lec9 - Introduction to Simulation - Lecture 9...

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Introduction to Simulation - Lecture 9 Multidimensional Newton Methods Jacob White Thanks to Deepak Ramaswamy, Jaime Peraire, Michal Rewienski, and Karen Veroy

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Outline • Quick Review of 1-D Newton – Convergence Testing • Multidimensonal Newton Method – Basic Algorithm – Description of the Jacobian. – Equation formulation. • Multidimensional Convergence Properties – Prove local convergence – Improving convergence
SMA-HPC ©2003 MIT Newton Idea 1-D Reminder ( ) ** Problem: Find such tha 0 t xf x = Use a Taylor Series Expansion () ( ) ( ) 2 2 * 2 fx f x f xx x x x ∂∂ =+ + ± 0 If x is close to the exact solution ( ) * f x x −≈

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SMA-HPC ©2003 MIT Newton Algorithm 1-D Reminder 0 Initial Guess, = 0 xk = Repeat { () ( ) 1 k kk k fx xx f x x + −= 1 =+ } Until ? ( ) 1 1 ? ? k f x threshold x x threshold + + < −<
SMA-HPC ©2003 MIT Newton Algorithm 1-D Reminder Algorithm Picture

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SMA-HPC ©2003 MIT Newton Algorithm 1-D Reminder Convergence Checks Need a "delta-x" check to avoid false convergence X f(x) 1 k x + k x * x () 1 a k f fx ε + < 11 ar kk k xx x εε + + −>+
SMA-HPC ©2003 MIT Newton Algorithm 1-D Reminder Convergence Checks ( ) Also need an " " check to avoid false convergence fx X f(x) 1 k x + k x * x ( ) 1 a k f ε + > 11 ar kk k xx x εε + + −<+

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SMA-HPC ©2003 MIT Newton Algorithm 1-D Reminder Local Convergence Convergence Depends on a Good Initial Guess X f(x) 0 x 1 x 2 x 0 x 1 x
SMA-HPC ©2003 MIT Example Problem Multidimensional Newton Method Strut and Joint 22 () o co o lxy ll FE A l ε =+ == xo yo xx f Fl l yy f l F L F 0 0 x y xL yL fF + = + = ( ) Fx = G 0 0 x y oL x llF l y l += OR

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SMA-HPC ©2003 MIT Example Problem Multidimensional Newton Method Nonlinear Resistors Nodal Analysis 1 b v + 1 i 2 i 2 b v 3 i +- 3 b v + 12 At Node 1: 0 ii + = - Nonlinear Resistors () ig v = 1 v 2 v ( ) ( ) 11 2 0 gv gv v ⇒+− = 32 At Node 2: 0 = ( ) ( ) 31 2 0 ⇒−− = - Two coupled nonlinear equations in two unknowns