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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 973 records · Page 54
All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

A geometric initial guess for localized electronic orbitals in modular biological systems

Recent first-principles molecular dynamics algorithms using localized electronic orbitals have achieved O(N) complexity and controlled accuracy in simulating systems with finite band gaps. However, accurately deter- mining the centers of these localized orbitals during simulation setup may require O(N 3 ) operations, which is computationally infeasible for many biological systems. We present an O(N) approach for approximating orbital centers in proteins, DNA, and RNA which uses non-localized solutions for a set of fixed-size subproblems to create a set of geometric maps applicable to larger systems. This scalable approach, used as an initial guess in the O(N) first-principles molecular dynamics code MGmol, facilitates first-principles simulations in biological systems of sizes which were previously impossible.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRSDate not supplied

Autonomous In-space Construction, Maintenance, and Reconfiguration Using Programmable Meta-Material

NASA ARC's Coded Structures Laboratory (CSL) is developing autonomous construction, maintenance, and reconfiguration technologies to meet long-duration and deep space infrastructure needs, in accordance with long-term NASA goals of "in-space reliance" and "mass-less exploration." We seek to achieve these capabilities by utilizing a "programmable meta-material" approach that integrates emerging advances in materials (mechanical meta-materials), manufacturing (cooperative mobile robotics), and autonomy (multi-agent planning algorithms). Through the ARMADAS project, we have shown assembly of high-performance engineered cellular materials using multiple cooperating mobile robotic assemblers. In this paper, we describe how such a programmable meta-material architecture may shift the paradigm of how we design, build, manufacture, and operate future space infrastructure and assets. The core of a programmable meta-material architecture consists of 3 main technology sub-areas: the structure, the assembly agents, and the assembly algorithms. We co-design these systems to ensure an adaptable system that can create and reconfigure structures from a base set of building block components. From this core technology, we can branch out and expand the capability of the system through additional secondary component types and robotic agents to perform activities such as inspections, maintenance, repair, payload installation, or perform power and communications interconnect. As these technologies mature, future designers will be able to utilize the system to rapidly integrate and operate assets in space or on planetary surfaces from a set of well-tested part library, or create their own modules to integrate into the system. A core trait to the development of this system is the automation approach. Because of the modular and functional discrete (pixel-like) nature of the structural system, a diverse set of powerful algorithms for analysis, planning, and simulation can be adapted and leveraged to optimize construction, maintenance, and dynamic reorganization (as hardware with programmable form and function). With an ability to free the design space from launch vehicle constraints and fundamentally shift how a mission is designed and conducted, we discuss the influence of a programmable meta-material architecture on mission design, build, and operations. For the "design phase", we discuss project lifecycle effects, costs, time, and performance. For the "build phase", we discuss reusability, ISRU, manufacturing, material logistics, and scalability. And for "operations", we discuss autonomy, maintenance and upgrades, reliability, and reconfiguration. Autonomy and modularity are the primary enabling traits of this system. Engineering systems that utilize a modular and reconfiguration building block approach such as digital communication and computation systems, currently lead all other areas of technology in size and complexity scalability. NASA is extending the benefits and flexibility of digital systems to hardware systems, to optimize materials lifecycle management and expand our space exploration mission capabilities.

in space assembly↗
DOE OSTI2021

Evaluation of Portable Acceleration Solutions for LArTPC Simulation Using Wire-Cell Toolkit

The Liquid Argon Time Projection Chamber (LArTPC) technology plays an essential role in many current and future neutrino experiments. Accurate and fast simulation is critical to developing efficient analysis algorithms and precise physics model projections. The speed of simulation becomes more important as Deep Learning algorithms are getting more widely used in LArTPC analysis and their training requires a large simulated dataset. Heterogeneous computing is an efficient way to delegate computationally intensive tasks to specialized hardware. However, as the landscape of compute accelerators quickly evolves, it becomes increasingly difficult to manually adapt the code to the latest hardware or software environments. A solution which is portable to multiple hardware architectures without substantially compromising performance would thus be very beneficial, especially for long-term projects such as the LArTPC simulations. In search of a portable, scalable and maintainable software solution for LArTPC simulations, we have started to explore high-level portable programming frameworks that support several hardware backends. In this paper, we present our experience porting the LArTPC simulation code in the Wire-Cell Toolkit to NVIDIA GPUs, first with the CUDA programming model and then with a portable library called Kokkos. Preliminary performance results on NVIDIA V100 GPUs and multi-core CPUs are presented, followed by a discussion of the factors affiecting the performance and plans for future improvements.

Yu, Haiwang↗
DOE OSTI2018

Development and validation of two-phase flow models and critical heat flux prediction for the highly-scalable CFD code NEK-2P

A project is underway to develop, verify and validate an advanced two-phase flow modeling capability for the highly-scalable high-performance CFD code NEK5000[1]. The goal of the project is to develop a new two-phase version of the NEK5000code, named NEK-2P, to simulate the two-phase flow and heat transfer phenomena that occur in a Boiling Water Reactor (BWR) fuel bundle under various operating conditions. The NEK-2P two-phase flow models follow the approach used for the Extended Boiling Framework [2-3] previously developed at Argonne, but include more fundamental physical models of boiling phenomena and advanced numerical algorithms for improved computational accuracy, robustness, and computational speed.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Basic Research Needs in Quantum Computing and Networking

Employing quantum mechanical resources in computing, information processing, and networking opens the door to potential exponential advantages over classical counterparts. However, quantifying and realizing such advantages poses extensive scientific and engineering challenges. Department of Energy (DOE) investments have driven steady progress in addressing such challenges. Recently developed quantum algorithms offer asymptotic exponential advantages in speed or accuracy for fundamental scientific problems. These problems include simulating physical systems, solving systems of linear equations, differential equations, and optimization problems. Empirical demonstrations on nascent quantum hardware suggest better performance on contrived computational tasks than classical analogs. However, the requirements for a quantum computer or network to demonstrate an end-to-end rigorously quantifiable performance improvement over classical analogs remains a grand challenge, especially for problems of practical value. In particular, what will be required for quantum technology to ultimately exhibit scalable, rigorous, and transformative performance advantages for practical applications? In July 2023, DOE’s Advanced Scientific Computing Research program in the Office of Science convened the Workshop on Basic Research Needs in Quantum Computing and Networking, where major opportunities and grand challenges were identified. The following five priority research directions (PRDs) were identified as a result of the workshop.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

A Comprehensive Comparison of Methods for Evaluating Dispatch of Long-Duration Energy Storage in Power Systems Models

Long-duration energy storage (LDES) could play a pivotal role in the transformation of electricity grids with high shares of variable renewable energy (VRE) such as solar and wind. However, the weather-dependent nature of VRE introduces challenges for grid balancing and stability, which LDES - along with short-duration energy storage (SDES) - can help address. However, modeling LDES in production cost models (PCMs) is particularly challenging due to the need for high temporal resolution over extended optimization windows while preserving chronology, which ensures the alignment of energy storage operations with VRE generation over multi-day periods. This report compares traditional dispatch methods with advanced LDES dispatch strategies, such as the extended horizon approach, across different PCM platforms and examines tradeoffs and scalability. The comparison reveals that the traditional 1-day optimization horizon within the PCM leads to inefficient utilization of LDES. In contrast, extending the optimization horizon as much as possible significantly reduces curtailment and improves storage dispatch, especially in renewable-dense systems. There is also promise in using state-of-charge or end volume targets set by an external model, however this requires an additional modeling set and generally increases computational burden. This paper presents a comparison of these various methods in a number of power systems, showing algorithms initially in small test systems and scaling up to large, country-wide simulations. Overall, the research presents the trade-offs of various computational methods and illustrates how LDES may play an essential role in power systems of the future.

14 SOLAR ENERGY↗
NASA NTRS2005

Discovering System Health Anomalies Using Data Mining Techniques

We present a data mining framework for the analysis and discovery of anomalies in high-dimensional time series of sensor measurements that would be found in an Integrated System Health Monitoring system. We specifically treat the problem of discovering anomalous features in the time series that may be indicative of a system anomaly, or in the case of a manned system, an anomaly due to the human. Identification of these anomalies is crucial to building stable, reusable, and cost-efficient systems. The framework consists of an analysis platform and new algorithms that can scale to thousands of sensor streams to discovers temporal anomalies. We discuss the mathematical framework that underlies the system and also describe in detail how this framework is general enough to encompass both discrete and continuous sensor measurements. We also describe a new set of data mining algorithms based on kernel methods and hidden Markov models that allow for the rapid assimilation, analysis, and discovery of system anomalies. We then describe the performance of the system on a real-world problem in the aircraft domain where we analyze the cockpit data from aircraft as well as data from the aircraft propulsion, control, and guidance systems. These data are discrete and continuous sensor measurements and are dealt with seamlessly in order to discover anomalous flights. We conclude with recommendations that describe the tradeoffs in building an integrated scalable platform for robust anomaly detection in ISHM applications.

Sriastava, Ashok, N.↗
NASA NTRS2018

A Blueprint for Demonstrating Quantum Supremacy with Superconducting Qubits

Long coherence times and high fidelity control recently achieved in scalable superconducting circuits paved the way for the growing number of experimental studies of many-qubit quantum coherent phenomena in these devices. Albeit full implementation of quantum error correction and fault tolerant quantum computation remains a challenge the near term pre-error correction devices could allow new fundamental experiments despite inevitable accumulation of errors. One such open question foundational for quantum computing is achieving the so called quantum supremacy, an experimental demonstration of a computational task that takes polynomial time on the quantum computer whereas the best classical algorithm would require exponential time and/or resources. It is possible to formulate such a task for a quantum computer consisting of less than a 100 qubits. The computational task we consider is to provide approximate samples from a non-trivial quantum distribution. This is a generalization for the case of superconducting circuits of ideas behind boson sampling protocol for quantum optics introduced by Arkhipov and Aaronson. In this presentation we discuss a proof-of-principle demonstration of such a sampling task on a 9-qubit chain of superconducting gmon qubits developed by Google. We discuss theoretical analysis of the driven evolution of the device resulting in output approximating samples from a uniform distribution in the Hilbert space, a quantum chaotic state. We analyze quantum chaotic characteristics of the output of the circuit and the time required to generate a sufficiently complex quantum distribution. We demonstrate that the classical simulation of the sampling output requires exponential resources by connecting the task of calculating the output amplitudes to the sign problem of the Quantum Monte Carlo method. We also discuss the detailed theoretical modeling required to achieve high fidelity control and calibration of the multi-qubit unitary evolution in the device. We use a novel cross-entropy statistical metric as a figure of merit to verify the output and calibrate the device controls. Finally, we demonstrate the statistics of the wave function amplitudes generated on the 9-gmon chain and verify the quantum chaotic nature of the generated quantum distribution. This verifies the implementation of the quantum supremacy protocol.

Kechedzhi, Kostyantyn↗
NASA NTRS2019

Prognostics As-A-Service: A Scalable Cloud Architecture for Prognostics

Comprehensive aircraft system health-state awareness is critical for maintaining safe, efficient growth in global operations, enabling higher levels of autonomy, and facilitating new forms of aviation. Maintainers, vehicle operators, air traffic controllers, dispatchers, pilots, autonomous systems, and other decision-makers must have reliable real-time knowledge of the vehicle health, the health of its critical composite systems, predictions of how health changes with time, and forecasts of how its capabilities change with health degradation to preserve safety and efficiency. Providing this information in a reliable manner in computationally constrained environments and across a wide range of vehicles and systems continues to be a challenge. This challenge can be partially resolved through cloud computing, where the execution of prognostic and diagnostic algorithms is performed on a network of remote servers hosted on the internet. NASA is developing a cloud computing service, Prognostics As-A-Service (PaaS), that explores the feasibility and challenges of cloud-enhanced prognostics. Though such a system has broad applicability, this research effort is focused on aviation applications.

Prognostics↗
DOE OSTI2025

Enriched immersed finite element and isogeometric analysis: algorithms and data structures

Immersed finite element methods provide a convenient analysis framework for problems involving geometrically complex domains, such as those found in topology optimization and microstructures for engineered materials. However, their implementation remains a major challenge due to, among other things, the need to apply nontrivial stabilization schemes and generate custom quadrature rules. This article introduces the robust and computationally efficient algorithms and data structures comprising an immersed finite element preprocessing framework. The input to the preprocessor consists of a background mesh and one or more geometries defined on its domain. The output is structured into groups of elements with custom quadrature rules formatted such that common finite element assembly routines may be used without or with only minimal modifications. The key to the preprocessing framework is the construction of material topology information, concurrently with the generation of a quadrature rule, which is then used to perform enrichment and generate stabilization rules. While the algorithmic framework applies to a wide range of immersed finite element methods using different types of meshes, integration, and stabilization schemes, the preprocessor is presented within the context of the extended isogeometric analysis. This method utilizes a structured B-spline mesh, a generalized Heaviside enrichment strategy considering the material layout within individual basis functions’ supports, and face-oriented ghost stabilization. Using a set of examples, the effectiveness of the enrichment and stabilization strategies is demonstrated alongside the preprocessor’s robustness in geometric edge cases. Additionally, the performance and parallel scalability of the implementation are evaluated.

Computer implementation↗
DOE OSTI2019

Development and Validation of a Conjugate Heat Transfer Model for the Two-Phase CFD Code NEK-2P

A project is underway to develop, verify and validate an advanced two-phase flow modeling capability for the highly-scalable, high-performance CFD code NEK5000. The goal of the project is to develop a new two-phase version of the NEK5000 code, named NEK-2P, to simulate the two-phase flow and heat transfer phenomena that occur in a Boiling Water Reactor (BWR) fuel bundle under various operating conditions. The NEK-2P two-phase flow models follow the approach used for the Extended Boiling Framework (EBF) [2-3] previously developed at Argonne, but include more fundamental physical models of boiling phenomena and advanced numerical algorithms for improved computational accuracy, robustness, and computational speed. The development of the NEK-2P two-phase solver and the implementation of the Extended Boiling Framework two-phase models were initially supported by Argonne National Laboratory (Argonne) through a Laboratory Directed Research and Development (LDRD) project during FY2014-2016. The development and validation of the two-phase models through analyses of selected two-phase boiling flow experiments was supported by the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program in FY2017-2019. This report focuses on the FY2019 development and validation of the Conjugate Heat Transfer (CHT) model for the NEK-2P Two-Phase, Computational Fluid Dynamics (CFD) code. The NEK-2P CHT model calculates the coupled behavior of the fluid and solid domains. In the fluid domain NEK-2P calculates the vapor and liquid phase temperatures, velocities and mass fractions. In the solid domain only the temperatures are calculated. The fluid and solid domains are coupled with a heat flux boundary condition at the fluid-solid interface. The heat flux and the wall temperatures at the fluid-solid interface are updated using the NEK-2P wall heat-flux partitioning model. A wall heat flux boundary condition is also applied to the exterior surface of the solid domain. The CHT model validation was illustrated through the analysis of three of the Becker Critical Heat Flux (CHF) tests. Reasonably good agreement with measured data was obtained in predicting the CHF location and post CHF wall temperature behavior illustrating the ability of the NEK-2P CHT model to simulate the CHF phenomena for a wide range of thermal-hydraulic conditions. Apart from the CHT model development and validation, initial simulations were performed for the Virginia Tech. (VT) 3x3 bare rod bundle experiments including both cold and boiling flow simulations.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2019

Overlapping localized exponential time differencing methods for diffusion problems

The localized exponential time differencing (ETD) based on overlapping domain decomposition has been recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in supercomputers. This paper serves as the first step toward building a solid mathematical foundation for this approach. We study the overlapping localized ETD schemes for a model time-dependent diffusion equation discretized in space by the standard central difference. Two methods are proposed and analyzed for solving the fully discrete localized ETD systems: the first one is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second one is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergences of the associated iterative solutions to the corresponding fully discrete localized ETD solution and to the exact semidiscrete solution are rigorously proved. Finally, numerical experiments are also carried out to confirm theoretical results and to compare the performance of the two methods.

58 GEOSCIENCES↗
NASA NTRS2013

ISSM: Ice Sheet System Model

In order to have the capability to use satellite data from its own missions to inform future sea-level rise projections, JPL needed a full-fledged ice-sheet/iceshelf flow model, capable of modeling the mass balance of Antarctica and Greenland into the near future. ISSM was developed with such a goal in mind, as a massively parallelized, multi-purpose finite-element framework dedicated to ice-sheet modeling. ISSM features unstructured meshes (Tria in 2D, and Penta in 3D) along with corresponding finite elements for both types of meshes. Each finite element can carry out diagnostic, prognostic, transient, thermal 3D, surface, and bed slope simulations. Anisotropic meshing enables adaptation of meshes to a certain metric, and the 2D Shelfy-Stream, 3D Blatter/Pattyn, and 3D Full-Stokes formulations capture the bulk of the ice-flow physics. These elements can be coupled together, based on the Arlequin method, so that on a large scale model such as Antarctica, each type of finite element is used in the most efficient manner. For each finite element referenced above, ISSM implements an adjoint. This adjoint can be used to carry out model inversions of unknown model parameters, typically ice rheology and basal drag at the ice/bedrock interface, using a metric such as the observed InSAR surface velocity. This data assimilation capability is crucial to allow spinning up of ice flow models using available satellite data. ISSM relies on the PETSc library for its vectors, matrices, and solvers. This allows ISSM to run efficiently on any parallel platform, whether shared or distrib- ISSM: Ice Sheet System Model NASA's Jet Propulsion Laboratory, Pasadena, California uted. It can run on the largest clusters, and is fully scalable. This allows ISSM to tackle models the size of continents. ISSM is embedded into MATLAB and Python, both open scientific platforms. This improves its outreach within the science community. It is entirely written in C/C++, which gives it flexibility in its design, and the power/speed that C/C++ allows. ISSM is svn (subversion) hosted, on a JPL repository, to facilitate its development and maintenance. ISSM can also model propagation of rifts using contact mechanics and mesh splitting, and can interface to the Dakota software. To carry out sensitivity analysis, mesh partitioning algorithms are available, based on the Scotch, Chaco, and Metis partitioners that ensure equal area mesh partitions can be done, which are then usable for sampling and local reliability methods.

Larour, Eric↗
NASA NTRSDate not supplied

Formalized Reasoning of Operational Volumes for Wildland Fire Fighting

This work is focused on the formalized reasoning of operational volumes as it relates to the current and future technologies developed by NASA to aid in wildfire fighting operations. One such technology is the unmanned aircraft system pilot kit (UASP-kit) developed by the Scalable Traffic Management for Emergency Response Operations (STEReO) project at NASA, which is used to increase situation awareness for a ground operator in the field. The UASP-kit utilizes operational volumes which represent mission areas and alerting volumes, to alert when another aircraft is within one of these volumes from received ADS-B data. This work is focused on developing a rigorous foundation for the concept of operational volumes for modeling and prototyping operations in such a tool as the UASP-kit. This includes establishing a class of algorithms to detect when an object is in an operational volume, and when an operational volume is intersecting or contained within another. Additionally, this work is focused on providing rigorous proof in an interactive theorem prover that the algorithms work as intended. Scenarios are presented that model current UASP-kit operations and extend past the current capabilities of the technology to modeling more complex scenarios such as mission planning.

Operational Volumes↗
DOE OSTI2020

A massively parallel and scalable multi-CPU material point method

Harnessing the power of modern multi-GPU architectures, we present a massively parallel simulation system based on the Material Point Method (MPM) for simulating physical behaviors of materials undergoing complex topological changes, self-collision, and large deformations. Our system makes three critical contributions. First, we introduce a new particle data structure that promotes coalesced memory access patterns on the GPU and eliminates the need for complex atomic operations on the memory hierarchy when writing particle data to the grid. Second, we propose a kernel fusion approach using a new Grid-to-Particles-to-Grid (G2P2G) scheme, which efficiently reduces GPU kernel launches, improves latency, and significantly reduces the amount of global memory needed to store particle data. Finally, we introduce optimized algorithmic designs that allow for efficient sparse grids in a shared memory context, enabling us to best utilize modern multi-GPU computational platforms for hybrid Lagrangian-Eulerian computational patterns. We demonstrate the effectiveness of our method with extensive benchmarks, evaluations, and dynamic simulations with elastoplasticity, granular media, and fluid dynamics. In comparisons against an open-source and heavily optimized CPU-based MPM codebase [Fang et al. 2019] on an elastic sphere colliding scene with particle counts ranging from 5 to 40 million, our GPU MPM achieves over 100x per-time-step speedup on a workstation with an Intel 8086K CPU and a single Quadro P6000 GPU, exposing exciting possibilities for future MPM simulations in computer graphics and computational science. Moreover, compared to the state-of-the-art GPU MPM method [Hu et al. 2019a], we not only achieve 2x acceleration on a single GPU but our kernel fusion strategy and Array-of-Structs-of-Array (AoSoA) data structure design also generalizes to multi-GPU systems. Our multi-GPU MPM exhibits near-perfect weak and strong scaling with 4 GPUs, enabling performant and large-scale simulations on a 10243 grid with close to 100 million particles with less than 4 minutes per frame on a single 4-GPU workstation and 134 million particles with less than 1 minute per frame on an 8-GPU workstation.

Wang, Xinlei↗
DOE OSTI2019

Tail queues: A multi-threaded matching architecture

As we approach exascale, computational parallelism will have to drastically increase in order to meet throughput targets. Many–core architectures have exacerbated this problem by trading reduced clock speeds, core complexity, and computation throughput for increasing parallelism. This presents two major challenges for communication libraries such as MPI: the library must leverage the performance advantages of thread level parallelism and avoid the scalability problems associated with increasing the number of processes to that scale. Hybrid programming models, such as MPI+X, have been proposed to address these challenges. MPI THREAD MULTIPLE is MPI's thread safe mode. While there has been work to optimize it, it largely remains non–performant in most implementations. While current applications avoid MPI multithreading due to performance concerns, it is expected to be utilized in future applications. One of the major synchronous data structures required by MPI is the matching engine. In this paper, we present a parallel matching algorithm that can improve MPI matching for multithreaded applications. We then perform a feasibility study to demonstrate the performance benefit of the technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

NEXT Generation Energy Technologies for Connected and Automated On-Road Vehicles (NEXTCAR Phase I & II)

The Ohio State University’s ARPA-E NEXTCAR project was a multi-phase, multi-year research, development, and demonstration program focused on improving the energy efficiency of connected and automated vehicles (CAVs). The team developed and validated advanced vehicle motion and powertrain control algorithms that coordinate propulsion and automation systems to optimize energy use. Key technologies included Dynamic Skip Fire engine control, predictive eco-driving functions such as Eco-Approach and Departure (Eco-AND) and Eco-Adaptive Cruise Control (Eco-ACC), and powertrain-agnostic optimization frameworks for hybrid, plug-in hybrid, and battery electric vehicles. The project successfully demonstrated up to 30% energy-efficiency improvement during real-world testing at the Transportation Research Center and the American Center for Mobility. The outcomes provide a foundation for scalable, cost-effective deployment of energy-optimized CAV technologies across the automotive industry.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2022

Scalable branching on dual decomposition of stochastic mixed-integer programming problems

In this work, we present a scalable branching method for the dual decomposition of stochastic mixed-integer programming. Our new branching method is based on the branching method proposed by Caroe and Schultz that creates branching disjunctions on first-stage variables only. We propose improvements to the process for creating branching disjunctions, including (1) branching on the optimal solutions of the Dantzig-Wolfe reformulation of the restricted master problem and (2) using a more comprehensive (yet simple) measure for the dispersions associated with subproblem solution infeasibility. We prove that the proposed branching process leads to an algorithm that terminates finitely, and we provide conditions under which globally optimal solutions can be identified after termination. We have implemented our new branching method, as well as the Caroe-Schultz method and a branch-and-price method, in the open-source software package DSP. Using SIPLIB test instances, we present extensive numerical results to demonstrate that the proposed branching method significantly reduces the number of node subproblems and solution times.

97 MATHEMATICS AND COMPUTING↗