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SMA-HPC ©1999 MIT

Laplace’s Equation in 3-D

Basis Function Approach

( )1

,

1i

i

n

c jj c

i j

panel j

x dSx x

A

α=

′Ψ =′−" !

"###$###%

( )

( )

11,1 1, 1

,1 ,n

cn

n n n n c

xA A

A A x

α

α

# $Ψ# $ # $ % &% & % & % &% & % & = % &% & % & % &% & % & % &% &% & Ψ' (' ( ' (

& &

' ( ' ''

' ( ' ''

& &

Put collocation points atpanel centroids

icx Collocationpoint

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Basis Function Approach

Panel j

icx Collocationpoint

,

1

i

i j

pa cnel j x xA dS ′

′−= !

,

i jc centr

j

id

i

o

Panel Area

x xA

−≈One point

quadratureApproximation

x

yz

t

4

,1 in

0.25*

i jc o

i jj p

Ar a

x xA

e

= −≈"

Four point quadrature

Approximation

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Normalized 1-D Problem

Basis Function Approach Collocation Discretization of

1-D Equation

( ) ( ) ( )1

0

,x g x x x dSσ′ ′ ′Ψ = " [ ]0,1x ∈

0 0x = 1nx =1x 1nx −2x

1σ nσ

1cx

2cxncx

( ) ( )1

1

,

to be evaluated

j

i

jx

j

x

n

cj

g x x dx Sσ−

=

Ψ ′ ′=! "!""#""$

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½

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Normalized 1-D Problem

Simple Quadrature Scheme

( )f x

x0 11

2

Area under the curve is

approximated by a rectangle

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Problem

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Use different polynomials

( ) ( )1

0 010

n

i ii

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i ii

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Each polynomial musticContain an termix

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Problem

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Normalized 1-D Problem

General Quadrature Scheme

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Normalized 1-D Problem

Simple Quadrature Scheme

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Gauss-Quad

Simple Quad

Number of Points

Error

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1

1

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Symmetrically Normalized 1-D Problem

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Evaluation Point Placement

-1 1

Simple-Quadrature Points

0

Gauss-Quadrature Points0-1 1

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Equation

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,

0

i i

i i

c c

Panel AreaA

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One point quadrature

Approximation

x

yz

, is an integrable singularity1

i

i i

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Basis Function Approach

,

1

i

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Symmetrized 1-D Exmaple

Note no 0ix =1

x

Quad Point

x

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