Envelope Formulation Research Articles

TRISO SiC Failure Probability for Reactivity Initiated Accidents in High-Temperature Gas-Cooled Reactors

This work analyzes the failure process of the silicon carbide (SiC) layer in tristructural isotropic (TRISO) during reactivity-initiated accident scenarios for a high-temperature gas-cooled reactor (HTGR) with BISON. Two cases are considered—a group control rod withdrawal (CRW) and a control rod ejection (CRE)—reproduced from a previous study. Failure probability is modeled using Weibull statistics, and worst-case scenario Weibull parameters are adopted to simulate the envelopes in BISON with a one-dimensional TRISO model. CRW scenario results are characterized by higher values of maximum energy deposition and final temperature and volumetric strain with respect to the CRE ones, but the latter have remarkably higher SiC failure probability, mainly due to the offset in strain rates between the two cases. This work also confirms the validity and conservatism of the performance envelopes produced in a previous work by replicating the envelope formulation using RELAP5-3D and RAVEN with a different sampling technique and obtaining consistent results. A sensitivity analysis using the Sobol variance decomposition method on SiC failure probability is then performed involving a set of inputs on both CRW and CRE. The two most important parameters are Weibull modulus and characteristic stress, and their relative importance depends on the specific case. The proposed interpretation of the results is that both energy deposition and strain rate influence the relative degree of importance of the failure parameters. Computation of 95% confidence intervals around worst-case scenario SiC failure probability values is also carried out for four different sets of Weibull parameters. A new criterion for SiC TRISO quality classification built upon safety-based ranges of Weibull parameters is proposed to be integrated in future Fuel-Production Quality Assurance Plans.

Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique

An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.

Stress path dependent hydromechanical behaviour of heterogeneous carbonate rock

The influence of stress paths, representative of reservoir conditions, on the hydromechanical behavior of a moderately heterogeneous carbonate has been investigated. Multiscale structural heterogeneities, common for instance in carbonate rocks, can strongly alter the mechanical response and significantly influence the evolution of flow properties with stress. Using a triaxial cell, the permeability evolutions during compression and the effects of brittle (fracture) and plastic (pore collapse) deformations at yield, were measured. A strong scattering was observed on the mechanical response both in term of compressibility and failure threshold. Using the porosity scaling predicted by an adapted effective medium theory (based on crack growth under Hertzian contact), we have rescaled the critical pressures by the normalized porosity deviation. This procedure reduces efficiently the scattering, revealing in the framework of proportional stress path loading, a linear relation between the critical pressures and the stress path parameter through all the deformation regimes. It leads to a new formulation for the critical state envelope in the 'mean stress, deviatoric stress' diagram. The attractive feature of this new yield envelope formulation relies on the fact that only the two most common different mechanical tests 'Uniaxial Compression' and 'Hydrostatic Compression', are needed to define entirely the yield envelope. Volumic strains and normalized permeabilities are finally mapped in the stresses diagram and correlated.

Time-Frequency Formulation for the Nonlinear Analysis of Coupled Phase-Locked Loops

In this work, harmonic balance (HB) and envelope- transient formulations of coupled phase-locked loops (CPLLs) are presented. The CPLL has the added difficulty of its autonomous behavior since no reference oscillator is present. The new formulation takes into account the autonomy of the system, introducing a special set of state variables, which depend on the autonomous frequencies. The hysteresis phenomenon in CPLLs is analyzed in detail, efficiently obtaining the pull-in and hold-in ranges through HB. The pole analysis of the perturbed HB system enables an accurate prediction of instabilities and resonances. Due to the CPLL autonomy, there exists an inherent noise accumulation effect. This effect is taken into account, analyzing the perturbation in terms of accumulation and deviation components. The envelope formulation allows simulating the CPLL behavior in the presence of modulation signals. The influence of the stability of the steady-state solution on the modulated signals is investigated. The simulation results have been successfully compared with the measurements in a manufactured CPLL system at 2 GHz.

Automated time domain modeling of linear and nonlinear microwave circuits using recurrent neural networks

In this article, a recurrent neural network (RNN) method is employed for dynamic time-domain modeling of both linear and nonlinear microwave circuits. An automated RNN modeling technique is proposed to efficiently determine the training waveform distribution and internal RNN structure during the offline training process. This technique extends a recent automatic model generation (AMG) algorithm from frequency-domain model generation to dynamic time-domain model generation. Two types of applications of the algorithm are presented, transient electromagnetic (EM) behavior modeling of microwave structures, and time-domain envelope modeling of power amplifiers (PA). For transient EM modeling, we consider EM structures with varying material and geometrical parameters. AMG automatically varies the EM structural parameters during training and drives time-domain EM simulators to generate necessary amount of data for RNN to learn. AMG aims to model the transient behavior with minimum RNN order while satisfying accuracy requirements. In modeling PA behavior, an envelope formulation is used to specifically learn the AM/AM and AM/PM distortions due to third-generation (3G) digital modulation input. The RNN PA model is able to model these time domain distortions after training and can accurately model the amplifier behavior in both time (AM/AM, AM/PM) and frequency (spectral re-growth). © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

Microscopic interface asymmetry and spin-splitting of electron subbands in semiconductor quantum structures

The microscopic interface asymmetry of (001)-grown semiconductor heterostructures that gives rise to heavy-light hole coupling even at zero in-plane wave vector k ‖, modifies also the subband dispersion of confined electrons. Starting from a multiband envelope formulation we apply matrix perturbation theory to derive explicit expressions caused by this interface asymmetry, which in the 2×2 conduction band Hamiltonian appear as a warping and a spin-splitting term. The warping term results in an inequivalence of the dispersion along [110] and [1 1 0] as required by the microscopic C 2v symmetry, while the spin-splitting term has the same structure as the corresponding term derived from the zinc-blende bulk inversion asymmetry. Implications with respect to spin-relaxation will be discussed.

TRANSIENT WAVE ENVELOPE ELEMENTS FOR WAVE PROBLEMS

A novel family of infinite wave envelope elements is described which can be used in conjunction with conventional finite elements to model the transient wave equation in unbounded regions. The elements are obtained by applying an inverse Fourier transformation to a mapped wave envelope formulation in the frequency domain. The discrete transient equations obtained in this way can be applied to two-dimensional and three-dimensional problems without restriction, being valid over a full range of excitation frequencies. The effectiveness and accuracy of the method is demonstrated in application to simple test cases which involve the calculation of transient sound fields generated by pulsating spheres and cylinders excited from rest in an unbounded region. Test solutions are compared to analytic solutions and to finite element solutions obtained by using large computational grids which extend beyond the region influenced by the transient disturbance.

Response of the lithosphere to in-plane force variations

The Earth's lithosphere deforms when subjected to vertical and horizontal tectonic forces because it is not perfectly rigid. The deformation has both elastic (flexural) and inelastic (brittle and ductile creep) components. We use the yield strength envelope formulation of a depth-dependent rheology to predict how the lithosphere will deform in response to applied in-plane force. Brittle deformation of the upper lithosphere is influenced by the distribution of existing faults and fractures in the crust and their orientation with respect to the applied tectonic force. The sense of fault reactivation indicates the sign of the applied in-plane force. The induced flexural deformation is governed by the pre-existing lithospheric deflection, the effective elastic thickness of the lithosphere when the force was applied, and the magnitude of the applied force. Using this framework, we compare the predicted response of the lithosphere to in-plane force variations with that observed in the Central Indian Ocean and the Jeanne d'Arc basin, offshore Newfoundland. The brittle deformation in both regions is readily observed in seismic reflection data. The upper oceanic crust in the Central Indian Ocean is broken into blocks bounded by high-angle reverse faults. The dip of these basement faults and their orientation parallel to the marine magnetic lineations suggest that they represent a reactivated crustal fabric imparted during the formation of oceanic crust. The reverse sense of fault reactivation is consistent with the north-south compression determined from earthquake focal mechanisms. In the Jeanne d'Arc basin, episodes of northwest-southeast extension reactivated the basin-bounding faults in a normal sense during late Barremian, early Aptian, and late Aptian time. In the late Albian, the same crustal blocks were reactivated and uplifted in response to north-northeast in-plane compression. Flexural deformation due to applied in-plane force is observed only in the Central Indian Ocean case, where the oceanic crust and most of the overlying Bengal Fan sediments were deformed into broad, east-west trending undulations with wavelengths of ≈ 100–300 km and amplitudes up to ≈ 2 km. In the Jeanne d'Arc basin, the flexural deformation which should accompany the fault reactivation is difficult to recognize from the preserved stratigraphy. Consequently, brittle deformation within extensional basins, which appears to be the most recognizable response of the lithosphere to in-plane force variations, obscures the effects of the flexural deformation. However, it is this flexural deformation that has been postulated as a tectonic alternative to glacial-eustasy in explaining the 3 rd-order sea level cycles. Because it is difficult to recognize the effects of the flexural deformation in the preserved stratigraphy, we surmise that it is not a viable alternative to glacial-eustasy in generating either regional or global unconformities. Instead, we propose that the flexural and brittle deformation due to applied in-plane force may locally enhance or diminish the effects of other processes responsible for unconformity generation.

Pressure drag and momentum fluxes due to the Alps. II: Representation in large‐scale atmospheric models

AbstractA three‐dimensional numerical model with interactive grid nesting is used to study the pressure drag and associated momentum fluxes due to the Alpine complex (for a specified upstream wind and static stability profile). Four basic questions are addressed which are relevant to the problems of flow description in the vicinity of mountains and the impact of orography in the context of mesoscale numerical weather prediction and general circulation models: What is the dependence of basic quantities such as the pressure drag on the model's horizontal resolution? Is there a need for non‐hydrostatic dynamics to account properly for the flow characteristics near steep mountains? Is there an effective role for enhanced orography such as an envelope formulation? Might a ‘gravity‐wave‐drag’ parametrization scheme be able to represent adequately the highest resolution fluxes and surface drag in a much coarser resolution model? A series of experiments with both hydrostatic and non‐hydrostatic models at resolutions of 80, 40, 20, 10 and 5 km is analysed, and two of these experiments are repeated with an envelope orography.It is concluded that hydrostatic simulations are sufficient to describe the basic dynamics even at 5 km resolution; however, there is a strong dependence of the drag and momentum fluxes on resolution, with the drag at 80 km resolution being almost half that at the highest resolution. It is also shown that the use of an envelope orography at coarser resolutions can compensate significantly for the drag under‐estimation, but is less effective in representing the vertical stress profiles or the direction of the stress vectors, which are systematically in error at the coarsest resolution. The potential role of gravity wave drag is confirmed and possible improvements considered. It is suggested that both envelope orography and gravity wave drag may make a significant contribution until resolutions are as high as 10 to 20 km.

Pressure drag and momentum fluxes due to the Alps. II: Representation in large-scale atmospheric models

A three-dimensional numerical model with interactive grid nesting is used to study the pressure drag and associated momentum fluxes due to the Alpine complex (for a specified upstream wind and static stability profile). Four basic questions are addressed which are relevant to the problems of flow description in the vicinity of mountains and the impact of orography in the context of mesoscale numerical weather prediction and general circulation models: (i) What is the dependence of basic quantities such as the pressure drag on the model's horizontal resolution? (ii) Is there a need for non-hydrostatic dynamics to account properly for the flow characteristics near steep mountains? (iii) Is there an effective role for enhanced orography such as an envelope formulation? (iv) Might a ‘gravity-wave-drag’ parametrization scheme be able to represent adequately the highest resolution fluxes and surface drag in a much coarser resolution model? A series of experiments with both hydrostatic and non-hydrostatic models at resolutions of 80, 40, 20, 10 and 5 km is analysed, and two of these experiments are repeated with an envelope orography. It is concluded that hydrostatic simulations are sufficient to describe the basic dynamics even at 5 km resolution; however, there is a strong dependence of the drag and momentum fluxes on resolution, with the drag at 80 km resolution being almost half that at the highest resolution. It is also shown that the use of an envelope orography at coarser resolutions can compensate significantly for the drag under-estimation, but is less effective in representing the vertical stress profiles or the direction of the stress vectors, which are systematically in error at the coarsest resolution. The potential role of gravity wave drag is confirmed and possible improvements considered. It is suggested that both envelope orography and gravity wave drag may make a significant contribution until resolutions are as high as 10 to 20 km.

Time-dependent wave envelope finite difference analysis of sound propagation

A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.

A finite element, wave envelope formulation for acoustical radiation in moving flows

An axisymmetric finite element model is presented for the prediction of radiation patterns generated by stationary acoustical sources in moving flows. The acoustical field is represented in an outer region by wave envelope elements which incorporate some features of ray acoustical behaviour. These are compatibly matched to a conventional finite element mesh in a region close to the generating mechanism. Results are presented for problems involving vibrating cylinders and spheres in subsonic mean flows. Good agreement is established between computed and exact solutions.