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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.

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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR.

Gallopoulos, E.↗
NASA NTRS1989

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block tridiagonal matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconstant coefficients. A method was recently proposed to parallelize and vectorize BCR. In this paper, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational compelxity lower than that of parallel BCR.

Gallopoulos, E.↗
NASA NTRS1991

Efficient use of direct solvers for the calculation of compressible flows

While the direct solution of systems of linear equations resulting from fluid dynamic problems has generally not been practical in the past, it is presently demonstrated that the direct method is often more efficient than the most popular iterative schemes when constructed in such a way as to take advantage of presently available vector processing capabilities and large memory. The vertical line Gauss-Seidel algorithm was chosen as the iterative method to be compared with the direct method. It is fond that the direct method becomes efficient only when large residual reductions are desired.

Riggins, David W.↗
DOE OSTI2022

Linear source method of characteristics in HELIOS2

The Studsvik Scandpower code HELIOS2 is a fast lattice transport code that has gained wide use for reactor fuel cycle analysis in commercial, regulatory, and research environments. It contains two transport solvers - Collision Probabilities (CP) and Method of Characteristics (MoC) - with a general geometry ability and generalized depletion for a variety of fuel types, experimental facilities, and isotope production applications. The HELIOS2 lattice code has been extended with the development of a linear source MoC option. The linear source equations are derived within the scope of the HELIOS2 general geometry and non-cyclic ray tracing. The implementation was tested for fine and coarse spatial mesh models of the 2D C5G7 MOX computational benchmark, which provides a pure and extreme test of a transport solver. The linear source yields at least comparable accuracy with reduced computational resource requirements relative to the flat-source approximation.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials

This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.

42 ENGINEERING↗
DOE OSTI2020

A computationally-efficient method for flamelet calculations

A new open-source code for the simulation of the diffusion flamelet equations is proposed. Emphasis is placed on using an approximate Jacobian to reduce the computational cost of the matrix operations. Performance of the proposed solvers is tested by performing flamelet calculations with kinetic mechanisms of varying sizes. For the unity Lewis number equations, the present iterative Newton solver using an approximate Jacobian greatly outperforms direct Newton solvers using exact Jacobians. The computation cost scales linearly with the number of species, leading to a reduction in solution times by two orders of magnitude for mechanisms containing thousands of species. The applicability of the Jacobian approximations to the solution of the non-unity Lewis number flamelet equations is assessed. The approximations are generally inadequate to solve the full non-unity Lewis number equations but can be used in some applications depending on the balance of terms in the flamelet equations. As an example, the flamelet solver is applied to the study of sooting tendencies in laminar co-flow diffusion flames where modified non-unity Lewis number flamelet equations, previously shown to accurately reproduce experimentally-measured Yield Sooting Indices (YSI), are solved. Here, the accelerated flamelet solver is well suited for sensitivity analysis and uncertainty quantification with large detailed kinetic mechanisms, tasks for which the computational cost was previously prohibitive.

42 ENGINEERING↗
DOE OSTI2020

An investigation of Newton-Sketch and subsampled Newton methods

Sketching, a dimensionality reduction technique, has received much attention in the statistics community. In this paper, we study sketching in the context of Newton's method for solving finite-sum optimization problems in which the number of variables and data points are both large. In this work, we study two forms of sketching that perform dimensionality reduction in data space: Hessian subsampling and randomized Hadamard transformations. Each has its own advantages, and their relative tradeoffs have not been investigated in the optimization literature. Additionally, our study focuses on practical versions of the two methods in which the resulting linear systems of equations are solved approximately, at every iteration, using an iterative solver. The advantages of using the conjugate gradient method vs. a stochastic gradient iteration are revealed through a set of numerical experiments, and a complexity analysis of the Hessian subsampling method is presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

hypre

hypre is a library of high performance preconditioners and solvers featuring multigrid methods for the solution of large, sparse linear systems of equations on massively parallel computers.

Falgout, RobertD↗
NASA NTRS1991

Parallel Methods on Large-Scale Structural Analysis and Physics Applications; Symposium, Hampton, VA, Feb. 5, 6, 1991, Selected Papers

Recent advances in parallel methods and algorithms integrated into large-scale codes are presented. Consideration is given to problem decomposition (substructuring), efficient matrix solution algorithms for shared memory architectures, dynamic and transient analysis algorithms for shared memory architectures, and algorithms for distributed and massively parallel architectures. Particular attention is given to partitioning of unstructured problems for parallel processing, parallel-vector computation for linear-structural analysis and nonlinear unconstraint optimization problems, a parallel-vector equation solver for unsymmetric matrices on supercomputers, parallel nonlinear finite element dynamic response, multigrid algorithms for solving structural mechanics problems on supercomputers, structural analysis on massively parallel computers, explicit finite element methods with contact-impact on SIMD computers, and the impact of mapping and sparsity on parallelized finite element method modules.

Storaasli, Olaf O.↗
NASA NTRS1993

Unsteady aerodynamics and flutter based on the potential equation

A time-domain three-dimensional full-potential solver is coupled with a linear structural dynamics model to investigate the unsteady aerodynamics and aeroelasticity of propfans. The solver allows calculations in multiple blade passages with independent blade motions. Aeroelastic calculations are performed in both frequency and time domains. Results are presented for two propfan configurations. Good agreement is seen between the full-potential results and results from linear theory since the flow is subsonic and the thickness of the propfan blades is small; the agreement is not as good for the cases in which the angle of attack is high. Some difficulty is encountered due to wave reflections from outer computational boundaries; however, this does not affect the results in the range of frequencies of interest.

Bakhle, Milind A.↗
DOE OSTI2023

Sparse Approximate Multifrontal Factorization with Composite Compression Methods

This article presents a fast and approximate multifrontal solver for large sparse linear systems. In a recent work by Liu et al., we showed the efficiency of a multifrontal solver leveraging the butterfly algorithm and its hierarchical matrix extension, HODBF (hierarchical off-diagonal butterfly) compression to compress large frontal matrices. The resulting multifrontal solver can attain quasi-linear computation and memory complexity when applied to sparse linear systems arising from spatial discretization of high-frequency wave equations. To further reduce the overall number of operations and especially the factorization memory usage to scale to larger problem sizes, in this article we develop a composite multifrontal solver that employs the HODBF format for large-sized fronts, a reduced-memory version of the nonhierarchical block low-rank format for medium-sized fronts, and a lossy compression format for small-sized fronts. This allows us to solve sparse linear systems of dimension up to 2.7 × larger than before and leads to a memory consumption that is reduced by 70% while ensuring the same execution time. The code is made publicly available in GitHub.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Navier-Stokes Solvers and Generalizations for Reacting Flow Problems

This is an overview of our accomplishments during the final term of this grant (1 September 2008 -- 30 June 2012). These fall mainly into three categories: fast algorithms for linear eigenvalue problems; solution algorithms and modeling methods for partial differential equations with uncertain coefficients; and preconditioning methods and solvers for models of computational fluid dynamics (CFD).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

SU-G-TeP1-15: Toward a Novel GPU Accelerated Deterministic Solution to the Linear Boltzmann Transport Equation

Purpose: To develop a Graphic Processor Unit (GPU) accelerated deterministic solution to the Linear Boltzmann Transport Equation (LBTE) for accurate dose calculations in radiotherapy (RT). A deterministic solution yields the potential for major speed improvements due to the sparse matrix-vector and vector-vector multiplications and would thus be of benefit to RT. Methods: In order to leverage the massively parallel architecture of GPUs, the first order LBTE was reformulated as a second order self-adjoint equation using the Least Squares Finite Element Method (LSFEM). This produces a symmetric positive-definite matrix which is efficiently solved using a parallelized conjugate gradient (CG) solver. The LSFEM formalism is applied in space, discrete ordinates is applied in angle, and the Multigroup method is applied in energy. The final linear system of equations produced is tightly coupled in space and angle. Our code written in CUDA-C was benchmarked on an Nvidia GeForce TITAN-X GPU against an Intel i7-6700K CPU. A spatial mesh of 30,950 tetrahedral elements was used with an S4 angular approximation. Results: To avoid repeating a full computationally intensive finite element matrix assembly at each Multigroup energy, a novel mapping algorithm was developed which minimized the operations required at each energy. Additionally, a parallelized memory mapping for the kronecker product between the sparse spatial and angular matrices, including Dirichlet boundary conditions, was created. Atomicity is preserved by graph-coloring overlapping nodes into separate kernel launches. The one-time mapping calculations for matrix assembly, kronecker product, and boundary condition application took 452±1ms on GPU. Matrix assembly for 16 energy groups took 556±3s on CPU, and 358±2ms on GPU using the mappings developed. The CG solver took 93±1s on CPU, and 468±2ms on GPU. Conclusion: Three computationally intensive subroutines in deterministically solving the LBTE have been formulated on GPU, resulting in two orders of magnitude speedup. Funding support from Natural Sciences and Engineering Research Council and Alberta Innovates Health Solutions. Dr. Fallone is a co-founder and CEO of MagnetTx Oncology Solutions (under discussions to license Alberta bi-planar linac MR for commercialization).

60 APPLIED LIFE SCIENCES↗
NASA NTRSDate not supplied

Implicit Preconditioning for Explicit Multigrid Solvers on Cut-Cell Cartesian Meshes

This work assesses the effectiveness of linearized implicit Euler preconditioning for multigrid solvers using an unpreconditioned, Jacobian-free Newton Krylov method to converge the linear system of equations. Multigrid convergence rates improve to approximately 0.75 across the cases tested including a Mach 2 supersonic wedge, transonic NACA 0012 airfoil, and ONERA M6 wing. While larger Krylov subspaces increase the convergence rate, they also increase the computational cost, such that 4-8 Krylov vectors often offers the fastest turnaround. Further reductions in computational cost are achieved with a sequential hybrid preconditioner that begins with the explicit multigrid solver before transitioning to the preconditioned algorithm later on. In addition, a novel implementation of dual time stepping is extended to include both common BDF methods as well as high-order implicit Runge-Kutta schemes. This particular formulation, which uses A −1 preconditioning, is amenable to matrix-free solvers, and the L-stable methods are especially suited for meshes with arbitrarily small cut-cells. Asymptotic order of convergence is demonstrated for BDF1, BDF2, SDIRK2, and 3rd-order Radau IIA time integration with unsteady 2D vortex simulations.

ARMD↗
DOE OSTI2014

A Flexible CUDA LU-based Solver for Small, Batched Linear Systems

This chapter presents the implementation of a batched CUDA solver based on LU factorization for small linear systems. This solver may be used in applications such as reactive flow transport models, which apply the Newton-Raphson technique to linearize and iteratively solve the sets of non linear equations that represent the reactions for ten of thousands to millions of physical locations. The implementation exploits somewhat counterintuitive GPGPU programming techniques: it assigns the solution of a matrix (representing a system) to a single CUDA thread, does not exploit shared memory and employs dynamic memory allocation on the GPUs. These techniques enable our implementation to simultaneously solve sets of systems with over 100 equations and to employ LU decomposition with complete pivoting, providing the higher numerical accuracy required by certain applications. Other currently available solutions for batched linear solvers are limited by size and only support partial pivoting, although they may result faster in certain conditions. We discuss the code of our implementation and present a comparison with the other implementations, discussing the various tradeoffs in terms of performance and flexibility. This work will enable developers that need batched linear solvers to choose whichever implementation is more appropriate to the features and the requirements of their applications, and even to implement dynamic switching approaches that can choose the best implementation depending on the input data.

bached linear solvers↗
DOE OSTI2016

PB-AM: An open-source, fully analytical linear poisson-boltzmann solver

Here, we present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized PB equation, for molecules represented as non-overlapping spherical cavities. The PB-AM software package includes the generation of outputs files appropriate for visualization using visual molecular dynamics, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators, and students that are more familiar with the APBS framework.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2006

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

Watson, Willie R.↗
DOE OSTI2014

A semi-analytical solver for the Grad-Shafranov equation

In toroidally confined plasmas, the Grad-Shafranov equation, in general, a non-linear partial differential equation, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta, and the Shafranov shift; then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and its parameters are adjusted with a non-linear fitting algorithm. This method is used to find hydromagnetic equilibria with normal and reversed magnetic shear and defined values of the elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal and poloidal beta, the safety factor at the 95% flux surface, and the plasma current are in agreement with usual experimental values for high beta discharges and the model can be used locally to describe reversed magnetic shear equilibria.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗