When solving steady-state or transient thermoelastic stress problems, boundary conditions must be known or determined in order to reveal the underlying thermal state. For direct problems where all boundary conditions are known, the procedure is relatively straightforward and mathematically tractable. On the other hand, the inverse problem where boundary conditions must be determined from remotely determined temperature and/or flux data is ill-posed and inherently sensitive to errors in the data. As a result, most analytical solutions to the inverse problem rely on a host of assumptions that usually restrict their utility to timeframes before the thermal wave reaches the natural boundaries of the structure. To help offset these limitations and at the same time investigate commonly encountered geometries, the inverse problem was solved using a least-squares determination of polynomial coefficients based on a generalized direct-solution to the Heat Equation. Once the inverse problem was solved in this fashion and the unknown boundary-condition determined, the resulting polynomial was used with the direct solution to determine the internal temperature and stress distributions as a function of time and coordinate. For a semi-infinite slab insulated on one boundary, excellent agreement was seen with various test cases including asymptotic-exponential and triangular temperature histories. Given the versatility of the integral- and half-order polynomials advocated, the method appears well suited for complicated thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties do not vary with temperature.
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