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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
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
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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
SMA-HPC ©2003 MIT
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.
Heat Flow 1-D Example
( )0T ( )1T
Unit Length RodNear End
TemperatureFar End
Temperature
Question: What is the temperature distribution along the bar
x
T
Incoming Heat
( )0T ( )1T
Heat Flow 1-D Example
Discrete Representation
(0)T (1)T
1) Cut the bar into short sections
1T 2T NT1NT −
2) Assign each cut a temperature
Heat Flow 1-D Example
Constitutive Relation
iT
Heat Flow through one section
1iT +
x∆
1,i ih +
11, heat flow i i
i iT Th
xκ +
+−
= =∆
0( )lim ( )xT xh xx
κ∆ →∂
=∂
Limit as the sections become vanishingly small
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
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 sh x T xh xx x x
κ∆ →∂ ∂ ∂
= =∂ ∂ ∂
Limit as the sections become vanishingly small
Heat in from left
Heat out from right
Incoming heat per unit length
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Heat Flow 1-D Example
Circuit Analogy
+-
+-
1R x
κ=∆
ssi xh= ∆(0)sv T= (1)sv T=
Temperature analogous to VoltageHeat Flow analogous to Current
1T NT
1-D Example
Normalized 1-D Equation
Normalized Poisson Equation
2
2
( ) ( ) ( )sT x u xh f x
x x xκ∂ ∂ ∂
= − ⇒ − =∂ ∂ ∂
Heat Flow
( ) ( )xxu x f x− =
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Residual EquationUsing Basis Functions
2
2
u fx∂
− =∂
Partial Differential Equation form
(0) 0 (1) 0u u= =
Basis Function Representation
Plug Basis Function Representation into the Equation
( ) ( ) ( )2
21
ni
ii
d xR x f x
dxϕ
ω=
= +∑
( ) ( ) ( )1 Basis Functions
n
h i ii
u x u x xω ϕ=
≈ =∑
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Using Basis Functions
( ) is a weighted sum of basis func o s nu tih x⇒
The basis functions define a space
1ϕ
2ϕ
3ϕ 5ϕ
4ϕ 6ϕ“Hat” basis functions Piecewise linear Space
Example1
| for some 's n
h h i i ii
X v X v β ϕ β=
= ∈ =
∑
( ) ( ) ( )1 Basis Functions
Introduce basis representation n
h i ii
u x u x xω ϕ=
≈ =∑
Example Basis functions
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Basis WeightsUsing Basis functions
Galerkin Scheme
Generates n equations in n unknowns
( ) ( ) ( )21
210
0n
il i
i
d xx f x dx
dxϕ
ϕ ω=
+ =
∑∫
Force the residual to be “orthogonal” to the basis functions
1,...,l n∈
( ) ( )1
0
0l x R x dtϕ =∫
SMA-HPC ©2003 MIT
Basis WeightsUsing Basis Functions Galerkin with integration by
parts
Only first derivatives of basis functions
( ) ( )( ) ( )1
1 1
0 0
0
n
i iil
i
xdd xdx x f x dx
dx dx
ωϕϕϕ= − =
∑∫ ∫
1,...,l n∈
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Convergence Analysis
huuThe question is
How does decrease with refinement?h
error
u u−123
• This time – Finite-element methods• Next time – Finite-difference methods
SMA-HPC ©2003 MIT
Convergence AnalysisHeat Equation
Overview of FEM
2
2
u fx∂
− =∂
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 v dx f v dx vx xa u v l v
Ω Ω
∂ ∂=
∂ ∂∫ ∫14243 14243
( , ) ( )a u v l v= for all v
SMA-HPC ©2003 MIT
Convergence AnalysisHeat Equation
Overview of FEM
( ) is a weighted sum of basis func o s nu tih x⇒
The basis functions define a space
1ϕ
2ϕ
3ϕ 5ϕ
4ϕ 6ϕ“Hat” basis functions Piecewise linear Space
Example1
| for some 's n
h h i i ii
X v X v β ϕ β=
= ∈ =
∑
( ) ( ) ( )1 Basis Functions
Introduce basis representation n
h i ii
u x u x xω ϕ=
≈ =∑
SMA-HPC ©2003 MIT
Convergence AnalysisHeat Equation
Overview of FEM
Using the norm properties, it is possible to show
Key Idea
( , ) ( )h i ia u lIf ϕ ϕ=
min
Prh hh w X hu u u w
ojectionSolutionError E
Then
rror
∈− = −14243 14243
U is restricted to be 0 at 0 and1!!
1 2, ,..for all .,i nϕ ϕ ϕ ϕ∈
defines a nor( , ) ( , )ma u u a u u u≡
SMA-HPC ©2003 MIT
Convergence AnalysisHeat Equation
Overview of FEM
huu
The question is only
How well can you fit u with a member of XhBut you must measure the error in the ||| ||| norm
1h
error
u u On
− = 14243
For piecewise linear:
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
SMA-HPC ©2003 MIT
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
