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Research Papers: Conduction

Quantifying Uncertainty in Multiscale Heat Conduction Calculations

[+] Author and Article Information
Prabhakar Marepalli

School of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0292
e-mail: pmarepalli@utexas.edu

Jayathi Y. Murthy

School of Mechanical Engineering,
The University of Texas at Austin,
Austin, TX 78712-0292
e-mail: jmurthy@me.utexas.edu

Bo Qiu

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Boston, MA 02139
e-mail: qiub@mit.edu

Xiulin Ruan

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: ruan@purdue.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 7, 2013; final manuscript received March 29, 2014; published online August 26, 2014. Assoc. Editor: Robert D. Tzou.

J. Heat Transfer 136(11), 111301 (Aug 26, 2014) (10 pages) Paper No: HT-13-1190; doi: 10.1115/1.4027348 History: Received April 07, 2013; Revised March 29, 2014

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 (here after referred to as MD noise), which is generated due to the possibility of multiple initial configurations corresponding to the same system temperature. When MD is used to compute 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 (PDFs) of the spatial temperature distribution resulting from the PDFs of thermal conductivity. 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.

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Figures

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Fig. 1

Phonon relaxation times for LA and TA branches at 300 K and 900 K

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Fig. 2

Thermal conductivity of silicon at different temperatures. The scatter in the data at each temperature is due to MD noise.

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Fig. 3

(a) and (b) Calibrated coefficients in the functional form of thermal conductivity surrogate model. (c) Thermal conductivity data reproduced by surrogate model. Data points in black represent the MD simulation data and the data points in red are those generated using surrogate model. The surrogate model captures the temperature dependence and the MD noise of thermal conductivity well.

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Fig. 4

Solution procedure to quantify uncertainty in multiscale heat conduction problem

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Fig. 5

Schematic and boundary conditions

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Fig. 6

Mean temperature contours for different orders of nonlinearity in thermal conductivity (k=aTb+ση)

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Fig. 7

Ratio of standard deviation to mean temperature for different orders of nonlinearity in thermal conductivity (k=aTb+ση)

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Fig. 8

PDF of temperature at a given spatial location for different degrees of nonlinearity in the temperature dependence of thermal conductivity

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Fig. 9

Mean temperature profile (degrees K) computed by sampling the gPC response surfaces at each finite volume cell in the domain

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Fig. 10

Ratio of standard deviation to mean of temperature computed by sampling the gPC response surfaces at each finite volume cell in the domain

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Fig. 11

PDFs of temperature at different spatial locations

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