lec12 (1)

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

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Simulation - Lecture 12 Thanks to Deepak Ramaswamy, Jaime Peraire, Michal Rewienski, and Karen Veroy Methods for Ordinary Differential Equations Jacob White
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 4
SMA-HPC ©2003 MIT Oscillations in a Space Frame Application Problems • What is the oscillation amplitude?
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Ground Bolts Struts Load Example Simplified for Illustration Application Problems Simplified Structure Oscillations in a Space Frame
Background image of page 6
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.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SMA-HPC ©2003 MIT Chemical Reaction Dynamics Application Problems • Does it explode? Crucible Reagent Strange green stuff • How fast is product produced?
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 10