Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: I. Computed Sensitivities

2022

JNE

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This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N), which enables the most efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in the model’s phase space. The mathematical framework underlying the nth-CASAM-N is proven to be correct by using mathematical induction. The nth-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 86%

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N): Mathematical Framework

2022

JNE

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“…A higherorder sensitivity analysis of this model has not been performed yet. On the other hand, solvers for the computation of first-, second-, third-and fourth-order sensitivities for neutron transport models are available, as described in the works by Fang and Cacuci [12,13], where it was found that the microscopic total cross-sections of isotopes 1 H and 239 Pu of the reactor physics benchmark analyzed therein are the most important parameters affecting that benchmark's leakage response. In particular, it was shown that the largest unmixed 4th-order sensitivity is S (4) = 2.720 × 10 6 , which is with respect to the total cross-section σ 30 t,6 for the 30th energy group (which comprises thermalized neutrons in the energy interval from 1.39 × 10 −4 eV to 0.152 eV) of isotope 6 (i.e., 1 H).…”

Section: Discussionmentioning

confidence: 99%

Illustrative Application of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems to the Nordheim–Fuchs Reactor Dynamics/Safety Model

2022

JNE

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The application of the recently developed “nth-order comprehensive sensitivity analysis methodology for nonlinear systems” (abbreviated as “nth-CASAM-N”) has been previously illustrated on paradigm nonlinear space-dependent problems. To complement these illustrative applications, this work illustrates the application of the nth-CASAM-N to a paradigm nonlinear time-dependent model chosen from the field of reactor dynamics/safety, namely the well-known Nordheim–Fuchs model. This phenomenological model describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This model is sufficiently complex to demonstrate all the important features of applying the nth-CASAM-N methodology yet admits exact closed-form solutions for the energy released in the transient, which is the most important system response. All of the expressions of the first- and second-level adjoint functions and, subsequently, the first- and second-order sensitivities of the released energy to the model’s parameters are obtained analytically in closed form. The principles underlying the application of the 3rd-CASAM-N methodology for the computation of the third-order sensitivities are demonstrated for both mixed and unmixed second-order sensitivities. For the Nordheim–Fuchs model, a single adjoint computation suffices to obtain the six 1st-order sensitivities, while two adjoint computations suffice to obtain all of the 36 second-order sensitivities (of which 21 are distinct). This illustrative example demonstrates that the number of (large-scale) adjoint computations increases at most linearly within the nth-CASAM-N methodology, as opposed to the exponential increase in the parameter-dimensional space which occurs when applying conventional statistical and/or finite difference schemes to compute higher-order sensitivities. For very large and complex models, the nth-CASAM-N is the only practical methodology for computing response sensitivities comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

“…The accompanying work [4], which is designated as Part I, has reported the exact computation of the most important 4th-order sensitivities of the PERP benchmark's total leakage response with respect to the benchmark's 180 group-averaged microscopic total cross sections. The 4th-order sensitivities obtained in [4] are used in this work to compute the contributions of the 4th-order terms, and compare their impact on the leakage response's expected value, variance and skewness with the corresponding impact stemming from the 1st-, 2nd-and 3rd-order sensitivities.…”

Section: Introductionmentioning

confidence: 99%

“…where JTX = G × I = 180 denotes the total number of microscopic total cross sections for G = 30 groups and I = 6 isotopes contained in the PERP benchmark [4].…”

mentioning

confidence: 99%

Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties

Fang

2022

JNE

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This work quantifies the impact of the most important 4th-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) reactor physics benchmark with respect to the benchmark’s 180 group-averaged microscopic total cross sections, on the expected value, variance and skewness of the benchmark’s leakage response. This work shows that, as the standard deviations of the cross sections increase, the contributions of the 4th-order sensitivities to the response’s expected value and variance become significantly larger than the corresponding contributions stemming from the 1st-, 2nd- and 3rd-order sensitivities. Considering a uniform 5% relative standard deviation for all microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value and variance of the PERP leakage response amount to 56% and 52%, respectively. Considering 10% uniform relative standard deviations for the microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value increase to nearly 90%. Consequently, if the computed value L(a) were considered to represent the actual expected value of the leakage response and the 4th-order sensitivities were neglected, the computed value would represent the actual expected value with an error of 3400%. Furthermore, uniform relative standard deviations of 5% and larger (10%) for the microscopic total cross sections cause the higher-order sensitivities to contribute increasingly higher amounts to the response standard deviation: the contributions stemming from the 4th-order sensitivities are larger than the contributions stemming from the 3rd-order sensitivities, which in turn are larger than those stemming from the 2nd-order sensitivities, which are themselves larger than the contributions stemming from the 1st-order sensitivities. This finding evidently underscores the need for computing sensitivities of order higher than first-order. The results obtained in this work also indicate that the 4th-order sensitivities produce a positive response skewness, causing the leakage response distribution to be skewed towards the positive direction from its expected value. Increasing the parameter standard deviations tends to decrease the value of the response skewness, causing the leakage response distribution to become more symmetrical about the mean value. The results presented in this work highlight the finding that the microscopic total cross section for hydrogen (H) in the lowest (“thermal”) energy group is the single most important parameter among the 180 microscopic total cross sections of the PERP benchmark, as it contributes most to the various response moments.