| The groundwork necessary for frequency domain analysis of solutions to the inverse heat conduction problem, using Laplace and z-transform techniques is presented. The solution is traced back through the direct solution methods used to generate the inverse solution, and draws heavily upon transfer function concepts. Transformation methods from continuous time ladder network models to discrete time models are developed. The mathematical relationship between ladder network models and finite difference models is derived. In addition, a new procedure for generating errorless discrete time models from Fourier series solutions is presented. The frequency response of direct discrete time solutions and the impact on the stability and noise performance of inverse algorithms arising from these direct solutions is presented. Inverse algorithm methods are generalized according to their frequency domain structure, rather than time domain, which effectively decouples the algorithm classification from a direct solution. Precompensation filter concepts are introduced, and a frequency domain test for inverse algorithm stability is developed. Examples of Stolz's method and Beck's method are included, and results show that noise is actually enhanced in some frequency bands under the proper circumstances. A new class of unconditionally stable inverse algorithms is derived, and three particular cases are developed and tested. Results showing the effects of the choice of direct model used to generate benchmark observations on the performance of the inverse algorithm are presented. |
