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Model and parameter identification through Bayesian inference in solid mechanics Hussein Rappel h.rappel@gmail.com September 07, 2018 Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 1 / 69 Introduction:

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Model and parameter identification through Bayesian inference in solid mechanics

Hussein Rappel

h.rappel@gmail.com

September 07, 2018

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 1 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4 a 1 y = -ax + b

Introduction to Gaussian Processes, Neil Lawrence Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 2 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 3 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 4 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 5 / 69

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Introduction:

Probabilistic modelling

Each point can be written as the model+ a corruption: y1 = ax + c + ω1 y2 = ax + c + ω2 y3 = ax + c + ω3

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 6 / 69

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Introduction:

Probabilistic modelling

Pierre-Simon Laplace 1749-1827

(source wikipedia)

The corruption term can be presented with a probability distribution

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 7 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4 How can we fit the y = ax + b line, having

  • nly one point?

Introduction to Gaussian Processes, Neil Lawrence Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 8 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4 If b is fixed = ⇒ a = y-b

x

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 9 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 10 / 69

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Introduction:

Probabilistic modelling

y x 1 2 3 4 1 2 3 4 b ∼ π1 = ⇒ a ∼ π2

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 11 / 69

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Introduction:

Probabilistic modelling

This is called Bayesian treatment. The model parameters are treated as random variables.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 12 / 69

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Introduction:

Bayesian perspective

Original belief Observations New belief

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 13 / 69

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Introduction:

Bayesian formula (inverse probability)

posterior

  • π(x|y) =

prior

π(x)×

likelihood

  • π(y|x)

π(y)

evidence

y := observation x := parameter π(x) := original belief π(y|x) := given by the mathematical model that relates y to x π(y) := is a constant number

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 14 / 69

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Introduction:

Bayesian formula (inverse probability)

π(x|y) ∝ π(x) × π(y|x)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 15 / 69

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BI in computational mechanics

σ ε

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 16 / 69

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Linear elasticity

σ = Eε σ ε E 1

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 17 / 69

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Linear elasticity

y = Eε + ω Ω ∼ πω(ω)

Capital letters denote a random variable Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 18 / 69

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Linear elasticity

ε σ πω(ω) =

1 √ 2πsω exp

(

  • ω2

2s2

ω

) Noise PDF is modelled through calibration test.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 19 / 69

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Linear elasticity

Bayes’ formula: π(E|y) = π(E)π(y|E)

π(y)

= π(E)π(y|E)

k

π(E|y) ∝ π(E)π(y|E)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 20 / 69

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Linear elasticity

y = Eε + ω Ω ∼ N(0, s2

ω)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 21 / 69

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Linear elasticity

π(y|E) = 1 √ 2πsω exp

(

  • (y - Eε)2

2s2

ω

)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 22 / 69

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Linear elasticity

Posterior: π(E|y) ∝ exp

(

  • (E-E)2

2s2

E

)

exp

(

  • (y-Eε)2

2s2

ω

)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 23 / 69

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Linear elasticity

Prediction interval: An estimate of an interval in which an observation will fall, with a certain probability. Credible region: A region of a distribution in which it is believed that a random variable lie with a certain probability.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 24 / 69

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Linear elasticity

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 25 / 69

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Prior effect

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 26 / 69

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Model uncertainty and input error

y = f(x, ε) + ω Ω ∼ πω(ω)

ε is the input variable and x is the parameter vector Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 27 / 69

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Model uncertainty and input error

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 28 / 69

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Model uncertainty and input error

Kennedy-O’Hagan (KOH) framework: y = f(x, ε) + d(xd, ε)

  • ytrue

+ ω

ε is the input variable and x is the parameter vector Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 29 / 69

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Model uncertainty and input error

Constant number d0 Deterministic function ∑l

i=0 aiεi

Random variable from normal distribution d ∼ N(m, s2

d)

Random variable from a normal distribution with input dependent mean and variance d ∼ N(m(ε), s2

d(ε))

Gaussian process (GP) ...

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 30 / 69

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Model uncertainty and input error

Bayes’ formula: π(x, xd|y) ∝ π(x)π(xd)π(y|x, xd)

Both material and model error parameters must be inferred.

If d(xd, ε) is deterministic (for simplicity): π(x, xd|y) ∝ π(x)π(xd)πω(y - f(x, ε) - d(xd, ε))

xd parameter vector of model uncertainty Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 31 / 69

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Model uncertainty and input error

Bayes’ formula: π(x, xd|y) ∝ π(x)π(xd)π(y|x, xd)

Both material and model error parameters must be inferred.

If d(xd, ε) is deterministic (for simplicity): π(x, xd|y) ∝ π(x)π(xd)πω(y - f(x, ε) - d(xd, ε))

xd parameter vector of model uncertainty Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 31 / 69

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Model uncertainty and input error

But what about the input error?

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 32 / 69

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Model uncertainty and input error

y = f(x, ε) + d(xd, ε) + ω ε∗ = ε + ωε Ω ∼ π(ω) Ωε ∼ π(ωε)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 33 / 69

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Model uncertainty and input error

Bayes’ formula: π(x, xd, ε|y, ε∗) ∝ π(y|x, xd, ε)π(ε|ε∗)π(x)π(xd) π(x, xd|y, ε∗) ∝

∫ b

0 π(y|x, xd, ε)π(ε|ε∗)dε π(x)π(xd)

π(y|x, xd, ε) = πω(y - f(x, ε) - d(xd, ε)) π(ε|ε∗) = πωε(ε∗ - ε)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 34 / 69

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Model uncertainty and input error

Bayes’ formula: π(x, xd, ε|y, ε∗) ∝ π(y|x, xd, ε)π(ε|ε∗)π(x)π(xd) π(x, xd|y, ε∗) ∝

∫ b

0 π(y|x, xd, ε)π(ε|ε∗)dε π(x)π(xd)

π(y|x, xd, ε) = πω(y - f(x, ε) - d(xd, ε)) π(ε|ε∗) = πωε(ε∗ - ε)

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 34 / 69

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Model uncertainty and input error:

Linear elasticity

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 35 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 36 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

Normal distribution with constant parameters

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 37 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

Normal distribution with an input-dependent mean

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 38 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

Gaussian process

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 39 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

without input error with input error Normal distribution with constant parameters

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 40 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

without input error with input error Normal distribution with an input-dependent mean

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 41 / 69

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Model uncertainty and input error:

Effect

  • f model uncertainty as well as input error

without input error with input error Gaussian process

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 42 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 43 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening

σ(ε, x) =

  

Eε if ε ≤ σy0

E

σy0 + HE

H+E

(

ε - σy0

E

)

if ε > σy0

E

x = [E, σy0, H] ε σ

σy0 E

σy0 E 1 1 HE

H+E

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 44 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, extrapolation Error in the stress measurements only Error in both the stress and the strain measurements

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 45 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, extrapolation Normal distribution with constant parameters Normal distribution with an input-dependent mean

Input error is considered for both cases. Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 46 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, extrapolation Gaussian process

Input error is also considered. Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 47 / 69

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Material parameter distribution

Geometrical randomness Material parameter randomness

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 48 / 69

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Material parameter distribution

Objective

Find the distribution from which the parameters of the specimens are coming with limited number of specimens, N=20.

Parameter π(Parameter) a c

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 49 / 69

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Identification scheme

π(x|y) ∝ π(x)π(y|x)

Parameters Material model Measurements (stress) Measurement noise Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 50 / 69

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Identification scheme

y ∼ π(y|xm) − → (1) xm ∼ π(xm|xPDF) − → (2) xPDF ∼ π(xPDF) − → prior π(xm, xPDF|y) ∝ π(y|xm)π(xm|xPDF)π(xPDF)

Parameters Material model Measurements (stress) Measurement noise

(1) (2)

PDF's parameters PDF

Hard to incorporate when the parameter distribution

  • r model is not standard!

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 51 / 69

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Identification scheme:

Bayesian updating-Least squares

(1) − → Least squares (2) − → Bayesian updating π(xPDF|xm) ∝ π(xPDF|xm)π(xPDF)

Parameters Material model Measurements (stress) Measurement noise

(1) (2)

PDF's parameters PDF

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 52 / 69

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Copula

Copulas are tools enable us to model dependence of several random variables in terms of their marginal distribution.

a1 c1 a2 c2 xPDF1 xPDF2

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 53 / 69

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Proposed framework

Identify copula parameter (ρ) using both xm1 and xm2 Identify (xPDF1)MAP using xm1 Find xm1 & xm2 using Least squares method Identify (xPDF2)MAP using xm2

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 54 / 69

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Identification results:

Brittle damage

σ ε E 1 εf

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 55 / 69

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Identification results:

Brittle damage True Identified Identified without correlation

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 56 / 69

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Identification results:

Liner elastic-linear hardening

ε σ

σy0 E

σy0 E 1 1

HE H+E

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 57 / 69

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Identification results:

Liner elastic-linear hardening

True Identified

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 58 / 69

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Identification results:

Liner elastic-linear hardening

Identified without correlation

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 59 / 69

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Influence of geometrical randomness

A: Regular B: Partially random C: Random

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 60 / 69

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Influence of geometrical randomness:

Brittle damage

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 61 / 69

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Influence of geometrical randomness:

Linear elastic-linear hardenig

Equivalent parameters scatter plots are given.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 62 / 69

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Influence of geometrical randomness

Including correlation has no significant influence in damage case. The effect of correlation becomes significant for the elastoplastic case. Increasing geometry randomness decreases the influence of correlation. Correlation is more important for long fibres and large fibre densities.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 63 / 69

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Influence of geometrical randomness

Including correlation has no significant influence in damage case. The effect of correlation becomes significant for the elastoplastic case. Increasing geometry randomness decreases the influence of correlation. Correlation is more important for long fibres and large fibre densities.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 63 / 69

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Influence of geometrical randomness

Including correlation has no significant influence in damage case. The effect of correlation becomes significant for the elastoplastic case. Increasing geometry randomness decreases the influence of correlation. Correlation is more important for long fibres and large fibre densities.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 63 / 69

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Influence of geometrical randomness

Including correlation has no significant influence in damage case. The effect of correlation becomes significant for the elastoplastic case. Increasing geometry randomness decreases the influence of correlation. Correlation is more important for long fibres and large fibre densities.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 63 / 69

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Summary and conclusion

Probability is the natural way of modelling lack of our knowledge (what Laplace calls it our ignorance). From Bayesian perspective (inverse probability) the parameters are treated as random variables. In addition to number of measurements the influence of the prior is dependent to the model (e.g. for viscoelastcity the prior has more significant effect than elastoplasticity). Incorporating model uncertainty as well as input error improves both identification results and probabilistic predictions.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 64 / 69

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Summary and conclusion

Probability is the natural way of modelling lack of our knowledge (what Laplace calls it our ignorance). From Bayesian perspective (inverse probability) the parameters are treated as random variables. In addition to number of measurements the influence of the prior is dependent to the model (e.g. for viscoelastcity the prior has more significant effect than elastoplasticity). Incorporating model uncertainty as well as input error improves both identification results and probabilistic predictions.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 64 / 69

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Summary and conclusion

Probability is the natural way of modelling lack of our knowledge (what Laplace calls it our ignorance). From Bayesian perspective (inverse probability) the parameters are treated as random variables. In addition to number of measurements the influence of the prior is dependent to the model (e.g. for viscoelastcity the prior has more significant effect than elastoplasticity). Incorporating model uncertainty as well as input error improves both identification results and probabilistic predictions.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 64 / 69

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Summary and conclusion

Probability is the natural way of modelling lack of our knowledge (what Laplace calls it our ignorance). From Bayesian perspective (inverse probability) the parameters are treated as random variables. In addition to number of measurements the influence of the prior is dependent to the model (e.g. for viscoelastcity the prior has more significant effect than elastoplasticity). Incorporating model uncertainty as well as input error improves both identification results and probabilistic predictions.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 64 / 69

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Summary and conclusion

Bayesian updating-Least squares framework proposed for material distribution identification. We tried to answer the question of “how accurate the material parameter PDF needs to be identified?” in presence of geometrical randomness.

Material correlation is not important for damage. Material correlation is important elastoplasticity. The correlation effects gets less as the geometrical randomness increases.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 65 / 69

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Summary and conclusion

Bayesian updating-Least squares framework proposed for material distribution identification. We tried to answer the question of “how accurate the material parameter PDF needs to be identified?” in presence of geometrical randomness.

Material correlation is not important for damage. Material correlation is important elastoplasticity. The correlation effects gets less as the geometrical randomness increases.

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 65 / 69

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The End

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 66 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, interpolation Error in the stress measurements only Error in both the stress and the strain measurements

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 67 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, interpolation Normal distribution with constant parameters Normal distribution with an input-dependent mean

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 68 / 69

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Model uncertainty and input error:

Linear elastic-linear hardening, interpolation Gaussian process

Hussein Rappel (UL-ULg) BI for parameter identification September 07, 2018 69 / 69