A Piecewise Algorithm for Improved Versatility and Accuracy of Least-Squares-Based Inverse Heat Conduction Solutions

An improved least-squares method based on a piecewise cubic formulation (splines) has been developed for solving transient inverse heat-conduction problems. A generalized solution was formulated by the convolution of a piecewise cubic spline representing the unknown surface temperature history and unit response using Duhamel's integral. When the resulting response or direct solution was used to fit remotely measured temperature data, the resulting coefficients in the original cubic spline determined the inverse for each time interval. Results indicated the versatility and accuracy of the method to predict the potentially complex excitation of the slab that included a common asymptotic exponential, 1−exp[−1/2t], as well as increasingly complex oscillatory behaviors from sin(t) and J₁(t), even with artificial errors; continuous polynomials were not able to handle such complex and oscillatory data. Since inverse problems are inherently ill-posed and sensitive to errors, smoothing techniques applied to the data and/or the convolution were found useful for improving the quality of the resulting inverse predictions. Provided a problem is linear and a unit response (or impulse) exists such that convolution is appropriate, an accurate, generalized, and modular solution for many complex inverse problems is now possible.