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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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

An automated and portable method for selecting an optimal GPU frequency

Power consumption poses a significant challenge in current and emerging graphics processing unit (GPU) enabled high-performance computing systems. In modern GPUs, dynamic voltage frequency scaling (DVFS) appears to be a reliable control to regulate power consumption and performance. However, the DVFS design space is large - hence, brute-force approaches are infeasible to select the optimal frequency. Furthermore, no single frequency can be universally optimal for applications with varying computational intensities. Thus, the application's complexity and the availability of a wide range of frequency settings are a challenge in selecting the optimal frequency configuration for a given GPU workload. To that end, this paper proposes a systematic approach that consists of three steps. The feature characterization study identifies the fine-grain GPU utilization metrics that influence the power consumption and execution time of a given workload. To understand the performance, power, and energy consumption behaviors of a workload across GPU's DVFS design space, we derived analytical power and performance models using the identified fine-grain features. Here, it is shown that the same set of GPU utilization metrics can estimate both the power consumption and execution time while being agnostic of changes to frequency and input sizes. Applying a power control with the single objective of reducing power may cause performance degradation, leading to more energy consumption. A multi-objective approach is proposed to select the optimal GPU DVFS configuration for a workload that reduces power consumption with negligible degradation in performance. The evaluation was conducted using SPEC ACCEL benchmarks and three real applications - NAMD LAMMPS, and LSTM on NVIDIA GV100, GA100, and AMD MI210 GPUs. On average, real applications showed 29.6% energy savings with a performance loss of 5.2% on GA100 and 22.6% energy savings with a performance loss of 4.7% on GV100. Moreover, the proposed models are portable to real applications, GPU architectures, and vendors, and require metric collection at only the default frequency rather than all supported DVFS configurations. Additionally, we conducted a comparison between our models and the GPU assembly instructions (PTX)-based static models. The results revealed a significant reduction in the average error rates, with a decrease from 19.7% to 3.1% for power models and from 29.4% to 5.2% for performance models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Machine learning for seismic low-frequency extrapolation

The cycle-skipping problem that plagues full waveform inversion (FWI) can be at least partially mitigated if low frequencies (which encode the kinematics of wave propagation in seismic data) are recorded. However, seismic sources and receivers are band-limited, so seismic data does not generally include signals down to 0 Hz. To improve our ability to solve the seismic inverse problem, one can synthesize this missing low-frequency (LF) content from the recorded high-frequency (HF) data using machine learning (ML) models. Deep learning models such as convolutional neural networks (CNNs) demonstrate impressive ability to perform low frequency extrapolation. However, such models require powerful hardware (GPU machines) and careful training. We assess the extrapolation capabilities of three different ML models that do not require GPU machines, namely, random forest, Gaussian process regression and gradient boosting, on both synthetic and real data. Experimental results on two synthetic data sets (generated from a low velocity lens embedded in a homogeneous medium, and the Marmousi model) demonstrate that FWI applied to the extrapolated data consistently improves inversion accuracy relative to FWI applied to the original data sets that do not contain low frequencies. Application of low-frequency extrapolation to real data from the Northwest Shelf of Australia demonstrates that tree-based ML models such as gradient boosting can outperform CNNs in terms of both accuracy and computational cost on non-GPU architectures.

58 GEOSCIENCES↗
DOE OSTI2024

Advanced Computing is at the Forefront of a New “Moonshot” Revolutionizing the North American Power Grid

In the 50+ years since the first humans landed on the moon, computing has grown at breakneck speed. We are faced with another challenge that is just as daunting, and just as important to overcome-modernizing the North American electric power grid-and high-performance computing (HPC) systems with specialized software will be an important element in rising to this challenge. We describe at a high level how software developed in the ExaSGD project addresses this "moonshot" goal by utilizing exascale computing and a novel high performance solver software stack to support the mission of decarbonizing power grid operations in an environment of uncertain weather and climate. To reach the exascale benchmark the team has made a number of first-of-their-kind innovations, including novel method for stochastic optimization, fine grained parallel methods for modeling power systems, and GPU resident sparse numerical linear solvers.

17 WIND ENERGY↗
DOE OSTI2026

Toward Energy-Efficient HPC: Insights from Power Profiling a Cloud-Resolving Earth System Model

Power is a fundamental constraint as supercomputing advances to exascale. Efficient operation within strict power budgets requires application-aware power management based on a detailed understanding of application-level power behavior. This work analyzes the Energy Exascale Earth System Model (E3SM) atmosphere component, SCREAM, on Perlmutter (NERSC) and Frontier (OLCF). We characterize power variation across inputs, concurrency levels, and power caps, evaluate the energy impact of code optimizations, and attribute energy within the code using a newly developed GPU energy model. Results show that SCREAM’s peak power remains stable during its core execution phase and decreases gradually as concurrency increases. Power capping experiments reveal a performance–energy "sweet spot". On Perlmutter, limiting GPU power to 50% of thermal design power (TDP) achieves up to 15% energy savings with a 7% performance penalty. On Frontier, a 40% TDP cap yields up to 10% energy savings with less than 10% performance loss. Code optimizations reduce SCREAM energy by shortening run time without increasing power. Modeling reveals a critical insight: data movement accounts for approximately 70% of SCREAM’s GPU energy. This fundamentally shifts the optimization focus from FLOPS to data transfer reduction for this class of applications, offering the most impactful strategy for improving energy efficiency. This work establishes a foundation for practical, application-aware power management at exascale.

Zhao, Zhengji [Lawrence Berkeley National Laborato↗
DOE OSTI2025

Harnessing distributed GPU computing for generalizable graph convolutional networks in power grid reliability assessments

Although machine learning (ML) has emerged as a powerful tool for rapidly assessing grid contingencies, prior studies have largely considered a static grid topology in their analyses. This limits their application, since they need to be re-trained for every new topology. Here, this paper explores the development of generalizable graph convolutional network (GCN) models by pre-training them across a range of grid topologies and contingency types. We found that a GCN model with auto-regressive moving average (ARMA) layers with a line graph representation of the grid offered the best predictive performance in predicting voltage magnitudes (VM) and voltage angles (VA). We introduced the concept of phantom nodes to consider disparate grid topologies with a varying number of nodes and lines. For pre-training the GCN ARMA model across a variety of topologies, distributed graphics processing unit (GPU) computing afforded us significant training scalability. The predictive performance of this model on grid topologies that were part of the training data is substantially better than the direct current (DC) approximation. Although direct application of the pre-trained model to topologies that are not part of the grid is not particularly satisfactory, fine-tuning with small amounts of data from a specific topology of interest significantly improves predictive performance. In general, this paper highlights the feasibility of training large-scale GNN models to assess the reliability of power grids by considering a wide variety of grid topologies and contingency types. With the advent of foundational models in ML and the exponential increase in GPU computing clusters, generalizable ML models will significantly enhance how utilities manage power systems and make decisions in real-time or near-real-time.

24 - POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Toward an AI-Powered Software Pipeline for Real-Time Tracking and Analysis of Wildfire and Smoke

Real-time tracking of wildfires and smoke is crucial for effective response, minimizing damage, protecting lives, and efficiently managing resources during fire emergencies. We develop a web-based AI-powered pipeline that detects wildfires in aerial video and estimates deployment-relevant behavior metrics, including cumulative burned area, burned-area growth rate, fire spread direction, and smoke dispersion. The system combines a YOLO-based detector with YCbCr-based fire segmentation, HSV-based smoke segmentation, Farneback optical flow, and centroid-based spatiotemporal tracking. Using ground sampling distance (GSD), pixel-level fire masks are converted to physical burned-area measurements by correlating fire pixel counts with camera altitude and tilt angle. We benchmark YOLO variants and non-YOLO baselines (GoogLeNet, CNN, DBN, Autoencoder, U-Net, and AlexNet) on the IEEE FLAME dataset and a newly created aerial frame dataset, Wildfire-DB. Cross-dataset evaluation uses a strict threshold-transfer protocol: decision thresholds are selected on FLAME validation and transferred unchanged to Wildfire-DB to quantify generalization under domain shift. YOLOv6 achieves the strongest cross-dataset frame-level fire detection on Wildfire-DB (ROC-AUC 0.8200, PR-AUC 0.8044, and transferred-threshold F1 0.7596). For tracking-oriented deployment requiring oriented localization, YOLO11-OBB provides the most reliable cross-dataset behavior among OBB-capable models while remaining computationally feasible. To analyze the feasibility of UAV deployment, we further measure inference efficiency using synchronized GPU and CPU power logs on a fixed workload of 1569 frames. YOLO-family models process the video in 5.73–12.47 seconds with net energy of 1247.28–1775.39 J, substantially lower latency and energy than heavier classification and reconstruction baselines. Overall, model optimality depends on operational objectives: YOLOv6 is best for cross-dataset detection robustness, whereas YOL...

Color segmentation↗
DOE OSTI2025

FPGA-accelerated SpeckleNN with SNL for real-time X-ray single-particle imaging

We present the implementation of a specialized version of our previously published unified embedding model, SpeckleNN, for real-time speckle pattern classification in X-ray Single-Particle Imaging (SPI), using the SLAC Neural Network Library (SNL) on an FPGA platform. This hardware realization transitions SpeckleNN from a prototypic model into a practical edge solution, optimized for running inference near the detector in high-throughput X-ray free-electron laser (XFEL) facilities, such as those found at the Linac Coherent Light Source (LCLS). To address the resource constraints inherent in FPGAs, we developed a more specialized version of SpeckleNN. The original model, which was designed for broader classification across multiple biological samples, comprised ~5.6 million parameters. The new implementation, while reducing the parameter count to 64.6K (a 98.8% reduction), focuses on maintaining the model's essential functionality for real-time operation, achieving an accuracy of 90%. Furthermore, we compressed the latent space from 128 to 50 dimensions. This implementation was demonstrated on the KCU1500 FPGA board, utilizing 71% of available DSPs, 75% of LUTs, and 48% of FFs, with an average power consumption of 9.4W according to the Vivado post-implementation report. The FPGA performed inference on a single image with a latency of 45.015 microseconds at a 200 MHz clock rate. In comparison, running the same inference on an NVIDIA A100 GPU resulted in an average power consumption of ~73W and an image processing latency of around 400 microseconds. Our FPGA-accelerated version of SpeckleNN demonstrated significant improvements, achieving an 8.9 × speedup and a 7.8 × reduction in power consumption compared to the GPU implementation. Key advancements include model specialization and dynamic weight loading through SNL, which eliminates the need for time-consuming FPGA design re-synthesis, allowing fast and continuous deployment of models (re)trained online. These innovations enable real-time adaptive classification and efficient vetoing of speckle patterns, making SpeckleNN more suited for deployment in XFEL facilities. This implementation has the potential to significantly accelerate SPI experiments and enhance adaptability to evolving experimental conditions.

47 OTHER INSTRUMENTATION↗
DOE OSTI2026

Cooling Matters: Benchmarking Large Language Models and Vision-Language Models on Liquid-Cooled Versus Air-Cooled H100 GPU Systems

The unprecedented growth in artificial intelligence (AI) workloads, recently dominated by large language models (LLMs) and vision-language models (VLMs), has intensified power and cooling demands in data centers. This study benchmarks LLMs and VLMs on two HGX nodes, each with 8× NVIDIA H100 graphics processing units (GPUs), using liquid and air cooling. Leveraging GPU Burn, Weights & Biases, and IPMItool, we collect detailed thermal, power, and computation data. Results show that the liquid-cooled systems maintain GPU temperatures between 41-50$^\circ$C, while the air-cooled counterparts fluctuate between 54-72$^\circ$C under load. This thermal stability of liquid-cooled systems yields 17% higher performance (54 TFLOPs/ GPU vs. 46 TFLOPs/GPU), performance-per-watt, reduced energy overhead, and greater system efficiency than the air-cooled counterparts. These findings underscore the energy and sustainability benefits of liquid cooling, offering a compelling path forward for hyperscale data centers seeking to optimize AI infrastructure. https://github.com/iscaas/Cooling-Matters.

Latif, Imran↗
DOE OSTI2023

CSPlib: A performance portable parallel software toolkit for analyzing complex kinetic mechanisms

Computational singular perturbation (CSP) is a method to analyze dynamical systems. It targets the decoupling of fast and slow dynamics using an alternate linear expansion of the right-hand side of the governing equations based on eigenanalysis of the associated Jacobian matrix. This representation facilitates diagnostic analysis, detection and control of stiffness, and the development of simplified models. For this work, we have implemented CSP in a C++ open-source library CSPlib using the Kokkos parallel programming model to address portability across diverse heterogeneous computing platforms, i.e., multi/many-core CPUs and GPUs. We describe the CSPlib implementation and present its computational performance across different computing platforms using several test problems. Specifically, we test the CSPlib performance for a constant pressure ignition reactor model on different architectures, including IBM Power 9, Intel Xeon Skylake, and NVIDIA V100 GPU. The size of the chemical kinetic mechanism is varied in these tests. As expected, the Jacobian matrix evaluation, the eigensolution of the Jacobian matrix, and matrix inversion are the most expensive computational tasks. When considering the higher throughput characteristic of GPUs, GPUs performs better for small matrices with higher occupancy rate. CPUs gain more advantages from the higher performance of well-tuned and optimized linear algebra libraries such as OpenBLAS.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

RAP: Resource-aware Automated GPU Sharing for Multi-GPU Recommendation Model Training and Input Preprocessing

Ensuring high-quality recommendations for newly onboarded users requires the continuous retraining of Deep Learning Recommendation Models (DLRMs) with freshly generated data. To serve the online DLRM retraining, existing solutions use hundreds of CPU computing nodes designated for input preprocessing, causing significant power consumption that surpasses even the power usage of GPU trainers. To this end, we propose RAP, an end-to-end DLRM training framework that supports Resource-aware Automated GPU sharing for DLRM input Preprocessing and Training. The core idea of RAP is to accurately capture the remaining GPU computing resources during DLRM training for input preprocessing, achieving superior training efficiency without requiring additional resources. Specifically, RAP utilizes a co-running cost model to efficiently assess the costs of various input preprocessing operations, and it implements a resource-aware horizontal fusion technique that adaptively merges smaller kernels according to GPU availability, circumventing any interference with DLRM training. In addition, RAP leverages a heuristic searching algorithm that jointly optimizes both the input preprocessing graph mapping and the co-running schedule to maximize the end-to-end DLRM training throughput. The comprehensive evaluation shows that RAP achieves 78.3× speedup on average over CPU-based DLRM input preprocessing frameworks. In addition, the end-to-end training throughput of RAP is only 2.04% lower than the ideal case, which has no input preprocessing overhead.

Wang, Zheng↗
DOE OSTI2025

$\mathrm{SageNet}$: Fast Neural Network Emulation of the Stiff-amplified Gravitational Waves from Inflation

Accurate modeling of the inflationary gravitational waves (GWs) requires time-consuming, iterative numerical integrations of differential equations to take into account their backreaction on the expansion history. To improve computational efficiency while preserving accuracy, we present the Stiff-amplified Gravitational-wave Emulator Network (SageNet), a deep learning framework designed to replace conventional numerical solvers (code available at https://github.com/YifangLuo/SageNet). SageNet employs a long short-term memory architecture to emulate the present-day energy density spectrum of the inflationary GWs with possible stiff amplification, Ω GW (f). Trained on a data set of 25,689 numerically generated solutions, SageNet allows accurate reconstructions of Ω GW (f) and generalizes well to a wide range of cosmological parameters; 90.9% of the test emulations with randomly distributed parameters exhibit errors of under 4%. In addition, SageNet demonstrates its ability to learn and reproduce the artificial, adaptive sampling patterns in numerical calculations, which implement denser sampling of frequencies around changes in spectral indices in Ω GW (f). The dual capability of learning both physical and artificial features of the numerical GW spectra establishes SageNet as a robust alternative to exact numerical methods. Finally, our benchmark tests show that SageNet reduces the computation time from tens of seconds to milliseconds, achieving a speedup of ∼10 4 times over standard CPU-based numerical solvers with the potential for further acceleration on GPU hardware. These capabilities make SageNet a powerful tool for accelerating Bayesian inference procedures for extended cosmological models. In a broad sense, the SageNet framework offers a fast, accurate, and generalizable solution to modeling cosmological observables whose theoretical predictions demand costly differential equation solvers.

Astronomy data modeling↗
DOE OSTI2024

A GPU accelerated mixed-precision Smoothed Particle Hydrodynamics framework with cell-based relative coordinates

Smoothed Particle Hydrodynamics (SPH) is essential for modeling complex large-deformation problems across various applications, requiring significant computational power. A major portion of SPH computation time is dedicated to the Nearest Neighboring Particle Search (NNPS) process. While advanced NNPS algorithms have been developed to enhance SPH efficiency, the potential efficiency gains from modern computation hardware remain underexplored. Here, this study investigates the impact of GPU parallel architecture, low-precision computing on GPUs, and GPU memory management on NNPS efficiency. Our approach employs a GPU-accelerated mixed-precision SPH framework, utilizing low precision float-point 16 (FP16) for NNPS while maintaining high precision for other components. To ensure FP16 accuracy in NNPS, we introduce a Relative Coordinated-based Link List (RCLL) algorithm, storing FP16 relative coordinates of particles within background cells. Our testing results show three significant speedup rounds for CPU-based NNPS algorithms. The first comes from parallel GPU computations, with up to a 1000x efficiency gain. The second is achieved through low-precision GPU computing, where the proposed FP16-based RCLL algorithm offers a 1.5x efficiency improvement over the FP64-based approach on GPUs. By optimizing GPU memory bandwidth utilization, the efficiency of the FP16 RCLL algorithm can be further boosted by 2.7x, as demonstrated in an example with 1 million particles. Our code is released at https://github.com/pnnl/lpNNPS4SPH.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

AI-accelerated protein-ligand docking for SARS-CoV-2 is 100-fold faster with no significant change in detection

Protein-ligand docking is a computational method for identifying drug leads. The method is capable of narrowing a vast library of compounds down to a tractable size for downstream simulation or experimental testing and is widely used in drug discovery. While there has been progress in accelerating scoring of compounds with artificial intelligence, few works have bridged these successes back to the virtual screening community in terms of utility and forward-looking development. We demonstrate the power of high-speed ML models by scoring 1 billion molecules in under a day (50 k predictions per GPU seconds). We showcase a workflow for docking utilizing surrogate AI-based models as a pre-filter to a standard docking workflow. Our workflow is ten times faster at screening a library of compounds than the standard technique, with an error rate less than 0.01% of detecting the underlying best scoring 0.1% of compounds. Our analysis of the speedup explains that another order of magnitude speedup must come from model accuracy rather than computing speed. In order to drive another order of magnitude of acceleration, we share a benchmark dataset consisting of 200 million 3D complex structures and 2D structure scores across a consistent set of 13 million “in-stock” molecules over 15 receptors, or binding sites, across the SARS-CoV-2 proteome. We believe this is strong evidence for the community to begin focusing on improving the accuracy of surrogate models to improve the ability to screen massive compound libraries 100 × or even 1000 × faster than current techniques and reduce missing top hits. The technique outlined aims to be a fast drop-in replacement for docking for screening billion-scale molecular libraries.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS1979

Initial comparison of single cylinder Stirling engine computer model predictions with test results

A NASA developed digital computer code for a Stirling engine, modelling the performance of a single cylinder rhombic drive ground performance unit (GPU), is presented and its predictions are compared to test results. The GPU engine incorporates eight regenerator/cooler units and the engine working space is modelled by thirteen control volumes. The model calculates indicated power and efficiency for a given engine speed, mean pressure, heater and expansion space metal temperatures and cooler water inlet temperature and flow rate. Comparison of predicted and observed powers implies that the reference pressure drop calculations underestimate actual pressure drop, possibly due to oil contamination in the regenerator/cooler units, methane contamination in the working gas or the underestimation of mechanical loss. For a working gas of hydrogen, the predicted values of brake power are from 0 to 6% higher than experimental values, and brake efficiency is 6 to 16% higher, while for helium the predicted brake power and efficiency are 2 to 15% higher than the experimental.

Tew, R. C., Jr.↗
DOE OSTI2025

GPU-friendly surface model for Monte-Carlo detector simulations

The demands for Monte-Carlo simulation are drastically increasing with the Large Hadron Collider’s high-luminosity upgrade, and are expected to exceed the currently available compute resources. At the same time, modern high-performance computing has adopted powerful hardware accelerators, particularly GPUs. The AdePT and Celeritas projects aim to address the demanding computational needs by leveraging these heterogeneous computing architectures. While both have successfully ported realistic detector simulations to GPUs using the VecGeom library, the complexity of geometry modeling emerged as a bottleneck. Thread divergence and high register usage were degrading the GPU performance. Therefore, a new, GPU-friendly surface-based model has been introduced in the VecGeom library that decomposes the divergent code of the 3D primitive solids into simpler and more balanced surface algorithms. In this work, we present the latest developments, focusing on the additions required to efficiently model complex setups like the CMS Phase-2 geometry. This includes memory reduction techniques, and adding accelerating structures for faster traversal.

Diederichs, Severin [CERN]↗
DOE OSTI2024

Early experiences evaluating the HPE/Cray ecosystem for AMD GPUs

Summary The Oak Ridge Leadership Computing Facility (OLCF) has a long history of supporting and promoting GPU‐accelerated computing starting with the deployment of the Titan supercomputer in 2021 and continuing with the Summit supercomputer which has a theoretical peak performance of approximately 200 petaflops. Because the majority of Summit's computational power comes from its 27,972 GPUs, users must port their applications to one of the supported programming models in order to make efficient use of the system. To prepare the transition to Frontier, the OLCF's exascale supercomputer, users will need to adapt to an entirely new ecosystem which will include new hardware and software technologies. First, users will need to familiarize themselves with the AMD Radeon GPU architecture. Furthermore, users who have been previously relying on CUDA will need to transition to the Heterogeneous‐Computing Interface for Portability (HIP) or one of the other supported programming models (e.g., OpenMP, OpenACC). In this work, we describe our initial experiences and lessons learned in porting three applications or proxy apps currently running on Summit to the HPE/Cray ecosystem to leverage the compute power from AMD GPUs: minisweep, GenASiS, and Sparkler. Each one is representative of current production workloads utilized at the OLCF, different programming languages, and different programming models.

Melesse Vergara, Verónica G.↗
DOE OSTI2023

Accelerating x-ray tracing for exascale systems using Kokkos

The upcoming exascale computing systems Frontier and Aurora will draw much of their computing power from GPU accelerators. The hardware for these systems will be provided by AMD and Intel, respectively, each supporting their own GPU programming model. The challenge for applications that harness one of these exascale systems will be to avoid lock-in and to preserve performance portability. We report here on our results of using Kokkos to accelerate a real-world application on NERSC's Perlmutter Phase 1 (using NVIDIA A100 accelerators) and Crusher, the testbed system for OLCF's Frontier (using AMD MI250X). By porting to Kokkos, we successfully ran the same X-ray tracing code on both systems and achieved speed-ups between 13 % and 66 % compared to the original CUDA code. Finally, these results are a highly encouraging demonstration of using Kokkos to accelerate production science code.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Data-scarce surrogate modeling of shock-induced pore collapse process

Understanding the mechanisms of shock-induced pore collapse is of great interest in various disciplines in sciences and engineering, including materials science, biological sciences, and geophysics. However, numerical modeling of the complex pore collapse processes can be costly. To this end, a strong need exists to develop surrogate models for generating economic predictions of pore collapse processes. Here, in this work, we study the use of a data-driven reduced-order model, namely dynamic mode decomposition, and a deep generative model, namely conditional generative adversarial networks, to resemble the numerical simulations of the pore collapse process at representative training shock pressures. Since the simulations are expensive, the training data are scarce, which makes training an accurate surrogate model challenging. To overcome the difficulties posed by the complex physics phenomena, we make several crucial treatments to the plain original form of the methods to increase the capability of approximating and predicting the dynamics. In particular, physics information is used as indicators or conditional inputs to guide the prediction. In realizing these methods, the training of each dynamic mode composition model takes only around 30 s on CPU. In contrast, training a generative adversarial network model takes 8 h on GPU. Moreover, using dynamic mode decomposition, the final-time relative error is around 0.3% in the reproductive cases. We also demonstrate the predictive power of the methods at unseen testing shock pressures, where the error ranges from 1.3 to 5% in the interpolatory cases and 8 to 9% in extrapolatory cases.

97 MATHEMATICS AND COMPUTING↗