Parameter identification for phase-field modeling of fracture: a Bayesian approach with sampling-free update
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ORIGINAL PAPER
Parameter identification for phase-field modeling of fracture: a Bayesian approach with sampling-free update T. Wu1 · B. Rosi´c2 · L. De Lorenzis1 · H. G. Matthies3 Received: 15 June 2020 / Accepted: 27 October 2020 © The Author(s) 2020
Abstract Phase-field modeling of fracture has gained popularity within the last decade due to the flexibility of the related computational framework in simulating three-dimensional arbitrarily complicated fracture processes. However, the numerical predictions are greatly affected by the presence of uncertainties in the mechanical properties of the material originating from unresolved heterogeneities and the use of noisy experimental data. The objective of this work is to apply the Bayesian approach to estimate bulk and shear moduli, tensile strength and fracture toughness of the phase-field model, thus improving accuracy of the simulations with the help of experimental data. Conventional approaches for estimating the Bayesian posterior probability density function adopt sampling schemes, which often require a large amount of model estimations to achieve the desired convergence, thus resulting in a high computational cost. In order to alleviate this problem, we employ a more efficient approach called sampling-free linear Bayesian update, which relies on the evaluation of the conditional expectation of parameters given experimental data. We identify the mechanical properties of cement mortar by conditioning on the experimental data of the three-point bending test (observations) in an online and offline manner. In the online approach the parameter values are sequentially updated on the fly as the new experimental information comes in. In contrast, the offline approach is used only when the whole history of experimental data is provided once the experiment is performed. Both versions of estimation are discussed and compared by validating the phase-field fracture model on an unused set of experimental data. Keywords Parameter identification · Phase-field model · Fracture · Bayesian approach · Linear Bayesian Update
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Phase-field modeling of brittle fracture . . . . . . . . . . . . 3 Bayesian updating . . . . . . . . . . . . . . . . . . . . . . . 3.1 Linear Bayesian update . . . . . . . . . . . . . . . . . . 3.2 PCE based sampling-free linear Bayesian update . . . . . 4 Online and offline identification . . . . . . . . . . . . . . . . 5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimental setup and simulation of the three-point bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Choice of the a priori . . . . . . . . . . . . . . . . . . .
B
T. Wu [email protected]
1
Institute of Applied Mechanics, Technische Universität Braunschweig, Pockelsstr. 3, 38106 Brunswick, Germany
2
Applied Mechanics and Data Analysis, Faculty of Engineering Technology, University of Twente, Horst - Ring N116, P.O. Box 217, 7500 AE Enschede, Netherlands
3
Instit
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