Isogeometric and stochastic collocation methods for nonlinear probabilistic multiscale problems in solid mechanics

• Applicants: Professorin Dr. Laura De Lorenzis; Professor Dr. Hermann Georg Matthies
• Subject Area: Mechanics
• Term: from 2014 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255747201

• Final Report Year: 2019

Final Report Abstract

The original aim of the project was to reduce the computational burden which arises from the numerical integration of the projection of residual terms in a Galerkin-like projection setting. At the start, the use of collocation methods looked like one promising avenue to reach this goal. But the numerical analysis and experiments have shown that the stochastic collocation method — essentially an interpolation — may produce instabilities when used for the propagation of uncertainties in nonlinear material models. To overcome this, the main focus of the project has changed from the collocation based to the regression/projection based mixed methods in a variational point of view. To allow efficient computations, we further focused on the reduction of the number of integration points by taking a Bayesian point of view on the considered numerical algorithms. In this manner the sparsity or lowrank approximations can be effectively introduced to the numerical procedure via suitable prior models. In this way the original aim of the project — reduction of evaluation points — was achieved, without insisting on the particular method of collocation. Indeed, the Bayesian approach in many cases allows even fewer evaluation points than collocation. In this regard, due of the instabilities observed for the collocation method when used for highly non-linear material models like plasticity, softening due to damage, etc., the initial plan of the project was abandoned. It was reformulated in such a way that we now wanted to compute proxy models with a few evaluations as possible of the full model. By using the the stochastic pseudo-spectral projection method, the approach was put again into a variational setting. The idea to reduce the number of evaluation points even further through a Bayesian — probabilistic numerics — point of view was then again integrated in the variational framework by noting that the conditional expectation can be formulated in this way. The prior was taken from the assumption that the coefficient tensor to be computed is sparse, and this has shown a surprising initial success. Considerable effort in the project was put into further development of Bayesian updating through the approximation of the conditional expectation, which has led to a number of new ‘filtering’ algorithms. However, even though the Bayesian perspective on probabilistic computations is natural, it is also expensive if the Bayesian updating is not done efficiently. To reduce the computational costs, we have studied different versions of approximate Bayesian rules that are based on conditional expectation. In this manner filtering procedures for the estimation of the posterior mean and covariance were developed. They could then further be used in other contexts like upscaling of random heterogeneous materials, and successfully used for the identification of material parameters describing reversible and irreversible phenomena.

Publications

• Comparison of Numerical Approaches to Bayesian Updating, In: A. Ibrahimbegović (Ed.), Multiscale Analysis, Probability Aspects, and Model Reduction, Computational Methods for Solids and Fluids, 41: 427–462, 2016
  B. Rosić, J. Sýkora, O. Pajonk, A. Kučerová, and H. G. Matthies
  
• Inverse Problems in a Bayesian Setting. In: A. Ibrahimbegović (Ed.), Multiscale Analysis, Probability Aspects, and Model Reduction, Computational Methods for Solids and Fluids, 41: 245–286, 2016
  H. G. Matthies, E. Zander, B. Rosić, A. Litvinenko, and O. Pajonk
  
• Parameter Estimation via Conditional Expectation — A Bayesian Inversion, Advanced Modeling and Simulation in Engineering Sciences, 3: 24, 2016
  H. G. Matthies, E. Zander, B. Rosić, and A. Litvinenko
  
• Sparse Bayesian polynomial chaos approximations of elasto-plastic material models, COMPLAS 2017 Proceedings, E. Oñate, D.R.J. Owen, D. Perić, and M. Chiumenti (Eds.), 256–267, CIMNE, Barcelona, 2017
  B. Rosić and H. G. Matthies
  
• Uncertainty Quantification and Bayesian Inversion. In: E. Stein, R. de Borst, and T. R. J. Hughes (eds.). Encyclopedia of Computational Mechanics 1 — Fundamentals, 2nd ed., John Wiley & Sons, Chichester, 2017
  H. G. Matthies
  
• Variational formulation with error estimates for uncertainty quantification via collocation, regression, and spectral projection. Proc. Appl. Math. and Mech., 17: 79–82, 2017
  J. Rang and H. G. Matthies