Structural Dynamics Validation Problem: An approximationtheoretic approach

1
Structural Dynamics Validation ProblemAn
approximation-theoretic approach

• Roger Ghanem
• Alireza Doostan
• University of Southern California
• Los Angeles, California

Validation Challenge Workshop, Sandia National
Laboratories, Albuquerque, NM, May 22-23, 2006
2
Outline

• Validation philosophy
• Uncertainty representation and estimation
• Develop intuition on simple problems
• Challenge problem

3
Validation philosophy
We develop an implementation of Validation with
AIAA document as guidelines

• Given a (physics model, data, and computational
  resources)
• compute limits on predictability
• i.e. which statements about system
  performance can be certified
• compute resource allocation (data/computing)
  along validation path

• Given a (physics model - with infinite
  data/computing resources)
• compute limits on predictability
  i.e. which statements about system performance
  can be certified

4
Error budget
limit on predictability, given a model
MUST BE QUANTIFIED !!!!
5
Motivation of approach

• Package information efficiently for intended
  purpose
• propagate information through large scale
  computational models.
• decide on a path for validation
• sensitivity to additional information
• sensitivity to uncertainty in model components
• sensitivity to numerical approximations

6
Representing uncertainty

• The random quantities are resolved as surfaces in
• a normalized space
• These could be, for example
• Parameters in a PDE
• Boundaries in a PDE (e.g. Geometry)
• Field Variable in a PDE

Independent random variables
Multidimensional Orthogonal Polynomials
Dimension of vector reflects complexity of
7
Error budget

• IF PREDICTION IS OBTAINED USING A WEAK FORM OF
  SOME GOVERNING EQUATION
• Joint error estimation is possible, for special
  cases
• infinite-dimesional gaussian measure Benth
  et.al, 1998
• tensorized identical independent measures
  Babuska et.al, 2004
• Joint error estimation is possible, for general
  measures, using nested approximating spaces
  (Doostan, Ghanem, Rozovsky, 2006)

8
Characterization of Uncertainty

• Galerkin Projections
• Efficient - unsuitable for dependent scales
• Maximum Likelihood
• Maximum Entropy
• Suitable for data-driven constraints
• Bayes Theorem

Characterize as random variables
9
Representing uncertainty
Starting with observations of process over a
limited points on the domain
Reduced order representation
Polynomial representation of KL variables
10
Characterizing UncertaintyMaximum Likelihood
Estimation
Physical object Linear Elasticity
Stochastic parameters
Beam with random heterogeneous material
properties. Observe realizations of system
response
Convergence as function of dimensionality
Reference Desceliers, Ghanem amd Soize, ,
IJNME, 2006.
11
Characterization of Uncertainty Bayesian
Inference
Posterior distributions of coefficients in
polynomial Expansion of
Distribution of the recovered process
Reference Doostan and Ghanem, , Journal of
Computational Physics, 2006.
12
Characterization of Uncertainty Maximum Entropy
Estimation with Moment Constraints
Reference Das, Ghanem, and Spall, SIAM Journal
on Scientific Computing, 2006.
13
Characterization of Uncertainty Maximum Entropy
Estimation / Spatio-Temporal Processes
Temperature time histories, , at
various depths.
14
Characterization of UncertaintyMaximum Entropy
Estimation with Histogram Constraints

• Reduced order model of
• KL expansion
  

Spearman Rank Correlation Coefficient is also
matched
A typical plot of marginal pdf for a
Karhunen-Loeve variable.
Reference Das, Ghanem, Finette, , Journal of
Geophysical Research, 2006.
15
Uncertainty modeling for system parameters
Approximate asymptotic representation
Representation on the set of observation
Remark Both intrinsic uncertainty and
uncertainty due to lack of data are represented.
Representation smoothed on the whole domain
Remark is formulated by spectral
decomposition of .
16
Additional information and sensitivity analysis

• Important remarks
• Asymptotically, the total uncertainty reduces
  to intrinsic uncertainty.
• Contribution of uncertainty due to limited
  information could be separated from that of the
  intrinsic uncertainty both at parameter level and
  response level.
• Sensitivity of the statistics of SRQ to
  parameters of can be quantified.

17
CDF of system parameters m1, c1, k1
Estimate 95 probability box
Remarks

• Confidence intervals are due to finite sample
  size.

18
Model accuracypredicting accelerations of
calibrated model
Frequency
Mean for calibrated linear model
Observation form actual system
Maximum acceleration of the top mass
Calibration Excitation Low
19
Validation path hypothesis test
System Response Quantity (SRQ)
Maximum acceleration of the top mass a3m
Propagation using calibrated stochastic linear
model
Stochastic Projection/ Monte Carlo
pdf

validation force
Equivalent hypothesis test
pdf
Remark Parameters are calibrated under .
mean of predicted from linear
model. observed acceleration on
validation specimen
95 confidence interval around
20
Validation path hypothesis test
Possible scenarios Repeat for all validation
data
pdf
pdf
95 confidence interval around
95 confidence interval around
No sufficient evidence to reject H0
H0 is rejected
Therefore
Validation metric
Model is considered validated if enough
validation specimens are deemed consistent with
the calibrated model. In the present case, 100
of specimens could not be rejected (i.e. were
consistent with the calibrated model).
21
Typical subsystem validation result
Calibration Excitation Medium Validation
Excitation High
Frequency
22
Typical subsystem validation resultneglecting
effect of finite sample
Calibration Excitation Medium Validation
Excitation High
Frequency
23
Subsystem validation outcome
24
Typical accreditation result
Frequency
Calibration Excitation Medium Accreditation
Excitation 2
25
System accreditation outcome
26
Prediction on target application
Remark Based on only 25 samples.
27
Conclusions

• Suitable Uncertainty Quantification can provide
  an integrated path for model validation.
• Current implementation is very demanding on
  function evaluations. This is a reflection of
  the validation criterion used. Comparison of
  CDFs will help manage this difficulty.