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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 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
NASA NTRS2016

An Introduction to Transient Engine Applications Using the Numerical Propulsion System Simulation (NPSS) and MATLAB

This document outlines methodologies designed to improve the interface between the Numerical Propulsion System Simulation framework and various control and dynamic analyses developed in the Matlab and Simulink environment. Although NPSS is most commonly used for steady-state modeling, this paper is intended to supplement the relatively sparse documentation on it's transient analysis functionality. Matlab has become an extremely popular engineering environment, and better methodologies are necessary to develop tools that leverage the benefits of these disparate frameworks. Transient analysis is not a new feature of the Numerical Propulsion System Simulation (NPSS), but transient considerations are becoming more pertinent as multidisciplinary trade-offs begin to play a larger role in advanced engine designs. This paper serves to supplement the relatively sparse documentation on transient modeling and cover the budding convergence between NPSS and Matlab based modeling toolsets. The following sections explore various design patterns to rapidly develop transient models. Each approach starts with a base model built with NPSS, and assumes the reader already has a basic understanding of how to construct a steady-state model. The second half of the paper focuses on further enhancements required to subsequently interface NPSS with Matlab codes. The first method being the simplest and most straightforward but performance constrained, and the last being the most abstract. These methods aren't mutually exclusive and the specific implementation details could vary greatly based on the designer's discretion. Basic recommendations are provided to organize model logic in a format most easily amenable to integration with existing Matlab control toolsets.

Matlab↗
NASA NTRS1993

Advanced radiometric and interferometric milimeter-wave scene simulations

Smart munitions and weapons utilize various imaging sensors (including passive IR, active and passive millimeter-wave, and visible wavebands) to detect/identify targets at short standoff ranges and in varied terrain backgrounds. In order to design and evaluate these sensors under a variety of conditions, a high-fidelity scene simulation capability is necessary. Such a capability for passive millimeter-wave scene simulation exists at TRW. TRW's Advanced Radiometric Millimeter-Wave Scene Simulation (ARMSS) code is a rigorous, benchmarked, end-to-end passive millimeter-wave scene simulation code for interpreting millimeter-wave data, establishing scene signatures and evaluating sensor performance. In passive millimeter-wave imaging, resolution is limited due to wavelength and aperture size. Where high resolution is required, the utility of passive millimeter-wave imaging is confined to short ranges. Recent developments in interferometry have made possible high resolution applications on military platforms. Interferometry or synthetic aperture radiometry allows the creation of a high resolution image with a sparsely filled aperture. Borrowing from research work in radio astronomy, we have developed and tested at TRW scene reconstruction algorithms that allow the recovery of the scene from a relatively small number of spatial frequency components. In this paper, the TRW modeling capability is described and numerical results are presented.

Hauss, B. I.↗
NASA NTRS1998

Parallel Implicit Algorithms for CFD

The main goal of this project was efficient distributed parallel and workstation cluster implementations of Newton-Krylov-Schwarz (NKS) solvers for implicit Computational Fluid Dynamics (CFD.) "Newton" refers to a quadratically convergent nonlinear iteration using gradient information based on the true residual, "Krylov" to an inner linear iteration that accesses the Jacobian matrix only through highly parallelizable sparse matrix-vector products, and "Schwarz" to a domain decomposition form of preconditioning the inner Krylov iterations with primarily neighbor-only exchange of data between the processors. Prior experience has established that Newton-Krylov methods are competitive solvers in the CFD context and that Krylov-Schwarz methods port well to distributed memory computers. The combination of the techniques into Newton-Krylov-Schwarz was implemented on 2D and 3D unstructured Euler codes on the parallel testbeds that used to be at LaRC and on several other parallel computers operated by other agencies or made available by the vendors. Early implementations were made directly in Massively Parallel Integration (MPI) with parallel solvers we adapted from legacy NASA codes and enhanced for full NKS functionality. Later implementations were made in the framework of the PETSC library from Argonne National Laboratory, which now includes pseudo-transient continuation Newton-Krylov-Schwarz solver capability (as a result of demands we made upon PETSC during our early porting experiences). A secondary project pursued with funding from this contract was parallel implicit solvers in acoustics, specifically in the Helmholtz formulation. A 2D acoustic inverse problem has been solved in parallel within the PETSC framework.

Keyes, David E.↗
DOE OSTI2020

Reduced Models for Chemical Kinetics derived from Parallel Ensemble Simulations of Stirred Reactors

Detailed chemical kinetic models are essential to improve the predictive capability of computational fluid dynamics (CFD) simulations and to impact the development of next generation combustion technologies. However, such detailed kinetic models are beyond the reach of CFD codes due to the computational cost of transporting additional chemical degrees of freedom. Reduced order models for combustion kinetics exploit the presence of lower dimensional manifolds in the state space and require far fewer number of scalars to be transported as part of CFD simulations. In this paper, we describe the software infrastructure developed to allow large ensembles of stirred reactor configurations with detailed chemical kinetics. The canonical reactor simulations are incorporated in a workflow with sparse grid and data reduction methodologies aimed at development of reduced order models. We present results from analyzing the results of the stirred reactor calculations for syngas mixture using principal component analysis (PCA) and neural network methodologies. The results show that nonlinear autoencoder model can surpass PCA in representing the original dataset with far fewer degrees of freedom.

Zhang, Pei↗
NASA NTRS1989

Sparse matrix methods research using the CSM testbed software system

Research is described on sparse matrix techniques for the Computational Structural Mechanics (CSM) Testbed. The primary objective was to compare the performance of state-of-the-art techniques for solving sparse systems with those that are currently available in the CSM Testbed. Thus, one of the first tasks was to become familiar with the structure of the testbed, and to install some or all of the SPARSPAK package in the testbed. A suite of subroutines to extract from the data base the relevant structural and numerical information about the matrix equations was written, and all the demonstration problems distributed with the testbed were successfully solved. These codes were documented, and performance studies comparing the SPARSPAK technology to the methods currently in the testbed were completed. In addition, some preliminary studies were done comparing some recently developed out-of-core techniques with the performance of the testbed processor INV.

Chu, Eleanor↗
DOE OSTI2022

Experimental Steps toward a Density Law for Chlorine-Crediting Criticality Models of Aqueous Plutonium solutions

Criticality accident prevention is an essential safety consideration for all operations involving fissionable and fissile materials. Predictive criticality calculations using advanced neutron transport codes, such as Monte Carlo N-Particle Transport (MCNP), are invaluable tools for designing and implementing operational limits. While indispensable, these tools are limited by the quality and accuracy of the inputs that the user provides to define the modeled system. Parameters such as atomic composition, shape, and material density must be accurately defined to obtain a meaningful result. In the case of material property details for a plutonium model, accurate material characterization data is sometimes sparse. Thus limitations in calculation accuracy can be reduced by improving our knowledge of material properties in the targeted fissile systems. One of the biggest remaining challenges to accurately defining fissile systems is the description of aqueous fissile solutions. Even simple properties, such as density, are not well-known for fissile solutions relevant to nuclear energy and security. Described here are initial efforts undertaken to improve criticality calculation inputs for fissile plutonium chloride solutions in water. This effort is focused on experimentally determining accurate solution characteristics for ternary plutonium chloride/hydrochloric acid/water systems, by measurement of water activity and solution density. The effect of inputting experimental densities for these solutions into MCNP criticality calculations is compared to the traditional approach of modeling an idealized (and fictitious) plutonium-water mixture. Expansion of these efforts to a working density law for aqueous plutonium chloride solutions is also discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Development of a Framework and Methodology for an Advanced Reactor Materials Environmental Effects Design Guide

Advanced non-light-water reactor components may operate at elevated temperature while experiencing cyclic loading, significant neutron irradiation, and exposure to reactor coolant. ASME Boiler and Pressure Vessel Code, Section III, Division 5, provides design rules for elevated-temperature service but does not include specific procedures to account for environmental effects on material properties. This report develops an initial framework and methodology for an Environmental Effects Design Guide (EEDG) focused on neutron irradiation; coolant-environment effects are reserved for future work. The proposed approach treats irradiation as a property-based overlay on the existing Division 5 design process, with two routes: a sparse-data route applying two reduction factors — FCR on creep-rupture strength and FF on fatigue life — for the creep-fatigue evaluations that typically control the design of advanced high-temperature reactor components, and a fuller framework developing the property-to-rule chain across the four Division 5 checks (primary load, strain limits and ratcheting, creep-fatigue, and buckling), together with swelling and weldments as scope items. Both routes are scoped by an in-pile qualification that restricts the use of post-irradiation-examination-derived properties in regimes where an in-pile mechanism could control the design outcome. Illustrative outputs derived on a compiled annealed Type 316 database — FCR ≈ 0.78–0.86 and FF ≈ 0.4 — demonstrate the calculation method within that specific dataset. The framework is an initial, testable design-rule concept; it identifies a practical path for preliminary design evaluations under sparse data and the material data and testing needed to develop the framework further.

Barua, Bipul (ORCID:0000000247184113)↗
DOE OSTI2015

brulilo, Version 0.x

brulilo is a Python package for building and evolving thermonuclear reaction networks, such as those used in astrophysical explosion calculations. It is very much a current work in progress, and much of the intended functionality is still under development. The main goal of brulilo is an easy-to-use code for quick calculations or post-processing, as well as a learning tool to test various implementations of solvers on both CPUs and GPUs. Currently, brulilo parses an input text file that specifies the reactions to be included in the network. The notation of the inputs are in the typical astrophysical notation; i.e. hydrogen burning to deuterium in the sun, "p + p + e --> d + neutrino", is given as "p(p,)d." Each reaction is added to a list comprising the reaction network, and once completed, the network contains a list of all isotopes and reactions involved. brulilo then parses publicly available nuclear data (binding energy, spin, partition function data, etc.) for each isotope from the WebNucleo group's data at nucleo.ces.clemson.edu. Fits to reaction rate data for each reaction are obtained from the publicly available data provided by the Joint Institute for Nuclear Astrophysics' Reaclib database. Reverse reactions are calculated from detailed balance. The reaction network then consists of a sparse system of mathematically stiff ordinary differential equations (ODEs). Physically, "stiffness" here means that there are some reaction rates that are occurring much faster than others; the ratio of the fast to slow often being greater than 10^15 in astrophysics. Numerical integration of such systems typically involves implicit integration schemes, which brulilo will leverage from the scientific computing Python package, Numpy, as well as implement its own. There has recently been work from researchers at Oak Ridge National Lab who have found good results from applying quasi-equilibrium and quasi-asymptotic conditions to the stiff system of ODEs, which effectively remove some of the stiffness and allow for efficient explicit integration techniques to be used. The original intent of brulilo was to implement these stiffness-alleviating techniques with explicit integrators and compare the performance to traditional implicit integrations of the full stiff system. This is still underway, as the code is very much in an alpha-release state. Furthermore, explicit integrators are often much easier to parallelize than their implicit counterparts. brulilo will implement parallelization of these techniques, leveraging both the Python implementation of MPI, mpi4py, as well as highly parallelized versions targeted at GPUs with PyOpenCL and/or PyCUDA.

Malone, Chris↗
DOE OSTI2015

brulilo, Version 0.x

brulilo is a Python package for building and evolving thermonuclear reaction networks, such as those used in astrophysical explosion calculations. It is very much a current work in progress, and much of the intended functionality is still under development. The main goal of brulilo is an easy-to-use code for quick calculations or post-processing, as well as a learning tool to test various implementations of solvers on both CPUs and GPUs. Currently, brulilo parses an input text file that specifies the reactions to be included in the network. The notation of the inputs are in the typical astrophysical notation; i.e. hydrogen burning to deuterium in the sun, "p + p + e --> d + neutrino", is given as "p(p,)d." Each reaction is added to a list comprising the reaction network, and once completed, the network contains a list of all isotopes and reactions involved. brulilo then parses publicly available nuclear data (binding energy, spin, partition function data, etc.) for each isotope from the WebNucleo group's data at nucleo.ces.clemson.edu. Fits to reaction rate data for each reaction are obtained from the publicly available data provided by the Joint Institute for Nuclear Astrophysics' Reaclib database. Reverse reactions are calculated from detailed balance. The reaction network then consists of a sparse system of mathematically stiff ordinary differential equations (ODEs). Physically, "stiffness" here means that there are some reaction rates that are occurring much faster than others; the ratio of the fast to slow often being greater than 10^15 in astrophysics. Numerical integration of such systems typically involves implicit integration schemes, which brulilo will leverage from the scientific computing Python package, Numpy, as well as implement its own. There has recently been work from researchers at Oak Ridge National Lab who have found good results from applying quasi-equilibrium and quasi-asymptotic conditions to the stiff system of ODEs, which effectively remove some of the stiffness and allow for efficient explicit integration techniques to be used. The original intent of brulilo was to implement these stiffness-alleviating techniques with explicit integrators and compare the performance to traditional implicit integrations of the full stiff system. This is still underway, as the code is very much in an alpha-release state. Furthermore, explicit integrators are often much easier to parallelize than their implicit counterparts. brulilo will implement parallelization of these techniques, leveraging both the Python implementation of MPI, mpi4py, as well as highly parallelized versions targeted at GPUs with PyOpenCL and/or PyCUDA.

Malone, Chris↗
DOE OSTI2021

Cognitive simulation models for inertial confinement fusion: Combining simulation and experimental data

The design space for inertial confinement fusion (ICF) experiments is vast, and experiments are extremely expensive. Researchers rely heavily on computer simulations to explore the design space in search of high-performing implosions. However, ICF multiphysics codes must make simplifying assumptions, and thus deviate from experimental measurements for complex implosions. For more effective design and investigation, simulations require input from past experimental data to better predict future performance. In this work, we describe a cognitive simulation method for combining simulation and experimental data into a common, predictive model. This method leverages a machine learning technique called “transfer learning,” the process of taking a model trained to solve one task, and partially retraining it on a sparse dataset to solve a different, but related task. In the context of ICF design, neural network models are trained on large simulation databases and partially retrained on experimental data, producing models that are far more accurate than simulations alone. Here, we demonstrate improved model performance for a range of ICF experiments at the National Ignition Facility and predict the outcome of recent experiments with less than 10% error for several key observables. We discuss how the methods might be used to carry out a data-driven experimental campaign to optimize performance, illustrating the key product—models that become increasingly accurate as data are acquired.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Uncertainty and Sensitivity Analysis of TREAT Transient Test no. 15

The U.S. Department of Energy (DOE) is preparing for the resumption of transient testing using the Transient Reactor Test Facility (TREAT) located at the Idaho National Laboratory (INL). In order to support the analysis of TREAT operation and experiment planning, the MAMMOTH reactor physics team has recently begun development of uncertainty and sensitivity analysis for the TREAT core simulated via MAMMOTH. MAMMOTH is a high-fidelity three-dimensional transient multi-physics code developed at INL under the DOE-NE Advanced Modeling and Simulation program (NEAMS). This research is part of the development presenting some initial uncertainty and sensitivity analysis of TREAT Transient Test no. 15. In this summary, we employ the Sobol sampling strategy implemented in RAVEN to efficiently perform the sensitivity and uncertainty analysis. The idea is using Sobol functional decomposition with generalized polynomial chaos (gPC) expansions to estimate the responses of interest. The gPC coefficients are then computed via the sparse grid spectral projection method. In addition, this strategy includes evaluation of the second or higher order sensitivity indices, allowing the study of interactions between several input variables. The principal focus of this analysis is to quantify the uncertainties in the total power and maximum fuel temperatures with respect to uncertainties in the initial transient conditions, such as rated power, fuel temperature, graphite fuel density and control rod boron concentration. In addition, both the first order and second order time-dependent sensitivity indices are computed and analyzed. The results show that the interactions between initial control rod boron concentration and the rated power will contribute at most 1% to the total uncertainty of the power peak.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2023

The Kokkos OpenMPTarget Backend: Implementation and Lessons Learned

As the supercomputing landscape diversifies, solutions such as Kokkos to write vendor agnostic applications and libraries have risen in popularity. Kokkos provides a programming model designed for performance portability, which allows developers to write a single source implementation that can run efficiently on various architectures. At its heart, Kokkos maps parallel algorithms to architecture and vendor specific backends written in lower level programming models such as CUDA and HIP. Another approach to writing vendor agnostic parallel code is using OpenMP’s directives based approach, which lets developers annotate code to express parallelism. It is implemented at the compiler level and is supported by all major high performance computing vendors, as well as the primary Open Source toolchains GNU and LLVM. Since its inception, Kokkos has used OpenMP to parallelize on CPU architectures. In this paper, we explore leveraging OpenMP for a GPU backend and discuss the challenges we encountered when mapping the Kokkos APIs and semantics to OpenMP target constructs. As an exemplar workload we chose a simple conjugate gradient solver for sparse matrices. We find that performance on NVIDIA and AMD GPUs varies widely based on details of the implementation strategy and the chosen compiler. Furthermore, the performance of the OpenMP implementations decreases with increasing complexity of the investigated algorithms.

Gayatri, Rahulkumar↗
DOE OSTI2022

Dissecting Tensor Cores via Microbenchmarks: Latency, Throughput and Numeric Behaviors

Tensor Cores have been an important unit to accelerate Fused Matrix Multiplication Accumulation (MMA) in all NVIDIA GPUs since Volta Architecture. To program Tensor Cores, users have to use either legacy wmma APIs or current mma APIs. Legacy wmma APIs are more easy-to-use but can only exploit limited features and power of Tensor Cores. Specifically, wmma APIs support fewer operand shapes and can not leverage the new sparse matrix multiplication feature of the newest Ampere Tensor Cores. However, the performance of current programming interface has not been well explored. Furthermore, the computation numeric behaviors of low-precision floating points (TF32, BF16, and FP16) supported by the newest Ampere Tensor Cores are also mysterious. In this paper, we explore the throughput and latency of current programming APIs. Further, we intuitively study the numeric behaviors of Tensor Cores MMA and profile the intermediate operations including multiplication, addition of inner product, and accumulation. All codes used in this work can be found in https://github.com/sunlex0717/DissectingTensorCores.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1992

Gravity and count probabilities in an expanding universe

The time evolution of nonlinear clustering on large scales in cold dark matter, hot dark matter, and white noise models of the universe is investigated using N-body simulations performed with a tree code. Count probabilities in cubic cells are determined as functions of the cell size and the clustering state (redshift), and comparisons are made with various theoretical models. We isolate the features that appear to be the result of gravitational instability, those that depend on the initial conditions, and those that are likely a consequence of numerical limitations. More specifically, we study the development of skewness, kurtosis, and the fifth moment in relation to variance, the dependence of the void probability on time as well as on sparseness of sampling, and the overall shape of the count probability distribution. Implications of our results for theoretical and observational studies are discussed.

Bouchet, Francois R.↗
DOE OSTI2017

Uncertainty and Sensitivity Analysis of TREAT Transient Test #15

The U.S. Department of Energy (DOE) is preparing for the resumption of transient testing using the Transient Reactor Test Facility (TREAT) located at the Idaho National Laboratory (INL). In order to support the analysis of TREAT operation and experiment planning, the MAMMOTH reactor physics team has recently begun development of uncertainty and sensitivity analysis for the TREAT core simulated via MAMMOTH. MAMMOTH is a high-fidelity three-dimensional transient multi-physics code developed at INL under the DOE-NE Advanced Modeling and Simulation program (NEAMS) [1]. This research is part of the development presenting some initial uncertainty and sensitivity analysis of TREAT Transient Test #15 [2]. In this summary, we employ the Sobol sampling strategy implemented in RAVEN [3] to efficiently perform the sensitivity and uncertainty analysis. The idea is using Sobol functional decomposition with generalized polynomial chaos (gPC) expansions to estimate the responses of interest [4]. The gPC coefficients are then computed via the spectral projection method with sparse grid approach. In addition, this strategy includes evaluation of the second or higher order sensitivity indices, allowing the study of interactions between several input variables. The principal focus of this analysis is to quantify the uncertainties in the total power and maximum fuel temperatures with respect to uncertainties in the initial transient conditions, such as rated power, fuel temperature, graphite fuel density and control rod boron concentration. In addition, both the first order and second order time-dependent sensitivity indices are computed and analyzed. The results show that the interactions between initial control rod boron concentration and the rated power will contribute at most 1% to the total uncertainty of the power peak.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

Sparse, iterative simulation methods for one-dimensional laminar flames

Sparse, iterative simulation methods for one-dimensional laminar flames are proposed. The resulting steady and unsteady flame solvers exploit approximate Jacobians to greatly reduce the computational cost associated with matrix operations. The constant, non-unity Lewis number assumption is introduced to further reduce the computational cost. The solvers are also parallelized to reduce time-to-solution on distributed memory computer systems. Computed laminar flame speeds and species profiles for a range of chemical mechanisms (from 10 to 2878 species) are compared against a well-validated commercial code and are found to be consistent within solver tolerances. The computation times of both the unsteady and steady solutions increase only linearly with the number of species, which is a significant improvement over the quadratic or cubic scaling of existing steady-state flame solvers. For the largest mechanism tested, the steady-state flame solver is two orders of magnitude faster than commonly-used codes. The use of an approximate Jacobian is shown to reduce the rate of convergence for the steady-state solver, but does not significantly affect the domain of convergence. The steady-state solver with approximate Jacobian is thus well suited for computationally efficient laminar flame speed sweeps with large kinetic mechanisms.

42 ENGINEERING↗
DOE OSTI2023

Analysing the effects of heating and gas puffing in Proto-MPEX helicon and auxiliary heated plasmas

The Material Plasma Exposure Experiment (MPEX) is being constructed at Oak Ridge National Laboratory to investigate critical fusion reactor issues, such as plasma–material interactions (PMIs) under reactor-relevant conditions and time scales. The linear device Proto-MPEX was used as a test bed to address anticipated research and development issues associated with heating scenarios and establish the physics basis for MPEX. The SOLPS-ITER code suite has been applied to understand plasma and neutral transport in Proto-MPEX and to increase confidence in predictive simulations for MPEX. Coupling between COMSOL and SOLPS is performed to implement a 2D electron heating profile of the helicon source. The simulations show reasonable agreement with the experimental data for plasma with helicon and auxiliary electron cyclotron heating (ECH). Both Bohm and constant diffusion (D$_\perp$: 0.5 m 2 s –1 and $\chi_\perp$: 1 m 2 s –1 ) simulations show similar levels of agreement with respect to the sparse experimental data available, assuming a few per cent impurity concentration. ECH significantly increases the target electron temperature, however, the target electron density is reduced compared to helicon-only heated plasmas due to an increase in flow velocity and radial losses. The simulations show that further increasing the ECH power results in an increase in the target electron density due to increased recycling flux and ionization. The results indicate that ECH significantly enhances the target heat flux, with ECH power of 50 kW increasing the target heat flux from 0.4 to 17 MW m –2 . It is found that a small amount of gas puffing (GP) near the target plate can further increase the target heat fluxes at the higher ECH power cases, but the target heat flux is reduced at higher GP conditions due to a significant reduction in electron temperature via radiation. ECH and GP scenarios can generate a higher target flux, facilitating improved PMI studies with more reactor-relevant plasma conditions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Porting AMG2013 to Heterogeneous CPU+GPU Nodes

LLNL's future advanced technology system SIERRA will feature heterogeneous compute nodes that consist of IBM PowerV9 CPUs and NVIDIA Volta GPUs. Conceptually, the motivation for such an architecture is quite straightforward: While GPUs are optimized for throughput on massively parallel workloads, CPUs strive to minimize latency for rather sequential operations. Yet, making optimal use of heterogeneous architectures raises new challenges for the development of scalable parallel software, e.g., with respect to work distribution. Porting LLNL's parallel numerical libraries to upcoming heterogeneous CPU+GPU architectures is therefore a critical factor for ensuring LLNL's future success in ful lling its national mission. One of these libraries, called HYPRE, provides parallel solvers and precondi- tioners for large, sparse linear systems of equations. In the context of this intern- ship project, I consider AMG2013 which is a proxy application for major parts of HYPRE that implements a benchmark for setting up and solving di erent systems of linear equations. In the following, I describe in detail how I ported multiple parts of AMG2013 to the GPU (Section 2) and present results for di erent experiments that demonstrate a successful parallel implementation on the heterogeneous ma- chines surface and ray (Section 3). In Section 4, I give guidelines on how my code should be used. Finally, I conclude and give an outlook for future work (Section 5).

97 MATHEMATICS AND COMPUTING↗