Inverse analysis of thermal states from transient histories with nonlinear thermophysical properties

When thermal-shock is an issue, the underlying thermal states are often difficult to determine because the boundary conditions must be known. The solution is to use an inverse approach where the boundary conditions are determined from remote data. Unfortunately, the inverse path is inherently ill-posed and therefore sensitive to errors. Moreover, many inverse models are limited to only constant thermophysical properties. Fortunately, generalized solutions based on measured temperature histories can be used to determine the underlying thermal excitation via a least-squares determination of coefficients, even with temperature dependent properties. With this in mind, two situations were studied, namely were the thermal conductivity/diffusivity had either a increasing or decreasing temperature dependence that could be described by a cubic polynomial. In each case, temperatures were determined at a remote location from a finite element simulation with the inverse prediction showing excellent agreement with the prescribed surface temperature. The derived solutions appear to be well suited for many thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial.