Abstract This work presents an illustrative application of the second-order adjoint sensitivity a... more Abstract This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-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 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most Nα large-scale computations using the first-level adjoint sensitivity system (1st-LASS) and second-level adjoint sensitivity system (2nd-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 2nd-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 2nd-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.
A phenomenological model has been developed to simulate the qualitative behavior of boiling water... more A phenomenological model has been developed to simulate the qualitative behavior of boiling water reactors (BWRs) in the nonlinear regime under deterministic and stochastic excitations. After the linear stability threshold is crossed, limit cycle oscillations appear ...
Abstract Dispensing with the traditional approach to solving the equations modeling multiplying c... more Abstract Dispensing with the traditional approach to solving the equations modeling multiplying critical nuclear systems as an eigenvalue system, this work proposes a new and comprehensive mathematical framework (C-Framework) that eliminates the need for solving eigenvalue problems when computing the forward and adjoint neutron flux distributions in critical reactors. Consequently, the C-Framework enables the mathematical and computational analysis of critical and noncritical multiplying systems, with or without external sources, in a unified manner. By eliminating the need for solving eigenvalue problems, the C-Framework also enables the use of more efficient numerical methods (than currently used) for computing the forward and adjoint neutron flux distributions in critical reactors. Furthermore, the C-Framework also enables the application of the Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM) as a replacement for the so-called generalized perturbation theory (GPT). The C-ASAM is much simpler to apply than the GPT, while not only yielding all of the results that the GPT can deliver, but also delivering results for all of the many—and not “GPT-allowable”—nonlinear responses of interest in reactor analysis that do not satisfy the very restrictive orthogonality relations required by the GPT’s underlying generalized adjoint equation. By dispensing with the need for solving eigenvalue problems involving the inversion of singular operators, the C-ASAM is vastly more general and more efficient than the GPT. These conclusions are underscored by exact analytical results presented for paradigm illustrative problems, which include problems that are solvable using the GPT (e.g., the system’s multiplication factor, ratios of reaction rates responses), and problems that are not solvable using the GPT (e.g., absolute reaction rates, equilibrium xenon concentration responses); all of these problems are shown to be solvable exactly and most efficiently within the C-ASAM framework.
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Sep 13, 1983
This paper describes sources of uncertainty in the data used for calculating dose estimates for t... more This paper describes sources of uncertainty in the data used for calculating dose estimates for the Hiroshima explosion and details a methodology for systematically obtaining best estimates and reduced uncertainties for the radiation doses received. (ACR)
Abstract This work presents an illustrative application of the second-order adjoint sensitivity a... more Abstract This work presents an illustrative application of the second-order adjoint sensitivity analysis methodology (2nd-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 2nd-ASAM indicates that for a physical system comprising Nα parameters, the computation of all of the first- and second-order response sensitivities requires (per response) at most Nα large-scale computations using the first-level adjoint sensitivity system (1st-LASS) and second-level adjoint sensitivity system (2nd-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 2nd-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 2nd-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.
A phenomenological model has been developed to simulate the qualitative behavior of boiling water... more A phenomenological model has been developed to simulate the qualitative behavior of boiling water reactors (BWRs) in the nonlinear regime under deterministic and stochastic excitations. After the linear stability threshold is crossed, limit cycle oscillations appear ...
Abstract Dispensing with the traditional approach to solving the equations modeling multiplying c... more Abstract Dispensing with the traditional approach to solving the equations modeling multiplying critical nuclear systems as an eigenvalue system, this work proposes a new and comprehensive mathematical framework (C-Framework) that eliminates the need for solving eigenvalue problems when computing the forward and adjoint neutron flux distributions in critical reactors. Consequently, the C-Framework enables the mathematical and computational analysis of critical and noncritical multiplying systems, with or without external sources, in a unified manner. By eliminating the need for solving eigenvalue problems, the C-Framework also enables the use of more efficient numerical methods (than currently used) for computing the forward and adjoint neutron flux distributions in critical reactors. Furthermore, the C-Framework also enables the application of the Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM) as a replacement for the so-called generalized perturbation theory (GPT). The C-ASAM is much simpler to apply than the GPT, while not only yielding all of the results that the GPT can deliver, but also delivering results for all of the many—and not “GPT-allowable”—nonlinear responses of interest in reactor analysis that do not satisfy the very restrictive orthogonality relations required by the GPT’s underlying generalized adjoint equation. By dispensing with the need for solving eigenvalue problems involving the inversion of singular operators, the C-ASAM is vastly more general and more efficient than the GPT. These conclusions are underscored by exact analytical results presented for paradigm illustrative problems, which include problems that are solvable using the GPT (e.g., the system’s multiplication factor, ratios of reaction rates responses), and problems that are not solvable using the GPT (e.g., absolute reaction rates, equilibrium xenon concentration responses); all of these problems are shown to be solvable exactly and most efficiently within the C-ASAM framework.
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Sep 13, 1983
This paper describes sources of uncertainty in the data used for calculating dose estimates for t... more This paper describes sources of uncertainty in the data used for calculating dose estimates for the Hiroshima explosion and details a methodology for systematically obtaining best estimates and reduced uncertainties for the radiation doses received. (ACR)
Uploads
Papers by Dan Cacuci