lec16 (1)

lec16 (1) - Introduction to Simulation - Lecture 16 Methods...

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Introduction to Simulation - Lecture 16 Methods for Computing Periodic Steady-State - Part II Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
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Outline Three Methods so far – Time integration until steady-state achieved – Finite difference methods – Shooting Methods Shooting Methods – State transition function – Sensitivity matrix – Matrix-Free Approach Spectral Methods – Galerkin and Collocation Methods
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SMA-HPC ©2003 MIT Basic Definition Periodic Steady-State Basics • Suppose the system has a periodic input • Many Systems eventually respond periodically () N N input state dx t Fxt u t dt ⎛⎞ ⎜⎟ =+ ⎝⎠ ( ) 0 xt T xt f o rt += > > t T2 T 3 T
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SMA-HPC ©2003 MIT Computing Steady State Periodic Steady-State Basics Time Integration Method • Time-Integrate Until Steady-State Achieved • Need many timepoints for lightly damped case! ( ) () () () 1 ˆˆ ˆ ( ) ll l dx t F xt ut x x tF x ul t dt =+ = + ∆+
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SMA-HPC ©2003 MIT Basic Formulation Boundary-Value Problem Differential Equation Solution Periodicity Constraint () () () N Differential Equations: ii d xt Fxt dt = ( ) ( ) N Periodicity Cons traints: 0 xT x =
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SMA-HPC ©2003 MIT Finite Difference Methods Boundary-Value Problem Nonlinear Problem ( ) () N [] () () periodicity constraint 0, input dx t F xt ut t T xT dt =+∈ = ±²³²´ Discretize with Backward-Euler ( ) ( ) 11 ˆˆ ˆ L xx t F x ut −∆ + ( ) ( ) 21 2 ˆ 2 xx t F x u t + ( ) ( ) 1 ˆ LL L t F x u L t + 1 2 ˆ ˆ ˆ FD L x x H x ⎛⎞ ⎡⎤ ⎜⎟ ⎢⎥ = ⎣⎦ ⎝⎠ # 0 = Solve Using Newton’s Method
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SMA-HPC ©2003 MIT Shooting Method Boundary-Value Problem Basic Definitions ( ) () dx t F xt ut dt =+ Start with And assume x(t) is unique given x(0). D.E. defines a State-Transition Function ( ) ( ) 01 1 ,, yt t Φ≡ ( ) 0 where ( ) is the D.E. solution given y =
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SMA-HPC ©2003 MIT Shooting Method Boundary-Value Problem Abstract Formulation Solve Use Newton’s method () ( ) ( ) ( ) 00 , 0 , 0 0 Hx x T x xT = Φ− = ±²²³ ² ²´ ( ) ,0, H Jx I x ∂Φ = ( ) ( ) 1 kk k k H Jx x x H x + −=
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