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An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs
It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. In particular, we consider stochastic elliptic PDEs with an affine dependence on the random variable. The goal is to recover the entire solution map from the parameter space to the finite element space. To this end, first, a reduced-basis solution using a weak greedy algorithm is constructed, then a DLS approximation is determined by evaluating the reduced-basis approximation instead of the full finite element approximation. The main advantage of the new approach is that one only need apply the DLS operator to the coefficients of the reduced-basis expansion, resulting in huge savings in both the storage of the DLS coefficients and the online cost of evaluating the DLS approximation. In addition, the recently developed quasi-optimal polynomial space is also adopted in the new approach, resulting in superior convergence rates for a wider class of problems than previous analyzed. Numerical experiments are provided that illustrate the theoretical results.
A computational model for molten corium spreading and solidification
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An Overview of the Null-Field Method. I. Formulation and Basic Results
In this paper we revisit the fundamentals of the null-field method with discrete sources. We prove the unique solvability of the null-field equations for the total field inside the particle and the internal field outside the particle, at all wavenumbers. For this purpose, we use the equivalence between the null-field and surface integral equations methods. Furthermore, we discuss the completeness property of different systems of discrete sources for a surface field approximation, and derive an infinite set of integral equations for the surface fields in a variety of discrete sources. Finally, we formulate the null-field scheme as an approach aiming to construct an approximate solution to the scattering problem. The way in which we introduce the matrix of a particle is different from the standard approach relying on the assumption that the system of regular vector spherical wave functions for the interior problem is a basis.
Initial Investigation of Coupled Electron-Photon Transport with SCEPTRE and RAPTURE (Rev.1.1)
The SCEPTRE1 code models coupled photon and electron transport in material media using discrete ordinates methods. The code can use different solvers in 1D and 3D in parallel, promising users the ability to run problems that are significant both in physics and computational scale. One capability that is perhaps unique at this time in the NNSA complex is SCEPTRE’s coupled photon– electron transport physics in a discrete ordinates approach. For example, incident x–rays can generate a significant electron population, whose transport can be simulated within the same code. In this context, fully–coupled means that photons can produce electrons and electrons can produce photons; uncoupled or no–coupling means photons do not produce electrons. Partial–coupling, an intermediate option in which photons can produce electrons but electrons do not produce photons, is not considered in the present work. The MCNP code can treat this physics with a Monte Carlo approach. MCNP should be viewed as complementary to SCEPTRE in both computational method (Monte Carlo versus discrete ordinates) as well as data libraries. SCEPTRE and MCNP should provide important checks on each other, especially from complementary modeling frameworks of a S n (discrete ordinates) mesh based method versus Monte Carlo particle based method. In order to learn more about SCEPTRE, and how it might inform engineering work at the Laboratory, we used it to carry out a set of initial studies. This was a series of 1D simulations of photon beams hitting Aluminum slabs. Our studies mainly used SCEPTRE directly, and made little use of the closely associated RAPTURE code (more on this below). The goals of these studies were first, to familiarize ourselves with operating the code, and second, to begin to assess the importance of coupled electron–photon transport. To support these studies, we began to develop a set of scripts to initialize inputs, run tools, process outputs, and plot results. An initial version of these scripts has been committed to the Lightning workflow repository. This memo relates the results of those studies.
The phase-integral method for radiative transfer problems with highly-peaked phase functions
Complete solutions to the radiative transfer equation, including both azimuth and depth dependence, are provided by the discrete-ordinate method of Chandrasekhar, but these solutions are often limited because of large computer requirements. This paper presents a 'phase-integral' method which greatly reduces the number of discrete ordinates needed in the solution for highly peaked phase functions. A composite quadrature method is shown to be effective in further reducing the number of discrete ordinates required for highly anisotropic phase functions. Examples are given to indicate convergence requirements and expected accuracy in the complete solution for Henyey-Greenstein and cloud-type phase functions.
A NEW MONTE CARLO METHOD FOR TIME-DEPENDENT NEUTRINO RADIATION TRANSPORT
Monte Carlo approaches to radiation transport have several attractive properties such as simplicity of implementation, high accuracy, and good parallel scaling. Moreover, Monte Carlo methods can handle complicated geometries and are relatively easy to extend to multiple spatial dimensions, which makes them potentially interesting in modeling complex multi-dimensional astrophysical phenomena such as core-collapse supernovae. The aim of this paper is to explore Monte Carlo methods for modeling neutrino transport in core-collapse supernovae. We generalize the Implicit Monte Carlo photon transport scheme of Fleck and Cummings and gray discrete-diffusion scheme of Densmore et al. to energy-, time-, and velocity-dependent neutrino transport. Using our 1D spherically-symmetric implementation, we show that, similar to the photon transport case, the implicit scheme enables significantly larger timesteps compared with explicit time discretization, without sacrificing accuracy, while the discrete-diffusion method leads to significant speed-ups at high optical depth. Our results suggest that a combination of spectral, velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo methods represents a robust approach for use in neutrino transport calculations in core-collapse supernovae. Our velocity-dependent scheme can easily be adapted to photon transport.
Non-gray gas radiation effect on mixed convection in lid driven square cavity
A numerical study is performed to investigate the effect of non-gray radiation on mixed convection in a vertical two sided lid driven square cavity filled with air-H{sub 2}O-CO{sub 2} gas mixture. The vertical moving walls of the enclosure are maintained at two different but uniform temperatures. The horizontal walls are thermally insulated and considered as adiabatic walls. The governing differential equations are solved by a finite-volume method and the SIMPLE algorithm was adopted to solve the pressure–velocity coupling. The radiative transfer equation (RTE) is solved by the discrete ordinates method (DOM). The spectral line weighted sum of gray gases model (SLW) is used to account for non-gray radiation properties. Simulations are performed in configurations where thermal and shear forces induce cooperating buoyancy forces. Streamlines, isotherms, and Nusselt number are analyzed for three different values of Richardson’s number (from 0.1 to 10) and by considering three different medium (transparent medium, gray medium using the Planck mean absorption coefficient, and non-gray medium assumption).
A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method
We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.
Analytical Models of Exoplanetary Atmospheres. IV. Improved Two-stream Radiative Transfer for the Treatment of Aerosols
We present a novel generalization of the two-stream method of radiative transfer, which allows for the accurate treatment of radiative transfer in the presence of strong infrared scattering by aerosols. We prove that this generalization involves only a simple modification of the coupling coefficients and transmission functions in the hemispheric two-stream method. This modification originates from allowing the ratio of the first Eddington coefficients to depart from unity. At the heart of the method is the fact that this ratio may be computed once and for all over the entire range of values of the single-scattering albedo and scattering asymmetry factor. We benchmark our improved two-stream method by calculating the fraction of flux reflected by a single atmospheric layer (the reflectivity) and comparing these calculations to those performed using a 32-stream discrete-ordinates method. We further compare our improved two-stream method to the two-stream source function (16 streams) and delta-Eddington methods, demonstrating that it is often more accurate at the order-of-magnitude level. Finally, we illustrate its accuracy using a toy model of the early Martian atmosphere hosting a cloud layer composed of carbon dioxide ice particles. The simplicity of implementation and accuracy of our improved two-stream method renders it suitable for implementation in three-dimensional general circulation models. In other words, our improved two-stream method has the ease of implementation of a standard two-stream method, but the accuracy of a 32-stream method.
SU-E-T-270: Diffusion Synthetic Acceleration for Linear Boltzmann Transport Equation
Purpose: Linear Boltzmann transport equation (LBTE) is as accurate as the Monte Carlo method (MC) for dose calculation in photon/particle therapy (LBTE is a deterministic and Eulerian formulation and MC is a statistical and Lagrangian description). An advantage of LBTE is that numerous acceleration techniques can be utilized for acceleration. This work is to explore the acceleration of LBTE via diffusion synthetic acceleration (DSA). Methods: For simplicity, two-dimensional, steady-state, and within-group LBTE is considered with two angular dimensions and two spatial dimensions. The discrete ordinate method is developed for solving this integro-differential equation. The angular variables are discretized using a level-symmetric quadrature set on the unit sphere. The spatial variables are discretized on the structured grid based on the diamond scheme. The source-iteration method (SI) is used to solve the discretized system.Since SI is slow in optically thick and highly scattering regime. DSA is developed to accelerate SI. The motivation for DSA is that diffusion equation (DE) is a good approximation of LBTE in the above regime. However, DE is much cheaper than LBTE computationally since DE only involves spatial variables. Thus, in each DSA iteration, DSA adds to the SI step a computationally-negligible DE step, i.e., to first solve DE with the SI residual as source term, and then compensate the SI solution with DE solution. Results: DSA was benchmarked and compared with SI. The difference between two methods was within 0.12% which verifies the accuracy of DSA, while DSA demonstrated the great advantage in speed, e.g., the reduction of iteration number to 6% and 4% respectively for cases with 100 and 1,000 scattering-absorption ratio that commonly occur in clinical dose calculation. Conclusion: DSA has been developed as one of many possible means for accelerating the numerical solver of LBTE for dose calculation. The authors were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500)
Development of a Novel Accelerator for Neutron Transport Solution Using the Galerkin Spectral Element Methods (Final Report)
This report summarizes the results of a three-year research project sponsored by the U.S. Department of Energy (DOE) Nuclear Energy University Program (NEUP) to develop and implement advanced acceleration schemes for the DOE NEAMS neutronics code PROTEUSSN. The project team included the University of Massachusetts Lowell, The Ohio State University, University of Michigan, and Argonne National Laboratory. The PROTEUS code is a high-fidelity capable deterministic neutron transport code based on unstructured finite element meshes, which solves the steady-state and transient neutron transport problem using the 2nd-order discrete ordinate method (SN2ND), the method of characteristics (MOC), and the advanced nodal transport method (NODAL). The DSA scheme has been implemented in PROTEUS to speed up SN transport calculations. The existing DSA scheme employs a consistent finite element formulation on the same fine mesh structure as the SN solution While the use of consistent discretizations makes the DSA method effective, it does not necessarily make it efficient because the numerical solution of a discretized elliptic diffusion problem itself can be costly, particularly when the problem size becomes large. We developed a new discontinuous Galerkin (DG) discretization of the diffusion equation, called DG-DSA, which can effectively and efficiently accelerate the SN transport iterations. As compared with the previous work, the novelty of our method is that the diffusion equation is solved on a coarse-mesh grid using the DG methods, and the DG diffusion discretization incorporates local hp adaptation, i.e., local adaptation of mesh size and/or polynomial degree, based on local total cross section (or optical thickness). Therefore, the resulting number of degrees of freedom (DOF) of the DG discretization is much less than the conventional consistent DSA discretizations, and thus DG-DSA can achieve significant improvement in computational efficiency. We implemented this scheme in PROTEUS-SN. In addition, we developed a new stabilization scheme named linear prolongation CMFD (lpCMFD). A novel feature of this scheme is that the conventional flat flux ratio–based scaling approach is replaced with a linear interpolation of the scalar flux differences at the coarse-mesh cell edges between the neutron transport and CMFD calculations. Fourier convergence analysis and numerical results show that lpCMFD is unconditionally stable and more effective than other CMFD based on acceleration schemes such as pCMFD, odCMFD. We also developed a new nonlinear diffusion acceleration scheme for solving neutron transport equations. This scheme, called LR-NDA, employs a local refinement approach on the framework of CMFD by solving a local boundary value problem of the scalar flux on the coarse-mesh structure to replace the piecewise constant scalar flux obtained by CMFD. The refined flux is then used to update the scalar flux in the neutron transport source iteration. It has been demonstrated that that LR-NDA is much more effective and stable than CMFD for a wide range of optical thicknesses. LR-NDA is a local adaptive method, which means LR-NDA does not necessarily require local refinement for all the coarse-mesh cells on the problem domain, i.e., it can be used only for relatively optically thick regions where the standard CMFD scheme would encounter the convergence problem.
Hermite WENO limiting for multi-moment finite-volume methods using the ADER-DT time discretization for 1-D systems of conservation laws
New Hermite Weighted Essentially Non-Oscillatory (HWENO) interpolants are developed and investigated within the Multi-Moment Finite-Volume (MMFV) formulation using the ADER-DT time discretization. Whereas traditional WENO methods interpolate pointwise, function-based WENO methods explicitly form a non-oscillatory, high-order polynomial over the cell in question. This study chooses a function-based approach and details how fast convergence to optimal weights for smooth flow is ensured. Methods of sixth-, eighth-, and tenth-order accuracy are developed. We compare these against traditional single-moment WENO methods of fifth-, seventh-, ninth-, and eleventh-order accuracy to compare against more familiar methods from literature. The new HWENO methods improve upon existing HWENO methods (1) by giving a better resolution of unreinforced contact discontinuities and (2) by only needing a single HWENO polynomial to update both the cell mean value and cell mean derivative. Test cases to validate and assess these methods include 1-D linear transport, the 1-D inviscid Burger's equation, and the 1-D inviscid Euler equations. Smooth and non-smooth flows are used for evaluation. These HWENO methods performed better than comparable literature-standard WENO methods for all regimes of discontinuity and smoothness in all tests herein. They exhibit improved optimal accuracy due to the use of derivatives, and they collapse to solutions similar to typical WENO methods when limiting is required. The study concludes that the new HWENO methods are robust and effective when used in the ADER-DT MMFV framework. Finally, these results are intended to demonstrate capability rather than exhaust all possible implementations.
Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods
Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law. It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.
Particle tracking approach for transport in three-dimensional discrete fracture networks: Particle tracking in 3-D DFNs
The discrete fracture network (DFN) model is a method to mimic discrete pathways for fluid flow through a fractured low-permeable rock mass, and may be combined with particle tracking simulations to address solute transport. However, experience has shown that it is challenging to obtain accurate transport results in three-dimensional DFNs because of the high computational burden and difficulty in constructing a high-quality unstructured computational mesh on simulated fractures. We present a new particle tracking capability, which is adapted to control volume (Voronoi polygons) flow solutions on unstructured grids (Delaunay triangulations) on three-dimensional DFNs. The locally mass-conserving finite-volume approach eliminates mass balance-related problems during particle tracking. The scalar fluxes calculated for each control volume face by the flow solver are used to reconstruct a Darcy velocity at each control volume centroid. The groundwater velocities can then be continuously interpolated to any point in the domain of interest. The control volumes at fracture intersections are split into four pieces, and the velocity is reconstructed independently on each piece, which results in multiple groundwater velocities at the intersection, one for each fracture on each side of the intersection line. This technique enables detailed particle transport representation through a complex DFN structure. Verified for small DFNs, the new simulation capability enables numerical experiments on advective transport in large DFNs to be performed. As a result, we demonstrate this particle transport approach on a DFN model using parameters similar to those of crystalline rock at a proposed geologic repository for spent nuclear fuel in Forsmark, Sweden.
Discrete Packet Analysis for Improved Atmospheric Rejection on Modulated Laser Signals
This slide-show discusses how the method of discrete packet analysis improves atmospheric compensation for quasi-CW fluorescence detection methods. This is key to improving remote sensing capabilities.
Qualitative numerical study of simultaneous high-G-intensified gas–solids contact, separation and segregation in a bi-disperse rotating fluidized bed in a vortex chamber
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Accelerating the solution of the S N equations with highly anisotropic scattering using the Fokker-Planck approximation
The discrete ordinates method can model forward-peaked transport problems accurately. However, convergence of discrete ordinates solution can become arbitrarily slow upon use of standard iterative procedures like source iteration. Standard zeroth and first moment-based acceleration methods like nonlinear diffusion acceleration and diffusion synthetic acceleration are ineffective in accelerating such problems because these methods do not correct higher order Legendre-moments of angular flux. In this paper, we explore the idea of using the Fokker-Planck approximation as a preconditioner to accelerate the convergence of solution of forward-peaked transport problems. Finally, we observe that such preconditioning can successfully reduce the iteration count by up to four orders of magnitude, and solver runtime by up to three orders of magnitude when compared to diffusion synthetic acceleration.