effect of total modeling uncertainty on the accuracy of numerical simulations · 1999-09-23 ·...

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


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.


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


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


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


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


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


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


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


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


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


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


NASA / LaRC EIGHT BAY TRUSS

Fixed end

Free end

�

�

�

�

�

�

�

�

�

��

��

��

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��

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��

��

��

��

��

��

��

��

��

��

;

<

=

Force input at Nodes 4 and 7

Acceleration response measured atNodes 1 through 32


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


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


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



101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce

Node 4 Z−Acceleration / 4 X−Force


101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce




COMPARISON OF SIMULATION ACCURACY FORSPECIFIC AND GENERIC MODELING UNCERTAINTY

101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce


ModelModel +/− 1 σ SpecificModel +/− 1 σ Generic

101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce



101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce



101

102

100

101

102

103

104

Frequency (Hz)

Acc

eler

atio

n/F

orce




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


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


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

)


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

)


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

)


Left Horizontal Quarter Point Upper Vertical Quarter Point

Center Point Lower Vertical Quarter Point


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

)


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

)


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

)


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

)


Left Horizontal Quarter Point Upper Vertical Quarter Point

Center Point Lower Vertical Quarter Point


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

)


0 0.01 0.02 0.03 0.04

0

2

4

6

8

10

12

14

Time (sec)

Dis

plac

emen

t (in

)


0 0.01 0.02 0.03 0.04

0

2

4

6

8

10

12

14

Time (sec)

Dis

plac

emen

t (in

)


0 0.01 0.02 0.03 0.04

0

2

4

6

8

10

12

14

Time (sec)

Dis

plac

emen

t (in

)


Gage 1 Gage 2

Gage 3 Gage 4


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


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.

effect of total modeling uncertainty on the accuracy of numerical simulations · 1999-09-23 ·...

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