A hierarchical Bayesian approach for calibration of stochastic material models

Nikolaos Papadimas,Timothy Dodwell

doi:10.1017/dce.2021.20

Abstract

AbstractThis article recasts the traditional challenge of calibrating a material constitutive model into a hierarchical probabilistic framework. We consider a Bayesian framework where material parameters are assigned distributions, which are then updated given experimental data. Importantly, in true engineering setting, we are not interested in inferring the parameters for a single experiment, but rather inferring the model parameters over the population of possible experimental samples. In doing so, we seek to also capture the inherent variability of the material from coupon-to-coupon, as well as uncertainties around the repeatability of the test. In this article, we address this problem using a hierarchical Bayesian model. However, a vanilla computational approach is prohibitively expensive. Our strategy marginalizes over each individual experiment, decreasing the dimension of our inference problem to only the hyperparameter—those parameter describing the population statistics of the material model only. Importantly, this marginalization step, requires us to derive an approximate likelihood, for which, we exploit an emulator (built offline prior to sampling) and Bayesian quadrature, allowing us to capture the uncertainty in this numerical approximation. Importantly, our approach renders hierarchical Bayesian calibration of material models computational feasible. The approach is tested in two different examples. The first is a compression test of simple spring model using synthetic data; the second, a more complex example using real experiment data to fit a stochastic elastoplastic model for 3D-printed steel.

Highlights

In engineering and across the physical sciences, both in academia and industry, the parameterization of constitutive models for materials from a set of experiments is a very common research question.Downloaded from https://www.cambridge.org/core. 23 Dec 2021 at 13:35:45, subject to the Cambridge Core terms of use.e20-2 Nikolaos Papadimas and Timothy DodwellParameterization of constitutive models is often used to classify or compare the response of different materials, or as the starting point for establishing mathematical models of the system, in which that material is used such as finite element analysis
We first set up a Bayesian formulation, from which we can estimate the distribution of material parameters given that we observe the outputs of only a single experiment
We use Markov Chain Monte Carlo (MCMC) methods to perform the Bayesian computations and briefly set out the detail here, but they are widely covered in the literature (Liu, 2008; Gelman et al, 2013). In this Rosenthal, contribution, 2009), where we we focus on generate aMcehtarionpoolfisn-Hsaasmtinpgle-sbaTse≔d ÈMψðC1ÞM, ψCð2Þa,l...go,rψitðhnmÞÉs (Roberts and of statistically dependent sets generated from oafkpnaorwamn eptreorps.oTsahledcishtariibnuitsiopnroqdÀuψc0ejdψ ðfirÞoÁm(Raocbuerrrtse,n2t 0s1ta1t)e

Summary

Introduction

In engineering and across the physical sciences, both in academia and industry, the parameterization of constitutive models for materials from a set of experiments is a very common research question. The approach addresses the limitations of standard, nonprobabilistic approaches (e.g., least square) and enables the capture of measurement, model uncertainties, and complex interdependencies between model parameters, including prior knowledge in cases where data are limited It provides a framework, in which the population statistics of the material parameters can be estimated, rather than snapshots of individual experiments under the assumption, they are independent and correlation between parameters is not significant. A HBM for Estimating Material Parameters from a set of Experiments We start this section by first recapping the parameterization of a material model from a single experimental test This single experiment approach is extended into a hierarchical Bayesian framework (Gelman et al, 2013), in which we infer the population statistics of the material parameters from a set of experimental tests. It considers computationally efficient strategies for implementing the hierarchical approach

Bayesian calibration of a material model given a single experiment

Conclusion