44th AIAA Aerospace Sciences Meeting and Exhibit 9 - 12 January 2006, Reno, Nevada
AIAA 2006-896
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deterministic exact mean Gram−Schmidt PC modes
1
Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution
0
10
−2
10
0.5
L1 error
position x
mode 0 mode 2
0
−4
10
mode 3 −6
10 mode 1 −0.5 0
−8
0.2
0.4
time t
0.6
0.8
1
10
0
1
2 3 polynomial order p
4
5
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1 sensor location x=0.5→
position x
0.8
temperature T
Pe=0→
velocity u
0.6
← Pe=1
0.4 Pe=5→
0.2
Pe=10→
sensor location xsl
0 0
0.2
0.4 0.6 position x
0.8
Pe=20 Pe=∞→ 1
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0
−2
10
10
−4
10
−6
10
0
L1 error
L1 error
10
GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr.
−2
−4
10
−6
1 2 polynomial order p
3
10
1
GSPC: mean GSPC: variance GSPC: distribution APC: mean APC: variance APC: distribution 2
3
4 5 # terms (P+1)
6
7
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3
Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution
0
10
2.5 2
L1 error
polynomial chaos expansion coefficient K
p
Gram−Schmidt PC Askey PC
1.5 1
−2
10
−4
10
0.5 −6
0 0
1
2 3 polynomial order p
4
10
5
0
1
2 3 polynomial order p
4
5
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1
0.8
Gram−Schmidt PC: mean Gram−Schmidt PC: variance Gram−Schmidt PC: distribution Askey PC: mean Askey PC: variance Askey PC: distribution
0
10
0.7 −2
0.6
L1 error
probability distribution function F
K
0.9
2
10
exact Gram−Schmidt PC expansion Askey PC expansion
0.5 0.4 0.3
−4
10
p=3−4
0.2
−6
10
p=0
0.1 0 0
10
p=1−2 −8
1
2 3 spring stiffness K
4
5
10
0
1
2 3 polynomial order p
4
5
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GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr.
0
10
0.8 p=4
0.6 0.4
L1 error
probability distribution function FT
1
p=1
−2
10
−4
10
0.2
0 −0.005
−6
0
0.005
0.01 0.015 0.02 temperature T
0.025
0.03
10
1
2
3
4 5 # terms (P+1)
6
7
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10
Legendre (p=1,4) exact
0.8 p=4
−2
10
0.6
L1 error
probability distribution function FT
1
p=1 0.4
−4
10 0.2
0 −0.005
−6
0
0.005
0.01 0.015 0.02 temperature T
0.025
0.03
10
0
GSPC: mean GSPC: var. GSPC: distr. APC: mean APC: var. APC: distr. 1
2 3 polynomial order p
4
5
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L Poisseuille flow
adiabatic wall 2h
L1
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350
GSPC (p=4) MC (N=1000)
probability distribution function FT
360
temperature Tw
340 mean
330 320
±σ
Tw
310 300 290 280 0
0.02
0.04 0.06 flow direction x
0.08
0.1
1
GSPC (p=1,4) MC (N=1000)
0.8 p=1
0.6
p=4
0.4 0.2 0 320
330
340 350 360 temperature T
370
380
out
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