Sensitivity information is often of interest in engineering applications (e.g., gradient-based optimization). Heat transfer problems frequently involve complicated geometries for which exact solutions cannot be easily derived. As such, it is common to resort to numerical solution methods such as the finite element method. The semi-analytic complex variable method (SACVM) is an accurate and efficient approach to computing sensitivities within a finite element framework. The method is introduced and a derivation is provided along with a detailed description of the algorithm which requires very minor changes to the analysis code. Three benchmark problems in steady-state heat transfer are studied including a nonlinear problem, an inverse shape determination problem, and a reliability analysis problem. It is shown that the SACVM is superior to the other methods considered in terms of computation time and sensitivity to perturbation size.

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In order to reduce the demanding computational requirements for the numerical solution of problems involving heat transfer problems of moving heat source deposition, we present an approach utilizing reduced order models based on proper orthogonal decomposition and associated Galerkin projection. We subsequently describe the finite element implementation of solution methodology for both the full order and the reduced order models, as well as the respective computational implementation details. Using this methodology, we performed a sensitivity analysis for a problem of a moving heat source to investigate the performance characteristics of the relevant reduced order model size and present the efficiency of the approach. We demonstrated the efficiency of the reduced models for performing inverse analysis.

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.