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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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DOE OSTI2022

EERC Final Topical Report – Findings on Subtask 3.2 – Produced Water Management Through Geologic Homogenization, Conditions, and Reuse

The Energy & Environmental Research Center (EERC) was awarded a contract by the North Dakota Industrial Commission (NDIC) Oil and Gas Research Program (NDIC No. G-051-101) to conduct a study on the recycling of water used in oil and gas operations, also known as produced water, from oil- and gas-producing regions of North Dakota as directed by Section 19 of North Dakota House Bill 1014. This final report provides a compilation of results of the study, which include regulatory, scientific, technological, and feasibility methods and considerations associated with North Dakota produced water management. The report provides a synopsis of this project’s previously submitted produced water assessment report entitled “Produced Water Management and Recycling Options in North Dakota” (Energy & Environmental Research Center, 2020), with updated values provided as appropriate. The report provides the results from the investigation of a novel produced water management strategy, referred to as geologic homogenization, conditioning, and reuse (GHCR), which aims to use a subsurface geologic formation as a natural medium for managing produced water recycling and reuse. Water management is a significant technical and economic challenge for sustainable oil and gas production, and water volumes are intrinsically linked to oil production volumes. North Dakota oil production rose to over 1.5 million barrels (MMbbl)/day in 2019, and despite a downturn in oil price in early 2020, North Dakota oil production has recovered to 1.1 MMbbl/day as of August 2021. Bakken petroleum system development between 2008 and 2020 has resulted in a nearly fourfold increase in produced water volumes to 642 MMbbl/yr in 2020 after peaking at 740 MMbbl/yr in 2019 and a fivefold increase in saltwater disposal (SWD) volumes to 565 MMbbl/yr in 2020 after a peak of 682 MMbbl/yr in 2019. Produced water and SWD volumes are forecasted to double by 2030. SWD is the primary method of produced water management used in North Dakota, with approximately 95% of the SWD volume occurring through subsurface injection into sandstones of the Dakota Group (Dakota). Localized pressurization of the Dakota resulting from SWD and projected increases in produced water volumes could impact the economics of North Dakota oil production. As a result, there is an emerging need to pursue alternative produced water management approaches, including recycling and reuse. While produced water recycling is not yet widespread, commercial operators are making strides in overcoming the technical challenges of using high salinity produced water in completion operations (Marathon Oil, 2020). As water management continues to be a key focal point in companies’ environmental, social, and governance (ESG) initiatives, focus on water management, including recycling, will likely continue to increase. Laboratory column testing, field sample collection, geologic modeling and numerical simulation, and techno-economic analysis all indicate that GHCR could feasibly be implemented as a potential water management option. Laboratory column testing and field sample collection indicate that the Inyan Kara sandstone and native formation fluid are capable of homogenizing with the Bakken produced water to a point where the fluid composition appears to stabilize. Extracting that stabilized fluid could be considered homogeneous and capable of providing individual batches of hydraulic fracturing fluid. Numerical simulation results indicate that extraction of fluids from the Inyan Kara in a GHCR implementation scenario is capable of reducing formation pressure, which would help ease localized pressurization of the Inyan Kara and extend the available capacity for nearby existing SWD wells. Economic analysis indicates that there are scenarios where GHCR implementation can be a competitive or even lower-cost option than a conventional water management approach. Site-specific conditions will dictate the economic potential of GHCR, but potentially attractive sites for GHCR implementation will be those that are located above a pressurized zone of the Inyan Kara, need six or more Bakken infill wells, and face high costs for conventional SWD and/or freshwater. Based on the regulatory review, drilling into the Inyan Kara for SWD and to harness as a source water for industrial use have precedent in North Dakota, and a workable regulatory solution for GHCR seems likely. However, restrictions in the state regarding surface storage and transport of produced fluids may limit some activities, which will affect how GHCR could ultimately be implemented. In summary, this study reveals pursuing GHCR can be a viable approach to water management in North Dakota. The GHCR concept addresses some of the challenges that hinder the more traditional approaches to recycling in the industry. Furthermore, an assessment of the current landscape of water management within the state reveals the ongoing trend of increasing volumes of produced water and SWD. Projections reveal that the volumes of produced water that need to be managed are expected to double over the next decade (Energy & Environmental Research Center, 2020). With the continued development of the Bakken and continuing driving factors related to ESG initiatives, implementing a practice such as GHCR is a feasible approach to adding recycling of produced water to industry within the state. This subtask was cofunded through the EERC–U.S. Department of Energy Joint Program on Research and Development for Fossil Energy-Related Resources Cooperative Agreement No. DE-FE0024233. Nonfederal funding was provided by the North Dakota Industrial Commission Oil and Gas Research Program. References Energy & Environmental Research Center, 2020, Produced water management and recycling options in North Dakota: Final Report for North Dakota Legislative Management Energy Development and Transmission Committee and North Dakota Industrial Commission. Marathon Oil, 2020, Sustainability report: https://cdn.sanity.io/files/ghcnw9z2/website/ 91744eb6ef8fbe59505a911c6b8d2e8dd9a537fa.pdf?dl (accessed November 2021).

02 PETROLEUM↗
DOE OSTI2020

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

In this paper, we present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Low-Temperature Geothermal Play Fairway Analysis for the Denver Basin

This project is part of a national initiative to showcase the benefits of incorporating low-temperature geothermal resource assessment into the deployment of geothermal heating, combined heat and power (CHP), and geothermal direct-use (GDU) technologies. The initiative was established to accelerate the country's decarbonization efforts by identifying potential for low-temperature geothermal resource utilization (<150 degrees C, e.g., CHP and GDU) in selected sedimentary basins with numerous population centers. The play fairway analysis (PFA) methodologies in this study were adapted from previous PFA investigations of sedimentary basin geothermal play types (SBGPTs) that evaluated the potential for low-temperature resources (<150 degrees C). Workflows, relevant datasets, a new Python library, and common and composite geological criteria maps are utilized to develop low-temperature geothermal resource favorability maps for the Denver Basin, a sedimentary basin spanning Colorado, Nebraska, and Wyoming. The replication of these methodologies in other SBGPTs can evaluate potential for low-temperature resources. To facilitate future assessment of low-temperature geothermal resources in SBGPTs, this project provides PFA workflows, data, tools, and favorability maps that will ultimately support the utilization of low-temperature geothermal resources in sedimentary basins.

combined heat and power↗
DOE OSTI2024

Low-Temperature Geothermal Play Fairway Analysis for the Denver Basin: Preprint

This project is part of a national initiative to showcase the benefits of incorporating low-temperature geothermal resource assessment into the deployment of geothermal heating and cooling (GHC), combined heat and power (CHP), and geothermal direct use (GDU) technologies. The initiative was established to accelerate the country's decarbonization efforts by identifying potential for low-temperature geothermal resource utilization (< 150 Degrees Celsius, i.e., GHC, CHP, and GDU) in selected sedimentary basins with numerous population centers. The Play Fairway Analysis (PFA) methodologies in this study were adapted from previous PFA investigations of sedimentary basin geothermal play types (SBGPTs) that evaluated the potential for low-temperature resources (< 150 Degrees Celsius). Workflows, relevant datasets, python code, common and composite geological criteria maps are utilized to develop low-temperature geothermal resource favorability maps for the Denver Basin, a sedimentary basin spanning Colorado, Nebraska, and Wyoming. The replication of these methodologies in other SBGPTs can evaluate potential for low-temperature resources. To facilitate future assessment of low-temperature geothermal resources in SBGPTs, this project provides PFA workflows, data, tools, and favorability maps that will ultimately support the utilization of low-temperature geothermal resources in sedimentary basins.

Denver Basin↗
DOE OSTI2021

Transitions between Rydberg states in two-color corotating circularly polarized laser pulses

We present a study of higher-order Raman (Λ, V, and S) transitions between Rydberg states involving the absorption and emission of at least three photons in the interaction of two-color corotating circularly polarized laser pulses with the hydrogen atom. Our analysis is based on results of numerical solutions of the corresponding time-dependent Schrödinger equation. Here, the results for the interaction with (ω,2ω) fields show that the simultaneous interaction with both fields results in an excited state distribution over a broad range of magnetic quantum numbers. Indications of the impact of the Λ, V, and S transitions are found via the analysis of final distributions resulting from the preparation of the atom in specific Rydberg states. Furthermore, we show that similar mechanisms for the redistribution of population between Rydberg states are present in two-color fields with larger differences in the central frequencies, i.e., (ω,3ω) and (ω,4ω) fields.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Circularity Futures Workshop Series: Summary Report

The aim of this report is to synthesize key feedback received from the three-part Circularity Futures workshop series held in Spring 2024. The workshop series was conducted by the National Renewable Energy Laboratory (NREL) on behalf of U.S. Department of Energy, Office Energy Efficiency and Renewable Energy (EERE), and was broken into three workshops: Workshop 1 - Circularity Analysis Needs and Priorities; Workshop 2 - Circularity Metrics and Indicators; and Workshop 3 - Circularity Data. Together, the workshops focused on identifying the existing priorities and gaps in the circularity modeling space, understanding different stakeholders' use and interpretation of circularity metrics and indicators, identifying common data gaps and data quality challenges, and assessing the robustness of available solutions. The workshop series brought a diverse group of stakeholders - including representatives from U.S. government offices, national labs, nonprofit organizations, industry, and academia - to collect first-hand feedback on needs, priorities, challenges and opportunities in the circularity modeling and analysis space. The workshop discussions highlighted numerous common needs, priorities and challenges among the interviewed groups. Several topics were frequently discussed, including: 1) Circularity as a pathway for sustainable economic growth: While circularity is generally defined in terms of resource conservation and reducing wasteful disposal of materials, participants agreed that circular strategies should serve broader economic, environmental, and social goals. It is therefore crucial for circularity analysis to look beyond waste reduction and instead evaluate a variety of impact metrics such as cost savings, job creation, air quality, and pollutant emissions. Mutli-criteria decision-making frameworks may be useful for making sense of disparate metrics and evaluating tradeoffs between impact categories.; 2) Economic and social factors are not well understood: Underdevelopment of existing end-of-life (EOL) management infrastructure, inconsistent standardization codes and policy space in reusing recycled content, and suboptimal collection and sorting strategies collectively contribute to uncertainty about the economic potential of circular pathways. The latter observation is consistent among all technologies but more emphasized for renewable energy systems. Social impacts of circularity practices are less understood and less researched than other sustainability aspects.; 3) Inconsistent methods for assessing emerging technologies: LCA and TEA results vary widely depending on the assumptions made with regards to market adoption of new technologies. Emerging technologies suffer limited availability of data needed to conduct a robust circularity analysis. Yet, understanding projected impacts of proposed nascent technology is a key need for different stakeholder groups.; and 4) Lack of temporally and geospatially explicit data: There is a need for open data that represents variations in circularity technologies over time and location. The lack thereof leads to aggregated and potentially misrepresented results in circularity analysis. Sensitivity analyses should be included to verify whether options perceived as more sustainable align with real-world practices.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

Wellbore integrity assessment with casing-based advanced sensing

Wellbore integrity is of paramount importance to subsurface resource extraction, energy storage 1 and waste disposal. After installation, well casing and cement are subject to mechanical stress due to near-well pressure changes and fluid induced corrosion. This is exacerbated for geothermal wells where produced fluid is at high temperature and corrosive. The current state-of-the-art technologies for wellbore integrity assessments are an array of cased hole logging tools. Wireline deployed acoustic, electromagnetic and mechanical tools are all available to inspect steel casing corrosion and casing-cement bond and these tools can provide high-resolution assessment of borehole conditions. They are intrusive, however, in terms of borehole preparation and interruption to the normal operation of the wells, and not suitable for high temperature or highly deviated well deployments. In addition, these measurements are performed infrequently due to high cost, and are therefore incapable of providing frequent data to better predict borehole degradation trajectory, which can help provide early warning of potential borehole failures. For this project we are developing a suite of novel, non-invasive, casing based tools for wellbore integrity assessment, combining fast/low cost screening with higher-precision investigation. Our approach is based on monitoring the response of the casing when energized at the wellhead, thereby interrogating the casing without well intervention. Lab, field and numerical approaches are used in our study. During the early stage of the research, we focus on numerical simulations, which have shown the sensitivity of the low frequency electromagnetic (EM) signals to changes in borehole depths and have successfully tested the concept at a field site with different length well casings. Initial seismic modeling efforts have also demonstrated our capability to simulate seismic tube wave and seismic field alterations due to borehole breakage and associated fluid leakage. Further numerical, laboratory and field experiments are underway for additional technology sensitivity analysis, particularly the transient EM/Seismic reflectometry methods, data acquisition optimization, and numerical simulation improvements.

Wilt, Michael↗
DOE OSTI2023

An introduction to the significance of sample size in particle analyses for nuclear forensics and radiological investigations

Particulate isotopic analysis in nuclear forensics has developed rapidly during the past two decades due to technical advances in determining the isotopic composition of individual particles. This paper introduces basic statistical concepts that can be applied by analysts to understand the importance of statistical adequacy when interpretating particle data. While these basic statistical methods provide a useful point-of-entry to particle data analysis, more sophisticated statistical and modeling approaches are needed to extract maximal information from such datasets in the future.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗
DOE OSTI2022

Demonstration of RELAP5-3D for transient analysis of a dual coolant lead lithium fusion blanket concept

Our research aims to build a multiphysics framework for transient analysis of the Dual- Coolant Lead-Lithium (DCLL) blanket design of the proposed Fusion Nuclear Science Facility (FNSF) for all Onticipated Operational Occurrences (AOOs) using Reduced Order Models (ROMs). This paper presents an assessment of using RELAP5-3D for transient thermal-hydraulic blanket analysis of the DCLL design. Preliminary validation studies for the RELAP5-3D properties were conducted using a representative vertical flow loop model and a comparison study using experimental data from the thermal convection corrosion loops at ORNL. These studies inspected the thermal hydraulic response of Lead-Lithium Eutectic (PbLi) systems within the RELAP5-3D code; mainly pressure drop and heat transfer, as compared to analytical calculations and experimental data. The Magnetohydrodynamic (MHD) pressure drop effect for liquid metal under the effects of a uniform magnetic field was implemented into our models using an equivalent forms loss coefficient based on the phasic momentum equations. We verified our implementation of MHD pressure drop within RELAP5-3D using a systematic approach consisting of two studies. Models were developed to perform comparison studies based on experimental data from the Argonne Liquid metal EXperiment (ALEX) facility and the Magnetohydrodynamic PbLi Experiment (MaPLE) facility. Using the developed validation basis, we built a simplified DCLL channel model within RELAP5-3D, inspected MHD pressure drop and performed transient temperature analysis. The DCLL channel model used radial heating data from the International Thermonuclear Experimental Reactor (ITER) which was normalized to the FNSF design. Here, a representative startup transient was performed using a relative power curve from the initial ramp up of a long-term power pulse of the Tore Supra reactor. This study demonstrated that RELAP5-3D ROMs are capable of accurate transient analysis of the DCLL blanket. We have since utilized MCNP to develop heating profiles based on the FNSF design and are working toward implementing them into our DCLL model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the 'mass' (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. Here we observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick's theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick's theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the 'frustrated' blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Spectral Analysis of Regular Material Point Method and its Application to Study High Pressure Reverse Osmosis Membrane Compaction and Embossing

Material Point Method (MPM) is gaining widespread interest in applied continuum mechanics. The fact that all the continuum properties are stored on the particles (or material points) and the governing equations are solved on these material points makes MPM extremely suited to problems involving severe material deformations, such as crack propagation, soil movement, and fluid flows. Despite its popularity, only a few studies have focused on the numerical properties of MPM. This presentation introduces a global spectral analysis of the regular material point method. Contrary to previous studies, the analysis focuses on the numerical properties of the method in the spectral space. The amplification factor is derived as a function of the non- dimensional wave numbers. It provides insights into the stability and dissipative properties of the method for various CFL and Fourier numbers. The effect of the grid shape functions, number of particles per cell and their locations inside the grid cell are also analyzed. The EXAGOOP MPM solver (https://github.com/NREL/Exagoop.git) is developed at the National Renewable Energy Laboratory as a part of the NAWI UHPRO project and is based on the AMReX framework. A single-level, uniform cartesian grid is used as the background mesh, while the particle class in AMReX is used to manage the material point operations. Linear hat and B-splines are used as grid shape functions, while the time integration is performed using explicit Euler time integration. EXAGOOP is both CPU and GPU compatible and has been demonstrated to work well on multiple compute architectures. The performance of EXAGOOP on various computing architectures is presented along with its application to study compaction and embossing of high-pressure reverse osmosis membranes. The MPM solution accurately reproduces the membrane deformation. The deformed pore size and structure simulated using MPM also agree well with experimental SEM images.

material point method↗
DOE OSTI2024

AutoCheck: Automatically Identifying Variables for Checkpointing by Data Dependency Analysis

Checkpoint/Restart (C/R) has been widely deployed in numerous HPC systems, Clouds, and industrial data centers, which are typically operated by system engineers. Nevertheless, there is no existing approach that helps system engineers without domain expertise and domain scientists without system fault tolerance knowledge identify those critical variables accounted for correct application execution restoration in a failure for C/R. To address this problem, we propose an analytical model and a tool (AutoCheck) that can automatically identify critical variables to checkpoint for C/R. AutoCheck relies on first, analytically tracking and optimizing data dependency between variables and other application execution state, and second, a set of heuristics that identify critical variables for checkpointing from the refined data dependency graph (DDG). AutoCheck allows programmers to pinpoint critical variables to checkpoint quickly within a few minutes. We evaluate AutoCheck on 13 representative HPC benchmarks, demonstrating that AutoCheck can efficiently identify correct critical variables to checkpoint.

HPC↗
DOE OSTI2020

Analysis into Asymptotic Convergence to Full Nonlinear Solutions and Exploration of the Implication of Numerical Operator Mutation of Differential Systems

A robust, sufficiently accurate and practical hydrodynamic simulation toolset is required as a key component of the modeling and simulation of air-gap electrostatic discharge events. This work was performed to complement these ongoing efforts. In particular, hydrodynamic simulations must be vetted to ensure they are robust and sufficiently accurate over relevant characteristic scales. Verification models were generated in order to cultivate the technical knowledge and expertise needed to properly create, implement and execute numerical simulations. Furthermore, this effort was utilized extensively to educate students on the mathematical and numerical principles underlying hydrodynamic simulations. This education opportunity, provided in a holistic and rigorous manner, has greatly benefited developing scientists and engineers with the necessary understandings and toolsets required to excel at accomplishing the task at hand, and, more generally, it has enabled them to generate key programmatic deliverables. This report articulates several subtilties; specifically, how perturbations, nonlinear behavior, and dissipative mechanisms influence numerical stability, how to properly structure mathematical and numerical solutions, and how to properly generate error estimation/assignment. A more rigorous discussion of the consequences of such topics can be found in the body of this report in Chapters 2 and 3 with qualitative findings discussed in Chapter 4.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Development and Validation of a Latent Thermal Energy Storage Model Using Modelica

An abundance of research has been performed to understand the physics of latent thermal energy storage with phase change material. Some analytical and numerical findings have been validated by experiments, but there are few free and open-source models available to the general public for use in systems simulation and analysis. The Modelica programming language is a good avenue to make such models available, because it is object-oriented, equation-based, declarative, and acausal. These characteristics have enabling the creation of component model libraries that can be used to build larger system simulations for design analysis. The authors have previously developed a numerical framework to model phase change thermal storage and have validated model predictions with experiments. The objectives of this paper are to describe the transfer of the numerical framework to an implementation in a Modelica component model and to validate the Modelica model with data from the experiment and the original numerical framework.

25 ENERGY STORAGE↗
DOE OSTI2022

Numerical study of advanced solar receiver tubes based on a coupled thermo-mechanical analysis for concentrated solar power tower plant

The search for more sustainable energy to match the growing energy demand begins with finding more dispatchable resources such as solar energy. As one of the promising solar technologies, concentrated solar power (CSP) has a full capacity to store thermal energy for extended operation. Nevertheless, some key components in CSP systems usually face extreme environment, such as uneven solar flux, cyclic thermal expansion, structural degradation on the solar absorber tubes in a Concentrated Solar Power Tower (CSPT) Plant. In this study, we applied Multiphysics simulation to explore the benefits of introducing optimized fins for heat transfer enhancement and uniform temperature distribution, the goal is to improve the thermal efficiency of such advanced solar absorber tubes. The results of this study can supply design guidance for the manufacturing process of absorber tubes, and eventually can benefit the solar energy community for the next generation of molten salt based CSP system.

14 SOLAR ENERGY↗
DOE OSTI2024

Extended Fayans energy density functional: optimization and analysis

The Fayans energy density functional (EDF) has been very successful in describing global nuclear properties (binding energies, charge radii, and especially differences of radii) within nuclear density functional theory. In a recent study, supervised machine learning methods were used to calibrate the Fayans EDF. Building on this experience, in this work we explore the effect of adding isovector pairing terms, which are responsible for different proton and neutron pairing fields, by comparing a 13D model without the isovector pairing term against the extended 14D model. At the heart of the calibration is a carefully selected heterogeneous dataset of experimental observables representing ground-state properties of spherical even–even nuclei. To quantify the impact of the calibration dataset on model parameters and the importance of the new terms, we carry out advanced sensitivity and correlation analysis on both models. The extension to 14D improves the overall quality of the model by about 30%. The enhanced degrees of freedom of the 14D model reduce correlations between model parameters and enhance sensitivity.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Discrete spherical harmonic functions for texture representation and analysis

A basis of discrete harmonic functions for efficient representation and analysis of crystallographic texture is presented. Discrete harmonics are a numerical representation of the harmonics on the sphere. A finite element formulation is utilized to calculate these orthonormal basis functions, which provides several advantageous features for quantitative texture analysis. These include high-precision numerical integration, a simple implementation of the non-negativity constraint and computational efficiency. Simple examples of pole figure and texture interpolation and of Fourier filtering using these basis sets are presented.

36 MATERIALS SCIENCE↗