The Second-Order Adjoint Sensitivity Analysis Methodology for Nonlinear Systems—II: Illustrative Application to a Nonlinear Heat Conduction Problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2^nd-ASAM) to a paradigm nonlinear heat conduction benchmark, which models a conceptual experimental test section containing heated rods immersed in liquid lead–bismuth eutectic (LBE). This benchmark admits an exact solution, thereby making transparent the underlying mathematical derivations. The general theory underlying 2^nd-ASAM indicates that for a physical system comprising N_α parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most N_α large-scale computations using the first-level adjoint sensitivity system (1^st-LASS) and second-level adjoint sensitivity system (2^nd-LASS), respectively. For this illustrative problem, six large-scale adjoint computations sufficed to compute exactly all 5 first-order and 15 distinct second-order derivatives of the temperature response to the five model parameters. The construction and solution of the 2^nd-LASS require very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Very significantly, only the sources on the right sides of the heat conduction differential operator needed to be modified; the left side of the differential equations (and hence the solver in large-scale practical applications) remains unchanged.

The second-order sensitivities play the following roles: (1) They cause the expected value of the response to differ from the computed nominal value of the response; for the nonlinear heat conduction benchmark, however, these differences were insignificant over the range of temperatures (400 to 900 K) considered. (2) They contribute to increasing the response variances and modifying the response covariances, but for the nonlinear heat conduction benchmark, their contribution was smaller than that stemming from the first-order response sensitivities, over the range of temperatures (400 to 900 K) considered. (3) They provide the leading contributions to causing asymmetries in the response distribution. For the benchmark test section considered in this work, the heat source, the boundary heat flux, and the temperature at the bottom boundary of the test section would cause the temperature distribution in the test section to be skewed significantly toward values lower than the mean temperature. On the other hand, the model parameters entering the nonlinear, temperature-dependent expression of the LBE conductivity would cause the temperature distribution in the test section to be skewed significantly toward values higher than the mean temperature. These opposite effects partially cancel each other. Consequently, the cumulative effects of model parameter uncertainties on the skewness of the temperature distribution in the test section are such that the temperature distribution in the LBE is skewed slightly toward higher temperatures in the cooler part of the test section but becomes increasingly skewed toward temperatures lower than the mean temperature in the hotter part of the test section. Notably, the influence of the model parameter that controls the strength of the nonlinearity in the heat conduction coefficient for this LBE test section benchmark would be strong if it were the only uncertain model parameter. However, if all of the other model parameters are also uncertain, all having equal relative standard deviations, the uncertainties in the heat source and boundary heat flux diminish the impact of uncertainties in the nonlinear heat conduction coefficient for the range of temperatures (400 to 900 K) considered for this LBE test section benchmark.

Ongoing work aims at generalizing the 2^nd-ASAM to enable the exact and efficient computation of higher-order response sensitivities. The availability of such higher-order sensitivities is expected to affect significantly the fields of optimization and predictive modeling, including uncertainty quantification, data assimilation, model calibration, and extrapolation.