lec03_Nodal_slides

lec03_Nodal_slides - Introduction to Simulation - Lecture 3...

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1 1 Introduction to Simulation - Lecture 3 Thanks to Jacob White Deepak Ramaswamy, Michal Rewienski, Karen Veroy Equation Formulation - Nodal Analysis Luca Daniel 2 Outline of today’s lecture • Matrix Construction From Schematics – Node-Branch “Stamping Procedure” • Circuit / flow / heat example • Struts and Joints – Nodal “Stamping Procedure” • Struts and Joints • Circuit / flow / heat example – Comparing Node-Branch vs. Nodal • Solution of Linear Systems – Existence and Uniqueness – Gaussian Elimination Basics (LU decomposition) 3 Two Struts Aligned with the X axis Conservation Law 12 At node 1: 0 xx ff += 2 At node 2: - 0 xL 1 f 11 xy 22 2 f L f Struts Example Nodal Formulation 4 Two Struts Aligned with the X axis Constitutive Equations () 20 1 2 x f Lxx ε =− 1 10 1 1 0 0 0 x x fL x x 1 f 2 f L f ) , ( * * * y x r = r ) , ( y x r = r ( ) r r L r r r r f r r r r r r r = * 0 * * * Nodal Formulation Struts Example 5 Two Struts Aligned with the X axis Reduced (Nodal) Equations 2 01 012 2 2 0 x x Lx L x x f x εε −+ −= 14444244443 2 0 x L x f f −− + = 0 2 1 = + x x f f 0 2 = + L x f f Nodal Formulation Struts Example 6 Outline of today’s lecture • Matrix Construction From Schematics – Node-Branch “Stamping Procedure” • Circuit • Struts and Joints – Nodal “Stamping Procedure” • Struts and Joints • Circuit – Comparing Node-Branch vs. Nodal • Solution of Linear Systems – Existence and Uniqueness – Gaussian Elimination Basics (LU decomposition)
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2 7 1) Number the nodes with one node as 0. 2) Write a conservation law at each node. except (0) in terms of the node voltages ! Nodal Formulation Generating Matrices Circuit Example 0 sA i sB i sC i () 0 1 1 2 1 2 = + + + V R V V R i i E B sC sB 0 1 4 3 = sC C i V V R 1 V 2 V 3 V 4 V E R A R B R C R D R 0 1 1 1 2 1 = + + V R V V R i A B sA 8 1 2 3 4 s I G v v v v ⎡⎤ ⎤⎡ ⎢⎥ ⎥⎢ = ⎦⎣ ⎣⎦ 1444424444 31 4 2 4 3 Nodal Formulation Generating Matrices Circuit Example One row per node, one column per node. For each resistor R 1 n 2 n sA i 0 1 2 3 sB i sA i sB i + sC i sC i E R A R B R C R D R sA i sB i sC i A R 1 B R 1 + B R 1 B R 1 B R 1 C R 1 C R 1 C R 1 C R 1 + E R 1 + D R 1 + 9 2 2 11 22 2 2 2 1 1 if ( 0) &( 0) else if ( 0) else , (,) (,) 1 1 Gn n RR R n n n R n Gn = + =+ = > + > > Nodal Matrix Stamping Algorithm Nodal Formulation Generating Matrices Circuit Example R n n G n n G 1 ) 1 , 1 ( ) 1 , 1 ( + = R n n G n n G 1 ) 2 , 1 ( ) 2 , 1 ( = R n n G n n G 1 ) 1 , 2 ( ) 1 , 2 ( = 10 N I V G N s n = b 2 2 jL JG u F J = b (Struts and Joints) (Resistor Networks) Nodal Formulation Generating Matrices 11 Outline of today’s lecture • Matrix Construction From Schematics – Node-Branch “Stamping Procedure” • Circuit • Struts and Joints – Nodal “Stamping Procedure” • Struts and Joints • Circuit – Comparing Node-Branch vs. Nodal • Solution of Linear Systems – Existence and Uniqueness – Gaussian Elimination Basics (LU decomposition) 12 1 V 2 V 3 V 4 V 99 V 100 V 101 V 102 V 103 V 200 V 901 V 902 V 903 V 1000
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This note was uploaded on 09/26/2010 for the course MECHANICAL 2.001 taught by Professor Prof.carollivermore during the Spring '06 term at MIT.

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lec03_Nodal_slides - Introduction to Simulation - Lecture 3...

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