Thermoelastic states as they pertain to thermal-shock are difficult to determine since the underlying boundary conditions must be known or measured. For direct problems where the boundary conditions such as temperature or flux, are known a priori, the procedure is mathematically tractable with many analytical solutions available. Although this is more practical from a measurement standpoint, the inverse problem where the boundary conditions must be determined from remotely determined temperature and/or flux data are ill-posed and therefore inherently sensitive to errors in the data. Moreover, the limited number of analytical solutions to the inverse problem rely on assumptions that usually restrict them to timeframes before the thermal wave reaches the natural boundaries of the structure. Fortunately, a generalized solution based on strain-histories can be used instead to determine the underlying thermal excitation via a least-squares determination of coefficients for generalized equations for strain. Once the inverse problem is solved and the unknown boundary condition on the opposing surface is 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 the two geometries explored, namely a thick-walled cylinder under an internal transient with external convection and a slab with one adiabatic surface, excellent agreement was seen with various test cases. The derived solutions appear to be well suited for many thermal scenarios provided that the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. While polynomials were employed for the current analysis, transcendental functions and/or combinations with polynomials can also be used.
Error estimate of a high accuracy difference scheme for Poisson equation with two integral boundary conditions
