Post on 24-May-2020
EFFECT OF TOTAL MODELING UNCERTAINTYON THE ACCURACY OF NUMERICAL SIMULATIONS
by
Timothy K. HasselmanACTA Incorporated
presented at the
IMA “Hot Topics” WorkshopDecision making Under Uncertainty:
Assessment of the Reliability of Mathematical Models Institute for Mathematics and its Applications
University of Minnesota
September 16-17, 1999
IMA Workshop Sep 99/2 of 25
OVERVIEW OF PRESENTATION
❑ Total Modeling Uncertainty - Definition
❑ A General Approach Suitable for All Simulations
❑ Specialization - Linear Structural Dynamics
❑ Quantification of Total Modeling Uncertainty - Examples• Frequency uncertainty• Mode shape uncertainty• Mass and stiffness uncertainty• Uncertainty expressed in terms of generalized “modal” metrics
❑ Accuracy of Numerical Simulations - Examples• Frequency response of a truss beam• Airblast response of a reinforced concrete wall
IMA Workshop Sep 99/3 of 25
TOTAL MODELING UNCERTAINTY
q Truth Assumption
That which we observe in the physical world
q Simulation Uncertainty
Statistical representation of the difference between simulated and observedphenomena, e.g. a covariance matrix of a discrete time series or frequencyresponse function (FRF).
q Total Modeling Uncertainty
Statistical representation of the difference between simulated and observedphenomena expressed in terms of generic, dimensionless metrics that may beaveraged over a family of generically similar phenomena, e.g. ratio of measuredto predicted natural frequency; normalized analysis-test eigenvector products.
IMA Workshop Sep 99/4 of 25
SOURCES OF VARIABILITY / UNCERTAINTY
q Test Article or Product Variability• Tolerances• Material properties• Construction
q Experimental Variability• Variability in experimental setup• Measurement variability• Variability in data processing
q Modeling Variability• Variability in modeling assumptions - parameterization• Parametric variability• Computational variability
q Total Modeling Uncertainty - All of the Above
IMA Workshop Sep 99/5 of 25
GENERAL APPROACH
q Observed Phenomena
q Corresponding Numerical Simulation
q Singular Value Decomposition
Y t
y t
y t
y t
y t y t y t
y t y t y t
y t y t y t
i
n
m
m
n n n m
0 50 50 5
0 5
0 5 0 5 0 50 5 0 5 0 5
0 5 0 5 0 5=
%&KK
'KK
()KK
*KK
=
�
!
"
$
####2
1 1 1 2 1
2 1 2 2 2
1 2
M
LL
M M ML
, ,
, ,
, ,
0
0
0
2
0
0
1 1
0
1 2
0
1
0
2 1
0
2 2
0
2
0
1
0
2
0
Y t
y t
y t
y t
y t y t y t
y t y t y t
y t y t y t
i
n
m
m
n n n m
0 50 50 5
0 5
0 5 0 5 0 50 5 0 5 0 5
0 5 0 5 0 5=
%&KK
'KK
()KK
*KK
=
�
!
"
$
####M
LL
M M ML
, ,
, ,
, ,
Y t U V D
Y t U V D
T
T
0 50 5
= == =
ΣΣ
φ ηφ η0 0 0 0 0 0 0
IMA Workshop Sep 99/6 of 25
PRINCIPAL COMPONENTS ANALYSIS
q Singular Values and Singular (Eigen) Vectors• Singular value (from the diagonal matrix, D)• Left (column) eigenvector• Right (row) eigenvector
q Orthonormal properties of Eigenvectors
q Principal Components : Energy-ordered “Modes”
Dii
=φ
i=
ηi=
YY D D D
y t Tr YY Tr D D
T T T T
i
n
i j
T T
iii
n
j
m
= =
= = =∑ ∑∑
φ η η φ φ φ
φ φ
0 51 62 7 1 6 1 6
2
2 2 2
φ φ δ η η δφ φ δ η η δ
i
T
j ij i j
T
ij
i
T
j ij i j
T
ij
= == =0 0 0 0
IMA Workshop Sep 99/7 of 25
PC METRICS FOR QUANTIFYING TOTAL MODELING UNCERTAINTY
q Express and as linear combinations of and
q Evaluate differences between observation and simulation
q Vectorize
φi’s 0η
i’s
φ φ ψ φ φ ψ
η ν η ηη ν
≅ =
≅ =
0 0
0 0
1 60 5
:
:
T
T
∆
∆
∆
∆
~ ~r
vec
vec D
vec
=
%&K
'K
()K
*K
ψ
ν
0 52 70 5
∆ ∆ ∆ψ ν, ~ ,D
∆ ∆
∆ ∆ ∆∆ ∆
φ φ φ φ ψ φ ψ
η η η ν η ν η
= − ≅ − =
= − == − ≅ − =
−
0 0 0
0 0 1
0 0 0
I
D D D D D D
I
0 5
0 5; ~
ηi’s 0φ
i’s
IMA Workshop Sep 99/8 of 25
EVALUATION OF TOTAL MODELING UNCERTAINTY
q Using Replicate Test Data for a Particular Test Article
q Using Test Data from Generically Similar Test Articles
S r rN
r r
rN
r
r r r r
T
ri
N
r
T
ri
N
~~ ~ ~ ~ ~
~
~ ~ ~ ~
~ ~
= − − =−
− −
= =
=
=
∑
∑
Ε ∆ ∆ ∆ ∆
Ε ∆ ∆
∆ ∆ ∆ ∆
∆
µ µ µ µ
µ
0 50 5 0 50 511
11
1
(Systematic Error)
S r rN
r rr r
T
i
NT
r
~~
~
~ ~ ~ ~= ==∑Ε ∆ ∆ ∆ ∆
∆
1 6 11
µ is assumed to be zero
IMA Workshop Sep 99/9 of 25
FAST RUNNING SIMULATION
q Recall Singular Value Decomposition
q Recall Difference Analysis
q Fast Running Simulation
φ φ φ φ ψ
η η η η ν
= + ≅ +
= + = +
= + ≅ +
0 0
0 0
0 0
∆ ∆
∆ ∆
∆ ∆
I
D D D D I D
I
0 52 70 5
~
Y t D
y t D t Di j
k
p
ik kk k jk
p
ik kk kj
0 52 7 2 7
=
= == =
∑ ∑
φ η
φ η φ η1 1
$ ~Y I D I D Iij
= + + +0 0 0φ ψ η ν∆ ∆ ∆0 5 2 7 0 5
IMA Workshop Sep 99/10 of 25
EVALUATING THE ACCURACY OFNUMERICAL SIMULATIONS
q Linear Covariance Propagation
q Interval Propagation - Vertex Method
q Monte Carlo Simulation• Diagonalize• Execute PC Model for randomly selected values of ∆s
∆Y Y c Y e Y c Y e
c
e
i j i j i j i j
i
j
= −
==
max , min ,, ,
0 5 2 7 0 5 2 7J Lcoordinates of vertices of rectangular hyperspace
coordinates of extrema within rectangular hyperspace
S S T S TYY YY Yr r r Yr
T≅ =$ $ $~ ~~
$~
T Y rYr$~
$ / ~= ∂ ∂
S rr r~~ : ~∆ Θ∆= s
IMA Workshop Sep 99/11 of 25
SPECIALIZATION TO LINEAR STRUCTURAL DYNAMICS
q Classical Normal Mode Analysis
q Modal Mass and Stiffness Matrices
q Assumed “True” Modal Mass and Stiffness Matrices
m m m I m
k k k k
= + = += + = +
0
0 0
∆ ∆∆ ∆λ
0 0 0 0
0 0 0 0 0
m M I
k K
T
T
= == =
φ φφ φ λ
0 0 0 0
0
0
0K M
M
k
j j
j
j
− =
=
===
λ φλφ
2 7Analytical eigenvalues
Analytical Eigenvectors
Analytical mass matrix
Analytical stiffness matrix
0
0
IMA Workshop Sep 99/12 of 25
MODAL METRICS FOR QUANTIFYINGMODELING UNCERTAINTY
q Normalization of Test Modes,
q Cross-orthogonality of Analysis and Test Modes
q Difference Between Analysis and Test Modes
q Modal Metrics for Quantifying Total Modeling Uncertainty
∆ ∆ ∆
∆ ∆ ∆ ∆
m
k
T
T
= − +
= − −− −
ψ ψ
λ λ λ ψ ψ λ λ
1 6
1 6~ / /0 1 2 0 0 0 1 2
φ
φ φj
T
jM0 1=
0 0 0 0φ φ φ φψ ψj
T
j j
TM M= =
∆∆ ∆
λ λ λφ φ φ φ ψ φ ψ
= −= − = − =
0
0 0 0I0 5
∆
∆
∆
∆
~ ~r
vec m
vec k
vec
=
%&K
'K
()K
*K
0 52 70 5ζ
IMA Workshop Sep 99/13 of 25
COMPARISON OF FAST-RUNNING SIMULATIONS
q General Approach
where are the principal component metrics corresponding to theenergy-ordered modes of a SVD.
q Linear Structural Dynamics Approach
where are the classical modal metrics from a real symmetriceigenvalue/eigenvector analysis.
y t D ti j ik kk k j
k
p2 7 2 7==
∑φ η1
y t ti j ik kk k j
k
p2 7 2 7==
∑φ ηΓ1
φ η, D and
φ η, Γ and
φ
η
k
kk
kt
==
=
Classical normal mode
Modal participation factor
Modal response time history
Γ0 5
IMA Workshop Sep 99/14 of 25
NASA / LaRC EIGHT BAY TRUSS
Fixed end
Free end
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Force input at Nodes 4 and 7
Acceleration response measured atNodes 1 through 32
IMA Workshop Sep 99/15 of 25
EXAMPLE OF TOTAL MODELING UNCERTAINTYCLASSICAL MODAL ANALYSIS
Normalized Frequency
0
0.02
0.04
0.06
0.08
0.1
1 2 3 4 5
Mode No.
RM
S E
rro
r
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Eigenvector Cross-orthogonality
Normalized Frequency
0
0.02
0.04
0.06
0.08
0.1
1 2 3 4 5
Mode No.
RM
S E
rro
r
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Eigenvector Cross-orthogonality
Variations of an 8-bay truss beam Generic family of space structures
IMA Workshop Sep 99/16 of 25
TOTAL MODELING UNCERTAINTY IN TERMS OFMODAL MASS AND STIFFNESS VARIABILITY
Variations of an 8-bay truss beam Generic family of space structures
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Modal Mass
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Normalized Modal Siffness
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Modal Mass
54
32
1 12
34
50
0.1
0.2
0.3
0.4
0.5
RM
S E
rro
r
Mode No.Mode No.
Normalized Modal Stiffness
IMA Workshop Sep 99/17 of 25
FRF SIMULATION ACCURACY BASED ONSTRUCTURE-SPECIFIC MODELING UNCERTAINTY
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 20 X−Acceleration / 4 X−Force
ModelModel +/− 2 σTest
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 4 X−Acceleration / 4 X−Force
ModelModel +/− 2 σTest
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 4 Z−Acceleration / 4 X−Force
ModelModel +/− 2 σTest
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 20 Z−Acceleration / 4 X−Force
ModelModel +/− 2 σTest
IMA Workshop Sep 99/18 of 25
COMPARISON OF SIMULATION ACCURACY FORSPECIFIC AND GENERIC MODELING UNCERTAINTY
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 4 Z−Acceleration / 4 X−Force
ModelModel +/− 1 σ SpecificModel +/− 1 σ Generic
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 20 X−Acceleration / 4 X−Force
ModelModel +/− 1 σ SpecificModel +/− 1 σ Generic
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 4 X−Acceleration / 4 X−Force
ModelModel +/− 1 σ SpecificModel +/− 1 σ Generic
101
102
100
101
102
103
104
Frequency (Hz)
Acc
eler
atio
n/F
orce
Node 20 Z−Acceleration / 4 X−Force
ModelModel +/− 1 σ SpecificModel +/− 1 σ Generic
IMA Workshop Sep 99/19 of 25
BLAST RESPONSE OF A BURIED R/C STRUCTURE
Finite Element Model
❑ ~ 80,000 continuum elements
(concrete and soil)
❑ ~ 20,000 rebar elements
Test Measurements
❑ 12 Pressure gages
(10 on interior walls)
❑ 12 Accelerometers on interior walls
❑ 5 Locations each wall
IMA Workshop Sep 99/20 of 25
EXAMPLE OF TOTAL MODELING UNCERTAINTYPRINCIPAL COMPONENTS ANALYSIS
Typical Singular Values
01020304050607080
1 2 3 4
Mode Number
Sin
gu
lar
Val
ue Test
Analysis
Normalized Singular Values
0
0.5
1
1.5
2
2.5
1 2 3 4
Mode Number
RM
S E
rro
r
12
34
12
34
0
0.2
0.4
0.6
0.8
1
RMS Error
Mode NumberMode Number
Right Eigenvector Cross-Orthogonality
12
34
12
34
0
0.2
0.4
0.6
0.8
1
1.2
1.4
RMS Error
Mode NumberMode Number
Left Eigenvector Cross-Orthogonality
IMA Workshop Sep 99/21 of 25
DISPLACEMENT SIMULATION UNCERTAINTY FOR R/C WALLPRE-UPDATE MODEL OF STRUCTURE 1
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
Left Horizontal Quarter Point Upper Vertical Quarter Point
Center Point Lower Vertical Quarter Point
IMA Workshop Sep 99/22 of 25
DISPLACEMENT SIMULATION UNCERTAINTY FOR R/C WALLPOST-UPDATE MODEL OF STRUCTURE 1
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04−1
0
1
2
3
4
5
6
7
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
Left Horizontal Quarter Point Upper Vertical Quarter Point
Center Point Lower Vertical Quarter Point
IMA Workshop Sep 99/23 of 25
DISPLACEMENT SIMULATION UNCERTAINTY FOR R/C WALLPRE-UPDATE MODEL OF STRUCTURE 2
0 0.01 0.02 0.03 0.04
0
2
4
6
8
10
12
14
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04
0
2
4
6
8
10
12
14
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04
0
2
4
6
8
10
12
14
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
0 0.01 0.02 0.03 0.04
0
2
4
6
8
10
12
14
Time (sec)
Dis
plac
emen
t (in
)
Model Model +/− 1 σTest
Gage 1 Gage 2
Gage 3 Gage 4
IMA Workshop Sep 99/24 of 25
SUMMARY
q Why total modeling uncertainty? The predictive accuracy of numerical simulations can be reliably estimated
only to the extent that total modeling uncertainty is quantified.
q How does one obtain the necessary data?By decomposing corresponding sets of simulated and measured response intotheir principal components, and statistically processing the differencesbetween corresponding PC (modal) metrics.
q How is the predictive accuracy of numerical simulations derived fromtotal modeling uncertainty?ì By first order covariance propagation,ì By interval propagation, and/orì By Monte Carlo simulation
IMA Workshop Sep 99/25 of 25
CONCLUSIONS
q Numerical simulations are increasingly relied upon for making criticaldecisions.
q The reliability of these simulations depends on uncertainty inherent in theunderlying mathematical models.
q A true representation of this uncertainty must include all possible sourcesof uncertainty, i.e. total modeling uncertainty.
q Decision making under uncertainty demands that total modelinguncertainty be quantified realistically.