Bayesian inference for material parameter identification

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Bayesian inference for material parameter identification

Hussein Rappel, Lars Beex, Jack Hale, Stephane Bordas

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Introduction

Mathematical model

Parameters

Computational

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Introduction

Mathematical model

Solver

Discretisation

Computational

model

_{Model reliability}

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Error minimisation

Least squares method(conventional

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Frequentist inference

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Frequentist inference

Pr (head ) =

10 ₁₆

_{Pr (tail ) =}

6

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Frequentist inference: Young’s modulus identification

Method of maximum likelihood

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Bayesian inference: Young’s modulus identification

Bayes’ formula:

σ

i

= E

i

+ Ω

Ω : noise in stress measurement

π(E |σ

i

) =

π(E )π(σ

_π(σ

i

|E )

i

)

=⇒ π(E |σ

i

) =

π(E )π(σ

i

|E )

C

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Why do we pick BI?

BI treats with the parameters as random variables

You probably will not test hundreds of specimens and then the prior

(π

prior

) may have a positive influence

For inverse problems, the prior (π

prior

) regularises the system (avoids

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What have we accomplished?

E (GPa) 0 100 200 300 400 500 : 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 :(E) :(Ej<m₎

Least squares method

0 #10!3 0 0.2 0.4 0.6 0.8 1 1.2 < (G P a) 0 0.05 0.1 0.15 0.2 0.25 0.3

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What have we accomplished?

240 220 E (GPa) 200 180 0.2 <y0(GPa) 0.25 0.4 1 0.8 0.6 0.2 0 0.3 #10!3 : #10-4 0 1 2 3 4 5 6 7 8 0 #10!3 0 0.5 1 1.5 2 2.5 < (G P a) 0 0.05 0.1 0.15 0.2 0.25 0.3

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What have we accomplished?

240 220 E (GPa) 200 180 0.2 <y0(GPa) 0.25 60 20 40 80 0.3 H (G P a) #10-3 0 0.5 1 1.5 0 #10!3 0 0.5 1 1.5 2 2.5 < (G P a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Bayesian inference for material parameter identification