Quantifying Uncertainty in Multiscale Heat Conduction Calculations

In recent years, there has been interest in employing atomistic computations to inform macroscale thermal transport analyses. In heat conduction simulations in semiconductors and dielectrics, for example, classical molecular dynamics (MD) is used to compute phonon relaxation times, from which material thermal conductivity may be inferred and used at the macroscale. A drawback of this method is the noise associated with MD simulation, which is generated due to the possibility of multiple initial configurations corresponding to the same system temperature; for phonon relaxation times, the spread may be as high as 20%. In this work we propose a method to quantify the uncertainty in thermal conductivity computations due to MD noise, and its effect on the computation of the temperature distribution in heat conduction simulations. Bayesian inference is used to construct a probabilistic surrogate model for thermal conductivity as a function of temperature, accounting for the statistical spread in MD relaxation times. The surrogate model is used in probabilistic computations of the temperature field in macroscale Fourier conduction simulations. These simulations yield probability density functions of the spatial temperature distribution. To allay the cost of probabilistic computations, a stochastic collocation technique based on generalized polynomial chaos (gPC) is used to construct a response surface for the variation of temperature (at each physical location in the domain) as a function of the random variables in the thermal conductivity model. Results are presented for the spatial variation of the probability density function of temperature as a function of spatial location in a typical heat conduction problem to establish the viability of the method.