lec12 (1)

# lec12 (1) - Introduction to Simulation - Lecture 12 Methods...

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Introduction to Simulation - Lecture 12 Thanks to Deepak Ramaswamy, Jaime Peraire, Michal Rewienski, and Karen Veroy Methods for Ordinary Differential Equations Jacob White

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Outline Initial Value problem examples Signal propagation (circuits with capacitors). Space frame dynamics (struts and masses). Chemical reaction dynamics. Investigate the simple finite-difference methods Forward-Euler, Backward-Euler, Trap Rule. Look at the approximations and algorithms Examine properties experimentally. Analyze Convergence for Forward-Euler
SMA-HPC ©2003 MIT Signal Transmission in an Integrated Circuit Application Problems Ground Plane Signal Wire Logic Gate Logic Gate • Metal Wires carry signals from gate to gate. • How long is the signal delayed? Wire and ground plane form a capacitor Wire has resistance

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SMA-HPC ©2003 MIT capacitor resistor Application Problems Circuit Model Signal Transmission in an Integrated Circuit • Model wire resistance with resistors. • Model wire-plane capacitance with capacitors. Constructing the Model • Cut the wire into sections.
SMA-HPC ©2003 MIT Oscillations in a Space Frame Application Problems • What is the oscillation amplitude?

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Ground Bolts Struts Load Example Simplified for Illustration Application Problems Simplified Structure Oscillations in a Space Frame
Application Problems Modeling with Struts, Joints and Point Masses Oscillations in a Space Frame Point Mass Strut • Replace cargo with point mass. Constructing the Model • Replace Metal Beams with Struts.

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SMA-HPC ©2003 MIT Chemical Reaction Dynamics Application Problems • Does it explode? Crucible Reagent Strange green stuff • How fast is product produced?
SMA-HPC ©2003 MIT Application Problems A 2x2 Example Signal Transmission in an Integrated Circuit Nodal Equations Yields 2x2 System C 1 R 2 R 1 R 3 C 2 Constitutive Equations c c dv iC dt = 1 RR iv R = Conservation Laws 112 0 CRR iii + += 232 0 + −= 1 12 2 11 22 2 23 2 1 0 0 1 dv R Cv dt dv R dt ⎡⎤ +− ⎢⎥ =− ⎣⎦ −+ 1 R i 1 C i 2 R i 2 C i 3 R i 1 v 2 v

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SMA-HPC ©2003 MIT eigenvectors Application Problems A 2x2 Example Signal Transmission in an Integrated Circuit 1 12 2 11 22 2 23 2 1 0 0 1 dv RR R Cv dt dv R dt ⎡⎤ +− ⎢⎥ =− ⎣⎦ −+ 13 2 Let 1, 10, 1 CC R == = 1.1 1.0 1.0 1.1 A dx x dt = ±²²³ ² ²´ 1 0 . 10 0 2 . 1 A −− = ⎜⎟ ⎝⎠ Eigenvalues and Eigenvectors Eigenvalues