1. Material Model Parameter Identification via Markov Chain Monte Carlo Christian Knipprath 1 Alexandros A. Skordos 2 www.bris.ac.uk/composites 1 – ACCIS, University of Bristol, UK 2 – Composites Centre, Cranfield University, UK

2. 2/19 Roadmap 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 5 th CompTest conference 14-16 February 2011

3. 3/19 Introduction 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. 5 th CompTest conference 14-16 February 2011

4. 4/19 Introduction 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 th CompTest conference 14-16 February 2011

5. 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 likelihood prior marginal 5 th CompTest conference 14-16 February 2011

6. 6/19 Numerical 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 5 th CompTest conference 14-16 February 2011

7. 7/19 Numerical Implementation 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 5 th CompTest conference 14-16 February 2011

8. 8/19 Numerical Implementation 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 5 th CompTest conference 14-16 February 2011

9. 9/19 Application 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 5 th CompTest conference 14-16 February 2011

10. 10/19 Application 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) 5 th CompTest conference 14-16 February 2011

11. 11/19 Application Conventional parameter identification method –Experiments required for in-plane parameter identification [0 º ] 8 in tension and compression [±45 º ] 2s, [+45 º ] 2s, [±67.5 º ] 2s under cyclic tensile loading for damage behaviour –Mode I & II delamination for cohesive interfaces –Information is extracted from experimental data (shown for shear) 5 th CompTest conference 14-16 February 2011

12. 12/19 Application 5 th CompTest conference 14-16 February 2011 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 convergence assessment methods

13. 13/19 Application Probability density plots –Elastic tensile modulus in fibre directions shows a single mode answer for the PDF –Additional modes were found for m and R 0 5 th CompTest conference 14-16 February 2011

14. 14/19 Application 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 5 th CompTest conference 14-16 February 2011

15. 15/19 Application 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 5 th CompTest conference 14-16 February 2011

16. 16/19 Comparison 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 5 th CompTest conference 14-16 February 2011

17. 17/19 Conclusion 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 5 th CompTest conference 14-16 February 2011

18. 18/19 Financial support from the CEC under the PreCarBi project (FP6-30848) is gratefully acknowledged Acknowledgement 5 th CompTest conference 14-16 February 2011

19. 19/19 Thank you for your attention. c.knipprath@bristol.ac.uk 5 th CompTest conference 14-16 February 2011