Search NASA⌕ Search

SEARCH · Search NASA

Results for “numerical integration”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Numerical integration of stochastic contact Hamiltonian systems via stochastic Herglotz variational principle

Within this work, we establish a stochastic contact variational integrator and its discrete version via stochastic Herglotz variational principle for stochastic contact Hamiltonian systems. A general structure-preserving stochastic contact method is provided to seek the stochastic contact variational integrators. Numerical experiments are performed to verify the validity of this approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

ARPIST: Provably accurate and stable numerical integration over spherical triangles

Numerical integration on spherical triangles, including the computation of their areas, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in computing the Jacobian of the transformation, even for poorly shaped triangles that may occur at poles in regular longitude-latitude meshes, by avoiding potential catastrophic rounding errors. We compare our proposed technique with L’Huilier’s Theorem for computing the area of spherical triangles, and also compare it with the recently developed LSQST method (Beckmann et al., 2014) and a radial-basis-function-based technique (Reeger and Fornberg, 2016) for integration of smooth functions on spherical triangulations. In conclusion, our results show that ARPIST enables better or comparable accuracy over previous methods while being easier to implement, significantly faster, and more tolerant of poor element shapes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Machine Learned Empirical Numerical Integrator from Simulated Data

Recently, a number of state-of-the-art surrogate machine learning (ML) models have been designed for global weather and climate prediction, which have been trained using reanalysis data products. Reanalysis data products are constructed using numerical model simulations that combine numerical integration of partial differential equations and parameterization schemes. These products are typically only archived and made available using coarsened spatial and temporal resolutions. This study explores the impact of the numerical generation methods used to produce the training datasets and the temporal resolution of those datasets on machine learning surrogate models. Using the nonlinear vector autoregression (NVAR) machine as an explainable ML technique, simple dynamical systems are emulated with ML models trained on data produced by three classical numerical integration schemes. NVAR is validated as a skillful ML method, capable of producing accurate predictions and, more importantly, reconstructing both the underlying dynamics and the numerical integration scheme used to generate the training data. However, the machine fails to generalize predictions on unseen test data generated by different numerical integration schemes, despite the underlying dynamical system being the same. This result provides a word of caution for the growing field of machine learning emulation of weather and climate dynamics. Furthermore, we illustrate using NVAR that training on temporally coarsened data may increase the required complexity of ML models and potentially introduce new numerical challenges. Finally, we discover that empirical integration schemes with arbitrary time-stepping sizes can be constructed directly from the data, which implies a potential for the development of empirical numerical integration schemes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Porting numerical integration codes from CUDA to oneAPI: a case study

We present our experience in porting optimized CUDA implementations to oneAPI. We focus on the use case of numerical integration, particularly the CUDA implementations of PAGANI and $m$-Cubes. We faced several challenges that caused performance degradation in the oneAPI ports. These include differences in utilized registers per thread, compiler optimizations, and mappings of CUDA library calls to oneAPI equivalents. After addressing those challenges, we tested both the PAGANI and m-Cubes integrators on numerous integrands of various characteristics. To evaluate the quality of the ports, we collected performance metrics of the CUDA and oneAPI implementations on the Nvidia V100 GPU. We found that the oneAPI ports often achieve comparable performance to the CUDA versions, and that they are at most 10% slower.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

PAGANI: a parallel adaptive GPU algorithm for numerical integration

We present a new adaptive parallel algorithm for the challenging problem of multi-dimensional numerical integration on massively parallel architectures. Adaptive algorithms have demonstrated the best performance, but efficient many-core utilization is difficult to achieve because the adaptive work-load can vary greatly across the integration space and is impossible to predict a priori. Existing parallel algorithms utilize sequential computations on independent processors, which results in bottlenecks due to the need for data redistribution and processor synchronization. Our algorithm employs a high-throughput approach in which all existing sub-regions are processed and sub-divided in parallel. Repeated sub-region classification and filtering improves upon a brute-force approach and allows the algorithm to make efficient use of computation and memory resources. A CUDA implementation shows orders of magnitude speedup over the fastest open-source CPU method and extends the achievable accuracy for difficult integrands. Our algorithm typically outperforms other existing deterministic parallel methods.

Sakiotis, Ioannis↗
DOE OSTI2020

Numerical integrators for Lagrangian oceanography

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

58 GEOSCIENCES↗
DOE OSTI2021

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. Finally, the SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

42 ENGINEERING↗
DOE OSTI2024

Numerical integration in the virtual element method with the scaled boundary cubature scheme

Abstract The virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Chin, Eric B.↗
DOE OSTI2021

Accurate numerical, integral methods for computing drift-kinetic Trubnikov-Rosenbluth potentials

A novel numerical method is employed to compute the integral form of the axi-symmetric Trubnikov-Rosenbluth potentials. Two methods for quadrature in pitch-angle are described and their convergence properties are studied. Careful attention is given to quadrature over a singular Green's function. Here it is shown that an infinite series representation of the Green's function can be used more efficiently than its closed form involving complete elliptic integrals. Then a collocation method in speed, with its associated quadrature scheme, is laid out and its convergence properties are studied. Using the proposed scheme, accurate low-order moments of the field collision operator are obtained using relatively few velocity space degrees of freedom. The scheme is showcased by solving for the equilibrium, axi-symmetric bootstrap current in tokamaks. A C 0 Gauss-Lobatto-Legendre finite element pitch-angle basis with vertex nodes at the trapped/passing boundary is shown, in the context of the integral methods used, to be much more efficient than the more common Legendre polynomial expansion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

An integrated numerical model for coupled poro-hydro-mechanics and fracture propagation using embedded meshes

Integrated models for fluid-driven fracture propagation and general multiphase flow in porous media are valuable to the study and engineering of several systems, including hydraulic fracturing, underground disposal of waste, and geohazard mitigation across such applications. This work extends the coupled model multiphase flow and poromechanical model of Ren et al. (2018) to admit fracture propagation (FP). The coupled XFEM-EDFM scheme utilizes a separate fracture mesh that is embedded on a static background mesh. The onset and dynamics of fracture propagation are governed by the equivalent stress intensity factor (SIF) criterion. A domain-integral method (J integral) is applied to compute this information. Additionally, an adaptive time-marching scheme is proposed to rapidly restrict and grow temporal resolution to match the underlying time-scales. The proposed model is verified with analytical solutions, and shows the capability to accurately and adaptively co-simulate fluid transport and deformation as well as the propagation of multiple fractures.

42 ENGINEERING↗
DOE OSTI2025

Surrogate models for development of unconventional shale reservoirs by an integrated numerical approach of hydraulic fracturing, flow and geomechanics, and machine learning

We develop well-completion surrogate models by taking an integrated workflow of hydraulic fracturing, flow, geomechanics, and machine learning simulation. There are three steps in the proposed workflow. First, history-matching processes are conducted with the field data including pumping and production data for characterization. Second, full-physics simulation is performed with various parameters of the field development (e.g., cluster spacing, clusters per stage, pumping rates and times, amount of proppant, and well spacing) to generate multiple simulation results by changing the parameters of the completion design with well-known hydraulic fracturing, reservoir, geomechanics simulators to calculate fracture geometry, reservoir depressurization, induced stress changes. The workflow is demonstrated over a field in the Southern Midland Basin. Here, we take two completion scenarios: a single well case followed by a multi-well case. Finally, a Long Short-Term Memory (LSTM) machine learning algorithm is employed to create surrogate models that can replicate the full-physics simulation results. Furthermore, results show that the trained models applied in the single well and multi-well cases for a particular geological system can provide good accuracy close to those provided by full-physics simulations. Specifically, the site-specific surrogate models can predict fracture parameters (length, height, and surface area) and cumulative production accurately with computational efficiency, suggesting our proposed workflow can be used as a pragmatic tool for expediting the well completion optimization process.

Geomechanics↗
DOE OSTI2023

Development of Analysis Methods that Integrate Numeric and Textual Equipment Reliability Data

Within the Light Water Reactor Sustainability (LWRS) program, the Risk-Informed Systems Analysis (RISA) Pathway is performing collaborative research on the development and deployment of technologies designed to assist operating nuclear power plants (NPPs) to reduce operating costs improve plant reliability and availability. One of the RISA research areas is focusing on the development of methods and tools designed to optimize plant operations (e.g., maintenance/replacement schedules, optimal maintenance postures for plant structures, systems, and components [SSCs]) in a manner that is more cost effective than current approaches and makes better use of available SSC health data. The Risk-Informed Asset Management (RIAM) project targets this research area by creating a direct bridge between component equipment reliability (ER) data and system engineer decision making regarding maintenance activity scheduling and component aging management. In this respect, one challenge that NPP system engineers are facing is that the amount of ER data being continuously generated is not only extremely large in size, but it comes in different forms: textual (e.g., condition or maintenance reports) and numeric (e.g., generated by monitoring systems). All these data elements provide them with valuable insights and information regarding: 1) the discovery of anomalous behaviors or degradation trends, 2) the identification of the possible causes behind such behaviors/trends, and 3) the prediction of their direct consequences. However, several challenges have proved to be roadblocks to this process. While some of these challenges are technical in nature (i.e., data are often distributed over several physical servers/databases), others are conceptual in nature: data elements come in different formats (e.g., numeric or textual), and measured values have different scales (e.g., vibration spectra and oil temperature). The activities performed by the RIAM project during FY23 directly tackles the need to simultaneously integrate the analysis of ER data in all its forms, numeric and textual. Note that such task has never been performed before due to the complexity of the systems under consideration but, most importantly, because of the technical challenges behind the harmonization of ER data formats and the lack of adequate computational methods to analyze them. Our approach borrows ideas and concepts from the medical field where integration of several data sources is vital to assist medical practitioners to perform correct diagnosis and indicate optimal treatments. In our view a NPP asset is equivalent to a patient in a medical context. The main difference is the complexity of a human body is a magnitude more complex when compared to typical assets commonly present in NPPs (e.g., centrifugal pumps, or motor operated valves). This simplifies our first requirement when analyzing heterogenous ER data formats: to put data into “context”. Context is here intended as the additional piece of information that is needed by ER data analysis tools to understand what these data elements are referring to, i.e., which king of knowledge they are generating. In our context, this knowledge can be translated into models that capture the form and functional architecture of assets/systems, their dependencies, and how they interact. These models actually emulate the knowledge that that NPP system engineers possess about assets and systems; this is their key of success when analyzing ER data, their challenge is ability to handle large amount of data. Here, we employ model-based system engineering (MBSE) models of systems and assets to represent and capture their architecture and functional, i.e. cause-effect, relations. Then, ER data elements are processed by identifying first of all which elements of the developed MBSE elements they are referring to. For numeric ER data this task is fairly easy since it is possible to precisely pinpoint what MBSE elements the corresponding sensor are observing (e.g., bearing temperature of a centrifugal pump). Task is much harder for textual data since the information contained in issue or maintenance reports needs to “be understood” by a computational tool. Here we called this process as “knowledge extraction”. Once again, we borrow the experience in the medical field where methods to extract knowledge from textual data have been developed in the past decade. The missing element for us is the availability of a complete dictionary of NPP related concepts (in addition to the MBSE models presented earlier) that can put “text into context”. In FY23, such dictionary has been developed along with all the computational elements required for knowledge extraction. Lastly, once numeric and textual ER data elements have been processed and “understood”, then the last step is the discovery of possible cause-effect relations among them. This is performed by observing if a logical connection through the MBSE models exists, and if the

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Approximation of periodic Green's operator in real space using numerical integration and its use in fast Fourier transform-based micromechanical models

In this work, we propose an expression for the periodic first derivative of Green's function in real space. The proposed expression allows an alternative way of computing the periodic Green's operator based on periodically summing the free-space Green's operator in terms of an appropriate quadrature rule. We provide computational examples, which show the accuracy of the proposed approach, together with reduced spurious oscillations in the solution fields.

42 ENGINEERING↗
DOE OSTI2021

Efficient numerical integration of thermal interaction rates

In many problems in particle cosmology, interaction rates are dominated by 2 ↔ 2 scatterings, or get a substantial contribution from them, given that 1 ↔ 2 and 1 ↔ 3 reactions are phase-space suppressed. We describe an algorithm to represent, regularize, and evaluate a class of thermal 2 ↔ 2 and 1 ↔ 3 interaction rates for general momenta, masses, chemical potentials, and helicity projections. A key ingredient is an automated inclusion of virtual corrections to 1 ↔ 2 scatterings, which eliminate logarithmic and double-logarithmic IR divergences from the real 2 ↔ 2 and 1 ↔ 3 processes. We also review thermal and chemical potential induced contributions that require resummation if plasma particles are ultrarelativistic.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Error Analysis on Numerical Integration Algorithms in a Hypoelasticity Framework

This report determines local truncation errors for common stress integration algorithms used in explicit finite element codes with hypoelastic material models. The hypoelastic integration algorithms in question utilize an operator splitting procedure in a rotation neutralized configuration, where the stress response is determined from de- coupling the total deformation into rotational and strain dependent components. This document analyzes the error in evolving the stress given a one-step time increment Δt and compares the errors associated with both the rotational and strain components of the operator splitting method. A slight modification to a traditional algorithm is proposed and studied, where the rate of deformation is appropriately rotated from the midstep configuration at t n+1/2 to the end step configuration at t n+1 before the constitutive evaluation. The proposed modification either completely eliminates the error associated with the rotation rate or is of the same order of magnitude as the original algorithm for the three test cases considered in this report. These cases consist of an unaxial stretch with a constant true strain rate with a rigid body rotation, an uniaxial stretch with a constant engineering strain rate with a rigid body rotation, and a simple shear deformation. All three cases are compared to a closed form solution, and in almost every test case the alternative algorithm yields the most accurate one-step local truncation error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Time-Lapse Integration at FWU: Fluids, Rock Physics, Numerical Model Integration, and Field Data Comparison

We present the current status of time-lapse seismic integration at the Farnsworth (FWU) CO2 WAG (water-alternating-gas) EOR (Enhanced Oil Recovery) project at Ochiltree County, northwest Texas. As a potential carbon sequestration mechanism, CO2 WAG projects will be subject to some degree of monitoring and verification, either as a regulatory requirement or to qualify for economic incentives. In order to evaluate the viability of time-lapse seismic as a monitoring method the Southwest Partnership (SWP) has conducted time-lapse seismic monitoring at FWU using the 3D Vertical Seismic Profiling (VSP) method. The efficacy of seismic time-lapse depends on a number of key factors, which vary widely from one application to another. Most important among these are the thermophysical properties of the original fluid in place and the displacing fluid, followed by the petrophysical properties of the rock matrix, which together determine the effective elastic properties of the rock fluid system. We present systematic analysis of fluid thermodynamics and resulting thermophysical properties, petrophysics and rock frame elastic properties, and elastic property modeling through fluid substitution using data collected at FWU. These analyses will be framed in realistic scenarios presented by the FWU CO2 WAG development. The resulting fluid/rock physics models will be applied to output from the calibrated FWU compositional reservoir simulation model to forward model the time-lapse seismic response. Modeled results are compared with field time-lapse seismic measurements and strategies for numerical model feedback/update are discussed. While mechanical effects are neglected in the work presented here, complementary parallel studies are underway in which laboratory measurements are introduced to introduce stress dependence of matrix elastic moduli.

4D↗