Inference and Uncertainty Propagation of Atomistically-Informed Continuum Constitutive Laws, Part 1: Bayesian Inference of Fixed Model Forms

Abstract

Uncertainty quantification techniques have the potential to play an important role in constructing constitutive relationships applicable to nanoscale physics. At these small scales, deviations from laws appropriate at the macroscale arise due to insufficient scale separation between the atomic and continuum length scales, as well as fluctuations due to thermal processes. In this work, we consider the problem of inferring the coefficients of an assumed constitutive model form using atomistic information and propagation of the associated uncertainty. A nanoscale heat transfer problem is taken as the model, and we use a polynomial chaos expansion to represent the thermal conductivity with a linear temperature dependence. A Bayesian inference method is developed to extract the coefficients in this expansion from molecular dynamics (MD) samples at prescribed temperatures. Importantly, the atomistic data are incompatible with the continuum model because of the finite probability of heat flowing in the opposite direction of the temperature gradient; we present a method to account for this in the model. The fidelity and uncertainty in these techniques are then examined. Validation is provided by comparing a continuum Fourier model against a larger all MD simulation representing the true solution.

Optimal Compressed Sensing and Reconstruction of Unstructured Mesh Datasets

“…We use a Bayesian regression method to perform the inference similar to the one used in [44]. By applying the ''log'' function to both sides of 19, the problem reduces to a linear multivariate inference problem which admits an analytical solution.…”

Inference and Uncertainty Propagation of Atomistically Informed Continuum Constitutive Laws, Part 2: Generalized Continuum Models Based on Gaussian Processes

“…1), and δ is the Kronecker symbol. The noise amplitude given by s 2 depends on the temperature in the MD simulation [12] but we assume that it is constant and equal to the variance of the MD heat flux data. There is no specific rule to choose the function h(·).…”

“…The proposed approach in this work builds on our previous effort [12] in which constitutive laws for large scale equations are statistically inferred from smaller scale realizations. In Part 1 of this series [12], Bayesian inference was used to estimate the coefficients of a polynomial chaos expansion (PCE) model of the thermal conductivity κ. An assumed form was used based on Fourier's law:…”

“…In the next section, we provide the mathematical formulation for the Gaussian Process (GP) [17][18][19] representation used for the heat flux, followed by the Bayesian inference methodology to infer the heat flux surface. We also discuss [12]), and (bottom row) from MD simulations with a global sampling approach, for L a = 53 nm when the statistical steady state is reached for different time averaging window widths t w , as indicated.…”

The MEAM parameter calibration tool: an explicit methodology for hierarchical bridging between ab initio and atomistic scales

“…Other authors have focused on uncertainty analysis methods to predict and manage potential reliability [21]. On the other hand, recent efforts have focused on making upscaling from atomistic potentials to course grain or continuum models more rigorous using similar uncertainty quantification [22][23][24][25]. The bridging process thus is usually complex and difficult to precisely quantify, but current progress suggests that it is possible to reduce user mediation in the bridging process to merely the specification of desired primary applications.…”