Abstract
The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that most literature focuses either on complex statistical and machine learning concepts and/or on relatively complex mechanical models. In order to introduce the approach as gently as possible, we only focus on stress–strain measurements coming from uniaxial tensile tests and we only treat elastic and elastoplastic material models. Furthermore, the stress–strain measurements are created artificially in order to allow a one-to-one comparison between the true parameter values and the identified parameter distributions.
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Abbreviations
- Conditional probability distribution:
-
Conditional probability distribution \(\pi (y|x)\) provides the plausibility of proposition y, given proposition x
- Correlation:
-
A general term for the dependence between pairs of random variables
- Correlation coefficient:
-
A measure for the strength of the dependence between pairs of random variables
- Covariance:
-
A measure that shows how two random variables depend on each other
- Covariance matrix:
-
A symmetric matrix in which the off-diagonal elements are covariances of pairs of random variables and the diagonal elements are variances of random variables
- Credible interval (region):
-
An interval (or a region in the multivariate case) of a distribution in which it is believed that one or more random variables (parameters in this study) lie with a certain probability
- Dependence and independence:
-
Two events are statistically independent if the occurrence of one has no influence on the probability of the occurrence of the other one (i.e. \(\pi (x)=\pi (x|y)\)). They are dependent if the occurrence of one has an influence on the probability of the occurrence of the other one (i.e. \(\pi (x)\ne \pi (x|y)\))
- Event:
-
A set of outcomes of an experiment
- Joint distribution:
-
A multivariate distribution
- Laplace approximation:
-
An approximation of a distribution with a Gaussian distribution centred at the MAP
- Likelihood function:
-
If the conditional probability distribution \(\pi (y|x)\) is regarded as a function of x for given fixed y, the function is called a likelihood function. The likelihood describes the plausibility of a parameter, given observations
- Marginal distribution:
-
A probability distribution as a function of a single variable or a combination of subsets of variables associated with a multivariate distribution (e.g. \(\pi (x)\), \(\pi (y)\), \(\pi (x,y)\), \(\pi (x,z)\) and \( \pi (y,z)\), for joint distribution \(\pi (x,y,z)\)). A marginal distribution is obtained by integrating a multivariate distribution over one or more (but not all) other variables
- Markov chain:
-
A stochastic model to describe a sequence of events in which the probability of each event only depends on the previous event
- Markov chain Monte Carlo (MCMC) methods:
-
A set of techniques to draw samples (i.e. simulate observations) from probability distributions by the construction of a Markov chain
- Maximum a posteriori probability (MAP) point:
-
A point at which the posterior distribution is (globally) maximum
- Mean (expected value):
-
A measure for the central value of the underlying distribution
- Multivariate distribution:
-
A probability distribution of two or more random variables
- Point estimate:
-
A scalar that measures a feature of a population, e.g. the mean value, the MAP point
- Population:
-
The total set of all possible observations that can be made
- Posterior distribution (posterior):
-
The probability distribution that describes one’s knowledge about a random variable (parameter in this study) after obtaining new measurements
- Posterior predictive distribution (PPD):
-
The distribution of unobserved measurements (observations), given the measured (observed) data
- Prior distribution (prior):
-
The probability distribution that describes one’s a-priori knowledge about a random variable (parameter in this study)
- Probability:
-
The likelihood (or plausibility) that a certain event occurs
- Probability density function (PDF):
-
The equation that describes a continuous probability distribution
- Probability distribution:
-
A function that provides the probabilities of the occurrence of the possible outcomes of an experiment
- Random sample:
-
A randomly chosen sample
- Random variable:
-
A variable of which the value depends on the outcome of a random experiment
- Realisation:
-
The value that a random variable takes or the outcome of an experiment after its occurrence
- Sample:
-
A set of observations from a population with the purpose of investigating particular properties of the population
- Standard deviation:
-
A measure for the possible deviation of a random variable from its mean. Large standard deviations indicate large possible differences; and vice versa
- Validation point:
-
A measurement (observation) used to assess the quality of a prediction based on the identified parameters, that is not used for the identification itself
- Variance:
-
The standard deviation squared
References
Everitt BS, Skrondal A (2010) The Cambridge dictionary of statistics. Cambridge University Press, Cambridge
Walpole RE, Myers RH, Myers SL, Ye K (2013) Probability andstatistics for engineers and scientists. Pearson Custom Library, Pearson, London
Gogu C, Haftka R, Riche RL, Molimard J, Vautrin A (2010) Introduction to the Bayesian approach applied to elastic constants identification. AIAA J 48(5):893–903
Higdon D, Lee H, Bi Z (2002) A Bayesian approach to characterizing uncertainty in inverse problems using coarse and fine scale information. IEEE Trans Signal Process 50:388–399
Wang J, Zabaras N (2004) A Bayesian inference approach to the inverse heat conduction problem. Int J Heat Mass Transf 47(17–18):3927–3941
Risholm P, Janoos F, Norton I, Golby AJ, Wells WM (2013) Bayesian characterization of uncertainty in intra-subject non-rigid registration. Med Image Anal 17(5):538–555
Lan S, Bui-Thanh T, Christie M, Girolami M (2016) Emulation of higher-order tensors in manifold Monte Carlo methods for Bayesian inverse problems. J Comput Phys 308:81–101
Beck JL, Katafygiotis LS (1998) Updating models and their uncertainties. I: Bayesian statistical framework. J Eng Mech 124(4):455–461
Oh CK, Beck JL, Yamada M (2008) Bayesian learning using automatic relevance determination prior with an application to earthquake early warning. J Eng Mech 134(12):1013–1020
Kennedy MC, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc Ser B (Stat Methodol) 63(3):425–464
Kaipio J, Somersalo E (2006) Statistical and computational inverse problems, vol 160. Springer, Dordrecht
Isenberg J (1979) Progressing from least squares to Bayesian estimation. In: Proceedings of the 1979 ASME design engineering technical conference, New York, pp 1–11
Alvin KF (1997) Finite element model update via Bayesian estimation and minimization of dynamic residuals. AIAA J 35(5):879–886
Marwala T, Sibusiso S (2005) Finite element model updating using Bayesian framework and modal properties. J Aircr 42(1):275–278
Lai TC, Ip KH (1996) Parameter estimation of orthotropic plates by Bayesian sensitivity analysis. Compos Struct 34(1):29–42
Daghia F, de Miranda S, Ubertini F, Viola E (2007) Estimation of elastic constants of thick laminated plates within a Bayesian framework. Compos Struct 80(3):461–473
Koutsourelakis PS (2012) A novel Bayesian strategy for the identification of spatially varying material properties and model validation: an application to static elastography. Int J Numer Methods Eng 91(3):249–268
Gogu C, Yin W, Haftka R, Ifju P, Molimard J, Le Riche R, Vautrin A (2013) Bayesian identification of elastic constants in multi-directional laminate from moiré interferometry displacement fields. Exp Mech 53(4):635–648
Muto M, Beck JL (2008) Bayesian updating and model class selection for hysteretic structural models using stochastic simulation. J Vib Control 14(1–2):7–34
Liu P, Au SK (2013) Bayesian parameter identification of hysteretic behavior of composite walls. Probab Eng Mech 34:101–109
Fitzenz DD, Jalobeanu A, Hickman SH (2007) Integrating laboratory creep compaction data with numerical fault models: a Bayesian framework. J Geophys Res Solid Earth 112(B8):B08410. https://doi.org/10.1029/2006JB004792
Most T (2010) Identification of the parameters of complex constitutive models: least squares minimization vs. Bayesian updating. In: Straub D (ed) Reliability and optimization of structural systems. CRC Press, New York, pp 119–130
Rosić BV, Kčerová A, Sýkora J, Pajonk O, Litvinenko A, Matthies HG (2013) Parameter identification in a probabilistic setting. Eng Struct 50:179–196
Hernandez WP, Borges FCL, Castello DA, Roitman N, Magluta C (2015) Bayesian inference applied on model calibration of fractional derivative viscoelastic model. In: Steffen Jr V, Rade DA, Bessa WM (eds) DINAME 2015-proceedings of the XVII international symposium on dynamic problems of mechanics, Natal
Rappel H, Beex LAA, Bordas SPA (2018) Bayesian inference to identify parameters in viscoelasticity. Mech Time-Depend Mater 22(2):221–258
Nichols JM, Link WA, Murphy KD, Olson CC (2010) A Bayesian approach to identifying structural nonlinearity using free-decay response: application to damage detection in composites. J Sound Vib 329(15):2995–3007
Abhinav S, Manohar CS (2015) Bayesian parameter identification in dynamic state space models using modified measurement equations. Int J Non-Linear Mech 71:89–103
Madireddy S, Sista B, Vemaganti K (2015) A Bayesian approach to selecting hyperelastic constitutive models of soft tissue. Comput Methods Appl Mech Eng 291:102–122
Oden JT, Prudencio EE, Hawkins-Daarud A (2013) Selection and assesment of phenomenological models of tumor growth. Math Models Methods Appl Sci 23(7):1309–1338
Chiachío J, Chiachío M, Saxena A, Sankararaman S, Rus G, Goebel K (2015) Bayesian model selection and parameter estimation for fatigue damage progression models in composites. Int J Fatigue 70:361–373
Babuška I, Sawlan Z, Scavino M, Szabó B, Tempone R (2016) Bayesian inference and model comparison for metallic fatigue data. Comput Methods Appl Mech Eng 304:171–196
Sarkar S, Kosson DS, Mahadevan S, Meeussen JCL, van der Sloot H, Arnold JR, Brown KG (2012) Bayesian calibration of thermodynamic parameters for geochemical speciation modeling of cementitious materials. Cement Concr Res 42(7):889–902
Cotter SL, Dashti M, Robinson JC, Stuart AM (2009) Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Probl 25(11):115008
Simo JC, Hughes TJR (2000) Computational inelasticity. Springer, New York
Ulrych TJ, Sacchi MD, Woodbury A (2001) A Bayes tour of inversion: a tutorial. Geophysics 66(1):55–69
Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis. Chapman & Hall/CRC texts in statistical science. Chapman & Hall/CRC, London
Beck JL, Au SK (2002) Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J Eng Mech 128(4):380–391
Marzouk YM, Najm HN, Rahn LA (2007) Stochastic spectral methods for efficient Bayesian solution of inverse problems. J Comput Phys 224(2):560–586
Kristensen J, Zabaras N (2014) Bayesian uncertainty quantification in the evaluation of alloy properties with the cluster expansion method. Comput Phys Commun 185(11):2885–2892
Andrieu C, De Freitas N, Doucet A, Jordan MI (2003) An introduction to MCMC for machine learning. Mach Learn 50(1–2):5–43
Brooks S, Gelman A, Jones G, Meng XL (2011) Handbook of Markov chain Monte Carlo. CRC Press, Boca Raton
Sinharay S (2003) Assessing convergence of the Markov chain Monte Carlo algorithms: ad review. ETS Res Rep Ser 2003(1):i-52
Gelman A, Roberts GO, Gilks WR (1996) Efficient Metropolis jumping rules. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 5. Oxford Science Publications. Oxford University Press, New York, pp 599–607
Haario H, Saksman E, Tamminen J (1999) Adaptive proposal distribution for random walk Metropolis algorithm. Comput Stat 14(3):375–396
Beck JL (2010) Bayesian system identification based on probability logic. Struct Control Health Monit 17(7):825–847
Prince SJD (2012) Computer vision: models learning and inference. Cambridge University Press, Cambridge
Rappel H, Beex LAA, Noels L, Bordas SPA (2018) Identifying elastoplastic parameters with Bayes’ theorem considering output error, input error and model uncertainty. Probab Eng Mech. https://doi.org/10.1016/j.probengmech.2018.08.004
Ling Y, Mullins J, Mahadevan S (2014) Selection of model discrepancy priors in Bayesian calibration. J Comput Phys 276(Supplement C):665–680
Bishop C (2006) Pattern recognition and machine learning. Information science and statistics. Springer, Berlin
Arhonditsis GB, Papantou D, Zhang W, Perhar G, Massos E, Shi M (2008) Bayesian calibration of mechanistic aquatic biogeochemical models and benefits for environmental management. J Mar Syst 73(1):8–30
Xiong Y, Chen W, Tsui KL, Apley DW (2009) A better understanding of model updating strategies in validating engineering models. Comput Methods Appl Mech Eng 198(15):1327–1337
Arendt PD, Apley DW, Chen W (2012) Quantification of model uncertainty: calibration, model discrepancy, and identifiability. J Mech Des 134(10):100908
Brynjarsdóttir J, O’Hagan A (2014) Learning about physical parameters: the importance of model discrepancy. Inverse Probl 30(11):114007
Acknowledgements
Hussein Rappel, Lars A.A. Beex and Stéphane P.A. Bordas would like to acknowledge the financial support from the University of Luxembourg. Hussein Rappel and Lars A.A. Beex are also grateful for the support of the Fonds National de la Recherche Luxembourg DFG-FNR grant INTER/DFG/16/1150192. Stéphane P.A. Bordas and Lars A.A. Beex are also grateful for the support of the Fonds National de la Recherche Luxembourg FNRS-FNR grant INTER/FNRS/15/11019432/EnLightenIt/Bordas. Jack S. Hale received funding from the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND Grant No. 6693582).
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Rappel, H., Beex, L.A.A., Hale, J.S. et al. A Tutorial on Bayesian Inference to Identify Material Parameters in Solid Mechanics. Arch Computat Methods Eng 27, 361–385 (2020). https://doi.org/10.1007/s11831-018-09311-x
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DOI: https://doi.org/10.1007/s11831-018-09311-x