Inverse modeling technology for parameter estimation

Computational simulation models are extensively used in the development, design, and analysis of an aircraft engine and its components to represent the physics of an underlying phenomenon. The use of such a model-based simulation in engineering often necessitates the need to estimate model parameters based on physical experiments or field data. This class of problems, referred to as inverse problems (Woodbury KA 2003 Inverse engineering handbook. CRC, Boca Raton) in the literature, can be classified as well-posed or ill-posed depending on the quality (uncertainty) and quantity (amount) of data that are available to the engineer. The development of a generic inverse modeling solver in a probabilistic design system (PEZ version 2.6 user-manual. Probabilistic design system at General Electric Aviation, Cincinnati) requires the ability to handle diverse characteristics in various models. These characteristics include (a) varying fidelity in model accuracy with simulation times from a couple of seconds to many hours; (b) models being black-box, with the engineer having access to only the input and output; (c) nonlinearity in the model; and (d) time-dependent model input and output. This paper demonstrates methods that have been implemented to handle these features, with emphasis on applications in heat transfer and applied mechanics. A practical issue faced in the application of inverse modeling for parameter estimation is ill-posedness, which is characterized by instability and nonuniqueness in the solution. Generic methods to deal with ill-posedness include (a) model development, (b) optimal experimental design, and (c) regularization methods. The purpose of this paper is to communicate the development and implementation of an inverse method that provides a solution for both well-posed and ill-posed problems using regularization based on the prior values of the parameters. In the case of an ill-posed problem, the method provides two solution schemes—a most probable solution closest to the prior, based on the singular value decomposition (SVD), and a maximum a posteriori probability (MAP) solution. The inverse problem is solved as a finite dimensional nonlinear optimization problem using the SVD and/or MAP techniques tailored to the specifics of the application. The objective of the paper is to demonstrate the development and validation of these inverse modeling techniques in several industrial applications, e.g., heat transfer coefficient estimation for disk quenching in process modeling, material model parameter estimation, sparse clearance data modeling, and steady state and transient engine high-pressure compressor heat transfer estimation.