lec19 (1)

# lec19 (1) - Introduction to Simulation Lecture 19 Laplaces...

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Introduction to Simulation - Lecture 19 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy, Jaime Peraire and Tony Patera Laplace’s Equation – FEM Methods Jacob White

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SMA-HPC ©2003 MIT Outline for Poisson Equation Section • Why Study Poisson’s equation – Heat Flow, Potential Flow, Electrostatics – Raises many issues common to solving PDEs. • Basic Numerical Techniques – basis functions (FEM) and finite-differences – Integral equation methods • Fast Methods for 3-D – Preconditioners for FEM and Finite-differences – Fast multipole techniques for integral equations
SMA-HPC ©2003 MIT Outline for Today • Why Poisson Equation – Reminder about heat conducting bar • Finite-Difference And Basis function methods – Key question of convergence • Convergence of Finite-Element methods – Key idea: solve Poisson by minimization – Demonstrate optimality in a carefully chosen norm

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SMA-HPC ©2003 MIT Drag Force Analysis of Aircraft • Potential Flow Equations – Poisson Partial Differential Equations.
SMA-HPC ©2003 MIT Engine Thermal Analysis • Thermal Conduction Equations – The Poisson Partial Differential Equation.

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SMA-HPC ©2003 MIT Capacitance on a microprocessor Signal Line • Electrostatic Analysis – The Laplace Partial Differential Equation.
Heat Flow 1-D Exam ple () 0 T 1 T Unit Length Rod Near End emperature Far End Temperature Question: What is the temperature distribution along the bar x T Incoming Heat 0 T 1 T

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Heat Flow 1-D Exam ple Discrete Representation (0) T (1) T 1) Cut the bar into short sections 1 T 2 T N T 1 N T 2) Assign each cut a temperature
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lec19 (1) - Introduction to Simulation Lecture 19 Laplaces...

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