Intro to Simulation

8/13/2019 Intro to Simulation

1/63

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


2/63

SMA-HPC 2003 MIT

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


3/63

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


4/63

SMA-HPC 2003 MIT

Drag Force Analysis

of Aircraft

Potential Flow Equations Poisson Partial Differential Equations.


5/63

SMA-HPC 2003 MIT

Engine Thermal

Analysis

Thermal Conduction Equations The Poisson Partial Differential Equation.


6/63SMA-HPC 2003 MIT

Capacitance on a microprocessor Signal Line

Electrostatic Analysis The Laplace Partial Differential Equation.


7/63

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


8/63


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


9/63


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


10/63


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


11/63


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




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


13/63

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 =



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 =

=



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 =

= =


Introduce basis representationn

h i ii

u x u x x =

=

Example Basis functions



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 =



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



ConvergenceAnalysis

huuThe question is

How does decrease with refinement?h

error

u u123

This time Finite-element methods Next time Finite-difference methods



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




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 =

= =


Introduce basis representationn

h i ii

u x u x x =

=


23/63

SMA-HPC 2003 MIT


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


24/63

SMA-HPC 2003 MIT


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:


25/63

SMA-HPC 2003 MIT


26/63

SMA-HPC 2003 MIT


27/63

SMA-HPC 2003 MIT


28/63

SMA-HPC 2003 MIT


29/63

SMA-HPC 2003 MIT


30/63

SMA-HPC 2003 MIT


31/63

SMA-HPC 2003 MIT


32/63

SMA-HPC 2003 MIT


33/63

SMA-HPC 2003 MIT


34/63

SMA-HPC 2003 MIT


35/63

SMA-HPC 2003 MIT


36/63

SMA-HPC 2003 MIT


37/63

SMA-HPC 2003 MIT


38/63

SMA-HPC 2003 MIT


39/63

SMA-HPC 2003 MIT


40/63

SMA-HPC 2003 MIT


41/63

SMA-HPC 2003 MIT


42/63

SMA-HPC 2003 MIT


43/63

SMA-HPC 2003 MIT


44/63

SMA-HPC 2003 MIT


45/63

SMA-HPC 2003 MIT


46/63

SMA-HPC 2003 MIT


47/63

SMA-HPC 2003 MIT


48/63

SMA-HPC 2003 MIT


49/63

SMA-HPC 2003 MIT


50/63

SMA-HPC 2003 MIT


51/63

SMA-HPC 2003 MIT


52/63

SMA-HPC 2003 MIT


53/63

SMA-HPC 2003 MIT


54/63

SMA-HPC 2003 MIT


55/63

SMA-HPC 2003 MIT


56/63

SMA-HPC 2003 MIT


57/63

SMA-HPC 2003 MIT


58/63

SMA-HPC 2003 MIT


59/63

SMA-HPC 2003 MIT


60/63

SMA-HPC 2003 MIT


61/63

SMA-HPC 2003 MIT


62/63

SMA-HPC 2003 MIT


63/63

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

Documents

Intro to Simulation