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

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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 00 (, ) xy 11 ) Given: x0, y0, x1, y1, W Find: x2, y2 22 ) Load force W 0 y fW . ¦ Need to Solve 0 x f ¦

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SMA-HPC ©2003 MIT Nonlinear Problems Struts Example Reminder: Strut Forces 11 , xy Y +, 0 0 0 1 1 22 c x y LL fE A L L L x ff L y L Lx y H 0 0 . L y f f x f (0,0) X
Nonlinear problems Strut Example 20 11 1 () xo xx f LL L H 0 0 +, + , 22 12 0 2 0 Lx x y y 0 . 0 Load force W 00 (, ) xy ) + , 1 y 0 . 0 21 2 f L 0 0 ) 1 f 2 f 0 ff . ¦ 0 yy W .. ¦

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Nonlinear problems Strut Example Why Nonlinear? 21 2 2 () o yy LL L H 0 0. 20 1 1 0 o W L 0 The strut forces change in both magnitude and direction Pull Hard on the Struts
Circuit Example Nonlinear problems Need to Solve 0 dr II . + - - + (1 ) 0 d t V V ds e 00 10v 10 d V 1 v 2 v 1 0 10 rr IV 0 0 src vr

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Solve Iteratively Nonlinear problems Hard to find analytical solution for () 0 fx Solve iteratively guess at a solution repeat for k = 0, 1, 2, …. until Ask • Does the iteration converge to correct solution ?
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## This note was uploaded on 11/08/2011 for the course AERO 16.872 taught by Professor Danielhastings during the Fall '03 term at MIT.

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

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