material model parameter identification via markov chain monte carlo christian knipprath 1...

Material Model Parameter Identification via Markov Chain Monte Carlo

Christian Knipprath1 Alexandros A. Skordos2

www.bris.ac.uk/composites

1 – ACCIS, University of Bristol, UK

2 – Composites Centre, Cranfield University, UK

2/19Roadmap

• Introduction– Origin of the MCMC method

– Motivation

• Numerical Implementation– Outline of the algorithm used

– Application to a simple problem

• Application to material models for composites – Description of constitutive model

– Conventional parameter identification method

– Employment of MCMC algorithm

– Comparison of conventional to MCMC

• Conclusions

5th CompTest conference14-16 February 2011

3/19Introduction

• Origins of Markov Chain Monte Carlo

– First presented in 1953 (Metropolis algorithm) by physicists from the Los Alamos National Laboratory to compute the potential fields of molecules in liquids

– Generalisation by Hastings lead to the Metropolis-Hasting.

– Conceptually able of solving inverse problems. Such problems are often subject to ill-posedness and benefit from the regularising properties of MCMC

– MCMC was a well established method in the context of statistical physics before the method was applied to other fields in the 1990s. Initially in econometric and financial modelling.


4/19Introduction

• Motivation

– Treatment of the material model as an inverse problem.

– Certainty of the parameters as an answer of the model response with relation to experimental data sets

– Uncertainty quantification

– Simultaneous analysis of large data sets


5/19

• Random walk Metropolis-Hastings algorithm– Iterative algorithm

– Employment of Bayes’ theorem of conditional probability to compute the joint posterior used in the acceptance probability

Numerical Implementation

posterior

likelihoodprior

marginal


6/19Numerical Implementation

• Joint likelihood distribution– Addresses the certainty of an

experimental data point in respect to the theoretical value

– All data points are considered

• Joint prior distribution– Addresses the certainty of a

proposed parameter considering its characteristic distribution

– Appropriate distribution to describe the parameter best is often difficult to identify, hence normal distribution is a “generic” choice

– Note: Traditional techniques imply of uniform prior

Log-Normal distribution

Normal distribution



• Application to a simple problem– Hooke’s law

– Setup:

• starting value 6 GPa

• 1000 iterations

• Tuning to reach acceptance ratio of 48%

• Burn-in range: 200 iterations



• Use of auto-correlation function (ACF) to determine thinning step size for sequence

• The resulting sample vector contains uncorrelated values gathered from the stationary sequence


9/19Application

• Ladevèze model for in-plane damage– Thermodynamic framework model

– Constitutive law used effective properties

– Damage evolution is determined using the energy dissipation threshold values

– Inelastic strains are computed via a Hill-type yield criterion


10/19Application

• Cohesive law for out-of-plane behaviour– Bilinear law

– Definition of stresses and energy limits

• For the use in MCMC both material models were implemented in an explicit manner

• Parameter set comprises 27 parameters (no rate effects)


11/19Application

• Conventional parameter identification method– Experiments required for in-plane parameter identification

• [0º]8 in tension and compression

• [±45º]2s, [+45º]2s, [±67.5º]2s under cyclic tensile loading for damage behaviour

– Mode I & II delamination for cohesive interfaces

– Information is extracted from experimental data (shown for shear)


12/19Application


• Employment of MCMC algorithm– Setup:

• Starting vector obtained from conventional method on single experiment

• 4,000,000 iterations

• Tuning to reach acceptance probability of around 25%

• Burn-in range: 1,000,000 iterations

• 4 chains in parallel

• Application of 3 convergenceassessment methods

13/19Application

• Probability density plots– Elastic tensile modulus in fibre

directions shows a single mode answer for the PDF

– Additional modes were found for m and R0


14/19Application

• Application of Parallel Tempering (PT)– Introduction of temperature parameter with

– Tempering parameter is used for numerical purposes:

– Higher values for T flatten the target distribution and allow the acceptance of a broader range of proposed parameters. These distributions are less likely to be trapped in local modes

– Parameter sets are swapped between chains based on a computed swapping probability

– Only the neutral (=1) can be used for the analysis


15/19Application

• Comparison for a single experiment– For transverse response the conventional method indicated

premature failure

– In the shear response stresses are over-predicted by the conventional method


16/19Comparison

• Comparison for a single experiment– Compression

- non-linear behaviour due to fibre buckling

– Conventional method leads to a value of 0.064

– MCMC method leads yields a value of 6.37±1.86


17/19Conclusion

• Parameters were identified with additional information provided by probability density functions

• This type of information can be the basis of stochastic simulations of the mechanical and damage response

• Although mean/median values are used in FE models the PDFs provide further information on parameters

• Further development will address– Tuning procedure

• Automated tuning algorithm

– Overall runtime

• Parallelisation


18/19

Financial support from the CEC under the PreCarBi project (FP6-30848) is gratefully acknowledged

Acknowledgement


19/19

Thank you for your attention.

[email protected]


material model parameter identification via markov chain monte carlo christian knipprath 1...

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