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High-Order Shock Fitting with Finite Element Methods

A moving-grid, shock-tracking, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. Three different methods for specifying the shock motion have been tested,and the order of accuracy of the resulting solutions was verified. It was found that one of the methods is applicable only for problems with a constant upstream flow state, whereas the other two methods converge to the expected solution for general shock problems. When using a finite element method with polynomial degree p, the two general shock motion methods converged with the expected order of accuracy of (p+1). Finally, accurate solutions are obtained for supersonic nozzle and blunt body bow shock problems. Based on the results, it is concluded that the shock tracking method can preserve high-order solution accuracy in the vicinity of a shock wave.

Luke D'Aquila↗

Culture-Independent Microbial Air Profiling using a Spaceflight-Compatible Nanopore Sequencing Method

Microbial monitoring of spacecraft air is critical toward assessing the efficacy of microbial controls within the environmental control and life support systems to protect the crew and vehicle environment. Currently, onboard the International Space Station (ISS), the air is monitored every quarter using an impaction air sampler. With this method, microbial cells and spores are pulled onto plates containing a growth medium. Following onboard incubation, the crew reports approximate microbial levels to the ground, but sample return is required for identification. Upon return of the plates, the isolates present are identified for crew health risk assessments. As NASA moves beyond low-Earth orbit, sample return will be impractical, and a near real-time monitoring capability is essential. Significant strides have been made in recent years to utilize a molecular-based method for microbial profiling of ISS surfaces. The developed method is independent of microbial culture, thus removing the bias toward detecting only culturable organisms, eliminating the need for sample return, and reducing the risk to crew health from exposure to high microbial levels. The work described here details the evaluation of three different air sampling platforms whose product is amenable to downstream molecular processing. The three samplers were compared in terms of mass and power requirements, ease of use, and the resulting data. For the two highest-ranking samplers, a basic concept of operations was developed to transfer the sample into the already established preparation and sequencing process. Using these concepts of operations, an in-depth comparison of the molecular data generated was compared to the historical culture-based method. Data from both methods detailed similar microbial profiles, while the molecular method detailed microbial identifications that were lacking from the culture data. The developed method will enable the generation of near real-time microbial profiles of the spacecraft atmosphere.

Brandon Dunbar↗

Harvesting Microgreens in Microgravity: Analysis of Six Different Methods

In long duration space missions, crops will be used to supplement the astronaut diet. One such proposed crop type is microgreens, the young seedlings of edible plants that are known for their high nutritional levels, intense flavors, colorful appearance, and variety of textures. While these characteristics make microgreens a great candidate for space crop production, their small size presents a unique challenge within the microgravity environment. To ensure that astronauts will be able to harvest microgreens in microgravity with ease while avoiding the introduction of debris to the spacecraft cabin, multiple harvesting methods were developed by the Space Crop Production Team at NASA’s Kennedy Space Center. Three parabolic flights were conducted in November and December of 2021 and during those flights three different microgreen cutting methods (guillotine, pepper grinder, scissors) as well as two different bagging methods (attached and manual) were tested. In each flight, the microgreens were contained inside of a glovebox and footage of all the microgreen harvests was recorded. The cutting and bagging method combination that introduced the lowest average number of particulates into the glovebox was the scissors with attached bagging, closely followed by the pepper grinder with attached bagging. However, the scissors with attached bagging may had introduced fewer particulates into the glovebox because on average, 27% of the microgreens were never cut using the scissors method, so there were fewer free-floating particulates generated. The cutting and bagging method combination that left the lowest average percentage of microgreens remaining on the hardware post-harvest was the pepper grinder with an attached bag. Future directions include involving microgreen harvests in analog environments and further development of the different microgreen cutting and bagging methods. This research was funded by multiple NASA grants at the Kennedy Space Center.

Haley O Boles↗

Micrometer to Atomic Scale Characterisation of Primitive Astromaterials Using A Novel Method, Metis-Fa: A Coordinated Atom Probe Tomography, Transmission Electron Microscopy and NanoSIMS Approach

Introduction: Presolar grains preserve isotopic, chemical and microstructural records of physical and chemical processing, and formation mechanisms within a vast range of evolved stellar systems, the interstellar medium, solar nebula and their parent bodies. These evolutionary records are preserved at the micrometric to atomic scale, requiring coordinated studies to expand our understanding of evolutionary processes occurringthroughout ours and external stellar systems [1]. NanoSIMS enabled rapid in situ identification and isotopic characterisation of presolar grains and their stellar origins using 17O/16O and 18O/16O, and 13C/12C isotopic ratios [1]. Coordination with transmission electron microscopy (TEM) revealed crystallographic and localised contextual relationships and quantitively constrained their major and minor compositions [1]. However, trace elements cannot be quantified, the most sensitive geochemical tracers of environmental conditions, essential to unravelling the chemical record of their evolutionary pathway and parent stellar systems [2-3] . Furthermore, owing to the combination of technical limitations (only 5 – 7 isotopes can be measured per NanoSIMS run) and their small grain sizes of 100 nm < 3 μm (with rare exceptions in nanodiamonds (2 nm ≤) and SiC (< 40 μm)), the number of measurable isotopes per grain volume is limited [1,3] . Through more comprehensive isotopic studies of presolar grains, NanoSIMS studies have shown the importance of the latter, identifying Fe and Mg as important indicators of nuclear synthetic processing and their stellar origins, respectively [4- 5]. Coordination of NanoSIMS and Atom Probe Tomography (APT) revealed morphological signatures, and isotopic and chemical signatures at major to trace levels without requirements for preselection of elements [6]. However, crystallographic signatures in localized contextual relationships cannot be measured. Consequently, coordination of NanoSIMS, TEM and APT is essential to gain access to almost all contextual, structural and geochemical signatures within each presolar grain.Transmission electron microscopy requires a 100 nm thin lamella which is unstable in APT and would not produce any viable data. Atom probe tomography requires a needle-shaped specimen which when measured in TEM removes the local context, impacts the quality of the TEM diffraction images due to the shank angle of the needle, and can alter the chemistry of beam sensitive materials from the higher degree of surface exposure at the tip. To address these issues, we developed METIS-Fa (Multi-technical measurements of Electron Transparent materials using an Indium Sandwich - a FIB approach). A novel method which enables coordination of NanoSIMS, TEM and APT for generalized and targeted studies of individual grains, including beam sensitive materials, without compromising sample preparation requirements for TEM and APT. This method requires only indium and a Focus Ion Beam (FIB), minimizing the movement of fragile materials while still enabling preparation of TEM lamella into APT needles. Samples: Initial experimental development and testing of the method occurred at Astromaterials Research and Exploration Science (ARES), Johnson Space Centre (JSC), NASA and APT measurements and needle preparation occurred at JdLC, Curtin University. Synthetic silicate samples were used as analogs for presolar silicates when performing a trial run of the method. Samples were extracted from a polished thin section created at JSC, NASA, comprised of 38 wt.% Si, 17 wt.% FeO, 13 wt.% MgO, 12 wt.% Al, 11 wt.% Ca based on electron microprobe analysis (EMPA) [8] . Experimental details, pressure and temperature conditions were presented in [8] and references therein. Testing of the capability to target individual grains in mineral matrices using this method for acquisition in APT, measured matrix regions in meteoritic thin sections of primitive meteorites. These meteorites and their identified presolar grains for future targeted studies are detailed in [9]. Techniques: The TEM-FIB lamella were prepared using a FIB. An e-beam assisted pt deposition was used as a protective coating for the synthetic and meteoritic samples. When targeting individual grains, a secondary e-beam assisted pt deposition button is placed over the desired grain before the protective coating to denote its location. A JEOL 2500SE field-emission TEM was used for high-resolution imaging, energy-dispersive X-ray (EDX) and electron diffraction data.TheMETIS-Fa method was experimentally designed, tested and executed using a FIB at ARES, JSCNASA. Needles for APT were prepared using the Tescan Lyra3 GM Dual Beam Focus Ion Beam (FIB) Field Emission SEM (FE-SEM) at the JdLC, Curtin University. Atom probe tomography measurements were conducted using a CAMECA Local Electrode Atom Probe, LEAP 4000X HR. Two pure indium needles were analyzed initially to constraining acquisition parameters and stability under the beam. Manual acquisition was required to maintain evaporation of specimen’s at the apex and monitor interactions with measurement parameters. Experimental Design: Indium foil is pressed onto an Al stub with a pneumatic press and mounted into the FIB adjacent to the TEM-FIB lamella of interest. Using a FIB, two indium slices (5 μm x ~300 nm x 3 μm) are extracted from indium foil and aligned with the TEM-FIB lamella before touching the TEM-FIB lamella. Each slice is then attached through cold welding to the FIB-TEM lamella. This approach eliminates the need for chemical treatments and proved effective for aligning the Indium within the region of interest for APT, holding it in place for up to 4 days during testing.Once both indium slices are attached within their pre-determined region per grain targeting requirements, they are gradually melted onto the FIB-TEM lamella.When targeting a specific grain, measurements should be taken of the pt button and its distance from edge to edge of the lamella before and after sandwiching. A secondary button should be placed over the same region after the Indium slices have been attached to improve precision when preparing APT needles. Results: Figure 1 shows two indium slices melted onto a FIB-TEM lamella, adding additional bulk for preparation into APT needles as shown in Figure 2 [7] . The latter was essential so samples could be measured in TEM and APT without compromising sample preparation requirements and consequently data quality and acquisition stability. METIS-Fa proved effective forimproving geometry. Figure 3 shows a successful APTrun of the synthetic silicate. EMPA, TEM and APTshowed no chemical alterations. During targetingtesting, a solar silicate grain was successfully identifiedand measured in TEM, and prepared into an APTneedle. However, the indium was melted too long during sample preparation, causing expansion andformation of internal porosity leading to sample loss.Conclusion: METIS-Fa greatly expands the number of isotopic and chemical signatures measured per grain volume, and enables measurements of contextual, structural, crystallographic, isotopic and geochemical signatures within individual grains. Gaining access to such a vast range of evolutionary signatures required for expanding our understanding of external stellar and planetary systems and the evolution of our solar system. This method was designed for application to a vast range of phases including being sensitive materials and thus provides a way for coordination of NanoSIMS, TEM and APT not just for the study of presolar grains and by extension primitive astromaterials, but studies in a vast range of other fields including the geosciences and material sciences.Acknowledgments: Thankyou to ARES, JSC, NASA; JdLC Curtin University and Space Science Technology Centre for the use of laboratory facilities and funding [confirm].

Nicole D Nevill↗

Towards Aerodynamic Shape Optimization Using an Immersed Boundary Overset Grid Method

Traditional Reynolds-averaged Navier-Stokes grid methods applied to aerodynamic shapeoptimization can struggle with the deformation of surface and volume grids at componentintersections, such as at wing-fuselage junctions. To overcome this, we propose an approachwhich utilizes curvilinear overset grids for the discretization of the domain, with the presenceof the body modeled using an immersed boundary method. This approach handles complexgeometries without the need for their explicit integration into the grid. The goal of this approachis to reduce grid generation time and allow for greater geometric freedom for component-basedaerodynamic shape optimization. Two different methods are presented: a source-term-basedand a ghost-node-based immersed boundary method. Flow analyses and adjoint solutionsobtained using the proposed methods show promising comparisons with standard body-fittedgrid methods. Preliminary aerodynamic shape optimization results obtained using one of theimmersed boundary methods are also presented.

TTT↗

Fail-safe reactivity compensation method for a nuclear reactor

The present invention relates generally to the field of compensation methods for nuclear reactors and, in particular to a method for fail-safe reactivity compensation in solution-type nuclear reactors. In one embodiment, the fail-safe reactivity compensation method of the present invention augments other control methods for a nuclear reactor. In still another embodiment, the fail-safe reactivity compensation method of the present invention permits one to control a nuclear reaction in a nuclear reactor through a method that does not rely on moving components into or out of a reactor core, nor does the method of the present invention rely on the constant repositioning of control rods within a nuclear reactor in order to maintain a critical state.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

System and method for analyzing oscillatory stability in electrical power transmission systems

A computer-based method for contingency analysis of oscillatory stability in an electrical power transmission system is provided. The method uses at least one processor. The method includes receiving, by the at least one processor, a plurality of component inputs from a plurality of system components within the electrical power transmission system. The method also includes generating a nominal matrix for the electrical power transmission system. The nominal matrix includes a set of equations at least partially modeling the electrical power transmission system. The method further includes calculating eigenvalues and eigenvectors of the nominal matrix. The method also includes identifying a contingency representing a postulated disturbance of the electrical power transmission system. The method further includes estimating a contingency eigenvalue for the contingency using the eigenvalues and eigenvectors of the nominal matrix.

Baone, Chaitanya Ashok↗

Synchrophasor measurement method for power systems

A synchrophasor measurement method for a device configured to take synchronized measurements in a power system is provided. The synchrophasor measurement method includes receiving global positioning system (GPS)-synchronized samples of a signal sensed by the device from the power system; determining a level of distortion of the signal; selecting, based on the level of distortion, a computation method, the computation method being one of an improved zero-crossing (IZC) method and an enhanced phase-lock-loop (EPLL) method; performing the selected computation method to determine at least one parameter of the signal at a reporting frequency, which is at least twice a line frequency of the power system; and outputting, at the reporting frequency, the at least one parameter to an operator of the power system to allow the operator to perform at least one of a monitoring and a controlling of at least one element of the power system.

Yao, Wenxuan↗

Influence of Coating Method on the Performance of Roll-to-Roll Coated PEM Fuel Cell Catalyst Layers

As production of fuel cell vehicles and power systems increases there is a need to understand the process science of fuel cell manufacturing. Industrial production of polymer electrolyte membrane fuel cell (PEMFC) electrodes will rely on continuous roll-to-roll (R2R) coating methods to meet the rates needed for mass production. There are a variety of coating methods that could be used to coat catalyst layers. The different physics of each method may result in different morphologies and performance. Here we present the results of a study comparing fuel cell catalyst layers coated using two R2R coating methods: slot-die and gravure. These two methods were selected because the physics of applying the liquid to the substrate is very different between the two cases. Pt/C catalysts layers were coated directly onto carbon-fiber gas diffusion media to create gas diffusion electrodes (GDEs), without the use of a decal transfer process. These GDEs were assembled into membrane electrode assemblies (MEAs) by hot pressing to Nafion membranes. In situ performance and other electrochemical diagnostics were used to determine the influence of coating method on catalyst layer electrochemical properties. Electron microscopy was utilized to understand the influence of coating method on electrode morphology and its influence on electrochemical properties. We also explored the influence of solvent ratio to understand the potential coupling between ink formulation and coating methodology.

DIRECT ENERGY CONVERSION,MATERIALS SCIENCE↗

Fail-safe reactivity compensation method for a nuclear reactor

The present invention relates generally to the field of compensation methods for nuclear reactors and, in particular to a method for fail-safe reactivity compensation in solution-type nuclear reactors. In one embodiment, the fail-safe reactivity compensation method of the present invention augments other control methods for a nuclear reactor. In still another embodiment, the fail-safe reactivity compensation method of the present invention permits one to control a nuclear reaction in a nuclear reactor through a method that does not rely on moving components into or out of a reactor core, nor does the method of the present invention rely on the constant repositioning of control rods within a nuclear reactor in order to maintain a critical state.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Scaling Building Energy Audits through Machine Learning Methods on Novel Drone Image Data

Building energy audits are time-consuming and labor-intensive. This paper describes a new method using machine learning (ML) techniques on novel data sources (drone images) to improve the identification of building characteristics and retrofit opportunities, and thereby reduce the effort for audits. The new ML method includes: (1) Building footprint extraction using line extraction, polygonization, and polygon-merging, (2) Building envelope extraction using PIX4d modeling software to reconstruct a building 3D model, (3) Visualization tool for viewing images from the 3D model, (4) Window-to-wall ratio (WWR) using state-of-art deep neural network semantic segmentation, (5) Envelope thermal anomaly detection using an unsupervised machine learning clustering algorithm, and (6) Rooftop energy equipment detection based on an object detection algorithm. The testing of this method involved a comparison of additional ML-generated information overlaid on current ‘state-of-practice’ audit and remote assessment baselines using evaluation metrics: labor time and associated cost, marginal benefits of using ML-generated information in workflows for audits and remote assessments, integration potential with existing processes and tools, and replicability/scalability of the method. In two test buildings in California that had comprehensive drawings and meter data available, the ML method effectively generated a building footprint, envelope, rooftop equipment, WWR, and locations of envelope thermal anomalies. Projected target segments of the ML method are sites with minimal drawings and energy data, and underserved sectors such as multistoried housing, disadvantaged communities, and schools for which the ML method can enable identification of building asset characteristics and prioritization of envelope retrofits and decentralized energy equipment retrofits.

Singh, Reshma↗

Ambient Mass Spectrometry Rapid Analysis Methods for Trace Inorganics

The ability to rapidly detect and characterize the chemistry of trace quantities of inorganic species has multiple potential applications, including nuclear safeguards, nuclear forensics, and environmental monitoring. Some mass spectrometric methods currently exist, but they are costly and time-intensive, often destroying chemical information in the process of ionization. Ambient mass spectrometry is a relatively new analytical method in which samples are introduced to the mass spectrometer from atmospheric pressure, rather than from high vacuum. Ambient MS can operate with gentler ionization methods such as electrospray ionization (ESI) and paper-spray ionization (PSI), reducing fragmentation and preserving the chemical information of the analyte. This may be useful for determining the intended purpose of material in a sample. Strontium, part of nuclear fallout, is analyzed here to test these new analytical methods for viability. Electrospray ionization mass spectrometry functions by spraying the sample as a solution through a needle charged to a few kV. A nebulizing gas flows out along with the sample to create the spray before droplets enter a drying chamber with heated drying gas to remove the solvent molecules. From there bare analyte ions enter the mass spectrometer. This method has the advantage of relatively fast sample preparation and stable ionization. More stable ionization produces higher-quality data. Paper-spray ionization mass spectrometry operates by applying sample to a paper triangle. The paper triangle is then electrified in a metal clamp to a few kV before spray solvent is applied. The spray solvent transports analyte in the sample to the tip, where it is ionized and enters the mass spectrometer. One advantage of this method is reduced sample preparation - an analyst can even swipe a surface and cut a triangle directly from the swipe. The data analysis serves to rapidly process tens of samples all at once. The mass spectrometer outputs a file containing a few hundred spectra for each sample. The pipeline processes each one to identify peaks of interest. A full palette of data analysis metrics are generated, including plots of the raw data and plots of calculated values such as isotopic ratios. The pipeline filters the raw data to remove noise and background peaks, as well as peaks representing detector saturation. A heat map is generated to show the overall spectrum in mass-time- intensity space. Post-processing, we obtained the desired calibration curves, demonstrating that both ESI-MS and PSI-MS are viable methods for quantifying trace amounts of inorganics while retaining their chemical speciation information. ESI-MS demonstrated a linear response region of 100 ppb to 20 ppm, with a limit-of-detection of around 100 ppb. At top left is an example spectrum with the {sup 88}SrNO{sub 3} peak identified to show the retention of chemical speciation. PSI-MS showed a linear response region of around 1 ppb to 1 ppm, with a limit-of-detection of near 1 ppb. PSI-MS produced more erratic results than ESI-MS, likely due to the variability in ionization resulting from manual positioning of the paper triangle near the MS inlet. Future efforts will focus on improving the repeatability of PSI-MS as well as analyzing other species of interest such as cerium and uranium. Improvements will also be made to create more reliable data extraction and intelligent data filtering.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Development of Methods for In-line Monitoring of Yield Stress During the Transfer of Radioactive Waste - 20435

Radioactive waste slurries at the Savannah River Site (SRS) and Hanford often behave as non-Newtonian fluids with a yield stress. The yield stress increases the size of equipment needed to transport or mix these slurries and can increase the risk of process upsets. This property is often measured by collection of samples and transportation to a laboratory, which can increase the risk of personnel exposure and even change the rheology. In addition, slurry rheology has been observed to change with time. Real-time in-line monitoring (RTIM) of rheological properties of slurries can have substantial benefits to critical processes which are currently in operation or will be part of future plans in Department of Energy (DOE) complex processes. The need for an appropriate RTIM technology that can potentially minimize or eliminate numerous sampling operations was recently identified. This research at Florida International University focuses on addressing the need for a reliable RTIM technology at US Department of Energy sites. The objective of this work is to investigate and develop appropriate and reliable technologies that can perform accurate and easy measurements of yield stress with minimal changes to the flow conditions at US Department of Energy sites. Use of ultrasound in longitudinal and shear modes was considered as a method for monitoring of yield stress variations with no disturbance to the flow. Measurement of shear wave velocity (in shear mode) and monitoring of sound velocity variations with start of flow from rest (longitudinal mode) are the underlying principles. Measurement of pressure loss and liquid rise (due to static differential pressure) during flow of slurries were two other methods that were considered and investigated. In addition, analysis of stability depth of a penetrometer in slurry medium (penetrometer method) revealed a potential suitability. This paper presents efforts associated with investigation of different methods mentioned earlier for kaolin-water mixtures at different concentrations. Theoretical analysis showed that all methods under investigation were capable of monitoring yield stress in a desirable range of 0 to 70 pascal. A bench scale test setup was constructed for investigation of pressure loss and liquid rise methods. Simulants at different concentrations were created for initial testing campaigns. For ultrasonic testing, effects of yield stress on sound propagation speed in longitudinal and shear modes were investigated for simulants placed inside PVC pipes. Parameters such as simulant concentration and pipe diameter were varied to find optimal values. It was found that spacing (or medium thickness) of 1/8 inch or less was necessary to observe echoes with sufficient strength. In addition, a significant effect of yield stress on sound propagation velocity was observed. For the liquid rise method, kaolin-water simulant at 1.17 specific gravity was pumped through a 1/4 inch pipe and liquid height was measured in a vertical riser branching off a three-way connection. Static gauge pressure was varied within [0.3 to 1.1] psi range by increase of pumping pressure and flow. Results showed consistent yield stress data within 10% deviation from values reported by a rheometer. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Improved Modelling for the Assembly Discontinuity Factors in the APEC Method

Modern reactor analysis of light water reactor depends largely on a two-step procedure based on the simplified equivalence theory (SET). Consequently, the resulting homogenized two-group cross-section (XS) and associated discontinuity factor (DF) are subject to noticeable, intrinsic uncertainties. In our previous study, we introduced the albedo-corrected parameterized equivalence constants (APEC) method, which is a new leakage correction method applicable to the fuel assembly homogenized two-group XS of the standard two-step procedure. In the APEC method, fuel assembly two-group XS is expressed as a simple polynomial function of assembly-wise current-to-flux ratio (CFR), and it was demonstrated that two-group XSs can be well corrected during a nodal calculation and the nodal accuracy is improved a lot. However, since the conventional assembly discontinuity factor (ADF) is still used in the current APEC method, a small but non-negligible error is unavoidable. Therefore, it is necessary to include improved DF model in the APEC method for a higher accuracy. The functionalized discontinuity factor (FDF) using the definition of CFR was investigated by authors before the development of the APEC method, where it was motivated by the boundary perturbation theory by F. Rahnema. However, in our previous study on FDF, lattice calculations were performed only with one-dimensional two-group diffusion equation and could not be applied to actual core analysis. In this paper, feasibility study of DF functionalization is performed based on a 2-D method of characteristics (MOC) lattice code, DeCART2D.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

Fail-safe reactivity compensation method for a nuclear reactor

The present invention relates generally to the field of compensation methods for nuclear reactors and, in particular to a method for fail-safe reactivity compensation in solution-type nuclear reactors. In one embodiment, the fail-safe reactivity compensation method of the present invention augments other control methods for a nuclear reactor. In still another embodiment, the fail-safe reactivity compensation method of the present invention permits one to control a nuclear reaction in a nuclear reactor through a method that does not rely on moving components into or out of a reactor core, nor does the method of the present invention rely on the constant repositioning of control rods within a nuclear reactor in order to maintain a critical state.

Nygaard, Erik T.↗

On the numerical accuracy in finite-volume methods to accurately capture turbulence in compressible flows

The goal of the present article is to understand the impact of numerical schemes for the reconstruction of data at cell faces in finite-volume methods, and to assess their interaction with the quadrature rule used to compute the average over the cell volume. Here, third-, fifth- and seventh-order WENO-Z schemes are investigated. On a problem with a smooth solution, the theoretical order of convergence rate for each method is retrieved, and changing the order of the reconstruction at cell faces does not impact the results, whereas for a shock-driven problem all the methods collapse to first-order. Here, study of the decay of compressible homogeneous isotropic turbulence reveals that using a high-order quadrature rule to compute the average over a finite-volume cell does not improve the spectral accuracy and that all methods present a second-order convergence rate. However the choice of the numerical method to reconstruct data at cell faces is found to be critical to correctly capture turbulent spectra. In the context of simulations with finite-volume methods of practical flows encountered in engineering applications, it becomes apparent that an efficient strategy is to perform the average integration with a low-order quadrature rule on a fine mesh resolution, whereas high-order schemes should be used to reconstruct data at cell faces.

97 MATHEMATICS AND COMPUTING↗

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗

Using Filter Methods to Guide Convergence for ADMM, with Applications to Nonnegative Matrix Factorization Problems

Nonconvex, nonlinear optimization problems arise naturally in parameter fitting and machine learning. While augmented Lagrangian methods have demonstrated robust convergence for classes of these problems, their convergence for block updates has been relatively unexplored outside of the context of the alternating direction method of multipliers (ADMM). ADMM has seen extensive use in these applications, but may exhibit uncertain convergence behavior in many practical nonconvex settings, and struggles with general nonlinear constraints. In contrast, filter methods have proved effective in enforcing convergence for sequential quadratic programming methods and interior point methods with feasibility criteria. We develop an ADMM-filter method for highly nonlinear and nonconvex problems. Here, we show convergence under mild assumptions for several types of coordinate descent schemes, and demonstrate our algorithm on nonnegative matrix factorization and completion problems in imaging and chemical spectrum analysis.

Nonconvex optimization↗