Direct Perturbation Method Research Articles

Nonlinear coupled vibration of an arch-beam structural system

Nonlinear dynamic analysis of a shallow arch coupled with a cantilever beam (resting on Winkler foundation) is investigated by direct perturbation method. Two key parameters, (beam over arch) density ratio and height of beam, recharacterize coupling intensity between arch and beam. Nonlinear dynamic responses of arch-beam coupled model are predicted by asymptotic analysis, which include hardening and softening cases. After numerical verification of arch-beam coupled model, the uncoupled model (i.e., skipping beam-arch coupling) is also investigated for comparison. Besides, two more related but degenerately coupled models, e.g., arch-oscillator coupled model [1] and arch-rigid body coupled model [2] are both compared with the current arch-beam coupled model, revealing one key coupling characteristic of elastic support/foundation, which is however overlooked in previous models, e.g., only partial mass is involved with dynamic coupling.

Development of an Adjoint Flux Calculation Technique for Exact Perturbation Theory in Monte Carlo Code MCS

A calculation technique computing the adjoint flux of perturbed system is developed for the exact perturbation theory in Monte Carlo transport. By using a correlated sampling and iterated fission probability methods together, the adjoint flux of perturbed system is calculated during a forward Mote Carlo simulation of an unperturbed system. In the method, no additional particle simulation is required to compute the adjoint flux of perturbed system. The exact perturbation method is implemented in the Monte Carlo code MCS. The results of perturbation method for k-eigenvalue change are compared against differential operator sampling, adjoint-weighted perturbation, and direct perturbation methods for the Godiva benchmark and PLUS7 fuel assembly.

Averaging over atom snapshots in linear-response TDDFT of disordered systems: A case study of warm dense hydrogen.

Linear-response time-dependent density functional theory (LR-TDDFT) simulations of disordered extended systems require averaging over different snapshots of ion configurations to minimize finite size effects due to the snapshot-dependence of the electronic density response function and related properties. We present a consistent scheme for the computation of the macroscopic Kohn-Sham (KS) density response function connecting an average over snapshot values of charge density perturbations to the averaged values of KS potential variations. This allows us to formulate the LR-TDDFT within the adiabatic (static) approximation for the exchange-correlation (XC) kernel for disordered systems, where the static XC kernel is computed using the direct perturbation method [Moldabekov et al., J. Chem. Theory Comput. 19, 1286 (2023)]. The presented approach allows one to compute the macroscopic dynamic density response function as well as the dielectric function with a static XC kernel generated for any available XC functional. The application of the developed workflow is demonstrated for the example of warm dense hydrogen. The presented approach is applicable for various types of extended disordered systems, such as warm dense matter, liquid metals, and dense plasmas.

Improvement of the CLUTCH method for sensitivity analysis of k-eigenvalue to continuous-energy nuclear data in NECP-MCX

The Iterated Fission Probability (IFP) method and Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Track-length importance Characterization (CLUTCH) method are commonly implemented in Monte-Carlo codes to calculate the sensitivity coefficients of keff to continuous-energy nuclear data. However, these methods are plagued by the problem of large variance of sensitivity coefficients of the scattering reactions with relatively small cross sections. This problem means that more particles are needed to accurately calculate these sensitivity coefficients. To address this problem, a novel technique called CLUTCH coupled with forced collisions (CLUTCH-FC) method is proposed to reduce the variance of sensitivity coefficients of those reactions. The forced collision method is performed with specified reactions rather than specified cells in tradition to increase the sampling of specified reactions and a new rule for the setting of cutoff weight of neutrons is also proposed to make the best use of forced collisions. Three methods, viz. IFP, CLUTCH and CLUTCH-FC methods, are implemented in the Monte-Carlo code NECP-MCX. The verification is conducted in Godiva, Jezebel and TMI-1 problems by comparing the sensitivity coefficients calculated by IFP and CLUTCH methods with those calculated by the direct numerical perturbation (DNP) method. The numerical results indicate that 1) the sensitivity coefficients calculated by the IFP and CLUTCH methods corroborate well with those calculated by the DNP method, and 2) the CLUTCH-FC method is efficient in improving the tally efficiency of sensitivity coefficients of scattering reactions with relatively small cross sections by comparing results of CLUTCH-FC and CLUTCH methods.

Nonlinear dynamics of traveling beam with longitudinally varying axial tension and variable velocity under parametric and internal resonances

In this investigation, the nonlinear dynamics of an axially accelerating viscoelastic beam on a pully mounting system have been analyzed. The axial tension of the beam is modeled as a function of the traveling velocity, support stiffness parameter as well as spatial coordinate. Geometric cubic nonlinearity in the equation of motion is due to elongation in the neutral axis of the beam. The integro-partial differential equation of motion of the axially accelerating beam associated with the simply supported end conditions is solved analytically by adopting the direct perturbation method of multiple time scales. As a result, a set of complex variable modulation equations is generated, which governs the modulation of amplitude and phase. This set of modulated equations is numerically solved to explore the influence of the support stiffness parameter upon the stability and bifurcation of the beam which has not been addressed in the existing literature. Apart from this, the impact of fluctuating velocity component, viscoelastic coefficient, longitudinal stiffness parameter, internal and parametric frequency detuning parameters on the stability and bifurcation analysis is studied, revealing significant dynamic characteristics of the traveling system. The fourth-order Runge–Kutta method is applied is to find the dynamic solution of the system. The system displays stable periodic, quasi-periodic, and mixed-mode dynamic responses along with the unstable chaotic behavior for a specific set of system parameters. The results obtained through an analytical–numerical approach may help the design and operation of an axially moving beam.

Parametrically excited axially accelerating viscoelastic beam subjected to time-varying axial speed, longitudinally varying axial tension and internal resonance

The present work investigates the nonlinear characteristics of an axially moving viscoelastic beam of the Kelvin–Voigt model on a pulley mounting system. The equation of motion is developed by including the terms like external and internal damping, geometric cubic nonlinearity, Coriolis, time-dependent axial speed, and longitudinally varying axial tension. The direct perturbation method of multiple time scales (MMS) is adopted to get an approximate solution for the nonlinear integro-partial differential equation of motion which leads to a set of complex variable modulation equations. These modulated equations are solved numerically to study the trivial state stability plot considering the effect of the support stiffness parameter for the first time. In addition to this, the impact of material, and viscous damping parameters on stability boundary is also studied. Continuation algorithm is implemented to analyse the influence of the support stiffness parameter, internal and parametric detuning parameters, fluctuating velocity component, and longitudinal stiffness parameter on the stability and bifurcation of the steady state solutions. Variation in internal frequency detuning parameter and variable axial speed component reveals different kinds of equilibrium solution curves in the nonlinear travelling system. The stability and bifurcation features observed in the present nonlinear system with change in the above control parameters are interesting and also not available in the existing literature.

Robust optimization of MTMD systems for the control of vibrations

Multiple Tuned Mass Dampers (MTMD) is one of the simplest and most reliable solution to control the vibrations of a structure, e.g. allowing to deal with a wide distribution of structural natural frequencies and usually showing inherent stability. Nevertheless, uncertainties in the behavior of structures under random dynamic input may induce implications on the MTMD effectiveness, e.g. detuning or amplifications. Typically, uncertainties in the input are exclusively considered, while uncertainties affect as much also the mechanical parameters.This paper proposes a robust optimum design of the MTMD, considering uncertainties both in the structural parameters and in the earthquake input. At this aim, a random vibration analysis of the response is adopted, and a direct linear perturbation method is applied on the uncertain parameters. Results show that a deterministic optimization is inappropriate for sensible systems as TMDs, while a robust optimization aims to better control the response with low computational efforts. Furthermore, the selection of the objective functions significantly affects the optimal design parameters, as the number of TMDs influences the robust designs.

Uncertainty propagation analysis of ionization chamber sensitivity in core power control system of a small PWR

Influence of the uncertainty of ionization chamber sensitivity on the core power control system of a small PWR is studied. Firstly, the model of each link in the core power control system is established with the transfer function. Then, a fuzzy PID controller with simulated annealing algorithm is designed to control the small PWR core power. Finally, under the large range load variation of reactor core, the uncertainty propagation of ionization chamber sensitivity is analyzed based on direct numerical perturbation method. The results show that under 100%FP-15%FP at the rate of 15%FP/min of power in the small PWR, the maximum uncertainty of core relative power is 6.03%. And the maximum uncertainty of coolant outlet temperature deviation and fuel temperature deviation are respectively 14.4% and 14.47%. The uncertainties of the output parameters can be stabilized in a small certain range by fuzzy PID controller.

Comparison of calculation methods for lead cooled fast reactor reactivity effects

This study analyzes three different reactivity feedback coefficients of the Advanced Lead-cooled Fast Reactor Demonstrator (ALFRED) reactor. During the evaluations, the central difference Direct Perturbation Method (DPM) and the Linear Perturbation Theory (LPT) were applied to determine the feedback coefficients and their uncertainties due to cross-section data. The continuous energy SERPENT Monte Carlo code, the multigroup TSUNAMI-3D sequence of the SCALE program package, and PARTISN discrete ordinate neutron transport code coupled with SEnTRi developed by the authors was applied, and the properties of these fundamentally different calculations are presented. In order to validate the methodologies, lead void coefficient calculations were performed for the Comet critical assembly, which confirmed that a good agreement could be reached between the different methods and measurement results. Besides the high-level transport approximation, the calculations indicated that the appropriate spatial resolution has outstanding importance in describing the high flux gradient.

Approximate solutions for the ion-laser interaction in the high intensity regime: matrix method perturbative analysis

This work provides an explicit expression for the second-order perturbative solution of a single trapped ion at the high intensity regime. Unlike other perturbative schemes, where the ion-laser dynamics has been explored using unitary transformations and the Lamb-Dicke regime, this analysis relies instead on a direct perturbation method, that may be implemented in simple manner and works especially well without resorting to additional approximations. Based on a matrix method and a final normalization of the perturbed solutions, the second-order perturbative analysis supply the probability to find the ion in its excited state; the resulting perturbative solution renders a high accuracy, comparable to the one based on the small rotation approximation.

Phase response approaches to neural activity models with distributed delay.

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

General perturbation correction: full-decomposition and physics-based elimination of non-secular terms

Discretized perturbation analysis based upon a structure's single (finite) mode Galerkin-truncated model may lead to erroneous results in comparison with full-basis discretization (or, equivalently, direct perturbation method). This error is due to the two involved analytical steps, i.e., multi-scale expansion and mode truncation, being non-commutative with each other. However, the key underlying physical origin for this error (or non-commutativity) is still unclear. The novelty of the current work is to propose a new physics-based error source interpretation and, accordingly, a general perturbation correction procedure. Explicitly, a general physics-based interpretation for the error source is referred to the incomplete characterization of low-order non-secular effects due to both spectrum mixture and single (finite) mode truncation. Three typical dynamical scenarios, i.e., hard external excitation, quadratic nonlinearity, and hard boundary motion, are addressed in the same unified framework. By introducing a full spectrum decomposition and a complete elimination of the non-secular term in a physically equivalent manner, a general correction procedure of the finite mode truncation often used in perturbation analysis is proposed and then validated through numerical examples.

Generalized Perturbation Theory Based Total Sensitivity and Uncertainty Analysis for High-Fidelity Neutronics Calculation

Neutronics calculation for nuclear reactor with high-fidelity technology can significantly reduce the uncertainties propagated from numerical approximation error and model error. However, the uncertainty of input parameters inevitably exists, especially for nuclear data. On the other hand, resonance self-shielding calculation is essential for multi-group assumption based high-fidelity neutronics calculation, which introduce the implicit effect for calculation responses. In order to fully consider the implicit effects in the process of uncertainty quantification, a generalized perturbation theory (GPT) based implicit sensitivity calculation method is proposed in this paper. Combining the explicit sensitivity coefficient, which can be quantified using classic perturbation theory, the total sensitivity coefficient of calculation responses is obtained. Then the total sensitivity and uncertainty module is established in self-developed neutron transport code with high-fidelity technology-HNET. To verify the accuracy of the sensitivity calculation methods proposed in this paper, a two-dimensional fuel pin problem is chosen to verify the sensitivity results, and the numerical results show good agreement with results calculated by a direct perturbation method. Finally, uncertainty analysis for two-dimensional fuel pin problem is performed and some general conclusions are obtained from the numerical results.

On the Effect of Angular and Spatial Discretization on Perturbation Calculations

In this article, different angular flux discretization options, namely discrete ordinates representation and spherical harmonics expansion are compared from the viewpoint of the accuracy of perturbation calculations. The PARTISN discrete ordinates neutron transport solver was coupled with the SEnTRi code, developed at BME, in order to perform perturbation theory calculations in different types of geometry descriptions and angular representations. With the help of the implemented code, the effect of the angular and spatial discretization on the results of perturbation theory calculations was investigated. Exact matches were observed in Cartesian geometries with the direct perturbation method when the discrete ordinates angular representation was used, and small discrepancies were found when the spherical harmonics expansion was applied. In cylindrical geometries, slight differences were observed with both angular expansions, which originate from the nature of the adjoint transport operator in curvilinear coordinate systems. The differences can be reduced to a negligible level with increased expansion order in both cases. Small discrepancies can have a significant effect in sensitivity, uncertainty and transient calculations, which that for high accuracy calculation the discrete ordinates representation of the angular dependent flux should be used or sufficiently high expansion order with the spherical harmonics must be applied.

Geometric sensitivity analysis of generalized response function with RMC code

GEneralized Adjoint Responses in Monte Carlo (GEAR-MC) method has been developed to perform sensitivity analysis of reaction rate ratios, which was to compute sensitivity coefficients of reaction rate ratios to nuclear data e.g. nuclear cross sections. In this work, the GEAR-MC method is extended to calculate sensitivity coefficients of reaction rate ratios to geometric parameters for the first time and is implemented in the Reactor Monte Carlo (RMC) code. This newly developed geometric sensitivity analysis capability is tested by four benchmark problems. The results are shown to agree within 4% and 3 σ with reference results obtained by the direct perturbation method for all cases. Moreover, the contributions from different terms to the sensitivity results are compared. It is shown that, for Godiva, Hemisphere Godiva and Jezebel benchmarks, the intergenerational term contributes much to sensitivity coefficients, and this term is necessarily considered correctly in the practical application.

Uncertainty quantification of proper orthogonal decomposition based online power-distribution reconstruction

In this paper, the function of uncertainty quantification is implemented in CORN, an online power-distribution reconstruction code based on the proper orthogonal decomposition (POD) method. The uncertainties in the detector signals are propagated to the reconstructed power distributions with the deterministic-based method in two steps. First, the sensitivity coefficients are calculated with the direct numerical perturbation (DNP) method. Then the uncertainty in the reconstructed power distribution based on the detector signals is carried out with the “sandwich” rule. The uncertainty quantification is carried out based on the core of BEAVRS benchmark cycle 1. Numerical results show that the maximum uncertainty of 3D reconstructed power distribution is less than 3% with the 1% uncertainties of detector signals for the POD method. At last, the uncertainties of power distributions for various expansion order are quantified. Eigenvalues can assist in the determination of the expansion order with a certain group of POD samples.

First-order effect of frequency-dependent selection on fixation probability in an age-structured population with application to a public goods game

In this paper, we deduce the first-order effect of frequency-dependent viability and fertility selection on the probability of fixation of a mutant in a large finite haploid population with a fixed age structure by applying a direct small perturbation method to the neutral two-timescale genealogical process. This effect is expressed in terms of fixation-fitness coefficients times ancestry coefficients that are related to the effective population size. In the case of constant selection, the fixation-fitness coefficients are functions of the coefficients of viability and fertility selection weighted by reproductive values and population-structure coefficients for the different age classes. This explains the difference between the effects of viability selection and fertility selection on fixation probability in age-structured populations. With frequency-dependent selection in the form of a public goods game, the fixation-fitness coefficients depend also on the public good allocation strategy of the population and the resource allocation strategies of the individuals. In this case, the results show that weak selection may favor the fixation of cooperation if the cooperators allocate more resources to reproduction versus survival than the defectors do.

Uncertainty quantification of sodium-cooled fast reactor based on the UAM-SFR benchmarks: From pin-cell to full core

To quantify the uncertainties in the neutronics calculations of sodium-cooled fast reactor (SFR), a series of sensitivity and uncertainty (S&U) analysis were performed on two typical SFRs, which were a large-size MOX-fueled core and a medium-size metallic fueled core. The analysis was done from the pin-cell level to full core level to investigate the uncertainty propagation in the two-step neutronics calculation scheme. The responses analyzed contained the kinf, homogenized cross-sections, sodium void reactivity (SVR), Doppler constant (KD) and control rod worth (CRW) at the sub-assembly level, and the keff and power distribution at the full core level. The direct numerical perturbation method was used to get the relative sensitivity coefficients and the ‘Sandwich Rule’ was used to get the relative uncertainties. The results showed that the relative uncertainties propagated linearly from pin-cell to full core for the kinf and keff. Relative uncertainties of reactivity like SVR were much bigger than the ones of kinf and keff. However, the relative uncertainty of power distribution was smaller than other responses. Differences were observed in the S&U analysis between the MOX fueled and metallic fueled SFRs. Softer spectrum of the large-size MOX fueled core caused bigger uncertainty compared to the metallic fueled core.

Sensitivity and uncertainty analysis for the PWR online power-distribution monitoring with NECP-ONION system

In this paper, the sensitivity and uncertainty analysis model has been proposed and implemented in the NECP-ONION system, which is capable of performing the 3D online power-distribution monitoring for PWR by applying the harmonics expansion method. A new method called the analytical method is used to quantify the relative sensitivity coefficients of the monitored power distributions with respect to the detector signals; then the “sandwich” rule is used to quantify the uncertainties of the monitored power distributions with the sensitivity-analysis results based on the uncertainty-propagation method. The BEAVRS benchmark is used to verify the sensitivity and uncertainty analysis capability in NECP-ONION. The results of the direct numerical perturbation method are used as reference to verify the sensitivity coefficients calculated with the analytical method. The results of the statistical sampling method and the analytical method are compared to each other for the uncertainty verification. Then, the sensitivity and uncertainty analyses with various expansion orders of harmonics and core burnup levels are carried out. Numerical results show that, the expansion order is the trade-off of the monitoring accuracy and uncertainty, while the sensitivity and uncertainty analysis is useful for the determination of the expansion order.

Sensitivity analysis for the weighting factor in the least-square method for the PWR on-line power-distribution monitoring

A 3D on-line power-distribution monitoring code has been developed based on the least-square method, due to the explicit relationship between the neutron-diffusion equation and detector signals. The BEAVRS benchmark is used for code verification and validation. The errors of the monitored detector-response distributions are small in the whole cycle. To validate the monitoring accuracy in nodes without detectors, measurements in one selected detector tube are neglected intentionally in the input of the monitoring code. The differences between the monitored detector responses “with” and “without” using the tube measurements as inputs are small and acceptable. Then monitoring calculations with different weighting factors from 10−3 to 104 are carried out at different burnup levels. The errors of monitored detector responses decrease with the weighting factor. The differences in the monitored detector responses “with” and “without” using the absent measurements increase with the weighting factor. After that, sensitivity analysis for the weighting factor is performed based on the direct numerical perturbation method. Similar tendency is observed for the sensitivity coefficients of monitored detector responses to detector measurements and the differences of monitored detector responses “with” and “without” using the certain tubes measurement. At last, based on the BEAVRS benchmark problem, a new definition for the “monitoring accuracy” to determine the optimum weighting factor is proposed.