Inverse Determination of Moving Heat Flux Distributions Using Reduced Order Models

Optimization-based solutions to inverse problems involve the coupling of an analysis model, such as a finite element model, with a numerical optimization method. The goal is to determine a set of parameters that minimize an objective function that is determined by solving the analysis model. In this paper, we present an approach that dramatically reduces the computational cost for solving this inverse problems in this way by replacing the original full order finite element model (FOM) with a reduced order model (ROM) that is both accurate and quick to compute. The reduced order model is constructed with basis functions generated using proper orthogonal decomposition of set of solutions from the FOM. A discrete Galerkin method is used to project the differential equation on the basis functions. This approach allows us to transform the linear full order finite element model into an equivalent discrete ROM with far fewer unknowns. The method is applied to a parameter estimation problem in heat transfer. Specifically, we determine the parameters governing the magnitude and distribution of an unknown surface heat flux moving at a constant velocity across the surface of a solid bar of material. A finite element model was implemented in the commercial package COMSOL and a corresponding ROM was constructed. The ROM was coupled with an optimization algorithm to determine the parameter values that minimized the distance between the computed surface temperatures and the target surface temperature. The target surface temperature was generated using simulated measurements produced from the full order finite element model. Several optimization methods were used. The results show the approach can recover the parameters with high accuracy with twenty seven FOM runs.