ) Sove Using Newon s Mehod

7 Boundary-Vaue Probem Shooing Mehod Basic Definiions dx( ) Sar wih = F ( x() ) + u() d And assume x() is unique given x(0). D.E. defines a Sae-ransiion Funcion Φ ( y,, ) x( ) 0 where x is he D.E. souion given x = y 0 SMA-HPC 2003 MI

8 Boundary-Vaue Probem Sove ( ) Use Newon s mehod Shooing Mehod Absrac Formuaion H x 0 = Φ x 0,0, x 0 = 0 x H ( x,0, ) Φ = J x I H x ( k )( k+ k ) = ( k ) J x x x H x SMA-HPC 2003 MI

9 Boundary-Vaue Probem Shooing Mehod Compuing Newon SMA-HPC 2003 MI ( ) o Compue Φ x 0,0, dx( ) Inegrae = F( x() ) + u() on [0,] d (,0, ) Φ x Wha is? x x ( 0) + ε x ( 0) x ε x Indicaes he sensiiviy of x() o changes in x(0)

10 Boundary-Vaue Probem Φ x,0, x SMA-HPC 2003 MI Shooing Mehod Sensiiviy Marix by Perurbaion ε x x x x ε N ε ε x x x x ε N N N N N ε ε ε N N

11 Boundary-Vaue Probem Shooing Mehod Efficien Sensiiviy Evauaion Differeniae he firs sep of Backward-Euer x ( 0) SMA-HPC 2003 MI ( ( 0 ) ( ( ) ) 0 ) xˆ x F xˆ + u = ( 0) ( 0) ( ˆ ) xˆ x 0 F x xˆ = x x x x I ( ˆ ) ( 0) x 0 F x xˆ = x x( 0) x 0 0 I

12 Boundary-Vaue Probem Shooing Mehod Efficien Sensiiviy Marix Con Appying he same rick on he -h sep I ( ˆ ) F x xˆ xˆ = x x x ( 0) ( 0) Φ x,0, x = I ( ˆ ) F x x SMA-HPC 2003 MI

13 Boundary-Vaue Probem Shooing Mehod Observaions on Sensiiviy Marix Newon a each imesep uses same marices Φ SMA-HPC 2003 MI x,0, x = ( ˆ ) F x I x imesep Newon Jacobian Formua simpifies in he inear case Φ x,0, x ( I A)

14 Shooing Mehod Marix-Free Approach Basic Seup Sar wih SMA-HPC 2003 MI dx d Use Newon s mehod ( ()) u () = F x + ( ) H x 0 = Φ x 0,0, x 0 = 0 H ( x,0, ) Φ = J x I H x ( k )( k+ k ) = ( k ) J x x x H x

15 Shooing Mehod Marix-Free Approach Marix-Vecor Produc Sove Newon equaion wih Kryov-subspace mehod SMA-HPC 2003 MI ( k x,0, ) Φ I x x = x Φ x x x b A ( k+ k) k ( k,0, ) Marix-Vecor Produc Compuaion ( x k,0, ) ( x k ε p j,0, ) ( x k,0, ) Φ Φ + Φ x ε j j I p p Kryov mehod search direcion

16 Shooing Mehod Marix-Free Approach Convergence for GCR Exampe dx Ax d = Shooing-Newon Jacobian 0 eig A rea and negaive Φ x,0, I = e A I x SMA-HPC 2003 MI

17 Shooing Mehod Marix-Free Approach Convergence for GCR-evas λ e A e I = S S λn e Many Fas Modes cuser a SMA-HPC 2003 MI Few Sow Modes arger han

18 Specra Mehods Fourier Represenaion runcaion Approximaion Periodic funcion fourier series x () Approximae a funcion wih runcaed series x () = = = Xe Xe i2π i2π SMA-HPC 2003 MI

19 Specra Mehods Fourier Represenaion Square Wave Exampe Copyrigh 997 by Aan V. Oppenheim and Aan S. Wisky SMA-HPC 2003 MI

20 Specra Mehods Fourier Represenaion Annoyance for Rea Funcions Rea x Fourier Coeffs compex conjugae X = X Can rewrie series wih fewer unknowns x () = Xe + X * e + X 0 = Rea = 0 * i2π + i2π SMA-HPC 2003 MI

21 Specra Mehods Fourier Represenaion Orhogonaiy erms in Fourier Series are orhogona 0 i2π i2π m e e d = 0 m Simpe formua for compuing coefficiens 0 e i2πm i2πm i2π () d e 0 = x = Xe d = X m SMA-HPC 2003 MI

22 Specra Mehods Fourier Represenaion Advanages For smooh funcions (infiniey con. diff) Fourier Coefficiens decay exponeniay fas im m > 0 e i2π m ( m ) x() d = im X = O c m > m Auomaicay saisfies periodiciy + i2π i2π = = + = = = x Xe Xe x () SMA-HPC 2003 MI

23 Specra Mehods Compuing Coefficiens Residua Pug represenaion ino differenia equaion d R X Xe F Xe u d, = () Residua 2 i π i2π = = Simpify by differeniaing represenaion 2 i2 i π π i2π R X Xe F Xe u = =, = () Residua SMA-HPC 2003 MI

24 Specra Mehods Compuing Coefficiens Coocaion and Gaerkin Coocaion Residua = 0 a es poins R( X, ) = 0 = {,...,2+ } Residua Gaerkin Residua orhog o Fourier erms 0 e i2π m R ( Xd, ) = 0 m {,...,0,... } Residua SMA-HPC 2003 MI

25 Specra Mehods Compuing Coefficiens Gaerkin Equaion Gaerkin Residua orhog o Fourier erms = i2π m 2 i2 i π π i2π Xe e F Xe u () d 0 = = 2 i πm i2π i2 m 2 π i π mx () + e F X e d + e u d = 0 0 = 0 m {,...,0,... } SMA-HPC 2003 MI

26 Specra Mehods Compuing Coefficiens inear Gaerkin F(x)=Ax 2 i π m i2π i2 m 2 π i π mx () + e A Xe d+ e u d = 0 0 = 0 i2π + A X U 2 ( ) X i ( ) U π ( ) 0 + A 0 0 = i2π A X U + SMA-HPC 2003 MI Diagona U m

27 Specra Mehods Compuing Coefficiens Coocaion Equaions Coocaion Residua zero a es imes 2 i2 i π π i2π R X X e F X e u = =, 0 = = Residua = {,...,2+ } SMA-HPC 2003 MI

28 Specra Mehods Compuing Coefficiens Discree Fourier ransform i2π i2π X x e e X x ( ) 2 = i2π 2+ i2π 2+ e e X x ( 2+ ) If Discree Fourier ransform(df) = hen DF Marix has orhog coumns 2 + SMA-HPC 2003 MI

29 Specra Mehods Compuing Coefficiens Coocaion using imepoins i2π x( ) F( x( )) u i2π ( x ) ( 2) F( x( 2 )) u DF ( DF ) = i2π x ( ) ( ( )) u F x 2+ Specra Differeniaion ( ) Convering imepoin ino Fourier Coeffs, Differeniaing, and hen reurning o ime SMA-HPC 2003 MI

30 Specra Mehods Compuing Coefficiens Specra Differeniaion Exampe Midde row, = 7 and 2+ = 7 SMA-HPC 2003 MI

31 Specra Mehods Compuing Coefficiens Specra Cooc vs. F-D 0 0 F( x( )) xˆ u( ) 2 xˆ F( x( 2 )) u( 2 ) = ˆ ( ) 0 0 x F x u ( 2+ ) ( 2+ ) DF i2π x( ) F( x( )) u( ) i2π ( x ) ( 2) F( x( 2 )) u( 2) ( DF ) = i2π x ( ) ( ( )) u F x ( ( 2 ) ) 2+ + SMA-HPC 2003 MI

32 Summary Four Mehods ime inegraion uni seady-sae achieved Finie difference mehods Shooing Mehods Specra Mehods Shooing Mehods Sae ransiion funcion Sensiiviy marix Marix-Free Approach Specra Mehods Gaerkin and Coocaion Mehods SMA-HPC 2003 MI