A Bayesian approach for parameter identification in elastoplasticity
Hussein Rappel, Lars Beex, Jack Hale, St´ephane Bordas
hussein.rappel@uni.lu
Introduction Mathematical model Parameters Solver Discretisation Computational
Introduction
Mathematical model
Computational
model Model reliability
Error minimisation
Least squares method
(conventional approach): σ = E J = 12 N P i =1 (σi− E i)2 E = argmin E J (E ) σ E 1
Frequentist inference
Frequentist inference
Frequentist inference: Young’s modulus identification
σ
Frequentist inference: Young’s modulus identification σ πnoise(ω) = √2πS1 noise exp− ω2 2S2 noise
Frequentist inference: Young’s modulus identification σm= E + Ω Ω ∼ πnoise(ω) σ σm
Frequentist inference: Young’s modulus identification
Method of maximum likelihood (ML):
π(σm|E ) = √ 1 2πSnoise exp−(σ m− E )2 2Snoise2
Frequentist inference: Young’s modulus identification
Method of maximum likelihood (ML):
π(σm|E ) = √ 1 2πSnoise exp−(σ m− E )2 2S2 noise
and for M measurments:
π(σm|E ) = 1 (2πSnoise2 )M2 exp− M P i =1 (σim− E i)2 2Snoise2 σm =hσ1, σ2, · · · , σM i
Bayesian inference Original belief Observations New belief π(cause|effect) = prior z }| { π(cause)× likelihood z }| { π(effect|cause) π(effect) | {z } evidence
Bayesian inference: Young’s modulus identification σ πnoise(ω) = √2πS1 noise exp− ω2 2S2 noise
Bayesian inference: Young’s modulus identification σm= E + Ω Ω ∼ πnoise(ω) σ σm
Bayesian inference: Young’s modulus identification
Bayes’ formula:
Bayesian inference: Young’s modulus identification
Bayes’ formula:
π(E |σm) = π(E )π(σπ(σmm)|E ) =⇒
π(E )π(σm|E )
Bayesian inference: Young’s modulus identification Bayes’ formula: π(E |σm) = π(E )π(σπ(σmm)|E ) =⇒ π(E )π(σm|E ) C π(E |σm) ∝ π(E )π(σm|E )
Bayesian inference: Young’s modulus identification π(E |σm) ∝ exp−(E − E ) 2 2S2 E exp−(σ m− E )2 2S2 noise
Bayesian inference: Young’s modulus identification π(E |σm) ∝ exp−(E − E ) 2 2SE2 exp−(σ m− E )2 2Snoise2
and for M measurements:
π(E |σm) ∝ M Y i =1 exp−(E − E ) 2 2SE2 exp−(σ m i − E i)2 2Snoise2
Bayesian inference: Young’s modulus identification π(E |σm) ∝ exp−(E − µ) 2 2S2 post with µ = f (σi, E , Snosie, SE, i) Spost = f (σi, E , Snosie, SE, i)
Bayesian inference: Young’s modulus identification : 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 :(E) :(Ej<m)
Least squares method
< (G P a) 0 0.05 0.1 0.15 0.2 0.25 0.3
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-perfectly plastic model identification 250 E (GPa) 200 0.2 <y0(GPa) 0.25 1 0.8 0.6 0.4 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
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-linear hardening model identification 240 220 0.25 90 20 30 40 50 60 70 80 0.3 H (G P a) #10-3 0.5 1 1.5 < (G P a) 0.05 0.1 0.15 0.2 0.25 0.3 0.35
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-nonlinear hardening model identification 0 #10!3 0 1 2 3 4 5 < (G P a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
95% credible region curves Measured value Mean
Bayesian inference: Young’s modulus identification with double uncertainty σm= E + Ωσ m= + Ω Ωσ and Ω ∼ πnoise(ω) σ πnoise(ω) σm
Bayesian inference: Young’s modulus identification with double uncertainty 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; 0m) :(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
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-perfectly plastic with double uncertainty < (G P a) 0.05 0.1 0.15 0.2 0.25 0.3
95% credible region curves Measured value Mean < (G P a) 0.05 0.1 0.15 0.2 0.25 0.3
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-linear hardening with double uncertainty 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
95% credible region curves Measured value Mean single uncertainty 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
95% credible region curves Measured value Mean
Bayesian inference: linear elastic-nonlinear hardening with double uncertainty < (G P a) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
95% credible region curves Measured value Mean < (G P a) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
95% credible region curves Measured value Mean
Conclousion
‘Closed form expression’ of the posterior for:
linear elasticity,
elastoplasticity with perfect plasticity, elastoplasticity with linear hardening, and elastoplasticity with nonlinear hardening.
Conclousion
‘Closed form expression’ of the posterior for:
linear elasticity,
elastoplasticity with perfect plasticity, elastoplasticity with linear hardening, and elastoplasticity with nonlinear hardening.
The results of BI cannot directly be compared to those of the least squares method.
Conclousion
‘Closed form expression’ of the posterior for:
linear elasticity,
elastoplasticity with perfect plasticity, elastoplasticity with linear hardening, and elastoplasticity with nonlinear hardening.
The results of BI cannot directly be compared to those of the least squares method.
Conclousion
‘Closed form expression’ of the posterior for:
linear elasticity,
elastoplasticity with perfect plasticity, elastoplasticity with linear hardening, and elastoplasticity with nonlinear hardening.
The results of BI cannot directly be compared to those of the least squares method.
The selected prior distribution has a significant effect on the results. BI leads to a distribution for the considered parameters, however the
Future work E (GPa) 160 180 200 220 240 260 : 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Specimen elastic modulus PDF Posterior PDF
A special acknowledgement
