An introduction to Bayesian inference for material parameter identification

An introduction to Bayesian inference for material parameter identification

Hussein Rappel, Lars Beex, Jack Hale, Stephane Bordas

[email protected]

Mathematical model

Parameters

Solver

Discretisation Computational

model Model reliability

Introduction

Mathematical model

Solver

Discretisation Computational

model Model reliability

Parameters

Error minimisation

Least squares method(conventional approach):

σ=E J = ¹₂ ^PN

i=1

(σ_i−E_i)² E =argmin

E

J(E)

σ

E

1

Example

Chance that a specific coin lands heads or tails

Frequentist inference

10 6

Pr(head) = ¹⁰₁₆

Frequentist inference

Pr(head) = ¹⁰₁₆ Pr(tail) = ₁₆⁶

Method of maximum likelihood(ML):

σ =E σi =Ei+ Ω

Ω :noise in stress measurement, it is a random variable

Frequentist inference: Young’s modulus identification

Method of maximum likelihood (ML):

Calibrateπ_noise(ω):

Method of maximum likelihood (ML):

Calibrateπ_noise(ω):

σ

Frequentist inference: Young’s modulus identification

Method of maximum likelihood (ML):

Calibrateπnoise(ω):

π_noise(ω) = ^{√ 1}

2πSnoise

exp− ^ω2

2S_noise²

σ

Method of maximum likelihood (ML):

σ_i =E_i + Ωwith π_noise(ω) = ^{√ 1}

2πSnoise

exp− ^ω2

2S_noise²

Frequentist inference: Young’s modulus identification

Method of maximum likelihood (ML):

σi =Ei + Ωwith π_noise(ω) = ^{√ 1}

2πSnoiseexp−_2S^ω2² noise

σi

_i

Method of maximum likelihood (ML):

σi =Ei+ Ω with πnoise(ω) = ^{√ 1}

2πSnoiseexp−_2S^ω2² noise

π(σi|E,Snoise) =

√ 1

2πS_noiseexp−(σ_i−E_i)² 2S_noise²

σi

_i

Frequentist inference: Young’s modulus identification

ML for M measurements:

σ^m =^hσ1, σ2,· · ·, σ_Mⁱ

^m =^h1, 2,· · · , M

i

π(σ^m|E,S_noise) =

1 (2πS_noise² )^M2

exp− PM i=1

(σ_i −E_i)² 2S_noise²

Bayesian inference

Given a coin is it biased or not?

If two rolls turn up head, do we have biased coin?

Original belief

Observqations

New belief

π(cause|effect) =

prior

z }| { π(cause)^×

likelihood

z }| { π(effect|cause) π(effect)

| {z }

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)

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_C ^i|E)

Bayesian inference: Young’s modulus identification

Bayes’ formula:

σi =Ei+ 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_C ^i|E) π(E|σ_i)∝π(E)π(σi|E)

BI for M measurment:

prior:π(E)π(σ1|E)π(σ2|E)· · ·π(σM−1|E) likelihood:π(σM|E)

π(E|σ_M)∝π(E)π(σ₁|E)π(σ₂|E)· · ·π(σM−1|E)π(σ_M|E)

Bayesian inference: Young’s modulus identification

π(E|σ_M)∝

M

Y

i=1

exp−(σ_i−E_i)² 2S_noise²

exp−(E−E)² 2S_E²

π(E|σ_M)∝

M

Y

i=1

exp−(σ_i−E_i)² 2S_noise²

exp−(E−E)² 2S_E²

π(E|σ_M)∝exp−(E−µ)² 2S_post²

with µ=f(σi,E,Snosie,S_E, i) S_post =f(σ_i,E,S_nosie,S_E, _i)

Frequentist vs Bayesian

Frequentist:

Data are a repeatable random sample there is a frequency Underlying parameters remain constant during this repeatable process

Parameters are fixed

Bayesian:

Data are observed from the realised sample

Parameters are unknown and described probabilistically Data are fixed

Error minimisation cannot take statistical info of measurement device into account

Why Bayesian?

Error minimisation cannot take statistical info of measurement device into account

You probably will not test hundreds of specimens and then the prior (πprior) may have a positive influence

Error minimisation cannot take statistical info of measurement device into account

You probably will not test hundreds of specimens and then the prior (π_prior) may have a positive influence

For inverse problems, the prior (πprior) regularises the system (avoids ill-posedness)

What have we accomplished?

Closed form expression of the posterior for:

elastoplasticity with perfect plasticity elastoplasticity with linear hardening elastoplasticity with nonlinear hardening

Closed form expression of the posterior for:

elastoplasticity with perfect plasticity elastoplasticity with linear hardening elastoplasticity with nonlinear hardening

The split between the purely elastic and elastoplastic problem is neatly incorporate

What have we accomplished?

‘Closed form’ expression of the posterior for:

elastoplasticity with perfect plasticity elastoplasticity with linear hardening elastoplasticity with nonlinear hardening

The split between the purely elastic and elastoplastic problem is neatly incorporate

The aformentioned ‘closed form posteriors’ have also been established when not only an stochastic error in the stress occurs but also in the strain

‘Closed form’ expression of the posterior for:

elastoplasticity with perfect plasticity elastoplasticity with linear hardening elastoplasticity with nonlinear hardening

The split between the purely elastic and elastoplastic problem is neatly incorporate

The aformentioned ‘closed form posteriors’ have also been established when not only an stochastic error in the stress occurs but also in the strain

What can be improved?

In all cases discussed, we search for parameters and we get a distribution of the parameters

In all cases discussed, we search for parameters and we get a distribution of the parameters

However, this is not the distribution of the heterogeneity in the material, but

a ‘certainty measure’ for the parameters

What can be improved?

In all cases discussed, we search for parameters and we get a distribution of the parameters

However, this is not the distribution of the heterogeneity in the material, but

a ‘certainty measure’ for the parameters

If heterogeneity is to be incorporated, we have to search for that as well, hence

search for parameters andthe distribution thereof→ future work...