Engineering topics

Yee, Ben C.

Publications and source records attributed to Yee, Ben C..

7 records

DOE OSTI2024

A Survey of Multimaterial Treatments for Thermal Radiative Transfer

Arbitrary Lagrangian-Eulerian methods are a popular choice for hydrodynamic modeling in radiation (rad-hydro) simulations. Because these methods involve a relaxation step that moves the mesh relative to material boundaries, multimaterial spatial zones are generally present. Accurate treatments of these zones are needed to resolve various physical phenomena of interest for inertial confinement fusion applications. However, these codes are often paired with single-material, deterministic thermal radiative transfer (TRT) codes that are oblivious to the material compositions of each zone. These single-material TRT codes can only accept homogenized material properties (opacities, specific heats, etc.) from the hydrodynamic code and output homogenized solutions. After each TRT time step, the multimaterial hydrodynamic code must dehomogenize the quantities computed by the TRT package in order to update subzonal material temperatures. The process by which hydrodynamic codes perform this dehomogenization has not been well documented in previous literature, and the methods can vary significantly from code to code. The purpose of this paper is to document, study, and compare existing techniques used for rad-hydro simulations as well as present a new method with potentially promising results. We summarize several methods and give comparisons on infinite-medium problems as well a finite-medium problem for two of the methods.

42 ENGINEERING↗

DOE OSTI2023

A family of independent Variable Eddington Factor methods with efficient preconditioned iterative solvers

We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with the Discontinuous Galerkin transport discretization from to form effective high-order, linear transport methods. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems presented by Arnold et al. to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners, which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. We demonstrate that the VEF solution converges to the S N transport solution as the mesh is refined on both problems with smooth and non-smooth behavior in angle. Parallel performance studies show that the interior penalty VEF discretization's linear solve weak scales out to 1024 processors and strong scales well on a single node. Particular attention is paid to the parallel performance of the VEF algorithm when used in combination with a parallel block Jacobi transport sweep.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Time Shifted Transport Tallies

When modeling experiments on platforms such as the National Ignition Facility (NIF), several different calculations are strung together to calculate the final diagnostic response of the detectors on the machine. For example, a highly detailed simulation may be performed of capsule implosion on NIF, where one of the key diagnostics is the history of the thermal radiation emission from the capsule. This simulation domain can be a few centimeters in size. The NIF target chamber is much larger, so a second calculation can be done to transport the energy from the capsule out to the diagnostics. Typically, the results from the first calculation are used as a point source in the second, since all the fine detail is not needed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A quadratic programming flux correction method for high-order DG discretizations of S transport

In this work, we present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the S N transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle. There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ~31% greater than performing one without fixup. The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are introduced, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the conclusions in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

3-D↗

DOE OSTI2017

Space-Dependent Wielandt Shifts for Multigroup Diffusion Eigenvalue Problems

We present a new concept—the space-dependent Wielandt shift (SDWS)—for accelerating the convergence of the power iteration (PI) scheme for multigroup diffusion k-eigenvalue problems. The SDWS improves on standard Wielandt shift (WS) techniques, which are empirical in nature and are typically effective only when the current estimate of the solution is reasonably converged. By accounting for the physics of the problem through SDWS, we are able to improve the acceleration for the initial iterates when the current estimate of the solution is not close to convergence. Numerical results from one-dimensional problems suggest that, compared to standard WS techniques, the new SDWS techniques can provide upward of a 46% reduction in the number of PIs required for convergence and a 40% reduction in the computational time required. This improvement is sensitive to several problem-dependent factors, such as the geometry and energy-dependence of the problem, the spatial discretization, and the initial guess. The reduction in computational time is also dependent on the linear solver in the PI scheme, as it is well known that WSs can significantly worsen the conditioning of the diffusion linear system. In this paper, we provide a detailed study of the impact of WSs on the performance of several iterative linear solvers. Results from our implementation of SDWS in the three-dimensional (3D) code MPACT show that SDWS can provide similar speedups for 3D multigroup diffusion eigenvalue problems. These results also show that moderate speedups can be obtained by applying SDWS to the coarse mesh finite difference (CMFD) solver in a CMFD-accelerated transport scheme. However, the benefit of doing this may be limited because all but the first few CMFD solves are relatively easy to converge, regardless of the WS used.

Nuclear Science & Technology↗

DOE OSTI2016

Space-Dependent Wielandt Shift Methods for Multigroup Diffusion Eigenvalue Problems

In the calculation of neutronic eigenfunctions and eigenvalues, it is often desirable to approximate the neutron transport equation by a diffusion equation that requires significantly less computational resources to solve. Even when a full transport solution is needed, the solution of the diffusion system can be used to accelerate iterative methods for calculating the transport eigenfunction and eigenvalue. The convergence rate of the basic power iteration method for eigenvalue problems is governed by the dominance ratio; realistic reactor core problems often have high dominance ratios and the power iteration method can converge slowly as a result. A common technique for accelerating the convergence of power iteration is the Wielandt shift (WS) method, which reduces the dominance ratio by 'shifting' the eigenvalue spectrum using an estimate of the desired eigenvalue. WS provides a significant speedup over power iteration if the eigenvalue estimate is close to the true eigenvalue, but it can be difficult to obtain such an estimate. In practice, the shift in the WS method can be dynamically updated in the iteration scheme, in a way that depends on the eigenvalue estimates from the most recent iterations. We present a physically-motivated Space-Dependent Wielandt Shift (SDWS) for solving multigroup diffusion eigenvalue problems. The space-dependence of the eigenvalue shift in SDWS enables a more effective acceleration of the power iteration scheme than WS at the beginning of the iteration scheme, when the current estimate of the eigenvalue is not close to the actual eigenvalue. Several variants of SDWS are discussed here, including an SDWS method that combines the strengths of SDWS (in the early iterations) with those of standard WS (in the later iterations). Numerical results from a 1-D multigroup diffusion problem demonstrate an order of magnitude speedup over power iteration and a 41% speedup over a standard WS technique. This speedup is problem-dependent and appears to be greater for 'more difficult' problems. In this summary, the term 'speedup' refers to a reduction in the number of iterations required to converge to the correct eigenvalue and eigenfunction. Because the scattering is treated fully implicitly (i.e., it is part of the left-hand side of the diffusion system and directly inverted), the computational cost of a single iteration does not vary significantly among the methods discussed in this summary. The number of iterations is therefore representative of the total computational cost for a method. We acknowledge that certain iterative solvers for linear systems (e.g., GMRES) require more computational time to converge when the system is nearly singular (i.e., when a very aggressive Wielandt shift is applied). Nonetheless, when using SDWS in the manner described in this summary, we have seen that the reduction in computational time from reducing the number of power iterations generally outweighs the extra computational time required by iterative solvers such as GMRES. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗