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Computational Methods in

Nondestructive Evaluation (NDE)

for Damage State Characterization

Mark Blodgett (PI), Jeremy Knopp, and John Aldrin

AFRL/RXLP

AFOSR Structural Mechanics

Annual Review

24-27 July 2012

AFOSR Program Manager: Dr. David Stargel

Computational Methods in NDE for Damage State Characterization

Challenge: Address Wide Variability and Uncertainty in NDE

Measurements, Part Geometry, Material Systems,

Flaw Characteristics and Environmental Factors

Problem: Characterize Cracks at Fastener Sites

Using Ultrasonic and Eddy Current NDE

Sensors

(Arrays)

Advanced

Models

Model-based

Inverse Methods

for Material State

Estimation

• multiscale damage characterization

• material properties measurement

Model-assisted

POD Assessment

NDE

Validation

Integrated

Probabilistic Models

• NDE/SHM

• life prediction

• risk assessment

• cost-benefit analysis

Optimal Service

Life Management

(Digital Twin)

Improve NDE Sensitivity / Characterization

(Depth, Small Cracks, Sizing) Payoff:

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

crack length (in)

PO

D

MAPOD

exp.

Approach (FY11):

Forward problems Inverse problems Mixed simulated

and empirical

evaluations

Efficient Methods for Propagating

Parameter Variability and Uncertainty

Objective (FY11-13)

Objective: To develop NDE characterization methods for damage in aircraft

structures and engine components using mechanical stress and

electromagnetic waves that integrate the development and

implementation of advanced computational methods

Proposed Research on New Capabilities:

• Efficient methods for propagating parameter variability and

uncertainty for forward problems in NDE,

• Statistical validation metrics incorporating stochastic

methods for robust inverse methods design,

(model-assisted POD / reliability assessment)

• Model-based crack characterization in aerospace applications,

• Develop model-based damage characterization and data fusion

methods for complex structural inspection problems,

• Integrate and optimize algorithms with emerging array sensor

technologies (MR arrays, 2D ultrasonic phased arrays)

Highlights - FY12

• Efficient methods for propagating parameter variability and uncertainty for

forward problems in NDE,

• PCM foundational methods

• KL-transform / ANOVA

• Smolyak Sparse grids

• Statistical validation metrics incorporating stochastic methods for robust

inverse methods design, (model-assisted POD / reliability assessment)

• Bayesian method in POD / MAPOD evaluation

• Traditional methods re-examined with Bayesian Tools

• Box-Cox / Bootstrap

• Physics-inspired model evaluations

• Bayesian POD model and uncertain parameter estimation (e.g. aspect ratio)

• Discussion of metrics for inverse methods (GP models)

• Model-based crack characterization in aerospace applications,

• Case studies for surface-breaking cracks [transition]

• Prelim. emerging array sensor technologies (MR arrays)

Uncertainty Propagation

• Motivation: Model evaluations are computationally expensive. There is a need for more efficient methods than Monte Carlo

• Objective: Efficiently propagate uncertain inputs through “black box” models and predict output probability density functions. (Non-intrusive approach)

• Approach: Surrogate models based on Polynomial Chaos Expansions meet this need.

Eddy Current NDE Model

[Deterministic]

n

2

1

Z~

UniformX

NormalX

n ~

~1

Eddy Current NDE Model

[Stochastic] ? ~

~Z

Input Parameters with Variation:

• Probe dimensions

(Liftoff / tilt)

• Flaw characteristics

(depth, length, shape)

Uncertainty Propagation

Uncertainty propagation for parametric NDE characterization problems:

• Probabilistic Collocation Method (PCM) approximates model

response with a polynomial function of the uncertain parameters.

• This reduced form model can then be used with

traditional uncertainty analysis approaches,

such as Monte Carlo.

Extensions of generalized polynomial chaos (gPC) to high-dimensional

(2D, 3D) damage characterization problems:

• Karhunen-Loeve expansion

• Analysis of variance (ANOVA)

• Smolyak Sparse Grids

N

Number of Unknowns

to Evaluate

1

Critical

Flaw Size

Key Damage and

Measurement

States (e.g. crack

length, probe liftoff)

Parameterized

Flaw

Localization

and Sizing

Full 3D

Damage and

Material State

Characterization

>1 >>1

N

i

iicfZ1

)()(ˆ xx

Approach (1): Karhunen-Loeve Expansion

• Address stochastic input variable reduction when number of

random variables (N) is large.

• Apply Karhunen-Loeve Expansion to map random variables into

a lower-dimensional random space (N').

Eddy Current Example:

• Correlation function (covariance model)

defines random conductivity map,

• Set choice of grid length to

– achieve model convergence and

– eliminate insignificant eigenvalues

for reduced order conductivity map.

Uncertainty Propagation and High Dimensional Model Representation

N

n

nnn

1

)()( xx )(x Karhunen-

Loéve

Expansion

N ...1

covariance

model

),( xx C

conductivity map with

N random variables

reduced order conductivity map

with N' random variables

N' random variables

Crystallites

(Grains)

=2.2*106 S/m

Coil

Approach (2): Analysis of Variance (ANOVA) Expansion

• Provides surrogate to represent high dimensional set of parameters

• Analogous to ANOVA decomposition in statistics

• Locally represent model output through

expansion at anchor point in -space

– Requires inverse problem

– Replace random surface with

equivalent 'homogeneous' surface


ANOVA Expansion

N

n

nnn

1


Loéve Expansion

N ...1

defined by

covariance

model

),( xx C

conductivity

map with

N random

variables

reduced order


N' random variables

N' random variables (1) Identify

unique

sources

of variance

(2) Identify

significant

factors

in model M ...1

M random variables

)(ξZ

N' >> M N >> N' ξξ

Approach (2): Analysis of Variance Expansion + Smolyak Sparse Grids

• Significant computational expense for high-dimensional integrals

• Can leverage sparse grids based on the Smolyak construction

[Smolyak, 1963; Xiu, 2010; Gao and Hesthaven, 2010]

– Provides weighted solutions at specific nodes and adds them to

reduce the amount of necessary solutions

– Sparse grid collocation provides subset of full

tensor grid for higher dimensional problems

– Approach can also be applied to gPC/PCM


ANOVA Expansion

N

n

nnn

1


Loéve Expansion

N ...1

defined by

covariance

model

),( xx C

conductivity

map with

N random

variables

reduced order


N' random variables

N' random variables (1) Identify

unique

sources

of variance

(2) Identify

significant

factors

in model M ...1

M random variables

)(ξZ

N' >> M N >> N' ξξ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1Sparse Grid and Full Tensor Product Grid

Bayes Factor in Hit-miss POD Evaluation

Example (1) [NTIAC 9002(3)L]:

• Application of wrong model with poor data

Approach: Investigate more sophisticated statistical models and

Bayes Factor for model evaluation and selection procedures

0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

80

90

100

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75PR

OB

AB

ILIT

Y O

F D

ET

EC

TIO

N (

%)

ACTUAL CRACK LENGTH - (Inch)

Data Set: A9002(3)L

Test Object : 2219

Aluminum,

Stringer

Stiffened Panels

Condition: After Etch

Method: Eddy Current,

Raster

Scan with

NTIAC, Nondestructive Evaluation (NDE) Capabilities Data Book 3rd ed., NTIAC DB-97-02, Nondestructive Testing Information Analysis

Center, November 1997

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bayes Factor in Hit-miss POD Evaluation

• Summary of Marginal Likelihood (ML) Results

Using Bayes Factor for POD Model Fits

a (mm)

â

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 parameter Probit 3 parameter

lower bound Probit

4 parameter Probit

(most likely)

3 parameter

upper bound Probit

ML

2 parameter Logit 3.70E-97

2 parameter Probit 9.08E-98

3 parameter lower bound Logit 7.29E-98

3 parameter lower bound Probit 1.02E-98

3 parameter upper bound Logit 3.27E-93

3 parameter upper bound Probit 1.64E-93

4 parameter Logit 7.24E-92

4 parameter Probit 2.49E-92

a (mm)

â

a (mm)

â

Knopp, J. and Zeng. L., “Statistical Analysis of Hit/Miss Data,” Materials Evaluation (submitted, 2012).

Box-Cox Transformations

• POD response data analysis (â vs a) assumes

homoscedasticity, and if that assumption is violated,

one must resort to hit/miss analysis. This was the

case for an early MAPOD study (Knopp 2007)

• Box-Cox transformation can remedy this problem.

0 1 2 3 4 5

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

sig

na

l re

sp

on

se

â

crack size (mm)

• Power transformation

• Common transformations

ââ

22ââ

5.0 ââ

0 ââ elog

5.0â

â1

0.1â

â1

Box-Cox Transformations

• Box-Cox enables â vs a analysis for data sets where

the variance is not constant but has some relationship

with the independent variable such as crack size.

analysis

method

λ left

censor

detection

threshold

false

calls

a90 (mm) a90/95 (mm) a90 - a90/95 %

difference

1st order

linear

0.45 0.13 0.23 0 2.176 2.327 6.9%

1st order

linear

0.5 0.14 0.195 1 2.102 2.257 7.3%

1st order

linear

0.5 0.195 0.195 1 2.269 2.53 11.5%

2nd order

linear

0.5 .14 0.195 1 2.277 2.472 8.5%

2nd order

linear

0.5 0.195 0.195 1 2.197 2.428 10.5%

hit/miss 1 0.187 1 1.72 2.04 18.6%

hit/miss 1 0.162 11 1.498 1.907 27.3%

Physics-inspired Models and Confidence Bound Evaluation

• Visual inspection reveals that a 2nd order

linear model may fit the data better than

the standard „â vs a‟ analysis.

• Evidence: p-value for a2 is 0.001 and

adjusted R-square value increases

slightly with inclusion of a2.

• Confidence bounds calculation on models more complicated than

„â vs a‟, especially transforming to probability of detection curve.

• Bootstrap methods are simple and flexible enough to provide

confidence bounds for a wide variety of models.

• Example for the previous

transformed data set with

λ = 0.5

– Results for a90 and a90/95

values converge well

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

Experiment

Simulation

a90 a90/95

Wald Method 2.102 mm 2.257 mm

Bootstrap 1,000 2.096 mm 2.281 mm

Bootstrap 10,000 2.099 mm 2.299 mm

Bootstrap 100,000 2.099 mm 2.297 mm

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.1

-0.05

0

0.05

0.1

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D

POD: physics-based model, Bayes/MCMC

• Example: Eddy Current Inspection of Cracks at Fastener Sites

• Case Study for Physics-based Model Evaluation:

•

• where f () is a function call for a physics-based model (i.e. VIC-3D)

• Bayesian POD Analysis Performed in Matlab + R:

• MCMC library in Matlab used for Bayesian Analysis

• Matlab Provides Option for Integration of Model Function in Bayesian Fit

• Compare Ahat-vs-a fit (MLE, Wald bounds) and a Physics-based Model Fit

Bayesian Methods with Physics-based Model POD / MAPOD Evaluation (1)

2110 ,ˆ aafa

zb

a

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.04

-0.02

0

0.02

0.04

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D

POD: ahat-vs-a model, MLE+WaldAhat-vs-a fit (MLE, Wald bounds) Physics-based model fit (Bayes/MCMC)

Physics-based

Model Fit

Provides

Better Match

and

Residuals Are

Reduced

),0(~ 2

N

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.1

-0.05

0

0.05

0.1

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.1

-0.05

0

0.05

0.1

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.04

-0.02

0

0.02

0.04

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D

POD: ahat-vs-a model, MLE+Wald



•

• where f () is a function call for a physics-based model (i.e. VIC-3D)

• Bayesian POD Analysis Performed in Matlab + R:

• MCMC library in Matlab used for Bayesian Analysis

• Matlab Provides Option for Integration of Model Function in Bayesian Fit

• Compare Ahat-vs-a fit (MLE, Wald bounds) and a Physics-based Model Fit

2110 ,ˆ aafa

zb

a

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.04

-0.02

0

0.02

0.04

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D

POD: ahat-vs-a model, MLE+Wald

Ahat-vs-a fit (MLE, Wald bounds) Physics-based model fit (Bayes/MCMC)

Result is

More Accurate

Representation

of the Data in

the POD Model

Fit

),0(~ 2

N


0.496 0.498 0.5 0.502 0.504 0.506 0.5080

2000

4000

6000

8000

theta1 (bias)

6.5 7 7.5 8 8.5 9 9.5 100

2000

4000

6000

theta2 (slope)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250

2000

4000

6000

1/theta3 (aspect ratio: a/b)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.05

0

0.05

0.1

0.15

a1 (in.)

ahat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.1

-0.05

0

0.05

0.1

a1 (in.)

resid

uals

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.5

1

a1 (in.)

PO

D




•

• where f () is a function call for a physics-based model

• 0 , 1 = model calibration parameters

• 2 = random variable associated with crack aspect ratio (b/a)

• 3 = random variable associated with liftoff variation

• Results: Fit POD Model and Estimate of Variation in Aspect Ratio

[use non-informative priors]

• Issues with „Naïve‟ Approach:

• Need true estimate of variance for

crack aspect ratio random variable

→ Use hierarchical models

• Address correlated / confounded

parameters in estimation problem

→ Use informative priors and

constraints from expert opinion

32110 ,;ˆ afa

zb

a

),0(~ 2

N


0 0.05 0.1 0.15 0.2-0.1

0

0.1

0.2

0.3

a1 (in.)

ahat


• Challenge: Address non-constant variance wrt flaw size

• Hierarchical NDE Measurement Models:

•

• where f () is a function call for a physics-based model

• 0 , 1 = model calibration parameters

• h = random variable (varying-slope model)

• 2h = variance in slope parameter

• 2 = random variable associated

with crack aspect ratio (b/a)

• 2h = variance in slope parameter

• Simple Test Case: Fit data from model with varying slope >> noise.

Hierarchical Models for Estimating Variance of a Random Variable

aaa ˆ110ˆ h

zb

a

),0(~ 2

ˆˆ aa N

1st

2nd

aafa ˆ2110 ;ˆ

hh ),0(~ 2

hh N ),(~ 2

2 22 N);,0(~ 2

ˆˆ aa N

physics-based model statistical model

• A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, 2003.

• A. Gelman and J. Hill, Data Analysis Using Regression and Multilevel/Hierarchical Models, 2007.

• Simple Test Case: Fit data from model with varying slope >> noise.

• Hierarchical NDE Measurement Models:

• Results: Ns = 100

Hierarchical Models for Estimating Variance of a Random Variable

aaa ˆ110ˆ h ),0(~ 2

ˆˆ aa N

hh ),0(~ 2

hh N

Ns = 100 0 1 h

True Value 0.0000 1.0000 0.3000 0.00100

WinBUGS

Mean -0.00024 1.0312 0.2580 0.00139

95% Credible Bds (-0.00119,0.00364) (0.9848,1.0810) (0.2286,0.2923) (0.00103,0.00210)

Matlab

Mean -0.00027 1.0361 0.2768 0.01166

95% Credible Bds (-0.00418,0.00071) (0.9862,1.0863) (0.2593,0.2974) (0.01012,0.01346)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.05

0.1

0.15

0.2

0.25

a1 (in.)

ahat

Bayesian Methods - Challenges

Uncertainty Quantification (UQ) community developing Bayesian framework

for the use of computational models with observational data

• SAMSI program on UQ (2012):

• SIAM UQ conference 2012: http://www.siam.org/meetings/uq12/

Key Insight / Research Directions:

• 1) Must include model discrepancy and not treat it as random error.

• Calibrating (inverting, tuning) a wrong model gives parameter estimates

that are wrong (not equal to their true physical values) [O‟Hagan, 2012]

• Gaussian Process (GP) models typically used to fit model discrepancy

[Kennedy/O‟Hagan 2002].

• 2) Use of prior information in Bayesian framework can greatly help.

• To learn about model parameters in the presence of discrepancy, better

prior information is needed [Bayarri, 2012]

• Elicitation of expert opinion is an active research topic [O‟Hagan, 2012]

• 3) Should leverage model form uncertainty (assessment) approaches.

• To identify best models and address limitations cited by UQ community

[Grandhi et al, Wright State University]

Compare NDE „Ahat-vs-a‟ POD and NDE Characterization Error (CE) Evaluations

Characterization Error Analysis:

• Build on Protocol for NDE/SHM

• Perform evaluation studies

• experimental sizing results

• simulated sizing results

• Evaluate characterization error

(êj) with respect to flaw size (ak)

• error model (êj= âj - aj)

• uncertainty bounds

Ahat-vs-A POD Analysis:

• Follow MIL-HDBK 1823A

• Perform evaluation studies

• experimental measurements

• simulated measurements

• Evaluate model of measurement

(âj) with respect to flaw size (ak)

• mean model

• confidence (uncert.) bounds

a1

â

a1

ê1

Model-assisted Process (Tools) for NDE Characterization Error (CE) Evaluation

2) Model-based

Uncertainty

Propagation

Model

Error

Input Parameter

Variability

(Distributions)

Stochastic

Models

Model

„Calibration„

(Error)

3) Revise Model Estimates

with Experimental Data

Using Bayesian Methods

Confidence Bounds

(Limited Samples)

Challenge: This is Complicated. Must Focus on Key Drivers of Error. [Annis]

Characterization Error

w/Uncertainty Bounds

1) Assess Key

Factors (Joint PDFs)

of Inversion Process

Build Experimental

Adjustments to

êj vs. ak

Build Theoretical

Adjustments to

êj vs. ak

Bayesian Methods:

Estimate unknown / uncertain

joint parameter distributions

Stochastic Numerical

Methods:

- Prob. Collocation Method

- KL Transform + ANOVA

Gaussian Processes:

Represent range of random

variables with multivariate

normal distribution

Bayesian Calibration:

Revise model (GP) using

Bayesian method with

Empirical model / data

Gaussian Process (GP)

• A Gaussian Process is a collection of random variables

with the property that the joint distribution of any finite

subset is a Gaussian function. It consists of:

– Mean function (model)

– Covariance function

– Learning equates to finding suitable

properties for the covariance function

– Hyperparameters: smoothness, characteristic length scales

[From MacKey, 2006,

“Gaussian Process Basics”]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Gaussian Process (GP)

Example: Ahat-vs-a Model Fit Using GP model

• GP model fit with

hyperparameter

estimation

• model structure

• meanfunc =

{@meanSum,

{@meanLinear,

@meanConst}};

• covfunc =

{@covPoly,2};

• likfunc =

@likGauss;

a1 (in.)

sample (prediction)

bounds (±2)

model bounds

(±2)

• Gaussian Processes Can be Used with Explicit Basis Functions (for example:

Physics-based Models)

â

covPoly = polynomial

covariance function (2nd order)

Model-based Crack Characterization in Aerospace Applications

To characterize a fatigue crack condition around a fastener site

through model-based damage characterization approaches,

a number of conditions must be simultaneously estimated:

Crack dimensions (a, b)

Crack location (z, q)

NDE source and sensor (EC, UT)

position and orientation (xk ,l )

Fastener and crack contact conditions (ai, j)

Morphology and residual stress across crack face

Enabling Technologies to Achieve Objectives:

• Model-based inverse methods (with validated model through benchmark studies)

• Efficient stochastic models addressing varying and uncertain conditions

• To evaluate the sensitivity of the inverse methods

• To optimize the design of the NDE procedure

• Discovery of ‘invariant‟ NDE data features (insensitive to varying conditions)

• Bayesian inverse methods (address refinement using a priori information)

Publications (FY12)

• Knopp, J., and Zeng, L., “Statistical Analysis of Hit/Miss Data”, Materials Evaluation, (submitted, 2012.)

• Knopp, J., Grandhi, R., Aldrin, J. C., Park, I., “Statistical Analysis of Eddy Current Data from Fastener

Site Inspections,” Journal of Nondestructive Evaluation, (submitted, 2012.)

• Aldrin, J. C., Knopp, J., Sabbagh, H. A., “Bayesian Methods in Probability of Detection Estimation and

Model-assisted Probability of Detection (MAPOD) Evaluation,” Review of Progress in QNDE, Vol. 31,

AIP, (to be published, 2013).

• Cherry, M. and Knopp, J., “Stochastic Collocation Method for Higher Dimensional NDE Problems with

Tensor Product Grids”, Review of Progress in QNDE, Vol. 31, AIP, (to be published, 2013).

• Aldrin, J. C., Knopp, J., “False Call Rate Estimation,” Review of Progress in QNDE, Vol. 31, AIP, (to be

published, 2013).

• Aldrin, J. C., Sabbagh, H. A., Murphy, R. K., Sabbagh, E. H., Knopp, J. S., Lindgren, E. A., Cherry, M.

R., “Demonstration of model-assisted probability of detection evaluation methodology for eddy current

nondestructive evaluation,” Review of Progress in QNDE, Vol. 31, AIP, pp.1733-1740, (2012).

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