DOI QR코드

DOI QR Code

Stochastic upscaling via linear Bayesian updating

  • Sarfaraz, Sadiq M. (Institute of Scientific Computing, Technische Universitat Braunschweig) ;
  • Rosic, Bojana V. (Institute of Scientific Computing, Technische Universitat Braunschweig) ;
  • Matthies, Hermann G. (Institute of Scientific Computing, Technische Universitat Braunschweig) ;
  • Ibrahimbegovic, Adnan (Lab. de Mecanique Roberval / Centre de Recherche Royallieu, Universite de Technologie de Compiegne)
  • Received : 2017.08.23
  • Accepted : 2017.09.28
  • Published : 2018.04.25

Abstract

In this work we present an upscaling technique for multi-scale computations based on a stochastic model calibration technique. We consider a coarse-scale continuum material model described in the framework of generalized standard materials. The model parameters are considered uncertain, and are determined in a Bayesian framework for the given fine scale data in a form of stored energy and dissipation potential. The proposed stochastic upscaling approach is independent w.r.t. the choice of models on coarse and fine scales. Simple numerical examples are shown to demonstrate the ability of the proposed approach to calibrate coarse scale elastic and inelastic material parameters.

Keywords

References

  1. Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2006), "Geometric means in a novel vector space structure on symmetric positive-definite matrices", SIAM J. Matr. Analy. Appl.
  2. Asokan, B. and Zabaras, N. (2006), "A stochastic variational multiscale method for diffusion in heterogeneous random media", J. Comput. Phys., 654-676.
  3. Bobrowski, A. (2005), Functional Analysis for Probability and Stochastic Processes, Cambridge, Cambridge University Press.
  4. Brady, L., Arwade, S., Corr, D., Gutierrez, M., Breysse, D., Grigoriu, M. and Zabaras, N. (2006), "Probability and materials: From nano-to macro-scale: A summary", Probab. Eng. Mech., 193-199.
  5. Clement, A., Soize, C. and Yvonnet, J. (2013), "Uncertainity quantification in computational stochastic multiscale analysis of nonlinear elastic materials", Comput. Meth. Appl. Mech. Eng., 61-82.
  6. Del Maso, G., De Simone, A. and Mora, M. (2006), "Quasistatic evolution problems for linearly elastic perfectly plastic materials", Arch. Rat. Mech. Analy., 237-291.
  7. Demmie, P. and Ostaja-Starzewski, M. (2015), "Local and non-local material models,spatial randomness and impact loading", Arch. Appl. Mech.
  8. Do, X.N., Ibrahimbegovic, A. and Brancherie, D. "Localized failure in damage dynamics", Coupled Syst. Mech., 4, 211-235.
  9. Do, X.N., Ibrahimbegovic, A. and Brancherie, D. (2015), "Combined hardening and localized failure with softening plasticity in dynamics", Coupled Syst. Mech., 4, 115-136. https://doi.org/10.12989/csm.2015.4.2.115
  10. Evensen, G. (2009), Data Assimilation-The Ensemble Kalman Filter, Berlin, Springer.
  11. Gelman, A., Carlin, J., Stern, H. and Rubin, D. (2014), Bayesian Data Analysis, Boca Raton, Taylor and Francis.
  12. Ghanem, R. and Das, S. (2011), Stochastic Upscaling for Inelastic Material Behavior from Limited Experimental Data, Computational Methods for Microstructure-Property Relationships, Springer, Berlin.
  13. Gorguluarslan, R. and Choi, S.K. (2014), "A simulation based upscaling technique for multiscale modeling of engineering systems under uncertainity", J. Multisc. Comput. Eng., 549-566.
  14. Halpen, B. and Nguyen, Q. (1974), "Plastic and visco-plastic materials with generalized potential", Mech. Res. Commun., 43-47.
  15. Halpen, B. and Nguyen, Q. (1975), "Sur les materiaux standard generalises", J. de Mecan., 39-63.
  16. Han, W. and Daya Reddy, B. (2013), Plasticity, Mathematical Theory and Numerical Analysis, Springer Verlag, New York, U.S.A.
  17. Hawkins-Daarud, A., Prudhomme, S., Van der Zee, K. and Oden, J. (2013), "Bayesian calibration, validation and uncertainity quantification of diffuse interface models of tumor growth", J. Math. Biol., 1457-1485.
  18. Ibrahimbegovic, A. (2009), Nonlinear Solid Mechanics, Springer, Berlin.
  19. Ibrahimbegovic, A. and Matthies, H.G. (2012), "Probabilistic multiscale analysis of inelastic localized failure in solid mechanics", Comput. Assist. Meth. Eng. Sci., 277-304.
  20. Ibrahimbegovic, A., Gharzeddine, F. and Chorfi, L. (1998), "Classical plasticity and viscoplasticity models reformulated: theoretical basis and numerical implementation", J. Numer. Meth. Eng., 1499-1535.
  21. Ibrahimbegovic, A., Markovic, D. and Gatuingt, F. (2003). "Constitutuve model of coupled damageplasticity and its finte element implementation", Rev. Europeenn. Des Elem., 381-405.
  22. Kaipio, J. and Somersalo, E. (2004), Statistical and Computational Inverse Problems, Springer, Berlin.
  23. Kennedy, M. and O'Hagan, A. (2001), "Bayesian calibration of computer models", J. Roy. Stat. Ser. B, 425-464.
  24. Koutsourelakis, P. (2007), "Stochastic upscaling in solid mechanics: An exercise in machine learning", J. Comput. Physi., 301-325.
  25. Liu, Y., Steven Greene, M., Chen, W., Dikin, D. and Liu, W. (2013), "Computational microstructure characterization and reconstruction for stochastc multiscale material design", Comput. Aid. Des., 65-76.
  26. Luenberger, D. (1969), Optimization by Vector Space Methods, John Wiley and Sons, Chichester.
  27. Markovic, D. and Ibrahimbegovic, A. (2006), "Complementary energy based FE modeling of coupled elasto-plastic and damage behavior for continuum microstructure compurtations", Comput. Meth. Appl. Mech. Eng., 5077-5093.
  28. Matthies, H.G. (1991), "Computation of constitutive response", In P. Wriggers, and W. Wagner, Nonlinear Computational Mechanics: State of the Art, Springer Verlag Berlin, Heidelberg.
  29. Matthies, H.G. (2007), "Uncertainity quantification with stochastic finite elements", Encyclop. Comput. Mech.
  30. Matthies, H.G. and Ibrahimbegovic, A. (2014), "Stochastic multiscale coupling of inelastic processes in solid mechanics. In M. Papadrakakis, and G. Stefanou", Multiscale Modelling and Uncertainity Quantification of Materials and Structures, Springer, Berlin.
  31. Matthies, H.G., Zander, E., Rosic, B., Litvinenko, A. and Pajonk, O. (2016), "Inverse problems in a Bayesian setting", In A. Ibrahimbegovic, Computational Methods for Solids and Fluids-Multiscale Analysis, Probability Aspects, and Model Reduction, Springer, Berlin.
  32. Ngo, V.M., Ibrahimbegovic, A. and Brancherie, D. (2014), "Stress-resultant model and finite element analysis of reinforced concrete frames under combined mechanical and thermal loads", Coupled Syst. Mech., 3, 111-144. https://doi.org/10.12989/csm.2014.3.1.111
  33. Ngo, V.M., Ibrahimbegovic, A. and Hajdo, E. (2014), "Nonlinear instability problems including localized plastic failure and large deformations for extreme thermomechanical load", Coupled Syst. Mech., 3, 89-110. https://doi.org/10.12989/csm.2014.3.1.089
  34. Nguyen, Q. (1977), "On the elastic plastic initial-boundary value problem and its numerical implementation", J. Numer. Meth. Eng., 817-832.
  35. Pajonk, O., Rosic, B., Litvinenko, A. and Matthies, H.M. (2012), "A deterministic filter for non- Gaussian Bayesian estimation-applications to dynamical system estimation with noisy measurements", Phys. D, 775-788.
  36. Papoulis, A. (1991), Probability Random Variables, and Stochastic Processes, McGraw-Hill, New York, U.S.A.
  37. Pian, T. and Sumihara, K. (1984), "Rational approach for assumed stress finite elements", J. Numer. Meth. Eng., 1685-1695.
  38. Rosic, B. and Matthies, H.G. (2014), "Variational theory and computations in stochastic plasticity", Arch. Comput. Meth. Eng., 457-509.
  39. Rosic, B., Litvinenko, A., Pajonk, O. and Matthies, H.G. (2012), "Sampling-free linear Bayesian update of polynomial chaos representation", J. Comput. Phys., 5761-5787.
  40. Rosic, B., Sykora, J., Pajonk, O., Kucerova, A. and Matthies, H.G. (2016), "Comparison of numerical approaches to Bayesian updating", In A. Ibrahimbegovic, Comutational Methods for Solids and Fluids-Multiscale Analysis, Probability Aspects and Model Reduction, Springer, Berlin.
  41. Starzewski, M. (2008), Microstructural Randomness and Scaling in Mechanics of Materials, Boca Raton, Chapman and Hall.
  42. Stefanou, G., Savvas, D. and Papadrakakis, M. (2015), "Stochastic finite element analysis of composite structure based on material microstructure", Compos. Struct., 384-392.
  43. Steven Greene, M., Liu, Y., Chen, W. and Liu, W. (2011), "Computational uncertainity analysis in multiresolution material via stochastic constitutive theory", Comput. Meth. Appl. Mech. Eng., 309-325.
  44. Suquet, P. and Lahellec, N. (2014), Elasto-Plasticity of Heterogeneous Materials at Different Scales, Procedia IUTAM.
  45. Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, U.S.A.
  46. Yvonnet, J. and Bonnet, G. (2014), "A consistent nonlocal scheme based on filter for the homogenenization of heterogeneous linear materials with non-separated scales", J. Sol. Struct., 196-209.

Cited by

  1. Review of Resilience-Based Design vol.9, pp.2, 2020, https://doi.org/10.12989/csm.2020.9.2.091
  2. Bayesian stochastic multi-scale analysis via energy considerations vol.7, pp.1, 2018, https://doi.org/10.1186/s40323-020-00185-y