Quadrature Sets Research Articles

A New Galerkin Quadrature Method Not Requiring a Matrix Inverse

We derive a new Galerkin quadrature (GQ) method for S n calculations that differs from the two methods preceding it in that a matrix inverse for an N d × N d matrix, where N d is the number of directions in the quadrature set, is no longer required. Galerkin quadrature methods are designed for calculations with highly anisotropic scattering. Such methods are not simply special angular quadratures but also are methods for representing the S n scattering source that offers several advantages relative to the standard scattering source representation when highly truncated Legendre cross-section expansions must be used. Galerkin quadrature methods are also useful when the scattering is moderately anisotropic, but the quadrature being used is not sufficiently accurate for the order of the scattering source expansion that is required. We derive the new method and present computational results showing that its performance for two challenging problems is comparable to those of the two GQ methods that preceded it.

Neutron and photon calculations of beam shaping assembly for BNCT based on Monte Carlo-discrete ordinates coupled method

The radiation field parameters at the beam shaping assembly (BSA) exit port are the key to boron neutron capture therapy (BNCT) treatment. However, obtaining calculation results with high efficiency and accuracy using only Monte Carlo (MC) method or discrete ordinates (SN) method is difficult. Based on NECP-MCX code and Marvin code, this paper implemented the Monte Carlo-discrete ordinates coupled method and applied this method to the calculation of the radiation field at the BSA exit port. The effect of mesh grid division, the order of quadrature set, and the bias of quadrature set on the results of the coupled calculation was studied. It was concluded that the order and bias of the quadrature set have not improved the coupled calculation results. It was found that the data library used in the SN calculation is the main cause. Therefore, the Monte Carlo-discrete ordinates coupled method can be quickly and accurately applied to the calculation of the radiation field at the BSA exit port by dividing the energy and coordinate intervals rationally and using a more suitable cross section library.

Another look at residual dynamic mode decomposition in the regime of fewer snapshots than dictionary size

Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues associated with finite truncations of Koopman operators (e.g., Extended Dynamic Mode Decomposition), such as spurious eigenvalues. Spectral properties computed by ResDMD include spectra, pseudospectra, spectral measures, Koopman mode decompositions, and dictionary verification. In scenarios where the number of snapshots is fewer than the dictionary size, particularly for exact DMD and kernelized Extended DMD, ResDMD has traditionally been applied by dividing snapshot data into a training set and a quadrature set. We demonstrate how to eliminate the need for two datasets through a novel computational approach of solving a dual least-squares problem. We analyze these new residuals for exact DMD and kernelized Extended DMD, demonstrating ResDMD’s versatility and broad applicability across various dynamical systems, including those modeled by high-dimensional and nonlinear observables. The utility of these new residuals is showcased through three diverse examples: the analysis of a cylinder wake, the study of airfoil cascades, and the compression of transient shockwave experimental data. This approach not only simplifies the application of ResDMD but also extends its potential for deeper insights into the dynamics of complex systems.

Goal-based angular adaptivity with contributon theory for the discrete ordinates transport equation

A goal-based angular adaptivity has been described for solving the discrete ordinates transport equation. This method uses the linear discontinuous finite element quadrature sets, allowing for local angular refinement across space-energy dimensions. Anisotropy quantified factor derived by contributon theory is introduced in this adaptivity to locate spatial regions that require more angular unknowns in every energy group for a given goal region. The goal-based error metric is considered to trigger local refinement in angle, which requires the solutions of the contributon transport equation to determine the importance to a user-defined functional. Several problems that include both one-group and multi-group transport calculations are displayed to demonstrate the effectiveness of this method. Results indicate that our goal-based adaptivity can significantly reduce computational expenses in terms of angular unknowns and computational times for a similar accuracy, as compared to uniform quadrature sets. It is expected that this adaptivity has the potential to enhance the efficiency of an even wider range of transport problems.

Bayesian reinforcement learning reliability analysis

A Bayesian reinforcement learning reliability method that combines Bayesian inference for the failure probability estimation and reinforcement learning-guided sequential experimental design is proposed. The reliability-oriented sequential experimental design is framed as a finite-horizon Markov decision process (MDP), with the associated utility function defined by a measure of epistemic uncertainty about Kriging-estimated failure probability, referred to as integrated probability of misclassification (IPM). On this basis, a one-step Bayes optimal learning function termed integrated probability of misclassification reduction (IPMR), along with a compatible convergence criterion, is defined. Three effective strategies are implemented to accelerate IPMR-informed sequential experimental design: (i) Analytical derivation of the inner expectation in IPMR, simplifying it to a single expectation. (ii) Substitution of IPMR with its upper bound IPMRU to avoid element-wise computation of its integrand. (iii) Rational pruning of both quadrature set and candidate pool in IPMRU to alleviate computer memory constraint. The efficacy of the proposed approach is demonstrated on two benchmark examples and two numerical examples. Results indicate that IPMRU facilitates a much more rapid reduction of IPM compared to other existing learning functions, while requiring much less computational time than IPMR itself. Therefore, the proposed reliability method offers a substantial advantage in both computational efficiency and accuracy, especially in complex dynamic reliability problems.

Investigation on uncertainty quantification of transonic airfoil using compressive sensing greedy reconstruction algorithms

Uncertainty quantification constructs the stochastic responses of system output under uncertainties. Traditional uncertainty quantification methods such as Monte Carlo and Full-Rank Polynomial Chaos Expansion need large amounts of samples. Compressive Sensing has the advantages of requiring few observation samples and sparse reconstruction. By combining Compressive Sensing with uncertainty quantification, the samples in demand can be reduced. Nevertheless, the sparse reconstruction algorithms directly affect predicting accuracy, especially in a practical aerodynamic scenario. To compare the effect of two l0-minimization based greedy reconstruction algorithms, Orthogonal Matching Pursuit and Subspace Pursuit, this study adopts them into uncertainty quantification of RAE2822 airfoil considering geometrical uncertainties. The Polynomial Chaos Expansion is used to realize sparse representation and Structured Random Observation Matrix is constructed by randomly selecting from Gaussian quadrature set. The two greedy reconstruction methods are compared to Monte Carlo and Full-Rank Probabilistic Collocations on convergence, required samples and accuracy. The results show that both methods could achieve similar accuracy as Monte Carlo with much fewer samples. Besides, Orthogonal Matching Pursuit is able to precisely reconstruct original signal with proper sparsity estimation, while Subspace Pursuit provides more robust prediction for different cases. This study presents an efficient approach that integrates compressive sensing into uncertainty quantification of airfoils.

Assessment of Computational Technicalities for 2D/1D Coupling Method and Its Comparison with 3D MOC

The 2D/1D coupling method is recognized as one of the preferred high‐fidelity calculation methods for reactors featured with well homogeneity in the axial direction. However, the choices of the technicalities, such as transverse leakage, axial solver, and transverse leakage splitting method, will lead to different calculational performance. This paper outlines the theory of the 2D/1D coupling method and describes the detailed implementations of the key technicalities. Based on numerical tests, the choices of the related calculational parameters are analyzed, such as the order of the quadrature set, ray spacing, and the axial mesh size. Then, the computational performance of six typical 2D/1D technicalities is compared and assessed. A comparison between 2D/1D coupling method and direct 3D MOC calculation is also made. For the transverse leakage, the Fourier expansion technique could significantly reduce the memory burden and computational cost, but with the similar accuracy as the anisotropic leakage term. It is recommended to use the axial DGFEM SN solver, which has better consistency and leads to higher computational efficiency. Additionally, the results also show that the 2D/1D coupling method can appropriately increase the axial mesh size, which only has slight effect on the accuracy. For nuclear reactors featured with nonstrong heterogeneity in axial direction, the 2D/1D coupling method has significant advantages than the direct 3D MOC calculation.

Entanglement boosts quantum synchronization between two oscillators in an optomechanical setup

We present an optomechanical model to explore how the entanglement can be associated with quantum synchronization of two mechanical oscillators. As both these entities can be characterized in terms of variances of a set of EPR-like conjugate quadratures, we investigate whether this leads to a specific condition for simultaneous existence of the both. In our model, one of the oscillators makes the cavity, while the other is kept suspended inside the cavity, and the always-on coupling between the two is mediated via the same cavity mode. We show that in presence of amplitude modulation with the same frequency as that of the oscillators, these oscillators get nearly complete quantum synchronized and entangled simultaneously in the steady state. We also show that entanglement always becomes accompanied by quantum synchronization, though the reverse is not necessarily true. Moreover, when entangled, the oscillators exhibits near-complete synchronization. Thus, entanglement delivers a catalytic effect on the quantum synchronization in the specific system we considered. This behaviour can be observed for a large range of relevant parameters.

Evaluation of directional quadrature schemes for simulating urban radiative transfer using the discrete ordinate method

In the last decades, different numerical methods have been applied to simulate radiative transfer in urban configurations. In cases where atmospheric air is treated as an absorbing gas, it has been shown in many cases that the use of the Discrete Ordinate Method has been shown in many situations to provide a good compromise between calculation cost and accuracy. In this paper, several quadrature schemes for directional integration in the DOM are compared for different urban conditions and types (visible, thermal infrared) of radiation. This allows for determining which angular quadrature scheme is the most appropriate for urban radiative transfer calculations. In conclusion, the Fibonacci quadrature set outperforms all other quadratures in the traditional canyon simulation with a transparent model. Meanwhile, the performance of Level Symmetric Odd and Hybrid quadrature (LSO/LSH) and Equal Weight Odd (EWO) catch up with Fibonacci's in a participative atmosphere model.

A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L2-stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.

Quantum synchronization and entanglement of indirectly coupled mechanical oscillators in cavity optomechanics: A numerical study

It is often conjectured that quantum synchronisation and entanglement are two independent properties which two coupled quantum systems may not exhibit at the same time. However, as both these properties can be understood in terms of the second order moments of a set of conjugate quadratures, there may exist specific conditions for simultaneous existence of entanglement and quantum synchronization. Here we present a theoretical scheme to achieve the same between two mechanical oscillators, which are indirectly coupled with each other via a coupling between two cavities. We show that in the presence of the cavity-oscillator coupling, quadratically varying with their displacements, these oscillators can be synchronized in the quantum sense and entangled as well, at times much longer than the decay time-scale of the cavity modes. Precisely speaking, we show that in the presence of quadratic coupling, entanglement criterion and quantum synchronization measure are simultaneously satisfied in steady state. This behaviour can be observed for a range of quadratic coupling, temperature, and frequency difference of the two oscillators.

An hp-angular adaptivity with the discrete ordinates method for Boltzmann transport equation

This paper describes an hp-angular adaptivity algorithm in the discrete ordinates method for Boltzmann transport applications with strong angular effects. This adaptivity uses discontinuous finite element quadrature sets with different degrees, which updates both angular mesh and the degree of the underlying discontinuous finite element basis functions, allowing different angular local refinement to be applied in space. The regular and goal-based error metrics are considered in this algorithm to locate some regions to be refined. A mapping algorithm derived by moment conservation is developed to pass the angular solution between spatial regions with different quadrature sets. The proposed method is applied to some test problems that demonstrate the ability of this hp-angular adaptivity to resolve complex fluxes with relatively few angular unknowns. Results illustrate that a reduction to approximately 1/50 in quadrature ordinates for a given accuracy compared with uniform angular discretization. This method therefore offers a highly efficient angular adaptivity for investigating difficult particle transport problems.

Modeling specular and diffuse reflection of UV LEDs for microbial inactivation in air ducts

A method for modeling UV LED irradiance in air sanitization ducts including non-uniform LED emission profile and surface reflection is described and used in conjunction with simple flow and pathogen inactivation models to calculate overall microbial inactivation. The Discrete Ordinates Method (DOM) with Legendre-Chebyshev quadrature sets is used to represent both non-uniform LED emission profiles and combined specular and diffuse surface reflection. Evaluation of DOM sensitivity to spatial and directional discretization indicated high order quadrature schemes (P36-T36) are required to accurately represent emission and reflection profiles. Predicted microbial inactivation compares well with limited measurements from a previous test system. Modeling shows that for a similar number of LEDs, highly reflective surfaces (diffuse or specular) produce higher inactivation and that for similar reflectivities, specular reflection produces higher inactivation than diffuse reflection. Positioning LEDs on two or four duct walls slightly increases inactivation compared to single wall positioning.

High-degree discontinuous finite element discrete quadrature sets for the Boltzmann transport equation

This paper proposes a family of high-degree discontinuous finite element (DFEM) quadrature sets designed to improve accuracy of integrals over local angular domain in the discrete ordinates method. The angular domain is divided into three types of spherical meshes by inscribing the octahedron, icosahedron and cube, respectively. High-degree discontinuous finite element basis functions are generated over different spherical meshes to resolve associated quadrature weights, producing high-degree DFEM quadrature sets. The performance and accuracy of the DFEM quadrature family was tested in the integration of spherical harmonics functions and some transport problems. Results demonstrate that kth-degree DFEM quadrature sets with any quadrature order can exactly integrate up to all kth-degree spherical harmonics functions in the direction cosines over local angular domain. Results also suggest that the performance of high-degree DFEM quadrature sets is comparable or better than that of traditional quadrature sets for transport problems with unsmooth or nearly discontinuous angular flux solutions. Locally-refined DFEM quadrature sets can obtain the same accuracy with less expense compared with uniform quadrature sets. DFEM quadrature sets with different degrees can offer two types of local refinement by increasing the quadrature order and changing the degree of discontinuous finite element basis functions, which can be further expected for use in an angular adaptive algorithm.

Convergence study of DGFEM SN based 2D/1D coupling method for solving neutron transport k-eigenvalue problems with Fourier analysis

2D/1D coupling method is widely used for solving 3D transport equation due to its high-fidelity and affordable computational burden. The convergence analysis is a key issue for 2D/1D coupling transport method. In this work, the convergence of an outstanding DGFEM SN based 2D/1D coupling method with CMFD acceleration is performed for neutron transport k-eigenvalue problem by Fourier analysis. In detail, the DGFEM based SN as 1D axial solver is coupled with the 2D MOC solver, accelerated by the 3D adCMFD. The Fourier analysis results show that adCMFD acceleration can significantly reduce the spectral radius and achieve excellent convergence. Moreover, the effects of the number of inner iterations on the convergence are analyzed. A combination of 4 times inner iteration for 1D axial solver and 4 times inner iteration for 2D planar solver is recommended. Additionally, the spectral radius of 2D/1D coupling method is insensitive to the order of DGFEM SN, the order of SN quadrature sets and the number of fine meshes per coarse mesh.

Goal-oriented multi-collision source algorithm for discrete ordinates transport calculation

Discretization errors are extremely challenging conundrums of discrete ordinates calculations for radiation transport problems with void regions. In previous work, we have presented a multi-collision source method (MCS) to overcome discretization errors, but the efficiency needs to be improved. This paper proposes a goal-oriented algorithm for the MCS method to adaptively determine the partitioning of the geometry and dynamically change the angular quadrature in remaining iterations. The importance factor based on the adjoint transport calculation obtains the response function to get a problem-dependent, goal-oriented spatial decomposition. The difference in the scalar fluxes from one high-order quadrature set to a lower one provides the error estimation as a driving force behind the dynamic quadrature. The goal-oriented algorithm allows optimizing by using ray-tracing technology or high-order quadrature sets in the first few iterations and arranging the integration order of the remaining iterations from high to low. The algorithm has been implemented in the 3D transport code ARES and was tested on the Kobayashi benchmarks. The numerical results show a reduction in computation time on these problems for the same desired level of accuracy as compared to the standard ARES code, and it has clear advantages over the traditional MCS method in solving radiation transport problems with reflective boundary conditions.

Double generalized spatial modulation

AbstractIn this article, motivated by the enhanced spatial modulation and quadrature spatial modulation (QSM) systems, we proposed a new design of double generalized spatial modulation (DGSM), which further exploits the spatial domain not only for the transmission of more extra information bits by increasing the number of antenna index (AI) vectors in an AI vector set, but also for the achievable transmit diversity gain. In DGSM system, we first design the in‐phase and quadrature AI vector sets, whose each AI vector having one or more than one nonzero element equaling to “1” is used for the activation of one or more than one transmit antenna. Then, a spatial antenna index vector (SIV) is generated by adding an AI vector from the in‐phase AI set and another AI vector from the quadrature AI set. Furthermore, a data symbol (eg, QAM/PSK) is modulated on the SIV for forming multiple versions of the data symbol and then transmitted, the maximum likelihood (ML) and sphere decoder (SD) algorithm are applied for the DGSM system. Finally, the performance analysis, including in the spectral efficiency, the computational complexity comparison between the DGSM system with SD algorithm and the diversity‐achieving QSM (DA‐QSM) with ML detector, are provided. Also, the average bit error probability is analyzed. Simulation results using Monte Carlo are presented to show the improvement of bit error ratio performance in comparison with DA‐QSM and other spatial modulation schemes.

Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

Improved memory management for solving neutron transport via a novel Modular Ray Tracing (MRT) approach embedded in parallel method of characteristic (MOC) framework

Poor memory management can be considered as the first restrictive barrier to the implementation of the neutron transport equation for large reactors. The current research presents a new MRT with the ability to reduce the required memory and increase cache coherency. Also, given the high impact of the angular quadrature set used in the transport calculation, the effects of the angular set are investigated. As a reliable trial for an angular quadrature set examination, comparisons between the Forward and Adjoint transport calculations are performed.Three types of angular quadrature sets are adopted for our optimized MOC steady-state calculation. Based on two known constraints for the quadrature set evaluation, the accuracy and computational cost of the presented sets are examined. Satisfying these two conditions, our implemented Legendre-Chebyshev quadrature set with S20 angular order results in 7.0000E-06 and the 9.0000E-06 difference with reference Keff of G5G7 benchmark in the forward and adjoint calculation respectively. As well, our optimized parallel algorithm for MOC results in 6.18 speedup with 16 threads.

DEVELOPMENT AND TESTING OF THE MODIFIED SIMPSON’S RULE FOR DISCRETE ORDINATES TRANSPORT APPLICATIONS

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.