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NASA NTRS2005

Small-Scale Drop Size Variability: Impact on Estimation of Cloud Optical Properties

Most cloud radiation models and conventional data processing techniques assume that the mean number of drops of a given radius is proportional to volume. The analysis of microphysical data on liquid water drop sizes shows that, for sufficiently small volumes, this proportionality breaks down; the number of cloud drops of a given radius is instead proportional to the volume raised to a drop size-dependent nonunit power. The coefficient of proportionality, a generalized drop concentration, is a function of the drop size. For abundant small drops the power is unity as assumed in the conventional approach. However, for rarer large drops, it falls increasingly below unity. This empirical fact leads to drop clustering, with the larger drops exhibiting a greater degree of clustering. The generalized drop concentration shows the mean number of drops per cluster, while the power characterizes the occurrence frequency of clusters. With a fixed total number of drops in a cloud, a decrease in frequency of clusters is accompanied by a corresponding increase in the generalized concentration. This initiates a competing process missed in the conventional models: an increase in the number of drops per cluster enhances the impact of rarer large drops on cloud radiation while a decrease in the frequency suppresses it. Because of the nonlinear relationship between the number of clustered drops and the volume, these two opposite tendencies do not necessarily compensate each other. The data analysis suggests that clustered drops likely have a stronger radiative impact compared to their unclustered counterpart; ignoring it results in underestimation of the contribution from large drops to cloud horizontal optical path.

Knyazikhin, Y.↗

NASA NTRS2009

Skill Assessment of a Spectral Ocean-Atmosphere Radiative Model

Ocean phytoplankton, detrital material, and water absorb and scatter light spectrally. The Ocean- Atmosphere Spectral Irradiance Model (OASIM) is intended to provide surface irradiance over the oceans with sufficient spectral resolution to support ocean ecology, biogeochemistry, and heat exchange investigations, and of sufficient duration to support inter-annual and decadal investigations. OASIM total surface irradiance (integrated 200 nm to 4 microns) was compared to in situ data and three publicly available global data products at monthly 1-degree resolution. OASIM spectrally-integrated surface irradiance had root mean square (RMS) difference= 20.1 W/sq m (about 11%), bias=1.6 W/sq m (about 0.8%), regression slope= 1.01 and correlation coefficient= 0.89, when compared to 2322 in situ observations. OASIM had the lowest bias of any of the global data products evaluated (ISCCP-FD, NCEP, and ISLSCP 11), and the best slope (nearest to unity). It had the second best RMS, and the third best correlation coefficient. OASIM total surface irradiance compared well with ISCCP-FD (RMS= 20.7 W/sq m; bias=-11.4 W/sq m, r=0.98) and ISLSCP II (RMS =25.2 W/sq m; bias= -13.8 W/sq m; r=0.97), but less well with NCEP (RMS =43.0 W/sq m ;bias=-22.6 W/sq m; x=0.91). Comparisons of OASIM photosynthetically available radiation (PAR) with PAR derived from SeaWiFS showed low bias (-1.8 mol photons /sq m/d, or about 5%), RMS (4.25 mol photons /sq m/d ' or about 12%), near unity slope (1.03) and high correlation coefficient (0.97). Coupled with previous estimates of clear sky spectral irradiance in OASIM (6.6% RMS at 1 nm resolution), these results suggest that OASIM provides reasonable estimates of surface broadband and spectral irradiance in the oceans, and can support studies on ocean ecosystems, carbon cycling, and heat exchange.

Gregg, Watson, W.↗

NASA NTRS2012

Does Ocean Color Data Assimilation Improve Estimates of Global Ocean Inorganic Carbon?

Ocean color data assimilation has been shown to dramatically improve chlorophyll abundances and distributions globally and regionally in the oceans. Chlorophyll is a proxy for phytoplankton biomass (which is explicitly defined in a model), and is related to the inorganic carbon cycle through the interactions of the organic carbon (particulate and dissolved) and through primary production where inorganic carbon is directly taken out of the system. Does ocean color data assimilation, whose effects on estimates of chlorophyll are demonstrable, trickle through the simulated ocean carbon system to produce improved estimates of inorganic carbon? Our emphasis here is dissolved inorganic carbon, pC02, and the air-sea flux. We use a sequential data assimilation method that assimilates chlorophyll directly and indirectly changes nutrient concentrations in a multi-variate approach. The results are decidedly mixed. Dissolved organic carbon estimates from the assimilation model are not meaningfully different from free-run, or unassimilated results, and comparisons with in situ data are similar. pC02 estimates are generally worse after data assimilation, with global estimates diverging 6.4% from in situ data, while free-run estimates are only 4.7% higher. Basin correlations are, however, slightly improved: r increase from 0.78 to 0.79, and slope closer to unity at 0.94 compared to 0.86. In contrast, air-sea flux of C02 is noticeably improved after data assimilation. Global differences decline from -0.635 mol/m2/y (stronger model sink from the atmosphere) to -0.202 mol/m2/y. Basin correlations are slightly improved from r=O.77 to r=0.78, with slope closer to unity (from 0.93 to 0.99). The Equatorial Atlantic appears as a slight sink in the free-run, but is correctly represented as a moderate source in the assimilation model. However, the assimilation model shows the Antarctic to be a source, rather than a modest sink and the North Indian basin is represented incorrectly as a sink rather than the source indicated by the free-run model and data estimates.

Gregg, Watson↗

NASA NTRS2012

On Spectral Invariance of Single Scattering Albedo for Weakly Absorbing Wavelengths

The single scattering albedo omega (sub 0 lambda) in atmospheric radiative transfer is the ratio of the scattering coefficient to the total extinction coefficient. For cloud water droplets both the scattering and absorption coefficients, thus the single scattering albedo, are functions of wavelength A and droplet size r. In this presentation we will show that for water droplets at weakly absorbing wavelengths, the ratio omega (sub 0 lambda)(r). The slope and intercept of the linear function are wavelength independent and sum to unity. This relationship allows for a representation of any single scattering albedo omega (sub 0 lambda) via one known spectrum omega (sub 0 lambda)(r(sub o)). We will provide a simple physical explanation of the discovered relationship. In addition to water droplets, similar linear relationships were found for the single scattering albedo of non-spherical ice crystals. The single scattering albedo $\omega _ {0\lambda }$ in atmospheric radiative transfer is the ratio of the scattering coefficient to the total extinction coefficient. For cloud water droplets both the scattering and absorption coefficients, and thus the single scattering albedo, are functions of wavelength $\lambda $ and droplet size $r$. We show that for water droplets at weakly absorbing wavelengths, the ratio $\omega _ {0\lambda } (r)$/$\omega _ {0\lambda } (r_{0})$ of two single scattering albedo spectra for two different droplet sizes is a linear function of $\omega _{0\lambda }(r)$. The slope and intercept of the linear function are wavelength independent and sum to unity. This relationship allows for a representation of any single scattering albedo $\omega_{0\lambda }(r)$ via one known spectrum $\omega_{0\lambda }(r_{0})$. We provide a simple physical explanation of the discovered relationship. Similar linear relationships characterize the single scattering albedo of non-spherical ice crystals.

Marshak, Alexander↗

NASA NTRS2012

A Search Model for Imperfectly Detected Targets

Under the assumptions that 1) the search region can be divided up into N non-overlapping sub-regions that are searched sequentially, 2) the probability of detection is unity if a sub-region is selected, and 3) no information is available to guide the search, there are two extreme case models. The search can be done perfectly, leading to a uniform distribution over the number of searches required, or the search can be done with no memory, leading to a geometric distribution for the number of searches required with a success probability of 1/N. If the probability of detection P is less than unity, but the search is done otherwise perfectly, the searcher will have to search the N regions repeatedly until detection occurs. The number of searches is thus the sum two random variables. One is N times the number of full searches (a geometric distribution with success probability P) and the other is the uniform distribution over the integers 1 to N. The first three moments of this distribution were computed, giving the mean, standard deviation, and the kurtosis of the distribution as a function of the two parameters. The model was fit to the data presented last year (Ahumada, Billington, & Kaiwi, 2 required to find a single pixel target on a simulated horizon. The model gave a good fit to the three moments for all three observers.

Ahumada, Albert↗

NASA NTRS1992

Effect of Index of Refraction on Radiation Characteristics in a Heated Absorbing, Emitting, and Scattering Layer

The index of refraction can considerably influence the temperature distribution and radiative heat flow in semitransparent materials such as some ceramics. For external radiant heating, the refractive index influences the amount of energy transmitted into the interior of the material. Emission within a material depends on the square of its refractive index, and hence this emission can be many times that for a biackbody radiating into a vacuum. Since radiation exiting through an interface into a vacuum cannot exceed that of a blackbody, there is extensive reflection at the internal surface of an interface, mostly by total internal reflection. This redistributes energy within the layer and tends to make its temperature distribution more uniform. The purpose of the present analysis is to show that, for radiative equilibrium in a gray layer with diffuse interfaces, the temperature distribution and radiative heat flux for any index of refraction can be obtained very simply from the results for an index of refraction of unity. For the situation studied here, the layer is subjected to external radiative heating incident on each of its surfaces. The material emits, absorbs, and isotropically scatters radiation. For simplicity the index of refraction is unity in the medium surrounding the layer. The surfaces of the layer are assumed diffuse. This is probably a reasonable approximation for a ceramic layer that has not been polished. When transmitted radiation or radiation emitted from the interior reaches the inner surface of an interface, the radiation is diffused and some of it thereby placed into angular directions for which there is total internal reflection. This provides a trapping effect for retaining energy within the layer and tends to equalize its temperature distribution. An analysis of temperature distributions in absorbing-emitting layers, including index of refraction effects, was developed by Gardon (1958) to predict cooling and heat treating of glass plates. The interfaces were optically smooth; the resulting specular reflections were computed from the Fresnel reflection laws. This provides a somewhat different behavior than for diffuse interfaces. A similar application was for heating that occurs in a window of a re-entry vehicle (Fowle et al., 1969). A number of recent papers (Rokhsaz and Dougherty, 1989; Ping and Lallemand, 1989; Crosbie and Shieh, 1990) further examined the effects of Fresnel boundary reflections and nonunity refractive index. Other examples of analyses of both steady and transient heat transfer to single or multiple plane layers (Amlin and Korpela, 1979; Tarshis et al., 1969) have used diffuse assumptions at the interfaces as in the present study

Siegel, R.↗

NASA NTRS1960

Kinetic Parameters for the Hydrogen and Ethylene Flames from Flashback Measurements

Values for the flame activation energy and an overall reaction order have been obtained for premixed hydrogen-oxygen and ethylene-oxygen flames with nitrogen and with argon as diluents; the values come from measurements of the change with pressure and flame temperature of the critical boundary velocity gradient for flashback. Measurements have been made for rich and stoichiometric hydrogen flames and for lean and stoichiometric ethylene flames. For hydrogen flames with nitrogen diluent an overall order of 2*3 is found; for ethylene flames with nitrogen diluent, the order obtained is 1*8. With argon diluent, values for the order are about 10 per cent lower for each flame. With nitrogen diluent the flame activation energy apparently increases with flame temperature; this increase is not found with argon diluent. For hydrocarbon flames the assumption that the initial equivalence ratio equals the effective equivalence ratio in the flame leads to an order of reaction with respect to oxygen that is considerably larger than unity. For hydrogen flames, that assumption is consistent with an oxygen order of unity.

Fine, B.↗

NASA NTRS2016

Soil Moisture Active Passive Mission L4_SM Data Product Assessment (Version 2 Validated Release)

During the post-launch SMAP calibration and validation (Cal/Val) phase there are two objectives for each science data product team: 1) calibrate, verify, and improve the performance of the science algorithm, and 2) validate the accuracy of the science data product as specified in the science requirements and according to the Cal/Val schedule. This report provides an assessment of the SMAP Level 4 Surface and Root Zone Soil Moisture Passive (L4_SM) product specifically for the product's public Version 2 validated release scheduled for 29 April 2016. The assessment of the Version 2 L4_SM data product includes comparisons of SMAP L4_SM soil moisture estimates with in situ soil moisture observations from core validation sites and sparse networks. The assessment further includes a global evaluation of the internal diagnostics from the ensemble-based data assimilation system that is used to generate the L4_SM product. This evaluation focuses on the statistics of the observation-minus-forecast (O-F) residuals and the analysis increments. Together, the core validation site comparisons and the statistics of the assimilation diagnostics are considered primary validation methodologies for the L4_SM product. Comparisons against in situ measurements from regional-scale sparse networks are considered a secondary validation methodology because such in situ measurements are subject to up-scaling errors from the point-scale to the grid cell scale of the data product. Based on the limited set of core validation sites, the wide geographic range of the sparse network sites, and the global assessment of the assimilation diagnostics, the assessment presented here meets the criteria established by the Committee on Earth Observing Satellites for Stage 2 validation and supports the validated release of the data. An analysis of the time average surface and root zone soil moisture shows that the global pattern of arid and humid regions are captured by the L4_SM estimates. Results from the core validation site comparisons indicate that "Version 2" of the L4_SM data product meets the self-imposed L4_SM accuracy requirement, which is formulated in terms of the ubRMSE: the RMSE (Root Mean Square Error) after removal of the long-term mean difference. The overall ubRMSE of the 3-hourly L4_SM surface soil moisture at the 9 km scale is 0.035 cubic meters per cubic meter requirement. The corresponding ubRMSE for L4_SM root zone soil moisture is 0.024 cubic meters per cubic meter requirement. Both of these metrics are comfortably below the 0.04 cubic meters per cubic meter requirement. The L4_SM estimates are an improvement over estimates from a model-only SMAP Nature Run version 4 (NRv4), which demonstrates the beneficial impact of the SMAP brightness temperature data. L4_SM surface soil moisture estimates are consistently more skillful than NRv4 estimates, although not by a statistically significant margin. The lack of statistical significance is not surprising given the limited data record available to date. Root zone soil moisture estimates from L4_SM and NRv4 have similar skill. Results from comparisons of the L4_SM product to in situ measurements from nearly 400 sparse network sites corroborate the core validation site results. The instantaneous soil moisture and soil temperature analysis increments are within a reasonable range and result in spatially smooth soil moisture analyses. The O-F residuals exhibit only small biases on the order of 1-3 degrees Kelvin between the (re-scaled) SMAP brightness temperature observations and the L4_SM model forecast, which indicates that the assimilation system is largely unbiased. The spatially averaged time series standard deviation of the O-F residuals is 5.9 degrees Kelvin, which reduces to 4.0 degrees Kelvin for the observation-minus-analysis (O-A) residuals, reflecting the impact of the SMAP observations on the L4_SM system. Averaged globally, the time series standard deviation of the normalized O-F residuals is close to unity, which would suggest that the magnitude of the modeled errors approximately reflects that of the actual errors. The assessment report also notes several limitations of the "Version 2" L4_SM data product and science algorithm calibration that will be addressed in future releases. Regionally, the time series standard deviation of the normalized O-F residuals deviates considerably from unity, which indicates that the L4_SM assimilation algorithm either over- or under-estimates the actual errors that are present in the system. Planned improvements include revised land model parameters, revised error parameters for the land model and the assimilated SMAP observations, and revised surface meteorological forcing data for the operational period and underlying climatological data. Moreover, a refined analysis of the impact of SMAP observations will be facilitated by the construction of additional variants of the model-only reference data. Nevertheless, the “Version 2” validated release of the L4_SM product is sufficiently mature and of adequate quality for distribution to and use by the larger science and application communities.

SMAP L4_SM↗

NASA NTRS2018

Rapid Model Import Tool (RMIT)

Our project is about developing a tool to implement conversion of 3D Computer Aided Design (CAD) models produced with software such as Delmia, 3DS Max, or Maya, into a size and format compatible with the Unity 3D environment. RMIT will be used to aid KSC engineering personnel in the design, development, testing, operations, and training on spacecraft, launch vehicles, facilities, and ground support equipment. For our project, we are using Blender, a free/open-source 3D graphics software, with the goal of developing, testing, and deploying a 3D CAD model converter tool. I worked on using Blender to import 3D CAD models exported from CATIA software into Collada file format. The Collada file format has file extension DAE. Importing the Collada DAE file as is into Blender, generates dots and dashes. In 3DS Max, there is an existing OpenCollada plugin and with that plugin, 3DS Max can import the DAE file successfully. But, Blender does not seem to have an OpenCollada plugin, so I worked on writing a new OpenCollada plugin for Blender. Since 3DS Max was able to display the image, I looked into comparing differences between the original DAE file and the DAE file exported from 3DS Max using the OpenCollada plugin. As Collada documents describing digital assets are XML files with file extension DAE, Collada files contains XML tags, making them easily modifiable. After some research, it appears that Blender does not like primitive 2D tags like tristrips and trifans. Changing those tags to polygons slightly improved the image, but the pieces were exploded. I found after further research comparing differences between the original file and the file exported from 3DS Max that the values inside the translate tags in the original file are scaled down by a factor of 25.4 in the exported file from 3DS Max, representing the millimeters to inches conversion (1 inch = 25.4 millimeters). After scaling down values inside all of the translate tags by 25.4, the exploded pieces stuck back in, but the image needed further improvement. I have been able to create a new plugin in Blender that takes the original DAE file, replaces the primitive 2D tags tristrips and trifans with polygons, scales down the values inside the translate tags by a factor of 25.4, and saves the changes into a temporary DAE file. After the temporary DAE file is imported into Blender, the temp file is then deleted, keeping the original DAE file intact. Starting with a DAE file that is exported using the NASA Enterprise Visualization Application (NEVA), a Collada exporter, from CATIA gives better results. NEVA is a Design Visualization product that is used for exporting 3D models from CATIA. With NEVA, the up axis is defined in the top-level node if navigation gravity is enabled. With that file, just replacing the primitive tags tristrips and trifans with polygons in yields a much improved image in Blender. As we identify more differences between the original DAE file and the DAE file exported from 3DS Max, this plugin can be improved further. Our goal is to have a model that is formatted and sized for import into Unity, and we are trying out different 3D programs to see which will work best.

Ayyangar, Arjun↗

NASA NTRS2020

RMIT AR Android Application Abstract

This summer, I've worked as an intern for IT-C1's Rapid Model Import Tool (RMIT) project under Bill Little. The goal of this project was to develop a mobile application for IT-C1's larger project, RMIT, so that it could be deployed in a convenient way. The current function of RMIT is to import file types that are not normally compatible with the Unity 3D software development engine and make them into files that can be placed into Unity for the specific purpose of viewing CAD models in virtual reality (VR). The mobile application that I worked on uses augmented reality (AR), using the camera on a mobile device, to show virtual models in the real world for the purpose of visualization and understanding. In addition to single-player capabilities, this application will include a multi-player mode so that collaborating engineers can see each other's work on their own devices.Working on this application required many steps. First, I hard to learn a lot about how mobile development is different from desktop development, and what the limitations of augmented reality are at its current state. From there, I was assigned new tasks and goals every two weeks, which included addingground plane and air plane functionality, designing a user interface, adding multi-player capability and allowing the user to manipulate models placed in a scene.

Dorian Thompson↗

NASA NTRS2021

Physical Characteristics of Unstructured Coronal Clouds

Active regions in the inner solar corona, when observed in X-ray emission, consist of bright, hot loops surrounded by unstructured clouds. The emission from the clouds extends to a height of≈4–5×104km at temperatures of ≈2–3 MK. These “hot clouds” are variable, but persist for many days and do not appear to connect directly to the active region streamers or other large-scale structures observed higher in the corona. We present an observational analysis of these diffuse structures to establish basic plasma parameters such as magnetic field strength, particle density, and temperature. The values ofβ, the ratio of the plasma pressure to the magnetic field pressure, were found to be generally less than unity, though often approaching unity in the upper portions of the active region, where the hot clouds are located. The magnetic field may therefore only partially confine these regions and inhibit flare-like instabilities that could otherwise be driven by gradients of plasma pressure and current density.

The Sun↗

DOE OSTI2020

Virtual/Augmented Reality (VR/AR):Tools for Nuclear Training, Operations and Design

Introduction: Problems: Mistakes in the field due to a lack of information (Ex: Pump repair failure rate 50%); Avoidable exposure due to multiple employees for procedure adherence and monitoring; Debates over 2D drawings can be more easily resolved. Purpose: Training: Employees can be trained more effectively. Operating: Operations can be carried out more efficiently with fewer workers required with AR. Design Review: Pull the designers and designs into VR. Design: VR: Unreal Engine, Unity, Custom. AR: HoloLens: Unity, Custom - DirectX 12.

42 ENGINEERING↗

DOE OSTI2024

Measurement of T-Dependence of the Beam Asymmetry (S) for the Reaction yp --> n'p AT GLUEX

We report on a measurement of the four-momentum transfer (-t) dependence of the Beam Asymmetry (Sigma) for the reaction yp --> n'p with the linearly polarized photon beam at GlueX. The observable is sensitive to the n' production mechanism. The Beam Asymmetry (Sigma) was extracted from azimuthal angular distributions (phi) combined to obtain yield asymmetries. This technique removes most contributions from instrumental asymmetries. These data are based on a data set that is five times more extensive than previous measurements. This allowed us to double the kinematic reach in the four-momentum transfer squared (-t) and reduce the statistical uncertainties. Up to a four-momentum transfer squared (-t) of 1.7 GeV2, the beam asymmetry is positive, close to unity, and shows little dependence on the four-momentum transfer -t. This indicates that the production process continues to be dominated by exchanges of the rho (p) and the omega (w) mesons. Further systematic cross-check was done by analyzing the decay mode nr^0r^0. This analysis also revealed positive asymmetries closer to unity providing an important systematic cross-check for the results obtained using n'-->nr+r-.

Paudel, Churamani↗

DOE OSTI2025

Immersive Scientific Visualization of Molten-Salt Reactor Waste Characteristics Using Virtual Reality

Immersive visualization is changing how we explore, communicate, and understand complex scientific systems. In nuclear energy, an area in which data are often multidimensional, time-dependent, and difficult to interpret, virtual reality (VR) represents a powerful and intuitive informational medium. This work introduces a VR-based platform that visualizes the post-shutdown behavior and waste management lifecycle of molten-salt reactors (MSRs), a next-generation reactor type with unique operational and safety characteristics. The platform, built in Unity, is streamed on the Meta Quest 3 headset. It transforms high-fidelity simulation data into an interactive, immersive experience. Users can explore time-dependent reactor characteristics such as nuclide decay, which is a key factor for evaluating reactor waste strategies. The datasets were generated using the MOOSE (Multiphysics Object-Oriented Simulation Environment) framework and then processed through ParaView scripting for smooth integration into Unity. From a visualization standpoint, the platform emphasizes spatial storytelling, temporal exploration, and user-centered interaction. Users can navigate 3D reactor geometries, slice through volumetric data, and manipulate time to observe how physical phenomena evolve. Real-scale rendering and embodied interaction make the experience feel tangible. The interface is designed to be accessible, even to those without nuclear or simulation expertise. This lowers the barrier for stakeholders, policymakers, and the general public, while still supporting expert analysis and collaborative decision-making. This work shows how immersive visualization can function as both a scientific tool and a communication interface. By integrating simulation, processing, and visualization into a cohesive workflow, we offer a scalable framework for immersive scientific storytelling. The modular design supports future extensions to other reactor types and lifecycle stages, from shutdown to long-term storage, making the platform adaptable for both research and outreach.

99 - GENERAL AND MISCELLANEOUS↗

DOE OSTI2021

An approach to characterizing spatial aspects of image system blur

Quantitative X-ray radiographic imaging systems that utilize a charged couple device (CCD) camera connected to a thick, monolithic scintillator can exhibit blur that varies spatially across the field of view, especially for thick scintillators used in pulse-power radiography of dynamically compressed objects. Here, a three-point approach to estimating and accounting for this effect is demonstrated by (a) using a local estimation technique to measure the effect of blurring a calibration object at key locations across the field of view, (b) combining each of the local estimates into a spatially varying blurring function via partitions of unity interpolation, and (c) resolving the effects of that blur on the image by solving an ill-posed inverse problem using a spatially varying regularization term. The technique is demonstrated on synthetic examples and actual radiographs collected at the Naval Research Laboratory’s (NRL)Mercury pulsed power facility.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Extended virtual element method for the Laplace problem with singularities and discontinuities

In this paper, we propose the extended virtual element method (X-VEM) to treat singularities and crack discontinuities that arise in the Laplace problem. The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes, wherein the basis functions are implicit (virtual)—they are not known explicitly nor do they need to be computed within the problem domain. In this work, suitable projection operators are used to decompose the bilinear form on each element into two parts: a consistent term that reproduces the first-order polynomial space and a correction term that ensures stability. A similar approach is pursued in the X-VEM with a few notable extensions. To capture singularities and discontinuities in the discrete space, we augment the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis (partition-of-unity) functions with enrichment functions. For discontinuities, basis functions are discontinuous across the crack and for singularities a weakly singular enrichment function that satisfies the Laplace equation is chosen. For the Laplace problem with a singularity, we devise an extended projector that maps functions that lie in the extended virtual element space onto linear polynomials and the enrichment function, whereas for the discontinuous problem, the consistent element stiffness matrix has a block-structure that is readily computed. An adaptive homogeneous numerical integration method is used to accurately and efficiently (no element-partitioning is required) compute integrals with integrands that are weakly singular. Once the element projection matrix is computed, the same steps as in the standard VEM are followed to compute the element stabilization matrix. Numerical experiments are performed on quadrilateral and polygonal (convex and nonconvex elements) meshes for the problem of an $\textit{L}$-shaped domain with a corner singularity and the problem of a cracked membrane under mode III loading, and results are presented that affirm the sound accuracy and demonstrate the optimal rates of convergence in the $\textit{L}^2$ norm and energy of the proposed method.

42 ENGINEERING↗

DOE OSTI2023

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

PhILMs: Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems

The landscape of computational science and engineering is continually evolving, with the challenge of high-dimensional regression problems standing as a significant hurdle in numerous scientific endeavors. Addressing this challenge, our research, funded by this award, has led to the development of an innovative computational framework known as Probabilistic Partition of Unity Networks (PPOU-Nets). This initiative represents a collaborative effort to harness the potential of mathematics and physics-informed machine learning in tackling multiscale and multiphysics problems prevalent in high-dimensional spaces. Through this work, we have proposed a novel methodology that seamlessly integrates adaptive dimensionality reduction and a mixture of experts model, thereby facilitating a more efficient and accurate approximation of complex functions. This research effort has not only advanced the state of computational science but also opened new avenues for exploration in quantum computing and beyond. This report outlines the motivation, methodology, key findings, and implications of our work, underscoring our contributions to the broader scientific community and the potential pathways for future research.

97 MATHEMATICS AND COMPUTING↗