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# lec8a - Introduction to Simulation Lecture 8 1-D Nonlinear...

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Introduction to Simulation - Lecture 8 1-D Nonlinear Solution Methods Jacob White Thanks to Deepak Ramaswamy Jaime Peraire, Michal Rewienski, and Karen Veroy

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Outline Nonlinear Problems Struts and Circuit Example Richardson and Linear Convergence Simple Linear Example Newton’s Method Derivation of Newton Quadratic Convergence – Examples Global Convergence Convergence Checks
Nonlinear problems Strut Example 0 0 ( , ) x y 1 1 ( , ) x y Given: x0, y0, x1, y1, W Find: x2, y2 2 2 ( , ) x y Load force W 0 y f W . ¦ Need to Solve 0 x f ¦

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SMA-HPC ©2003 MIT Nonlinear Problems Struts Example Reminder: Strut Forces 1 1 , x y Y + , 0 0 0 1 1 2 2 1 1 c x y L L f EA L L L x f f L y f f L L x y H 0 0 . L y f f x f (0,0) X
Nonlinear problems Strut Example 2 0 1 1 1 ( ) x o x x f L L L H 0 0 + , + , 2 2 1 2 0 2 0 L x x y y 0 . 0 Load force W 0 0 ( , ) x y 1 1 ( , ) x y + , + , 2 2 2 2 1 2 1 L x x y y 0 . 0 2 1 2 2 2 ( ) x o x x f L L L H 0 0 2 2 ( , ) x y 1 f 2 f 1 2 0 x x f f . ¦ 1 2 0 y y f f W . . ¦

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Nonlinear problems Strut Example Why Nonlinear? 2 1 2 2 ( ) o y y L L L H 0 0 . 2 0 1 1 ( ) 0 o y y L L W L H 0 0 . The strut forces change in both magnitude and direction Pull Hard on the Struts
Circuit Example Nonlinear problems Need to Solve 0 d r I I . + - - + ( 1) 0 d t V V d s I I e 0 0 10v 10 d V 1 v 2 v 1 0 10 r r I V 0 0 src v r I I 0

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Solve Iteratively Nonlinear problems Hard to find analytical solution for ( ) 0 f x Solve iteratively guess at a solution repeat for k = 0, 1, 2, ….
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• Fall '03
• DanielHastings
• Convergence, Tier One, Scaled Composites White Knight, 2004 in spaceflight, Sub-orbital spaceflight, SpaceShipOne flight 16P

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lec8a - Introduction to Simulation Lecture 8 1-D Nonlinear...

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