SEARCH · Search NASA

Results for “Linear Equation Solver”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

637 records · Page 36

All sources NASA NTRS DOE OSTI

DOE OSTI2015

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)

60 APPLIED LIFE SCIENCES↗

DOE OSTI2012

Study on the well-posedness, convergence and the stability of the semi-implicit upwind numerical solver for the multi-fluid model

The well-posedness, convergence and the stability of the two-fluid code has been studied for a long time. Most of the investigations concern the semi-implicit upwind solution scheme for the six equation two-fluid model such as used in RELAP5 3 or TRACE 21. Since the system code, SPACE 2, adopts one more field, a droplet field, it consists of nine equations (3 mass, 3 momentum and 3 energy balance equations) and thus more involved investigations are necessary to confirm the stability and convergence. For this objective, the old issue of the well-posedness, convergence and the stability is revisited and some general guidelines to develop a well-posed numerical multi-fluid model are derived as follows; (1) Hyperbolicity of the corresponding system of partial differential equations is not a necessary condition for the development of a numerical model for multi-phase flow, but whether or not it is hyperbolic can provide guidance relative to initial conditions, boundary conditions, and expected high frequency behavior of the model. (2) A necessary condition for a well-posed numerical model is stability in the von Neumann sense, i.e. growth factor less than 1.0 for the shortest wave-length, 2{Delta}x. (3) The smallest node size used for convergence studies should be of the order of the characteristic dimension of the average description, i.e. smaller nodes can be used so long as they do not result in unphysical growth of wave-lengths less than the characteristic dimension. The usual mathematical definition of convergence i.e. the behavior of the solution as the node size approaches zero, is not appropriate for the discrete averaged numerical model, since there is diminished physical meaning to behavior at wavelengths less than the characteristic dimension of the average description. Under these guidelines, dispersion analysis and von Neumann stability analysis are performed for the three field multi-fluid, semi-implicit, upwind numerical model to show that the necessary conditions for well-posedness are met. To study the non-linear stability and the convergence, various runs with the torus problem using the SPACE code are utilized to confirm the frequency cascading effects that augment non-linear stability. A robust mechanism for the flow regime change is also a very important factor for developing non-linearly stable and convergent code. A open pipe flow problem is also simulated to investigate the non-linear stability effects in a flow geometry more typical of real safety code applications. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2014

Development and Benchmarking of a Hybrid PIC Code For Dense Plasmas and Fast Ignition (Final Report)

Radiation processes play an important role in the study of both fast ignition and other inertial confinement schemes, such as plasma jet driven magneto-inertial fusion, both in their effect on energy balance, and in generating diagnostic signals. In the latter case, warm and hot dense matter may be produced by the convergence of a plasma shell formed by the merging of an assembly of high Mach number plasma jets. This innovative approach has the potential advantage of creating matter of high energy densities in voluminous amount compared with high power lasers or particle beams. An important application of this technology is as a plasma liner for the flux compression of magnetized plasma to create ultra-high magnetic fields and burning plasmas. HyperV Technologies Corp. has been developing plasma jet accelerator technology in both coaxial and linear railgun geometries to produce plasma jets of sufficient mass, density, and velocity to create such imploding plasma liners. An enabling tool for the development of this technology is the ability to model the plasma dynamics, not only in the accelerators themselves, but also in the resulting magnetized target plasma and within the merging/interacting plasma jets during transport to the target. Welch pioneered numerical modeling of such plasmas (including for fast ignition) using the LSP simulation code. Lsp is an electromagnetic, parallelized, plasma simulation code under development since 1995. It has a number of innovative features making it uniquely suitable for modeling high energy density plasmas including a hybrid fluid model for electrons that allows electrons in dense plasmas to be modeled with a kinetic or fluid treatment as appropriate. In addition to in-house use at Voss Scientific, several groups carrying out research in Fast Ignition (LLNL, SNL, UCSD, AWE (UK), and Imperial College (UK)) also use LSP. A collaborative team consisting of HyperV Technologies Corp., Voss Scientific LLC, FAR-TECH, Inc., Prism Computational Sciences, Inc. and Advanced Energy Systems Inc. joined efforts to develop new physics and numerical models for LSP in several key areas to enhance the ability of LSP to model high energy density plasmas (HEDP). This final report details those efforts. Areas addressed in this research effort include: adding radiation transport to LSP, first in 2D and then fully 3D, extending the EMHD model to 3D, implementing more advanced radiation and electrode plasma boundary conditions, and installing more efficient implicit numerical algorithms to speed complex 2-D and 3-D computations. The new capabilities allow modeling of the dominant processes in high energy density plasmas, and further assist the development and optimization of plasma jet accelerators, with particular attention to MHD instabilities and plasma/wall interaction (based on physical models for ion drag friction and ablation/erosion of the electrodes). In the first funding cycle we implemented a solver for the radiation diffusion equation. To solve this equation in 2-D, we used finite-differencing and applied the parallelized sparse-matrix solvers in the PETSc library (Argonne National Laboratory) to the resulting system of equations. A database of the necessary coefficients for materials of interest was assembled using the PROPACEOS and ATBASE codes from Prism. The model was benchmarked against Prism's 1-D radiation hydrodynamics code HELIOS, and against experimental data obtained from HyperV's separately funded plasma jet accelerator development program. Work in the second funding cycle focused on extending the radiation diffusion model to full 3-D, continued development of the EMHD model, optimizing the direct-implicit model to speed up calculations, add in multiply ionized atoms, and improved the way boundary conditions are handled in LSP. These new LSP capabilities were then used, along with analytic calculations and Mach2 runs, to investigate plasma jet merging, plasma detachment and transport, restrike and advanced jet accelerator design. In addition, a strong linkage to diagnostic measurements was made by modeling plasma jet experiments on PLX to support benchmarking of the code. A large number of upgrades and improvements advancing hybrid PIC algorithms were implemented in LSP during the second funding cycle. These include development of fully 3D radiation transport algorithms, new boundary conditions for plasma-electrode interactions, and a charge conserving equation of state that permits multiply ionized high-Z ions. The final funding cycle focused on 1) mitigating the effects of a slow-growing grid instability which is most pronounced in plasma jet frame expansion problems using the two-fluid Eulerian remap algorithm, 2) extension of the Eulerian Smoothing Algorithm to allow EOS/Radiation modeling, 3) simulations of collisionless shocks formed by jet merging, 4) simulations of merging jets using high-Z gases, 5) generation of PROPACEOS EOS/Opacity databases, 6) simulations of plasma jet transport experiments, 7) simulations of plasma jet penetration through transverse magnetic fields, and 8) GPU PIC code development. The tools developed during this project are applicable not only to the study of plasma jets, but also to a wide variety of HEDP plasmas of interest to DOE, including plasmas created in short-pulse laser experiments performed to study fast ignition concepts for inertial confinement fusion.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2023

CFD Integrated Transition Modeling for High-Speed Flows via Coupled OVERFLOW-LASTRAC Analysis

This work details ongoing efforts at the NASA Langley Research Center to develop and validate a general-use CFD tool that includes built-in predictions of boundary-layer transition in high-speed flows. Existing tools for the coupling of the NASA OVERFLOW structured overset RANS solver and LASTRAC stability analysis code have been extended to capture boundary-layer transition in high-speed flows driven by either Mack’s first-mode or secondmode instabilities and/or crossflow instabilities. The efficacy of this coupled approach has been demonstrated by examining a variety of supersonic and hypersonic test cases for which experimental validation data is available. The intermittency prescription parameters inherited from prior low-speed applications are found to be suitable for some, but not all high-speed flow scenarios. The method is found to be robust for straight cone configurations and improvements necessary to accurately capture transition on geometries with spatially complex amplification factor envelopes with intermediate regions of slow N-factor variation are examined. The automated, coupled analysis is demonstrated for a 3d supersonic test case and found to perform well within the limits of the linear stability analysis on which it relies. Finally, a preliminary investigation of the method’s robustness to non-ideal CFD meshes is conducted.

High-speed flow↗

NASA NTRS2014

Adjoint Methods for Adjusting Three-Dimensional Atmosphere and Surface Properties to Fit Multi-Angle/Multi-Pixel Polarimetric Measurements

This paper derives an efficient procedure for using the three-dimensional (3D) vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties and improve their fit with multi-angle/multi-pixel radiometric and polarimetric measurements of scattered sunlight. The proposed adjoint method uses the 3D VRTE to compute the measurement misfit function and the adjoint 3D VRTE to compute its gradient with respect to all unknown parameters. In the remote sensing problems of interest, the scalar-valued misfit function quantifies agreement with data as a function of atmosphere and surface properties, and its gradient guides the search through this parameter space. Remote sensing of the atmosphere and surface in a three-dimensional region may require thousands of unknown parameters and millions of data points. Many approaches would require calls to the 3D VRTE solver in proportion to the number of unknown parameters or measurements. To avoid this issue of scale, we focus on computing the gradient of the misfit function as an alternative to the Jacobian of the measurement operator. The resulting adjoint method provides a way to adjust 3D atmosphere and surface properties with only two calls to the 3D VRTE solver for each spectral channel, regardless of the number of retrieval parameters, measurement view angles or pixels. This gives a procedure for adjusting atmosphere and surface parameters that will scale to the large problems of 3D remote sensing. For certain types of multi-angle/multi-pixel polarimetric measurements, this encourages the development of a new class of three-dimensional retrieval algorithms with more flexible parametrizations of spatial heterogeneity, less reliance on data screening procedures, and improved coverage in terms of the resolved physical processes in the Earth's atmosphere.

linearization↗

NASA NTRS2014

Adjoint Methods for Adjusting Three-Dimensional Atmosphere and Surface Properties to Fit Multi-Angle Multi-Pixel Polarimetric Measurements

adjoint methods↗

NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗