Inverse determination of thermal boundary conditions from transient surface temperatures and strains in slabs and tubes

For most problems where the thermal boundary condition (temperature or flux) is known a priori, the analytical procedure for determining internal temperatures, strains, and/or stresses is tractable with many solutions available. Although more practical from a measurement standpoint, the inverse problem where the boundary conditions must be determined from remotely determined temperature and/or flux data is ill posed and inherently sensitive to data errors. Moreover, only a limited number of solutions are available, and they are usually restricted to timeframes before the thermal wave reaches any boundaries. Fortunately, a generalized direct solution based on strain histories can be used to determine thermal boundary conditions via a least squares determination of coefficients. Once the inverse problem is solved and the unknown boundary condition determined, the resulting polynomial can then be used with the generalized direct solution to determine the thermal and stress states as a function of time and position. For thick slabs and tubes, excellent agreement was seen with various test cases. 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 and the thermophysical properties that do not vary with temperature.