Abstract
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. Moreover, this approach leads to significant speed up over traditional methods by orders of magnitude. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity.
Highlights
Many measurements are made indirectly: quantities of interest (QoIs) are estimated by measuring one or more other quantities and calculating the required values from a model that links the QoIs to the measured quantities
We describe how a Markov chain Monte Carlo (MCMC) algorithm that incorporates a surrogate forward model, here based on a stochastic Galerkin finite element method (SGFEM), Figure 1: Diagram of the laser flash experiment set up
In this paper we formulated the determination of the thermal diffusivity of a material given laser flash experiment data as a Bayesian inverse problem
Summary
Many measurements are made indirectly: quantities of interest (QoIs) are estimated by measuring one or more other quantities and calculating the required values from a model that links the QoIs to the measured quantities. Examples include scatterometry, where surface profile is inferred from intensity measurements, determination of Young’s modulus, where the measured quantities are force and displacement, and determination of thermal diffusivity from measurements of temperature and time. Where λ is thermal conductivity, is density, and cp is specific heat capacity. Density and specific heat capacity can be measured independently, so measurement of thermal diffusivity is often used to evaluate thermal conductivity. Thermal conductivity is a key property in understanding heat transport in solids. Accurate characterisation of thermal conductivity supports design of thermal behaviour in products ranging from protective coatings for turbine blades in high temperature gas streams to insulation for low-temperature carriers for transport of vaccines. Reliable uncertainty evaluation of thermal conductivity enables designers to have confidence that their products will meet the required specification under all circumstances
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