Dan G. Cacuci

Second-order adjoint sensitivity analysis methodology (2nd-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems: I. Computational methodology

Abstract This work presents the second-order forward and adjoint sensitivity analysis methodologies ( 2nd - FSAM and 2nd - ASAM ) for computing exactly and efficiently the second-order functional derivatives of physical (engineering, biological, etc.) system responses (i.e., “system performance parameters”) to the systems model parameters. The definition of “system parameters” used in this work includes all computational input data, correlations, initial and/or boundary conditions, etc. For a physical system comprising N α parameters and N r responses, we note that the 2nd - FSAM requires a total of ( N α 2 / 2 + 3 N α / 2 ) large-scale computations for obtaining all of the first- and second-order sensitivities, for all N r system responses. On the other hand, for one functional-type system response, the 2nd - ASAM requires one large-scale computation using the first-level adjoint sensitivity system for obtaining all of the first-order sensitivities, followed by at most N α large-scale computations using the second-level adjoint sensitivity systems for obtaining exactly all of the second-order sensitivities. Therefore, the 2nd - FSAM should be used when N r ≫ N α , while the 2nd - ASAM should be used when N α ≫ N r . The original 2nd - ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities (i.e., functional Gateaux-derivatives) for large-systems involving many parameters, as usually encountered in practice. Very importantly, the implementation of the 2nd - ASAM requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities.

Reducing Uncertainties via Predictive Modeling: FLICA4 Calibration Using BFBT Benchmarks

Abstract This work applies the predictive modeling procedure formulated by Cacuci and Ionescu-Bujor [Nucl. Sci. Eng., Vol. 165, p. 18 (2010)] to assimilate experimental data from the international Organisation for Economic Co-operation and Development/U.S. Nuclear Regulatory Commission boiling water reactor full-size fine-mesh bundle test (BFBT) benchmarks to calibrate and reduce systematically and significantly the uncertainties in the predictions of the light water reactor thermal-hydraulic code FLICA4. The BFBT benchmarks were designed by the Nuclear Power Engineering Corporation of Japan for enabling systematic validation of thermal-hydraulic codes by using full-scale experimental data. This work specifically uses BFBT experimental data for the “pump trip for a high-burnup assembly” in the predictive modeling formalism to calibrate parameters and time-dependent boundary conditions (power, mass flow rates, and outlet pressure distributions) in FLICA4, yielding best-estimate predictions of axial void fraction distributions. The resulting uncertainties for the best-estimate time-dependent model parameters and void fraction response distributions are shown to be smaller than the a priori experimental and computed uncertainties, thus demonstrating the successful use of predictive modeling for the large-scale reactor analysis code FLICA4 using BFBT benchmark-grade experiments.

The Second-Order Adjoint Sensitivity Analysis Methodology for Nonlinear Systems—I: Theory

Abstract The use of adjoint methods for computing first-order sensitivities (i.e., functional derivatives) of results (responses) produced by a computational model to the model’s parameters was initiated in the nuclear engineering sciences in the 1940s. The field of nuclear science and engineering also provided pioneering works, during the 1970s, for computing second-order response sensitivities of responses associated with the adjoint neutron and radiation transport and/or diffusion equations. These works generally indicated that the second-order sensitivities of responses such as reaction rates and the system’s effective multiplication factor to cross sections were computationally intensive to obtain, requiring O(Nα2) large-scale computations per response, for a system comprising Nα model parameters, and were considerably smaller than the corresponding first-order sensitivities. These results likely gave rise to the generally held opinion that second-order sensitivities are generally insignificant in reactor physics, which may, in turn, have led to diminishing interest in developing efficient methods for computing second-order sensitivities for nuclear engineering systems. This work presents the second-order adjoint sensitivity analysis methodology (2nd-ASAM) for nonlinear systems, which yields exactly and efficiently the second-order functional derivatives of physical system responses (i.e., system performance parameters) to the system’s model parameters. For a physical system comprising Nα parameters, forward methods require a total of (Nα2/2+3Nα/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all system responses. In contradistinction, the 2nd-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1st-LASS) for obtaining all of the first-order sensitivities, followed by at most Nα large-scale computations using the second-level adjoint sensitivity systems, for obtaining exactly all of the second-order sensitivities of a functional-type response. The construction, implementation, and solution of the 2nd-ASAM require very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Furthermore, because of the symmetry properties of the second-order sensitivities, the 2nd-ASAM comprises the inherent automatic solution verification of the correctness and accuracy of the second-level adjoint functions used for the efficient and exact computation of the second-order sensitivities. The use of the 2nd-ASAM to compute exactly all of the second-order response sensitivities to model input parameters is expected to enable significant advances in related scientific disciplines, particularly the areas of uncertainty quantification and predictive modeling, including model validation, reduced-order modeling, data assimilation, model calibration, and extrapolation.

Predictive Modeling of Coupled Systems: Uncertainty Reduction Using Multiple Reactor Physics Benchmarks

Abstract This work illustrates reactor physics applications of the predictive modeling of coupled multiphysics systems (PMCMPS), formulated by Cacuci (2014), by means of the benchmarks Godiva (a bare uranium sphere) and Jezebel-239 and Jezebel-240 (bare plutonium spheres). The PMCMPS methodology was ab initio developed in the response space, to reduce as much as possible the computational memory requirements for predictive modeling of very large systems involving not only many model parameters but also many model responses. The model parameters considered in this work include individual cross sections for each material, nuclide, reaction type, and energy group, giving the following totals: 2241 parameters for Jezebel-239, 1458 parameters for Jezebel-240, and 2916 parameters for Godiva. Eight responses were considered for Jezebel-239 (the effective multiplication factor; the center core fission rates for 233U, 238U, 237Np, and 239Pu; and the center core radiative capture rates for 55Mn, 93Nb, and 63Cu). Three responses (the effective multiplication factor and the center core fission rates for 233U and 237Np) were selected for Jezebel-240, and eleven responses were selected for Godiva (the reaction rate types listed for Jezebel-239, along with the radiative capture rates for 107Ag, 127I, and 81Br). The PMCMPS methodology ensures that increasing the amount of information yields more accurate predictions, with smaller predicted uncertainties, as long as the considered information is consistent. This fact is amply illustrated in this work, which shows that the interdependence of responses that were measured in more than one benchmark is stronger than for responses that were measured in a single benchmark. More generally, the consideration of the complete information, including couplings, provided jointly by all three benchmarks (as opposed to consideration of the benchmarks as separate systems) leads to more accurate predictions of nominal values for responses and model parameters, yielding larger reductions in the predicted uncertainties that accompany the predicted mean values of responses and model parameters.

On chaotic dynamics in nuclear engineering systems

The mapping [rho][sub n+1] = [var phi][[rho][sub n] + kQ[sub o](exp [rho][sub n] [minus] 1)] is shown to belong to the universal class of quadratic mappings with a negative Schwarzian derivative, thus rigorously providing the reasons underlying this mappings ability to follow the well-known Feigenbaum scenario to deterministic chaos. This scenario proceeds through an infinite cascade of period-doubling bifurcations, as noted in numerical experiments by Shabalin in a recent paper on power instabilities in periodically pulsed reactors. An analysis of this paper is also presented together with an overall perspective of the current state of research on chaotic dynamics in nuclear engineering systems.

A Paradigm-Shifting Methodology for Sensitivity Analysis of Critical Multiplying Nuclear Systems

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.

Predictive Modeling of a Buoyancy-Operated Cooling Tower Under Saturated Conditions—II: Optimal Best-Estimate Results with Reduced Predicted Uncertainties

Abstract This paper provides the results of the adjoint sensitivity model developed in the accompanying Part I for a natural draft counter-flow cooling tower. The selected responses are (1) outlet air temperature, (2) outlet water temperature, (3) outlet water mass flow rate, (4) air outlet relative humidity, and (5) air mass flow rate. Explicit expressions for the best-estimate nominal values of the model parameters and responses are also provided, together with the best-estimate reduced standard deviations of the predicted model parameters and responses. The results stemming from this work show that the PM_CMPS procedure reduces the predicted standard deviations of all responses and model parameters.

Second-Order Sensitivity Analysis of Uncollided Particle Contributions to Radiation Detector Responses

Abstract This work presents an application of Cacuci’s Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the simplified Boltzmann equation that models the transport of uncollided particles through a medium to compute efficiently and exactly all of the first- and second-order derivatives (sensitivities) of a detector’s response with respect to the system’s isotopic number densities, microscopic cross sections, source emission rates, and detector response function. The off-the-shelf PARTISN multigroup discrete ordinates code is employed to solve the equations underlying the 2nd-ASAM. The accuracy of the results produced using PARTISN is verified by using the results of three test configurations: (1) a homogeneous sphere, for which the response is the exactly known total uncollided leakage, (2) a multiregion two-dimensional (r-z) cylinder, and (3) a two-region sphere for which the response is a reaction rate. For the homogeneous sphere, results for the total leakage as well as for the respective first- and second-order sensitivities are in excellent agreement with the exact benchmark values. For the nonanalytic problems, the results obtained by applying the 2nd-ASAM to compute sensitivities are in excellent agreement with central-difference estimates. The efficiency of the 2nd-ASAM is underscored by the fact that, for the cylinder, only 12 adjoint PARTISN computations were required by the 2nd-ASAM to compute all of the benchmark’s 18 first-order sensitivities and 224 second-order sensitivities, in contrast to the 877 PARTISN calculations needed to compute the respective sensitivities using central finite differences, and this number does not include the additional calculations that were required to find appropriate values of the perturbations to use for the central differences.

MULTI-PRED: A Software Module for Predictive Modeling of Coupled Multi-Physics Systems

Abstract The software module MULTI-PRED implements the methodology for predictive modeling of coupled multi-physics systems (PM-CMPS) formulated by Cacuci [Ann. Nucl. Energy, Vol. 70, p, 266 (2014)]. This methodology fully takes into account the coupling terms between the systems but requires only the computational resources that would be needed to perform predictive modeling on each system separately. The PM-CMPS methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution based on a priori known mean values and uncertainties characterizing the experimental and computational parameters and results of interest responses called for the multi-physics models under consideration. This maximum entropy a priori distribution is combined, using Bayes’ theorem, with the likelihood provided by the multi-physics simulation models to obtain a formal posterior distribution. Subsequently, the posterior distribution thus obtained is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the multi-physics model parameters and responses along with corresponding reduced uncertainties. Noteworthy, the predictive modeling methodology for the coupled systems is constructed such that the systems can be considered sequentially rather than simultaneously, while preserving exactly the same results as if the systems were treated simultaneously. Consequently, very large coupled systems, which could perhaps exceed available computational resources if treated simultaneously, can be treated with the PM-CMPS methodology presented in this work sequentially and without any loss of generality or information, requiring just the resources that would be needed if the systems were treated sequentially. The PM-CMPS methodology can be applied to reduce uncertainties in both forward and inverse problems. Three demonstration problems are provided to illustrate the application of the PM-CMPS methodology. The first problem presents the application of the PM-CMPS methodology to a simple particle diffusion problem which admits a closed-form analytical solution which facilitates a rapid understanding of this methodology and its predicted results. The second demonstration problem presents the application of the PM-CMPS methodology to the problem of inverse prediction, from detector responses in the presence of counting uncertainties, of the thickness of a homogeneous slab of material containing uniformly distributed gamma-emitting sources for optically thin and thick slabs. This problem highlights the essential role played by the relative uncertainties (or, conversely, accuracies) of measured and computed responses. The third demonstration problem presents the application of the PM-CMPS methodology to the F-Area cooling towers at the Savannah River National Lab. This problem demonstrates that the PM-CMPS methodology reduces the predicted response uncertainties not only at locations where measurements are available, but also at locations where measurements are not available. MULTI-PRED is written in Fortran and runs on Linux and Windows systems. A C++ version will also become available.

Sensitivity and Uncertainty Analysis of Counter-Flow Mechanical Draft Cooling Towers—I: Adjoint Sensitivity Analysis

Abstract For counter-flow mechanical draft cooling towers, the air in the fill can reach the point of saturation before leaving the fill section. The heat and mass transfer to the saturated air by evaporative cooling inside the fill are modeled with some assumptions and with over 50 parameters for boundary conditions, cooling tower geometries, heat and mass transfer correlations, water and air thermal properties, etc. Because of the parameter uncertainties and modeling assumptions, the accuracy and reliability of the cooling tower model need to be evaluated by quantifying the uncertainties associated with the model output. First, sensitivities of the model output with respect to all the model parameters need to be analyzed. Based on the cooling tower model, this work developed adjoint sensitivity models for the saturated case to compute efficiently and exactly the sensitivities of the model responses to all model parameters by applying the general adjoint sensitivity analysis methodology for nonlinear systems. The solution of the adjoint sensitivity models are independently verified. With the sensitivities known, the model parameters can be ranked in their importance for contributing to response uncertainties. The propagation of the uncertainties in the model parameters to the uncertainties in the model outputs can be evaluated. By further applying the predictive modeling for coupled multiphysics systems methodology, the cooling tower model for the saturated case can be improved by reducing the model prediction uncertainties through assimilation of experimental measurements and calibration of model parameters.