Quantifying Uncertainty in Multiscale Heat Conduction Calculations

Abstract

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.

Uncertainty quantification in three dimensional natural convection using polynomial chaos expansion and deep neural networks

“…Thus in the recent years, there has been a growing interest in analysis of the effects of stochastic variations in the inputs on the outputs. There are multiple examples in the literature in which the uncertainty propagation techniques are combined with the deterministic numerical simulations [12][13][14][15][16][17][18][19][20][21][22][23]. It is popular to use the polynomial chaos expansion (PCE) for uncertainty prop-agation in which the output is approximated as a summation of polynomial basis which are functions of the stochastic inputs.…”

“…It is popular to use the polynomial chaos expansion (PCE) for uncertainty prop-agation in which the output is approximated as a summation of polynomial basis which are functions of the stochastic inputs. The two main classes of methods to estimate the coefficients of the PCE are stochastic Galerkin projection [12][13][14][15] and collocation [17][18][19]. Stochastic Galerkin method requires modification of the underlying deterministic code since it requires solution of a new set of equations and thus, is called as intrusive method.…”

Finite volume simulation framework for die casting with uncertainty quantification

“…Use of deterministic simulations alone to analyze the engineering systems is incomplete due to the lack of precisely defined input data. Thus, there has been a growing interest [29][30][31][32][33][34][35][36][37][38] in coupling uncertainty propagation techniques with the deterministic numerical simulations to estimate the effects of stochastic variations in the input process parameters on the outputs. The polynomial chaos expansion is a popular method used to estimate the relation between input and output parameters.…”

“…The polynomial chaos expansion is a popular method used to estimate the relation between input and output parameters. Stochastic Galerkin projection [29][30][31] and collocation [32][33][34] are two strategies to estimate the coefficients of the polynomial chaos expansion. Stochastic Galerkin method is an intrusive method since it requires solution of a new set of equations and thus, modification of the underlying deterministic code which becomes a significant additional effort.…”

Uncertainty quantification in non-equilibrium molecular dynamics simulations of thermal transport

“…Rizzi et al focused on the effect of uncertainties associated with the force-field parameters on bulk water properties using MD simulations [16]. Marepalli et al in [17] considered a stochastic model for thermal conductivity to account for inherent noise in MD simulations, and study its impact on spatial temperature distribution during heat conduction. Jacobson et al in [18] implemented an uncertainty quantification framework to optimize a coarse-grained model for predicting the properties of monoatomic water.…”