A multi-thermogram-based Bayesian model for the determination of the thermal diffusivity of a material

Allard, Alexandre; Fischer, Nicolas; Ebrard, Géraldine; Hay, Bruno; Harris, P M; Wright, L; Rochais, Denis; Mattout, Jérémie

doi:10.1088/0026-1394/53/1/s1

Cited by 10 publications

(16 citation statements)

References 18 publications

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“…This is an example of an uninformative prior. Another common choice is the Jeffreys' prior [19], which is based on the Fischer information matrix [2,22]. We choose an inverse gamma prior distribution for σ 2 to ensure positivity and so that we can exploit it's conjugacy with the Gaussian distribution.…”

Section: Prior Distributionmentioning

confidence: 99%

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Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

et al. 2019

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Determination of the thermal properties of a material is an important task in many scientific and engineering applications. How a material behaves when subjected to high or fluctuating temperatures can be critical to the safety and longevity of a system's essential components. The laser flash experiment is a well-established technique for indirectly measuring the thermal diffusivity, and hence the thermal conductivity, of a material. In previous works, optimization schemes have been used to find estimates of the thermal conductivity and other quantities of interest which best fit a given model to experimental data. Adopting a Bayesian approach allows for prior beliefs about uncertain model inputs to be conditioned on experimental data to determine a posterior distribution, but probing this distribution using sampling techniques such as Markov chain Monte Carlo methods can be incredibly computationally intensive. This difficulty is especially true for forward models consisting of time-dependent partial differential equations. We pose the problem of determining the thermal conductivity of a material via the laser flash experiment as a Bayesian inverse problem in which the laser intensity is also treated as uncertain. We introduce a parametric surrogate model that takes the form of a stochastic Galerkin finite element approximation, also known as a generalized polynomial chaos expansion, and show how it can be used to sample efficiently from the approximate posterior distribution. This approach gives access not only to the sought-after estimate of the thermal conductivity but also important information about its relationship to the laser intensity, and information for uncertainty quantification. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity.

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Section: Prior Distributionmentioning

confidence: 99%

“…We note that the laser flash experiment was also considered in a Bayesian setting by Allard et al in [2]. Hence, since the set up of the physical experiment is similar, our discussion in Sections 1.1 and 1.2 below follows [2, Section 1].…”

Section: Introductionmentioning

confidence: 99%

Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

et al. 2019

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“…1−4 The thermal diffusivity of a material can be derived from a model that includes the differential equations governing heat transfer. 5 We calculate properties such as diffusion, viscosity, density, and the elastic modulus from molecular dynamics simulations and further modeling such as Arrhenius equation, adjustments for finite size effects, extrapolating to alternate conditions. 6−8 When reporting the physical property, it is important to also report its sensitivity to design choices in model and data selection.…”

Section: Introductionmentioning

confidence: 99%

“…Many physical properties are derived from models. For example, reaction rates and enthalpy in a chemical reaction network can be determined from a kinetics model. − The thermal diffusivity of a material can be derived from a model that includes the differential equations governing heat transfer . We calculate properties such as diffusion, viscosity, density, and the elastic modulus from molecular dynamics simulations and further modeling such as Arrhenius equation, adjustments for finite size effects, extrapolating to alternate conditions. − …”

Section: Introductionmentioning

confidence: 99%

Model-Specific to Model-General Uncertainty for Physical Properties

Zhan

Kitchin

2022

Ind. Eng. Chem. Res.

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Many physical properties are derived from models, and we would like to be able to report the property with its uncertainty in an interpretable way. Different frameworks for interpreting uncertainty have been proposed. For example, “aleatoric” refers to randomness in the experiment or observations, while “epistemic” refers to ignorance about the best model. In addition, there are many ways to calculate uncertainty, including the frequentist confidence interval and the Bayesian probability distribution. In this work, we improve the understanding of which sources of uncertainty are captured by different methods. We use an example of obtaining physical properties based on function derivatives from an equation of state for Pd and Au. We obtain uncertainties using three methods: the delta method and nonlinear regression, Bayesian nonlinear regression, and Gaussian process regression. We develop a Gaussian process with joint covariance over a function and its first and second derivatives. The delta method and Bayesian regression uncertainties are consistent with each other and capture epistemic uncertainty, specifically model parameter uncertainty. The Gaussian process extends this to uncertainties arising from model selection.

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“…Several papers dealing with uncertainty analysis associated with thermal diffusivity measurements performed using the laser flash method have been published during the last two decades. They present the identification and quantification of influencing parameters and sources of measurement errors [9,10], the establishment of detailed uncertainty budgets [11,12] according to the ISO/BIPM Guide to the expression of uncertainty in measurement [13], or the development of alternative approaches (e.g., Bayesian or multi-convolutional approaches) to evaluate the uncertainty on thermal diffusivity measurements [14][15][16]. All these published works are limited to measurements performed from room temperature to 1000 °C at maximum, and sometimes do not take into account the contribution of some significant uncertainty factors (spatially non-uniform heating, non-linearity of the infrared detector output with respect to temperature, variation of the specimen thickness due to the thermal expansion of the tested material…).…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Assessment for Very High Temperature Thermal Diffusivity Measurements on Molybdenum, Tungsten and Isotropic Graphite

et al. 2021

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The French National Metrology Institute LNE has improved its homemade laser flash apparatus in order to perform accurate and reliable measurements of thermal diffusivity of homogeneous solid materials at very high temperature. The inductive furnace and the associated infrared (IR) detection systems have been modified and a specific procedure for the in situ calibration of the used radiation thermometers has been developed. This new configuration of the LNE’s diffusivimeter has been then applied for measuring the thermal diffusivity of molybdenum up to 2200 °C, tungsten up to 2400 °C and isotropic graphite up to 3000 °C. Uncertainties associated with these high temperature thermal diffusivity measurements have been assessed for the first time according to the principles of the “Guide to the Expression of Uncertainty in Measurement” (GUM). Detailed uncertainty budgets are here presented in the case of the isotropic graphite for measurements performed at 1000 °C, 2000 °C and 3000 °C. The relative expanded uncertainty (coverage factor k = 2) of the thermal diffusivity measurement is estimated to be between 3 % and 5 % in the whole temperature range for the three investigated refractory materials.

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A multi-thermogram-based Bayesian model for the determination of the thermal diffusivity of a material

Cited by 10 publications

References 18 publications

Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

Surrogate accelerated Bayesian inversion for the determination of the thermal diffusivity of a material

Model-Specific to Model-General Uncertainty for Physical Properties

Uncertainty Assessment for Very High Temperature Thermal Diffusivity Measurements on Molybdenum, Tungsten and Isotropic Graphite

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