intro to simulation
TRANSCRIPT
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Introduction to Simulation - Lecture 19
Thanks to Deepak Ramaswamy, Michal Rewienski,and Karen Veroy, Jaime Peraire and Tony Patera
Laplaces Equation FEM Methods
Jacob White
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Outline for Poisson Equation Section
Why Study Poissons 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
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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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Drag Force Analysis
of Aircraft
Potential Flow Equations Poisson Partial Differential Equations.
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Engine Thermal
Analysis
Thermal Conduction Equations The Poisson Partial Differential Equation.
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Capacitance on a microprocessor Signal Line
Electrostatic Analysis The Laplace Partial Differential Equation.
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Heat Flow 1 D Example
( )0T ( )1T
Unit Length Rod
Near EndTemperature
Far End
Temperature
Question: What is the temperature distribution along the bar
x
T
Incoming Heat
( )0T ( )1T
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Heat Flow 1 D Example
Discrete Representation
(0)T (1)T
1) Cut the bar into short sections
1T 2T N T 1 N T
2) Assign each cut a temperature
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Heat Flow 1 D Example
Constitutive Relation
iT
Heat Flow through one section
1iT + x
1,i ih +
11, heat flow
i ii i
T T h
x ++
= =
0
( )lim ( ) x
T xh x x
=
Limit as the sections become vanishingly small
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Heat Flow 1 D Example
Conservation Law
1iT
Two Adjacent Sections
iT
1iT +
1,i ih +
, 1i ih x
control volume
1, , 1i i i sih hh x+ = Heat Flows into Control Volume Sums to zero
Incoming Heat ( ) sh
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Heat Flow 1 D Example
Conservation Law
1, , 1i i i sih hh x+ =
Heat Flows into Control Volume Sums to zero
1iT iT 1iT +1,i ih +, 1i ih x
Incom ing H eat ( ) sh
0 ( ) ( )lim ( ) x s h x T xh x x x x
= =
Limit as the sections become vanishingly small
Heat infrom left
Heat outfrom right
Incomingheat perunit length
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Heat Flow 1 D Example
Circuit Analogy
+-
+-
1 R x
=
s si xh= (0) sv T = (1) sv T =
Temperature analogous to VoltageHeat Flow analogous to Current
1T N T
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1 D Example
Normalized 1 D Equation
Normalized Poisson Equation
2
2
( ) ( )( ) s
T x u xh f x
x x x
= =
Heat Flow
( ) ( ) xxu x f x =
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Residual EquationUsing BasisFunctions
2
2
u f x
=
Partial Differential Equation form
(0) 0 (1) 0u u= =Basis Function Representation
Plug Basis Function Representation into the Equation
( ) ( ) ( )2
21
n
iii
d x R x f xdx
== +
( ) ( ) ( ){1 Basis Functions
n
h i ii
u x u x x =
=
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Using BasisFunctions
( ) is a weighted sum of basis func o s nu tih xThe basis functions define a space
1
2
3 5
4 6 Hat basis functions Piecewise linear Space
Example1
| for some 'sn
h h i i ii
X v X v =
= =
( ) ( ) ( ){1 Basis Functions
Introduce basis representationn
h i ii
u x u x x =
=
Example Basis functions
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Basis WeightsUsing Basisfunctions
Galerkin Scheme
Generates n equations in n unknowns
( ) ( ) ( )21
210
0n
il i
i
d x x f x dxdx
= + =
Force the residual to be orthogonal to the basis functions
{ }1,...,l n
( ) ( )1
0
0l x R x dt =
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Basis WeightsUsing BasisFunctions Galerkin with integration by
parts
Only first derivatives of basis functions
( ) ( ) ( ) ( )11 1
0 0
0
n
i iil
i xd d x dx x f x dx
dx dx
= =
{ }1,...,l n
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ConvergenceAnalysis
huuThe question is
How does decrease with refinement?h
error
u u123
This time Finite-element methods Next time Finite-difference methods
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Convergence AnalysisHeat Equation
Overview of FEM
2
2
u f x
=
Partial Differential Equation form
(0) 0 (1) 0u u= =Nearly Equivalent weak form
Introduced an abstract notation for the equation u must satisfy
for all
( , ) ( )
u vdx f v dx v
x x
a u v l v
=
142 43 14243
( , ) ( )a u v l v= for all v
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Convergence AnalysisHeat Equation
Overview of FEM
( ) is a weighted sum of basis func o s nu tih xThe basis functions define a space
1
2
3 5
4 6 Hat basis functions Piecewise linear Space
Example1
| for some 'sn
h h i i ii
X v X v =
= =
( ) ( ) ( ){1 Basis Functions
Introduce basis representationn
h i ii
u x u x x =
=
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Convergence AnalysisHeat Equation
Overview of FEM
Using the norm properties, it is possible to show
Key Idea
( , ) ( )h i ia u l If =
min
Pr h hh w X h
u u u w
ojectionSolution Error E
Then
rror
=
14243 14243
U is restricted to be 0 at 0 and1!!
{ }1 2, ,..for all .,i n
defines a nor ( , ) ( , )ma u u a u u u
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Convergence AnalysisHeat Equation
Overview of FEM
huu
The question is only
How well can you fit u with a member of X hBut you must measure the error in the ||| ||| norm
1herror
u u O n = 142 43For piecewise linear:
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Summary
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