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Metallic Powder Core Tubular Wire (PCTW) Development for Additive Manufacturing

NASA Langley Research Center (LaRC) in Hampton, VA, and the Center for Welding, Joining, and Coating Research (CWJCR) in the Department of Metallurgical and Materials Engineering at the Colorado School of Mines (CSM) in Golden, CO, have collaborated for nearly fifteen years to develop metallic powder core tubular wire (PCTW) feedstock for additive manufacturing (AM) using electron beam deposition. The motivation for this development was to offset dealloying observed during deposition of titanium and aluminum alloy solid wire feedstock in the LaRC electron beam freeform fabrication (EBF3) system and to explore novel alloy compositions by tailoring the powder fill. EBF3 is a fusion-based AM process that is performed under vacuum and is consequently prone to vaporization loss of low vapor pressure alloying elements. Research spanning four graduate student research projects succeeded in developing the basic methodology to fabricate PCTW, the mass balance equations used to define the powder fill, microstructure control through particulate inoculation, and the feasibility of producing metal matrix composite materials. Metallic PCTW is comprised of a thin foil sheath wrapped around a powder blend. Fabrication of PCTW at CSM was accomplished using a tubular wire mill system to first bend a flat strip into a u-shaped cross section, deposit a powder fill, create a lap joint to seal the wire, and draw to final diameter. Powder management systems were developed to ensure uniform mixing of powder constituents and identify feed rates for optimum fill ratios. Processing parameters were successfully developed for titanium and aluminum alloy sheaths to account for material work hardening during drawing. Mass balance equations were developed to account for compositional contributions of the sheath and powder fill. PCTW composition can be adjusted through selection of the sheath or powder materials. PCTWs were successfully developed to offset Al loss in deposits of Ti-6-4 and Mg loss in Al 6061, resulting in deposits that were within the composition limits for each material. Ti-6-4 deposits were fabricated using PCTW comprised of commercially pure (CP) titanium sheath and a fill of pre-alloyed Ti-6-4 powder with additions of elemental Al and V powders. Additions of both elements were required to account for the CP Ti from the sheath, and Al was further enhanced to offset documented losses in EBF3 deposits. For Al 6061 deposits, composition was modulated through the choice of sheath material. The PCTW was made using an Al 5052 sheath to supplement Mg content and a fill composition of pre-alloyed 6061 powder with additions of Si and Cu.The slower solidification rate of the EBF3 process compared with laser powder bed fusion processes can lead to coarse-grained microstructures and mechanical property anisotropy. EBF3deposits of Ti-6-4 exhibited large epitaxially grown β grains, which result in degraded mechanical properties in the deposition direction. Microstructure refinement was achieved through a combination of electron beam modulation and the use of a PCTW designed for Ti-6-4 deposits and modified by the addition of powder made from Fe and B. Resulting deposits exhibited reductions in β grain size, α phase colony intercept length and lath width, and an increased volume fraction of β phase. The mechanism of microstructure refinement was related to precipitation of boride particles. Hardness testing confirmed that mechanical property anisotropy was reduced and an overall increase in hardness was achieved.Metal matrix composite (MMC) materials offer strength and stiffness benefits over unreinforced materials. Two methods were investigated using PCTW technology for producing Al 6061 basedMMC materials; the direct addition of SiC reinforcing particles and the in-situ generation of reinforcements through reaction synthesis of precursor powders. Al 6061 based MMCs reinforced with SiC exhibited particle clustering and formation of aluminum carbides during deposition. The application of a Ni coating to the SiC particles prior to incorporation into the powder fill eliminated aluminum carbide formation and aided particle distribution during deposition. Modulation of beam focus and power further reduced aluminum carbide formation and aided particle distribution. In reaction synthesis, precursor additions undergo an exothermic reaction during deposition to form new ceramic products, generally of finer scale than the initial precursor components. The product phases provide reinforcement in the MMC and act as nucleation sites during solidification, resulting in grain refinement, improved mechanical properties, and reduced susceptibility to solidification cracking. Higher tensile strength and stiffness were demonstrated in Al 6061 based MMCs with 2 vol% reinforcement. These combined research efforts demonstrate the potential of PCTW technology for customizing of alloy composition and achieving microstructure control. A manufacturing process for titanium- and aluminum-based PCTWs was developed based on iterative design, theoretical calculations, and empirical data. PCTW technology successfully mitigated Al loss from Ti-6-4 and Mg loss in Al 6061, resulted in grain refinement in Ti-6-4, and demonstrated fabrication of Al based MMCs

Marcia S Domack↗

DOE OSTI2024

Impact of Higher Fidelity Design Iterations on Critical System Criteria

Nuclear criticality experiments are effective at informing the performance of nuclear data libraries across many applications. This work explores the implications of refining critical experiment MCNP models from their low fidelity optimization phase to penultimate neutronic models. Specifically, this work is focused on two series of plutonium fueled experiments funded through internal programs at Los Alamos National Laboratory building off previous efforts under the EUCLID (Experiments Underpinned by Computational Learning for Improvements in Nuclear Data) collaboration. Thales, the first of the two collaborations, is a fast spectrum Ta-reflected plutonium experiment to support operations at PF-4. The second experiment are twin configurations designed to target the intermediate energy cross sections in 239 Pu. Motivation for this experiment stems from the PARallel Approach of Differential and InteGral Measurements (PARADIGM) collaboration which hopes to achieve a significant reduction in 239 Pu cross section uncertainties in the intermediate region.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A Two-Dimensional Non-Linear Magnetic Equivalent Circuit Model to Facilitate the Preliminary Design of a Normal Conducting Quadrupole

Normal conducting quadrupoles have been used to focus charged particle beams in synchrotrons, beam transfer lines, medical linacs, etc. for a long time. Optimization techniques based on analytical expressions combined with the use of a numerical field analysis tool exist in the literature for the design of an optimal pole tip shape. However, the initial shape and dimensions of the remaining yoke (including the pole itself and yoke base that act as the return path) are usually less well defined. This article discusses a design methodology for a normal conducting quadrupole based on a two-dimensional magnetic equivalent circuit. Here, this approach considers the geometry of the entire magnet and the non-linear behavior of the yoke material, thereby eliminating the initial iterations. The design outcome of this exercise serves as a competent starting point and can then be used to refine the pole tip, pole taper, yoke size, and add other geometrical features to achieve the required field quality, gradient, coil considerations, etc. by employing a finite element analysis tool. An example design study to demonstrate the proposed methodology is presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

3D Play Fairway Analysis for Examining of Superhot Reservoir Production Scenarios

The DEEPEN (DE-risking Exploration for geothermal Plays in magmatic ENvironments) project was a multi-laboratory, international effort to reduce uncertainty and improve resource characterization in superhot geothermal systems. Building on this foundation, this work advances open-source tools designed to lower the exploration risk and cost of superhot geothermal projects while promoting transparency, reproducibility, and efficiency in exploration workflows. These tools are being tested at two key sites: (1) the Nesjavellir Geothermal Area in Iceland, where the Icelandic Deep Drilling Project (IDDP) will drill its third well, and (2) Newberry Volcano in Oregon, USA, where Mazama Energy will pilot the first superhot enhanced geothermal system (EGS). A major outcome is the creation of a modular, open-source Python framework for play fairway analysis (PFA) in 2D and 3D, called geoPFA. The PFA workflow has been expanded to produce pseudo conceptual models, and will soon be refined to assess reservoir components through integration with the thermo-hydraulic-mechanical-chemical (THMC) simulator TReactMech, to enable iterative coupling between PFA and THMC models, improving characterization of superhot systems. All three of the Icelandic Deep Drilling Project's production scenarios were analyzed via this framework: (1) a superhot deep injection well paired with conventional production wells at Nesjavellir, (2) a superhot deep production well at Nesjavellir, and (3) superhot enhanced geothermal system at Newberry Volcano. This analysis provides useful insights around conceptual modeling of these production scenarios, helping to inform decisions around which scenario is best suited for which types of environments.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

3D Play Fairway Analysis for Examining of Superhot Drilling Production Scenarios: Preprint

15 GEOTHERMAL ENERGY↗

DOE OSTI2025

A staged deep learning approach to spatial refinement in 3D temporal atmospheric transport

High-resolution spatiotemporal simulations effectively capture the complexities of atmospheric plume dispersion in complex terrain. However, their high computational cost makes them impractical for applications requiring rapid responses or iterative processes, such as optimization, uncertainty quantification, or inverse modeling. To address this challenge, this work introduces the Dual-Stage Temporal Three-dimensional UNet Super-resolution (DST3D-UNet-SR) model, a highly efficient deep learning model for plume dispersion predictions. DST3D-UNet-SR is composed of two sequential modules: the temporal module (TM), which predicts the transient evolution of a plume in complex terrain from low-resolution temporal data, and the spatial refinement module (SRM), which subsequently enhances the spatial resolution of the TM predictions. We train DST3D-UNet-SR using a comprehensive dataset derived from high-resolution large eddy simulations (LES) of plume transport. We propose the DST3D-UNet-SR model to significantly accelerate LES of three-dimensional (3D) plume dispersion by three orders of magnitude. Additionally, the model demonstrates the ability to dynamically adapt to evolving conditions through the incorporation of new observational data, substantially improving prediction accuracy in high-concentration regions near the source.

3D temporal sequences↗

DOE OSTI2025

A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas

We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following Du Toit et al. (2018), the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a lower-order approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2005

Combined Radar and Radiometer Analysis of Precipitation Profiles for a Parametric Retrieval Algorithm

A methodology to analyze precipitation profiles using the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) and precipitation radar (PR) is proposed. Rainfall profiles are retrieved from PR measurements, defined as the best-fit solution selected from precalculated profiles by cloud-resolving models (CRMs), under explicitly defined assumptions of drop size distribution (DSD) and ice hydrometeor models. The PR path-integrated attenuation (PIA), where available, is further used to adjust DSD in a manner that is similar to the PR operational algorithm. Combined with the TMI-retrieved nonraining geophysical parameters, the three-dimensional structure of the geophysical parameters is obtained across the satellite-observed domains. Microwave brightness temperatures are then computed for a comparison with TMI observations to examine if the radar-retrieved rainfall is consistent in the radiometric measurement space. The inconsistency in microwave brightness temperatures is reduced by iterating the retrieval procedure with updated assumptions of the DSD and ice-density models. The proposed methodology is expected to refine the a priori rain profile database and error models for use by parametric passive microwave algorithms, aimed at the Global Precipitation Measurement (GPM) mission, as well as a future TRMM algorithms.

Masunaga, Hirohiko↗

NASA NTRS2014

Explaining Earths Energy Budget: CERES-Based NASA Resources for K-12 Education and Public Outreach

Among atmospheric scientists, the importance of the Earth radiation budget concept is well understood. Papers have addressed the topic for over 100 years, and the large Clouds and the Earth's Radiant Energy System (CERES) science team (among others), with its multiple on-orbit instruments, is working hard to quantify the details of its various parts. In education, Earth's energy budget is a concept that generally appears in middle school and Earth science curricula, but its treatment in textbooks leaves much to be desired. Students and the public hold many misconceptions, and very few people have an appreciation for the importance of this energy balance to the conditions on Earth. More importantly, few have a correct mental model that allows them to make predictions and understand the effect of changes such as increasing greenhouse gas concentrations. As an outreach element of the core CERES team at NASA Langley, a multi-disciplinary group of scientists, educators, graphic artists, writers, and web developers has been developing and refining graphics and resources to explain the Earth's Energy budget over the last few decades. Resources have developed through an iterative process involving ongoing use in front of a variety of audiences, including students and teachers from 3rd to 12th grade as well as public audiences.

Chambers, L. H.↗

DOE OSTI2024

Real space iterative reconstruction for vector tomography (RESIRE-V)

Tomography has had an important impact on the physical, biological, and medical sciences. To date, most tomographic applications have been focused on 3D scalar reconstructions. However, in some crucial applications, vector tomography is required to reconstruct 3D vector fields such as the electric and magnetic fields. Over the years, several vector tomography methods have been developed. Here, we present the mathematical foundation and algorithmic implementation of REal Space Iterative REconstruction for Vector tomography, termed RESIRE-V. RESIRE-V uses multiple tilt series of projections and iterates between the projections and a 3D reconstruction. Each iteration consists of a forward step using the Radon transform and a backward step using its transpose, then updates the object via gradient descent. Incorporating with a 3D support constraint, the algorithm iteratively minimizes an error metric, defined as the difference between the measured and calculated projections. The algorithm can also be used to refine the tilt angles and further improve the 3D reconstruction. To validate RESIRE-V, we first apply it to a simulated data set of the 3D magnetization vector field, consisting of two orthogonal tilt series, each with a missing wedge. Our quantitative analysis shows that the three components of the reconstructed magnetization vector field agree well with the ground-truth counterparts. We then use RESIRE-V to reconstruct the 3D magnetization vector field of a ferromagnetic meta-lattice consisting of three tilt series. Our 3D vector reconstruction reveals the existence of topological magnetic defects with positive and negative charges. We expect that RESIRE-V can be incorporated into different imaging modalities as a general vector tomography method. To make the algorithm accessible to a broad user community, we have made our RESIRE-V MATLAB source codes and the data freely available at https://github.com/minhpham0309/RESIRE-V.

47 OTHER INSTRUMENTATION↗

DOE OSTI2025

Seismic Contingency Auto Generator

This code takes in premade earthquake scenario XML files from USGS, power grid data, and converts them into a contingency file (.con file) that can be used by power grid solvers. Within the .con file are a number (Specified by the user) of contingencies that have randomly failed power transformers based on their likelihood of failure and peak ground acceleration (PGA) value around the transformer. The transformers' likelihood of failure was calculated based on a variety of finite element modeling on various transformer designed for specific transformer voltage classes. Parameters from these FEM were used to create generic fragility curves for transformers within a specific voltage class, which correspond with earthquake PGA values to produced a probability of failure for a given earthquake scenario. More refined versions of this process, such as specifying specific transformer design categories within a voltage class, could also be applied in future iterations of the software.

Vaagensmith, Bjorn [Idaho National Laboratory (INL↗

DOE OSTI2021

Assessing Self-Definition and Relatedness in Level of Personality Functioning

The two polarities model (TPM) of personality organizes psychological assessment and psychotherapy and connects to personality disorder diagnosis using the DSM-5 Alternative Model for Personality Disorders (AMPD). The authors developed scales assessing the TPM from an existing self-report measure for level of personality functioning (LPF), a core component of the AMPD. Iterative content analyses of the LPF measure yielded scales for Autonomy and Communion corresponding to dimensions of the TPM. The scales were refined via internal consistency analyses using a measure of psychological attachment and studied in development and validation samples. Associations with relevant external criteria were explored in a series of multiple regressions. Furthermore, the new content-based LPF scales were illustrated with a case vignette. Although the new Autonomy/Communion scales await further validation prior to clinical use, initial evidence suggests that they may bridge the nomological nets of the TPM and AMPD and potentially offer clinical utility in assessment and treatment planning.

60 APPLIED LIFE SCIENCES↗

NASA NTRS2004

Processing of High Resolution, Multiparametric Radar Data for the Airborne Dual-Frequency Precipitation Radar APR-2

Following the successful Precipitation Radar (PR) of the Tropical Rainfall Measuring Mission, a new airborne, 14/35 GHz rain profiling radar, known as Airborne Precipitation Radar - 2 (APR-2), has been developed as a prototype for an advanced, dual-frequency spaceborne radar for a future spaceborne precipitation measurement mission. . This airborne instrument is capable of making simultaneous measurements of rainfall parameters, including co-pol and cross-pol rain reflectivities and vertical Doppler velocities, at 14 and 35 GHz. furthermore, it also features several advanced technologies for performance improvement, including real-time data processing, low-sidelobe dual-frequency pulse compression, and dual-frequency scanning antenna. Since August 2001, APR-2 has been deployed on the NASA P3 and DC8 aircrafts in four experiments including CAMEX-4 and the Wakasa Bay Experiment. Raw radar data are first processed to obtain reflectivity, LDR (linear depolarization ratio), and Doppler velocity measurements. The dataset is then processed iteratively to accurately estimate the true aircraft navigation parameters and to classify the surface return. These intermediate products are then used to refine reflectivity and LDR calibrations (by analyzing clear air ocean surface returns), and to correct Doppler measurements for the aircraft motion. Finally, the the melting layer of precipitation is detected and its boundaries and characteristics are identifIed at the APR-2 range resolution of 30m. The resulting 3D dataset will be used for validation of other airborne and spaceborne instruments, development of multiparametric rain/snow retrieval algorithms and melting layer characterization and statistics.

airborne radar↗

DOE OSTI2024

A New Reduced Order Model For The Mechanistic Creep Behavior Of UO 2

This manuscript describes an ongoing NEAMS effort to better determine the performance of advanced nuclear fuels, in particular the creep behavior of doped UO$_2$ for light water reactors. In our previous work, we outlined a method to utilize data generated from lower length scale simulations and implement it into the engineering scale fuel performance analysis. This process has been further refined, and in addition, new data has been used to train the surrogate model which has also been substantially improved since the previous iteration. The new model is compared against the current empirical model used in BISON using both scoping calculations to define the performance over the parameter space and using integral instrumented fuel assessment cases to determine the impact of these models on the overall fuel performance. Suggestions and guidance for future improvements to this method are provided to ensure the model covers relevant parameter space and phenomena.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1982

Profiles of electrified drops and bubbles

Axisymmetric equilibrium shapes of conducting drops and bubbles, (1) pendant or sessile on one face of a circular parallel-plate capacitor or (2) free and surface-charged, are found by solving simultaneously the free boundary problem consisting of the augmented Young-Laplace equation for surface shape and the Laplace equation for electrostatic field, given the surface potential. The problem is nonlinear and the method is a finite element algorithm employing Newton iteration, a modified frontal solver, and triangular as well as quadrilateral tessellations of the domain exterior to the drop in order to facilitate refined analysis of sharply curved drop tips seen in experiments. The stability limit predicted by this computer-aided theoretical analysis agrees well with experiments.

Basaran, O. A.↗

NASA NTRS1989

Prediction of turbine rotor-stator interaction using Navier-Stokes methods

Flows in turbomachinery are generally complex and do not easily lend themselves to numerical computation. The flows are three-dimensional and inherently unsteady. Complicated blade geometries and flow phenomena such as separation and periodic transition from laminar to turbulent flow add to the numerical complexity. Nevertheless, the accurate numerical analysis of such flows is a problem of considerable interest and practical importance to the turbomachinery community. Much of the early work in turbomachinery flow prediction focussed on airfoil cascades. While such analyses of flows in isolated airfoil rows have helped improve understanding of the flow phenomena and have gained widespread acceptance in the industrial community as a design tool, they do not yield any information regarding the unsteady effects arising out of rotor-stator aerodynamic interaction. These interaction effects become increasingly important as the distance between successive stator and rotor rows is decreased. Thus, the need exists for analytical tools that treat the rotor and stator airfoils as a system and provide information regarding the magnitude and the impact of the unsteady effects. The focus a three-dimensional, time-accurate, thin-layer Navier-Stokes code that was recently developed to study rotor-stator interaction problems. A system of patched and overlaid grids that move relative to each other is used to discretize the flow field and the governing equations are integrated using a third-order upwind scheme set in an iterative, implicit framework. The code was used to simulate subsonic flow through an axial turbine configuration for which considerable experimental data exists. Grid refinement studies were also conducted as part of the code validation process. The current status of the research, along with planned future directions, are also discussed.

Madavan, Nateri K.↗

NASA NTRS2017

CisLunar Habitat Internal Architecture Design Criteria

BACKGROUND: In preparation for human exploration to Mars, there is a need to define the development and test program that will validate deep space operations and systems. In that context, a Proving Grounds CisLunar habitat spacecraft is being defined as the next step towards this goal. This spacecraft will operate differently from the ISS or other spacecraft in human history. The performance envelope of this spacecraft (mass, volume, power, specifications, etc.) is being defined by the Future Capabilities Study Team. This team has recognized the need for a human-centered approach for the internal architecture of this spacecraft and has commissioned a CisLunar Phase-1 Habitat Internal Architecture Study Team to develop a NASA reference configuration, providing the Agency with a "smart buyer" approach for future acquisition. THE CISLUNAR HABITAT INTERNAL ARCHITECTURE STUDY: Overall, the CisLunar Habitat Internal Architecture study will address the most significant questions and risks in the current CisLunar architecture, habitation, and operations concept development. This effort is achieved through definition of design criteria, evaluation criteria and process, design of the CisLunar Habitat Phase-1 internal architecture, and the development and fabrication of internal architecture concepts combined with rigorous and methodical Human-in-the-Loop (HITL) evaluations and testing of the conceptual innovations in a controlled test environment. The vision of the CisLunar Habitat Internal Architecture Study is to design, build, and test a CisLunar Phase-1 Habitat Internal Architecture that will be used for habitation (e.g. habitability and human factors) evaluations. The evaluations will mature CisLunar habitat evaluation tools, guidelines, and standards, and will interface with other projects such as the Advanced Exploration Systems (AES) Program integrated Power, Avionics, Software (iPAS), and Logistics for integrated human-in-the-loop testing. The mission of the CisLunar Habitat Internal Architecture Study is to become a forcing function to establish a common understanding of CisLunar Phase-1 Habitation Internal Architecture design criteria, processes, and tools. The scope of the CisLunar Habitat Internal Architecture study is to design, develop, demonstrate, and evaluate a Phase-1 CisLunar Habitat common module internal architecture based on design criteria agreed to by NASA, the International Partners, and Commercial Exploration teams. This task is to define the CisLunar Phase-1 Internal Architecture Government Reference Design, assist NASA in becoming a "smart buyer" for Phase-1 Habitat Concepts, and ultimately to derive standards and requirements from the Internal Architecture Design Process. The first step was to define a Habitat Internal Architecture Design Criteria and create a structured philosophy to be used by design teams as a filter by which critical aspects of consideration would be identified for the purpose of organizing and utilizing interior spaces. With design criteria in place, the team will develop a series of iterative internal architecture concept designs which will be assessed by means of an evaluation criteria and process. These assessments will successively drive and refine the design, leading to the combination and down-selection of design concepts. A single refined reference design configuration will be developed into in a medium-to-high fidelity mockup. A multi-day human-in-the-loop mission test will fully evaluate the reference design and validate its configuration. Lessons learned from the design and evaluation will enable the team to identify appropriate standards for Phase-1 CisLunar Habitat Internal Architecture and will enable NASA to develop derived requirements in support of maturing CisLunar Habitation capabilities. This paper will describe the criteria definition process, workshop event, and resulting CisLunar Phase-1 Habitat Internal Architecture Design Criteria.

Jones, R.↗

DOE OSTI2020

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L 2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H -1 -type norm, but not L 2 -coercive. Nevertheless, the L 2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L 2 norm, or by properly choosing the finite element spaces, providing implicit control of the L 2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L 2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. Finally, we believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

97 MATHEMATICS AND COMPUTING↗