Bayesian inference of non-linear multiscale model parameters accelerated by a Deep Neural Network

Abstract We develop a Bayesian Inference (BI) of the parameters of a non-linear multiscale model and of its material constitutive laws using experimental composite coupon tests as observation data. In particular we consider non-aligned Short Fibers Reinforced Polymer (SFRP) as a composite material system and Mean-Field Homogenization (MFH) as a multiscale model. Although MFH is computationally efficient, when considering non-aligned inclusions, the evaluation cost of a non-linear response for a given set of model and material parameters remains too prohibitive to be coupled with the sampling process required by the BI. Therefore, a Neural-Network (NNW) is first trained using the MFH model, and is then used as a surrogate model during the BI process, making the identification process affordable.

[1]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[2]  Charles L. Tucker,et al.  Stiffness and thermal expansion predictions for hybrid short fiber composites , 1990 .

[3]  Stéphane Bordas,et al.  A Tutorial on Bayesian Inference to Identify Material Parameters in Solid Mechanics , 2019, Archives of Computational Methods in Engineering.

[4]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[5]  Hussein Rappel,et al.  Estimating fibres’ material parameter distributions from limited data with the help of Bayesian inference , 2019, European Journal of Mechanics - A/Solids.

[6]  Andrea Vigliotti,et al.  Bayesian inference of the spatial distributions of material properties , 2018, Journal of the Mechanics and Physics of Solids.

[7]  J. Willis,et al.  Variational Principles for Inhomogeneous Non-linear Media , 1985 .

[8]  Rodney Hill,et al.  Continuum micro-mechanics of elastoplastic polycrystals , 1965 .

[9]  Thomas Most,et al.  Identification of the parameters of complex constitutive models: Least squares minimization vs. Bayesian updating , 2010 .

[10]  Heikki Haario,et al.  Adaptive proposal distribution for random walk Metropolis algorithm , 1999, Comput. Stat..

[11]  Hedi Nouri.  Modélisation et identification de lois de comportement avec endommagement en fatigue polycyclique de matériaux composite a matrice thermoplastique , 2009 .

[12]  Marc G. D. Geers,et al.  A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials , 2017, J. Comput. Phys..

[13]  Pedro Ponte Castañeda.  A new variational principle and its application to nonlinear heterogeneous systems , 1992 .

[14]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[15]  B. Kenmeugne,et al.  Improvements of multiaxial fatigue criteria computation for a strong reduction of calculation duration , 1999 .

[16]  Ludovic Noels,et al.  Bayesian identification of Mean-Field Homogenization model parameters and uncertain matrix behavior in non-aligned short fiber composites , 2019, Composite Structures.

[17]  Wei Chen,et al.  A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality , 2017 .

[18]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[19]  I. Doghri,et al.  A combined incremental-secant mean-field homogenization scheme with per-phase residual strains for elasto-plastic composites , 2013 .

[20]  Kumar Vemaganti,et al.  A Bayesian approach to selecting hyperelastic constitutive models of soft tissue , 2015 .

[21]  M. Vincent,et al.  Description and modeling of fiber orientation in injection molding of fiber reinforced thermoplastics , 2005 .

[22]  Fredrik Larsson,et al.  On-the-Fly Adaptivity for Nonlinear Twoscale Simulations Using Artificial Neural Networks and Reduced Order Modeling , 2019, Front. Mater..

[23]  Carsten Könke,et al.  Coupling of scales in a multiscale simulation using neural networks , 2008 .

[24]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[25]  Gaël Varoquaux,et al.  Scikit-learn: Machine Learning in Python , 2011, J. Mach. Learn. Res..

[26]  John R. Willis,et al.  Bounds and self-consistent estimates for the overall properties of nonlinear composites , 1987 .

[27]  Francesco Ubertini,et al.  Estimation of elastic constants of thick laminated plates within a Bayesian framework , 2007 .

[28]  B. A. Le,et al.  Computational homogenization of nonlinear elastic materials using neural networks , 2015 .

[30]  I. Doghri,et al.  Micromechanics of inelastic composites with misaligned inclusions: Numerical treatment of orientation , 2006 .

[31]  K. Ip,et al.  Parameter estimation of orthotropic plates by Bayesian sensitivity analysis , 1996 .