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At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Implementation of a Mesh refinement algorithm into the quasi-static PIC code QuickPIC

Plasma-based acceleration (PBA) has emerged as a promising candidate for the accelerator technology used to build a future linear collider and/or an advanced light source. In PBA, a trailing or witness particle beam is accelerated in the plasma wave wakefield (WF) created by a laser or particle beam driver. The WF is often nonlinear and involves the crossing of plasma particle trajectories in real space and thus particle-in-cell methods are used. The distance over which the drive beam evolves is several orders of magnitude larger than the wake wavelength. This large disparity in length scales is amenable to the quasi-static approach. Three-dimensional (3D), quasi-static (QS), particle-in-cell (PIC) codes, e.g., QuickPIC, have been shown to provide high fidelity simulation capability with 2-4 orders of magnitude speedup over 3D fully explicit PIC codes. In PBA, the witness beam needs to be matched to the focusing forces of the WF to reduce the emittance growth. In some linear collider designs, the matched spot size of the witness beam can be 2 to 3 orders of magnitude smaller than the spot size (and wavelength) of the wakefield. Such an additional disparity in length scales is ideal for mesh refinement where the WF within the witness beam is described on a finer mesh than the rest of the WF. A mesh refinement scheme is described that has been implemented into the 3D QS PIC code, QuickPIC. Very fine (high) resolution is used in a small spatial region that includes the witness beam and progressively coarser resolutions in the rest of the simulation domain. A fast multigrid Poisson solver has been implemented for the field solve on the refined meshes and a Fast Fourier Transform (FFT) based Poisson solver is used for the coarse mesh. The code has been parallelized with both MPI and OpenMP, and the parallel scalability has also been improved by using pipelining. A preliminary adaptive mesh refinement technique is described to optimize the computational time for simulations with an evolving witness beam size. Several test problems are used to verify that the mesh refinement algorithm provides accurate results. Additionally, the results are benchmarked against highly resolved simulations exhibiting near-azimuthal symmetry, performed using QPAD—a novel hybrid QS PIC code that uses a PIC description in the coordinates (r, ct – z) and a gridless description in the azimuthal angle, Φ.

Linear collider↗
DOE OSTI2025

Accelerating high-order continuum kinetic plasma simulations using multiple GPUs

Kinetic plasma simulations solve the Vlasov-Poisson or Vlasov-Maxwell equations to evolve scalar-variable distribution functions in position-velocity phase space and vector-variable electromagnetic fields in configuration space. The immense computational cost of evolving high-dimensional variables, and their large number of degrees of freedom, often limits the utility of continuum kinetic simulations and presents a challenge when it comes to accurately simulating real-world physical phenomena. To address this challenge, we present techniques that accelerate and minimize the computational work required for a scalable Vlasov-Poisson solver. We show theoretical hardware compute and communication bounds for solving a fourth-order finite-volume Vlasov-Poisson system. These bounds are then used to inform and evaluate the design of performance portable algorithms for a multiple graphics processing unit (GPU) accelerated version of the Vlasov-Poisson solver VCK-CPU [1]. We demonstrate that the multi-GPU Vlasov solver implementation, VCK-GPU, simultaneously minimizes required inter-process data transfer while also being bounded by the machine network performance limits. This results in an overall strong scaling speedup per timestep of up to 40x in three-dimensional phase space (one position, two velocity coordinates) and 54x in four dimensional phase space (two position, two velocity coordinates) and a 341x increase in simulation throughput of the GPU accelerated code over the existing CPU code. The GPU code is also able to weak scale up to 256 compute nodes and 1024 GPUs. In conclusion, we demonstrate that the improved compute performance enables exploring configurations which were previously computationally infeasible, including resolving fine-scale distribution function filamentation and multi-species dynamics with realistic electron-proton mass ratios.

Continuum kinetics↗
NASA NTRS1981

A stochastic-dynamic model for global atmospheric mass-field statistics

Global atmospheric mass field error correlations based on satellite observations and on numerical forecasts show strong and systematic latitude dependence. A model for the latitude dependent spatial correlation structure of mass field forecast errors is derived from dynamical considerations. Three methods of solution were tested. In the first method, the equation was solved by expansion in spherical harmonics, and the correlation function was computed analytically using the expansion coefficients. In the second method, the finite difference equivalent of the equation was solved using a fast poisson solver. The correlation function was computed using stratified sampling of the individual realizations. In the third method, a higher order equation was derived, and solved directly in finite differences by two successive applications of the fast poisson solver. The three methods were compared for accuracy and efficiency, and the third method was chosen as clearly superior.

Ghil, M.↗
NASA NTRS1981

A stochastic-dynamic model for global atmospheric mass field statistics

A model that yields the spatial correlation structure of atmospheric mass field forecast errors was developed. The model is governed by the potential vorticity equation forced by random noise. Expansion in spherical harmonics and correlation function was computed analytically using the expansion coefficients. The finite difference equivalent was solved using a fast Poisson solver and the correlation function was computed using stratified sampling of the individual realization of F(omega) and hence of phi(omega). A higher order equation for gamma was derived and solved directly in finite differences by two successive applications of the fast Poisson solver. The methods were compared for accuracy and efficiency and the third method was chosen as clearly superior. The results agree well with the latitude dependence of observed atmospheric correlation data. The value of the parameter c sub o which gives the best fit to the data is close to the value expected from dynamical considerations.

Ghil, M.↗
DOE OSTI2024

Modeling and Simulation of Electrostatics of Ge$_{\text{1-x}}$Sn$_{\text{x}}$ Layers Grown on Ge Substrates

This work introduces a comprehensive simulation tool that provides a robust 1D Schrödinger – Poisson solver for modeling the electrostatics of heterostructures with an arbitrary number of layers, and non-uniform doping profiles along with the treatment of partial ionization of dopants at low temperatures. The effective masses are derived from the first-principles calculations. The solver is used to characterize three Ge 1-x Sn x /Ge heterostructures with non-uniform doping profiles and determine the subband structure at various temperatures. Here, the simulation results of the sheet carrier densities show excellent agreement with the experimentally extracted data, thus demonstrating the capabilities of the solver.

42 ENGINEERING↗
NASA NTRS1998

Massively Parallel Solution of Poisson Equation on Coarse Grain MIMD Architectures

In this paper a new algorithm, designated as Fast Invariant Imbedding algorithm, for solution of Poisson equation on vector and massively parallel MIMD architectures is presented. This algorithm achieves the same optimal computational efficiency as other Fast Poisson solvers while offering a much better structure for vector and parallel implementation. Our implementation on the Intel Delta and Paragon shows that a speedup of over two orders of magnitude can be achieved even for moderate size problems.

parallel algorithms MIMD parallel architectures↗
NASA NTRS1992

Multigrid acceleration of a fractional-step solver in generalized curvilinear coordinate systems

The efficiency of the INS3D-fractional step (FS) code has been significantly enhanced by accelerating the Poisson solver with a multigrid (MG) method. This is apparently the first implementation of the MG method for solving the 3D discrete Poisson-like equation obtained in the derivation of FS solution methods for the incompressible Navier-Stokes equations in generalized nonorthogonal coordinate systems using a staggered arrangement of the variables. The MG solver is insensitive to the geometry, even in cases of highly nonorthogonal clustered meshes, as well as to the free parameters of MG methods.

Rosenfeld, Moshe↗
DOE OSTI2021

HTR-1.3 solver: Predicting electrified combustion using the hypersonic task-based research solver

Here this manuscript presents an updated open-source version of the Hypersonics Task-based Research (HTR) solver. The solver, whose main features are presented in Di Renzo et al. (2020) and Di Renzo & Pirozzoli (2021), is designed for direct numerical simulation of reacting flows at high Reynolds numbers. This new version extends the applications of the HTR solver to turbulent combustion in the presence of external electric fields. In particular, a new distributed Poisson solver compatible with heterogeneous architectures has been incorporated in the algorithm to compute the electric potential distribution in bi-periodic configurations. The drift fluxes of the electrically charged species are now included in the transport equations using a targeted essentially non-oscillatory scheme. A verification of these new features of the solver is provided using one-dimensional burner stabilized flames, whereas a three dimensional turbulent flame is utilized to discuss the scalability of the proposed numerical tool.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1992

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗
NASA NTRS1993

Implementing abstract multigrid or multilevel methods

Multigrid methods can be formulated as an algorithm for an abstract problem that is independent of the partial differential equation, domain, and discretization method. In such an abstract setting, problems not arising from partial differential equations can be treated. A general theory exists for linear problems. The general theory was motivated by a series of abstract solvers (Madpack). The latest version was motivated by the theory. Madpack now allows for a wide variety of iterative and direct solvers, preconditioners, and interpolation and projection schemes, including user callback ones. It allows for sparse, dense, and stencil matrices. Mildly nonlinear problems can be handled. Also, there is a fast, multigrid Poisson solver (two and three dimensions). The type of solvers and design decisions (including language, data structures, external library support, and callbacks) are discussed. Based on the author's experiences with two versions of Madpack, a better approach is proposed. This is based on a mixed language formulation (C and FORTRAN + preprocessor). Reasons for not using FORTRAN, C, or C++ (individually) are given. Implementing the proposed strategy is not difficult.

Douglas, Craig C.↗
NASA NTRS1973

Direct numerical solution of three-dimensional equations containing elliptic operators.

A direct three-dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic-Reduction Poisson solver to three dimensions. Both a 'most direct' cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.

Martin, E. D.↗
NASA NTRS1988

A solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A solution method based on a fractional step approach is developed for obtaining time-dependent solutions of the three-dimensional, incompressible Navier-Stokes equations in generalized coordinate systems. The governing equations are discretized conservatively by finite volumes using a staggered mesh system. The primitive variable formulation uses the volume fluxes across the faces of each computational cell as dependent variables. This procedure, combined with accurate and consistent approximations of geometric parameters, is done to satisfy the discretized mass conservation equation to machine accuracy as well as to gain favorable convergence properties of the Poisson solver. The discretized equations are second-order-accurate in time and space and no smoothing terms are added. An approximate-factorization scheme is implemented in solving the momentum equations. A novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two and three-dimensional solutions are compared with other numerical and experimental results to validate the present method.

Rosenfeld, Moshe↗
NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗
DOE OSTI2022

GenASiS Mathematics: Object-oriented manifolds, operations, and solvers for large-scale physics simulations (version 2)

We report GenASiS Mathematics provides modern Fortran classes furnishing extensible object-oriented functionality for the solution of fields governed by selected partial differential equations. The initial release included extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon. This revision - Version 2 of Mathematics - includes significant reorganization and streamlining of these classes, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A sixth order Mehrstellen scheme with an application to the Method of Local Corrections for the 3D Poisson equation

We present a sixth order finite difference scheme for Poisson’s equation when discretized with the compact 27-point stencil based on Mehrstellen corrections of the forcing function term f. Our approach results in a sixth order accurate solution error as opposed to a fourth-order error imposed by the classical Mehrstellen correction for the 19-point and 27-point stencils. The present study is a continuation of former work of Spotz and Carey (1996) on compact finite difference schemes for Poisson’s equation where sixth order convergence may be obtained under the assumption that the fourth order derivatives of f are determined analytically. Specifically, we show that sixth order convergence can still be attained when only values of f at grid points are available. The sixth order Mehrstellen scheme is further coupled with a Method of Local Corrections (MLC) 3D Poisson solver improving to sixth order accuracy the results reported in Kavouklis and Colella (2019). The MLC test case considered involves an adaptive grid that comprises 7.5 billion cells.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

Parallel tridiagonal equation solvers

Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.

Stone, H. S.↗
DOE OSTI2020

Uncertainty-based weighted least squares density integration for background-oriented schlieren

We propose an improved density integration methodology for Background-Oriented Schlieren (BOS) measurements that overcomes the noise sensitivity of the commonly used Poisson solver. Here, the method employs a weighted least-squares (WLS) optimization of the 2D integration of the density gradient field by solving an over-determined system of equations. Weights are assigned to the grid points based on density gradient uncertainties to ensure that a less reliable measurement point has less effect on the integration procedure. Synthetic image analysis with a Gaussian density field shows that WLS constrains the propagation of random error and reduces it by 80% in comparison to Poisson for the highest noise level. Using WLS with experimental BOS measurements of flow induced by a spark plasma discharge shows a 30% reduction in density uncertainty in comparison to Poisson, thereby increasing the overall precision of the BOS density measurements.

42 ENGINEERING↗
NASA NTRS1997

Efficient Parallel Kernel Solvers for Computational Fluid Dynamics Applications

Distributed-memory parallel computers dominate today's parallel computing arena. These machines, such as Intel Paragon, IBM SP2, and Cray Origin2OO, have successfully delivered high performance computing power for solving some of the so-called "grand-challenge" problems. Despite initial success, parallel machines have not been widely accepted in production engineering environments due to the complexity of parallel programming. On a parallel computing system, a task has to be partitioned and distributed appropriately among processors to reduce communication cost and to attain load balance. More importantly, even with careful partitioning and mapping, the performance of an algorithm may still be unsatisfactory, since conventional sequential algorithms may be serial in nature and may not be implemented efficiently on parallel machines. In many cases, new algorithms have to be introduced to increase parallel performance. In order to achieve optimal performance, in addition to partitioning and mapping, a careful performance study should be conducted for a given application to find a good algorithm-machine combination. This process, however, is usually painful and elusive. The goal of this project is to design and develop efficient parallel algorithms for highly accurate Computational Fluid Dynamics (CFD) simulations and other engineering applications. The work plan is 1) developing highly accurate parallel numerical algorithms, 2) conduct preliminary testing to verify the effectiveness and potential of these algorithms, 3) incorporate newly developed algorithms into actual simulation packages. The work plan has well achieved. Two highly accurate, efficient Poisson solvers have been developed and tested based on two different approaches: (1) Adopting a mathematical geometry which has a better capacity to describe the fluid, (2) Using compact scheme to gain high order accuracy in numerical discretization. The previously developed Parallel Diagonal Dominant (PDD) algorithm and Reduced Parallel Diagonal Dominant (RPDD) algorithm have been carefully studied on different parallel platforms for different applications, and a NASA simulation code developed by Man M. Rai and his colleagues has been parallelized and implemented based on data dependency analysis. These achievements are addressed in detail in the paper.

Sun, Xian-He↗