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SLIDE 1

ULg, Liège, Belgium Nonintrusive probabilistic UQ 1 / 33

IE-net Managing, handling, and modeling uncertainty in mechanical design

Nonintrusive probabilistic quantification of uncertainties with application to the management of manufacturing tolerances

Maarten Arnst

May 21, 2015

SLIDE 2

Motivation

ULg, Liège, Belgium Nonintrusive probabilistic UQ 2 / 33

Manufacturing tolerances in metal forming Raw materials variability:

Material properties.

. . .

Process variability:

Blank holder force.
Initial dimensions.
Friction.

. . .

Modeling limitations:

Constitutive model.
FE discretization.

. . .

Input variables.

→ →

Product variability:

Final dimensions.
Springback.

. . .

Prediction limitations:

Numerical noise.

. . .

Output variables.

SLIDE 3

Outline

ULg, Liège, Belgium Nonintrusive probabilistic UQ 3 / 33

■

Motivation.

■

Outline.

■

Context and current practice.

■

New methods.

■

Example: Metal forming.

■

Conclusion and outlook.

■

References.

■

Contact information.

SLIDE 4

Selected elements from context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 4 / 33

SLIDE 5

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 5 / 33

Example: Bending of a beam

ℓ

Young’s modulus y moment of inertia j

p u

utput variable

tip displacement

= g

model

(gy, j, p, ℓg

input variables

) = pℓ3 3yj .

Let y be uncertain (e.g., imperfect knowledge at design time, imperfect manufacturing when compared to the design,. . . ). Given uncertainty in y, what is the resulting uncertainty in u?

SLIDE 6

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33

Example: Bending of a beam (continued)

■

A probabilistic context effects the propagation of uncertainty from y to u as follows:

y u y ρY ρU u u = g(y) = pℓ3

3yj

P

U ≤ u
= P
g−1(u) ≤ Y
because g is a decreasing function.

SLIDE 7

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33

Example: Bending of a beam (continued)

■

A probabilistic context effects the propagation of uncertainty from y to u as follows:

y u y ρY ρU u u = g(y) = pℓ3

3yj

P

U ≤ u
= P
g−1(u) ≤ Y
because g is a decreasing function.

SLIDE 8

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33

Example: Bending of a beam (continued)

■

A probabilistic context effects the propagation of uncertainty from y to u as follows:

y u y ρY ρU u u = g(y) = pℓ3

3yj

P

U ≤ u
= P
g−1(u) ≤ Y
because g is a decreasing function.

SLIDE 9

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 6 / 33

Example: Bending of a beam (continued)

■

A probabilistic context effects the propagation of uncertainty from y to u as follows:

y u y ρY ρU u u = g(y) = pℓ3

3yj

P

U ≤ u
= P
g−1(u) ≤ Y
because g is a decreasing function.

SLIDE 10

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 7 / 33

Example: Bending of a beam (continued)

■

Elaborating this expression by means of the “changes of variables” formula,

u ρU(u)du = +∞

g−1(u)

ρY (y)dy =

u

ρY

g−1(u)

dg−1 du (u)du = u ρY

g−1(u)
dg−1

du (u)

du,

we find the following relationship between the probability density functions of the input and output variables, that is, of the Young’s modulus and the tip displacement:

ρU(u) = ρY

g−1(u)
dg−1

du (u)

.

SLIDE 11

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 8 / 33

Example: Bending of a beam (continued)

■

Probability density function, mean, variance, and confidence interval:

u ρU(u) u − kσU u + kσU u σU

◆ Mean u =

R uρU(u)du,

◆ Variance σ2

U =

R(u − u)2ρU(u)du,

◆ Pc-Confidence interval [u − kσU, u + kσU] such that u−kσU

u−kσU ρU(u)du ≥ Pc.

SLIDE 12

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 9 / 33

Example: Bending of a beam (continued)

■

Using the change-of-variables formula, we can deduce the following expression for the mean u:

u =

R

uρU(u)du =

R

uρY

g−1(u)
dg

dy

g−1(u)
−1

du =

R

g(y)ρY (y) dg dy (y) −1 dg dy (y)dy =

R

g(y)ρY (y)dy,

and we can deduce the following expression for the variance σ2

U:

σ2

U =

R

(u − u)2ρU(u)du =

R

(u − u)2ρU

g−1(u)
dg

dy

g−1(u)
−1

du =

R

(g(y) − u)2ρY (y) dg dy (y) −1 dg dy (y)dy =

R

(g(y) − u)2ρY (y)dy. ■

In conclusion, to determine the mean and the variance of the output, knowledge of the probability density function of the input and an integration method are required.

SLIDE 13

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 10 / 33

Example: Bending of a beam (continued)

■

If the model is linearized,

u = g(y) ≈ g(y) + dg dy (y)

(y − y).

then the expression for the mean u can be simplified as follows:

u =

R

g(y)ρY (y)dy ≈

R
g(y) +

dg dy (y)

(y − y)
ρY (y)dy = g(y),

and the expression for the variance σ2

U can be simplified as follows:

σ2

U =

R

(g(y) − u)2ρY (y)dy ≈

R

dg dy (y)

(y − y)

2 ρY (y)dy = dg dy (y) 2 σ2

Y .

■

In conclusion, linearising the model makes things much simpler!! Now, to approximate the mean and the variance of the output, knowledge of only the mean and variance of the input suffices.

SLIDE 14

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 11 / 33

Example: ISO 98 ISO 98: Guide to the expression of uncertainty in measurement.

SLIDE 15

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 12 / 33

Example: ISO 98 (continued) ISO 98: Guide to the expression of uncertainty in measurement.

SLIDE 16

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 13 / 33

Example: Robust design in aerospace engineering From: A. Karl, B. Farris, L. Brown, and N. Metzger (Rolls-Royce). Robust design and optimization: Key methods and applications. Stanford, 2011.

SLIDE 17

Context and current practice

ULg, Liège, Belgium Nonintrusive probabilistic UQ 14 / 33

Some limitations associated with the approaches described so far...

■

Engineering problem

◆ Limited in scope to scalar uncertain quantities.

◆ However, more complex uncertainties can be encountered in engineering problems, such as uncertain geometries, uncertain processes and fields, and uncertain matrices.

■

Characterization of uncertainties

◆ Limited to mean and variance.

◆ No emphasis on constraints that can be imposed by mechanics and physics.

■

Propagation of uncertainties

◆ Approximation entailed by linearization of the model.

■

Sensitivity analysis of uncertainties

◆ Limited to local sensitivity analysis that is also encountered in deterministic problems.

◆ However, global sensitivity analysis can also be of interest; and many new interesting questions can be asked in an uncertainty-quantification-enabled context.

SLIDE 18

Selected elements from new methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 15 / 33

SLIDE 19

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 16 / 33

Overview

Engineering

problem

Sensitivity analysis.

Process optimization. Design optimization. Model validation.

Analysis
f uncertainties
✓

✌ ☎ ② ♣

UQ

Characterization
f uncertainties
Mechanical modeling.

Statistics.

Propagation
f uncertainties
Monte Carlo sampling.

Stochastic expansion (polynomial chaos). Scientific computing.

·

Gaussian.
Γ-distribution.

. . .

SLIDE 20

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 17 / 33

Characterization of uncertainties

■

The objective of the characterization of uncertainties is to assign an appropriate probability distribution to the uncertain input variables.

■

An appropriate probability distribution can be obtained by applying methods from mathematical statistics to the available information. In engineering, this available information typically consists not only of observed samples but also of applicable mechanical and physical laws.

◆ Catalogs of probability distributions.

◆ Principles of construction.

◆ Methods for parameter estimation.

◆ Methods for model selection.

◆ . . .

■

If a sufficient amount of data is available, much of this can be automated.

■

Current ressearch allows to consider as uncertain not only scalar input variables but also geometries, fields of mechanical and physical properties, matrix-valued input variables, etc.

SLIDE 21

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 18 / 33

Characterization of uncertainties (continued)

−3 −1.5 1.5 3 x 10

−8

−1 −0.5 0.5 1x 10

−9

Position [m] Displacement [m]

Random geometry. From: M. Arnst and R. Ghanem. Probabilistic electromechanical modeling of nanostructures with random geometry. Journal of Computational and Theoretical Nanoscience, 6:2256–2272, 2009.

SLIDE 22

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 18 / 33

Characterization of uncertainties (continued)

−3 −1.5 1.5 3 x 10

−8

−1 −0.5 0.5 1x 10

−9

Position [m] Displacement [m]

Random geometry. From: M. Arnst and R. Ghanem. Probabilistic electromechanical modeling of nanostructures with random geometry. Journal of Computational and Theoretical Nanoscience, 6:2256–2272, 2009.

SLIDE 23

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 18 / 33

Characterization of uncertainties (continued)

−3 −1.5 1.5 3 x 10

−8

−1 −0.5 0.5 1x 10

−9

Position [m] Displacement [m]

Random geometry. From: M. Arnst and R. Ghanem. Probabilistic electromechanical modeling of nanostructures with random geometry. Journal of Computational and Theoretical Nanoscience, 6:2256–2272, 2009.

SLIDE 24

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 18 / 33

Characterization of uncertainties (continued)

−3 −1.5 1.5 3 x 10

−8

−1 −0.5 0.5 1x 10

−9

Position [m] Displacement [m]

Random geometry. From: M. Arnst and R. Ghanem. Probabilistic electromechanical modeling of nanostructures with random geometry. Journal of Computational and Theoretical Nanoscience, 6:2256–2272, 2009.

SLIDE 25

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 19 / 33

Characterization of uncertainties (continued) Random fields. From: M. Arnst. Inversion of probabilistic models of structures using measured transfer functions. Thèse de Doctorat, Ecole Centrale Paris, France, 2007.

SLIDE 26

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 20 / 33

Characterization of uncertainties (continued)

[K + iωD − ω2M]u(ω) = f(ω).

Random matrices. From: F. Nyssen, M. Arnst, and J.-C. Golinval. Experimental modal identification of mistuning in an academic bladed disk and comparison with the blades geometry variations. ASME Turbo Expo, 2015.

SLIDE 27

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 20 / 33

Characterization of uncertainties (continued)

[K + iωD − ω2M]u(ω) = f(ω).

Random matrices. From: F. Nyssen, M. Arnst, and J.-C. Golinval. Experimental modal identification of mistuning in an academic bladed disk and comparison with the blades geometry variations. ASME Turbo Expo, 2015.

SLIDE 28

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 20 / 33

Characterization of uncertainties (continued)

[K + iωD − ω2M]u(ω) = f(ω).

Random matrices. From: F. Nyssen, M. Arnst, and J.-C. Golinval. Experimental modal identification of mistuning in an academic bladed disk and comparison with the blades geometry variations. ASME Turbo Expo, 2015.

SLIDE 29

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 21 / 33

Propagation of uncertainties

■

The next step is to propagate the uncertainties introduced in the input variables through the model to the output variable to quantify the impact of these uncertainties on this output variable.

■

We apply the Monte Carlo method: blanc

◆ First, the Monte Carlo method involves generating an ensemble of i.i.d. samples with probability distribution ρX:

{xℓ, 1 ≤ ℓ ≤ ν}.

◆ Then, the computational model is used to map each sample of X into a sample of Y , that is,

yℓ = g(xℓ).

to obtain the corresponding ensemble of i.i.d. samples of Y , written as follows:

{yℓ, 1 ≤ ℓ ≤ ν}.

◆ Finally, the second-order statistical descriptors of Y (if they exist) are approximated as

y ≈ yν = 1 ν

ν

ℓ=1

yℓ

and

σ2

Y ≈ (σν Y )2 = 1

ν

ℓ=1

(yℓ − yν)2.

◆ This can be extended to approximating the PDF (if it exists), quantiles, . . . of Y .

SLIDE 30

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 22 / 33

Propagation of uncertainties (continued)

■

The Monte Carlo method has the following advantages:

◆ it is nonintrusive, that is, it requires only the repeated solution of the computational model for different values assigned to its input variables; the computational model need not be modified.

◆ it is adapted to parallel computation.

◆ convergence can be monitored during the computation.

◆ the rate of convergence is independent of the number of input variables.

■

Note that the Monte Carlo method can be improved using

◆ advanced simulation procedures,

◆ importance sampling,

◆ multilevel approaches,

◆ . . .

■

To gain efficiency, g can also be replaced by a surrogate model in the Monte Carlo method. This is the principle of stochastic expansion methods.

SLIDE 31

New methods

ULg, Liège, Belgium Nonintrusive probabilistic UQ 23 / 33

Sensitivity analysis of uncertainties

■

The objective of the next step is to gain useful insight into how uncertainty in the input variables induces uncertainty in the output variable.

■

There exist several types of sensitivity analysis of uncertainties:

◆ methods involving scatter plots,

◆ regression, correlation, and elementary effect analysis,

◆ variance-based sensitivity analysis,

◆ differentiation-based sensitivity analysis,

◆ . . .

■

Variance-based sensitivity analysis leads to an “uncertainty budget:”

σ2

Y

variance
f output variable Y

= sX1

contribution

from input variable X1

+ . . . + sXm

contribution

from input variable Xm

+

remainder

contribution

from interaction of X1,. . . ,Xm

.

SLIDE 32

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 24 / 33

SLIDE 33

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 25 / 33

Engineering problem Raw materials variability:

Material properties.

. . .

Process variability:

Blank holder force.
Initial dimensions.
Friction.

. . .

Modeling limitations:

Constitutive model.
FE discretization.

. . .

Input variables.

→ →

Product variability:

Final dimensions.
Springback.

. . .

Prediction limitations:

Numerical noise.

. . .

Output variables.

SLIDE 34

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 26 / 33

Engineering problem (continued)

■

blanc Material properties

(h, s)

Model

y = g(h, s) Springback angle y

■

Observed samples (hobs

1 , sobs 1 ), (hobs 2 , sobs 2 ), . . . , (hobs n , sobs n ).

h [MPa] s [MPa]

1488 375 1485 403 1514 407 1500 377 . . . . . .

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

■

Mechanics and physics impose that h and s be positive.

SLIDE 35

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 36

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 37

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 38

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 39

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 40

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 41

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 42

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 43

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 44

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 45

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 46

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 47

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 48

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 49

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 27 / 33

Characterization of uncertainties

■

We select the bivariate gamma probability distribution

ρ(H,S)(h, s; h, σ2

H, s, σ2 S, ρ

parameters of the PDF

) = ρΓ(h; h, σ2

H)

gamma marginal

ρΓ(s; s, σ2

S)

gamma marginal

σ

cΓ(h; h, σ2

H)cΓ(s; s, σ2 S); ρ

Gaussian copula

.

blanc

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

■

This probability distribution assigns vanishing probability to negative values of h and s.

SLIDE 50

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 28 / 33

Characterization of uncertainties (continued)

■

We estimate adequate values for the parameters of the bivariate gamma probability distribution by using the method of maximum likelihood as follows:

(ˆ h, ˆ σ2

H, ˆ

s, ˆ σ2

S, ˆ

ρ) = solution of max

(h,σ2

H,s,σ2 S,ρ)

l(h, σ2

H, s, σ2 S, ρ),

where the likelihood of the parameters h, σ2

H, s, σ2 S, and ρ is given by

l(h, σ2

H, s, σ2 S, ρ) = n

ℓ=1

ρ(H,S)(hobs

ℓ , sobs ℓ ; h, σ2 H, s, σ2 S, ρ).

300 350 400 450 500 1100 1300 1500 1700 1900

Yield stress [MPa] Hardening modulus [MPa]

SLIDE 51

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 52

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 53

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 54

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 55

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 56

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 57

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

SLIDE 58

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

■

Stochastic expansion method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ ↓ →

blanc Computationally inexpensive Computationally inexpensive surrogate model

→

300 350 400 450 500 1300 1400 1500 1600 1700 0.04 0.05 0.06 0.07 s [MPa] k [MPa] a [rad]

Computationally inexpensive surrogate model

→

Computationally inexpensive Computationally inexpensive surrogate model

SLIDE 59

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 29 / 33

Propagation of uncertainties

■

Monte Carlo method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ →

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

■

Stochastic expansion method:

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

→ ↓ →

blanc

300 350 400 450 500 1100 1300 1500 1700 1900 Yield stress [MPa] Hardening modulus [MPa]

Computationally inexpensive surrogate model

→

300 350 400 450 500 1300 1400 1500 1600 1700 0.04 0.05 0.06 0.07 s [MPa] k [MPa] a [rad]

Computationally inexpensive surrogate model

→

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

Computationally inexpensive surrogate model

SLIDE 60

Example: Metal forming

ULg, Liège, Belgium Nonintrusive probabilistic UQ 30 / 33

Sensitivity analysis of uncertainties

0.04 0.05 0.06 0.07 50 100 150 200 Angle [rad] Probability density [1/rad]

solid: PDF of output.

blanc

[0.046 rad, 0.059 rad]. 95%-confidence interval.

blanc blanc Propagation.

s_H s_S

sH, sS represent significance of inputs

in inducing uncertainties in output. blanc blanc blanc Sensitivity analysis.

SLIDE 61

Conclusion and outlook

ULg, Liège, Belgium Nonintrusive probabilistic UQ 31 / 33

■

Context and current practice are rich but entail limitations.

■

New methods are emerging for characterizing, propagating, and analyzing uncertainty:

◆ characterization of uncertain geometries, uncertain fields, and uncertain matrices.

◆ propagation of uncertainties by using Monte Carlo sampling.

◆ sensitivity analysis to guide resource allocation towards reducing uncertainty, robust design, robust control,. . .

■

These new methods are easily usable by engineers. Because they are nonintrusive, these new methods can be easily and effectively integrated with key tools (CAD, FEM,. . . ) used in industry.

SLIDE 62

References

ULg, Liège, Belgium Nonintrusive probabilistic UQ 32 / 33

■

The methods in this presentation are described in greater detail in the following paper:

■

Recent books on uncertainty quantification:

◆

R. Ghanem and P

. Spanos. Stochastic finite elements: a spectral approach. Dover, 2003.

◆

O. Le Maître and O. Knio. Spectral methods for uncertainty quantification: with application to

computational fluid dynamics. Springer, 2010.

◆

D. Xiu. Numerical methods for stochastic computations: a spectral method approach. Princeton University

Press, 2010.

◆

M. Grigoriu. Stochastic systems: uncertainty quantification and propagation. Springer, 2012.

◆

R. Smith. Uncertainty Quantification: Theory, Implementation, and Applications. SIAM, 2013.

◆

C. Soize. Stochastic models of uncertainties in computational mechanics. ASCE, 2014.

SLIDE 63

Contact information

ULg, Liège, Belgium Nonintrusive probabilistic UQ 33 / 33

Maarten Arnst Department of Aerospace and Mechanical Engineering University of Liège Quartier Polytech 1 B52/3 (office 0/419) Allée de la Découverte 13A B-4000 Liège, Belgium Email: maarten.arnst@ulg.ac.be Publications repository: http://orbi.ulg.ac.be/simple-search?query=arnst