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SIERRA/Aero Theory Manual - Version 4.48

SIERRA/Aero is a two and three dimensional, node-centered, edge-based finite volume code that approximates the compressible Navier-Stokes equations on unstructured meshes. It is applicable to inviscid and high Reynolds number laminar and turbulent flows. Currently, two classes of turbulence models are provided: Reynolds Averaged Navier-Stokes (RANS) and hybrid methods such as Detached Eddy Simulation (DES). Large Eddy Simulation (LES) models are currently under development. The gas may be modeled either as ideal, or as a non-equilibrium, chemically reacting mixture of ideal gases. This document describes the mathematical models contained in the code, as well as certain implementation details. First, the governing equations are presented, followed by a description of the spatial discretization. Next, the time discretization is described, and finally the boundary conditions. Throughout the document, SIERRA/ Aero is referred to simply as Aero for brevity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Shape Optimization for Navier-Stokes Equations with Algebraic Turbulence Model: Numerical Analysis and Computation

We study the shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to the optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by the generalized Navier-Stokes system with nontrivial boundary conditions. This paper deals with numerical aspects of the problem.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Multi-solid and multi-fluid diffuse interface model: Applications to dynamic fracture and fragmentation

We extend the model of diffuse solid–fluid interfaces developed earlier by authors of this paper to the case of arbitrary number of interacting hyperelastic solids. Plastic transformations of solids are taken into account through a Maxwell type model. The specific energy of each solid is given in separable form: it is the sum of a hydrodynamic part of the energy depending only on the density and the entropy, and an elastic part of the energy which is unaffected by the volume change. It allows us to naturally pass to the fluid description in the limit of vanishing shear modulus. In spite of a large number of governing equations, the model has a quite simple mathematical structure: it is a duplication of a single visco-elastic model. The model is well posed both mathematically and thermodynamically: it is hyperbolic and compatible with the second law of thermodynamics. The resulting model can be applied in the situations involving an arbitrary number of fluids and solids. In particular, we show the ability of the model to describe spallation and penetration phenomena occurring during high velocity impacts.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

GALE 3.1 Verification Report

This report is documentation of the verification activities that have been performed on GALE-3.1 to ensure that the codes meet the requirements and specifications set forth by NRC and that the code meets its intended purpose. Additional benchmark tests for GALE-3.1 are documented in Appendix A of NUREG-0016, Revision 2 for GALE-BWR 3.1 and Appendix A of NUREG-0017, Revision 2 for GALE-PWR 3.1. These activities are done in accordance with the GALE Software Quality Assurance Plan (PNNL-24249) and the GALE Configuration Management & Maintenance Plan (PNNL-24250). These two plans meet the requirements set for by both PNNL software quality assurance requirements and the NRC’s Software Quality Assurance Program and Guidelines (NUREG/BR-0167). The verification activities are documented to conform to Level 2 requirements of NUREG/BR-0167. Verifications were performed for both GALE-3.1 BWR and GALE-3.1 PWR Codes. The GALE 3.1 BWR and PWR codes are computerized mathematical models for calculating the releases of radioactive material in gaseous and liquid effluents. These computer codes are tools which can be used to determine conformance with the requirements of Appendix I to Title 10, Part 50 of the Code of Federal Regulations (10 CFR 50).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Some Mathematical Problems of Control of Quantum Systems

Currently, quantum technologies are actively developing; these technologies are based on quantum effects in individual quantum systems—atoms or molecules. The mathematical study of problems of control of quantum systems is of particular importance. In this paper, we consider certain problems associated with control of quantum systems: extrema of target functionals for population transfer problems and generation of unitary processes and incoherent control and generation of arbitrary density matrices for open quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Compressive sensing optical design and simulation tool

A system or method for designing optical systems includes generating a mathematical model of an imaging system; tracing a plurality of light rays from a source through the model of the imaging system in a forward direction and a backward direction incident upon an arbitrarily defined detector location; calculating the total light rays incident upon the arbitrarily defined detector location in response to the plurality of light rays; tracing a series of calibration images first through the computational or compressive imager mathematical model; measuring an output set of parameters based on a plurality of detected rays calibration images at the defined detector location; storing the output set of parameters in software, and determining an estimated system transfer function.

42 ENGINEERING↗

Cubature on Wiener Space: Pathwise Convergence

Cubature on Wiener space (Lyons and Victoir in Proc. R. Soc. Lond. A 460(2041):169-198, 2004) provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast computation of European option prices in generic diffusion models.We give a random walk interpretation of cubature and similar (e.g. the Ninomiya-Victoir) weak approximation schemes. By using rough path analysis, we are able to establish weak convergence for general path-dependent option prices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Stabilization of the stochastically forced equilibria for nonlinear discrete-time systems with incomplete information

A stabilization problem of the equilibrium of stochastically forced nonlinear discrete-time system with incomplete information is considered. Our approach uses a regulator which synthesizes the required stochastic sensitivity of the equilibrium. Mathematically, this problem is reduced to the solution of some quadratic matrix equations. A description of attainability sets and algorithms for regulators design is given. The general results are applied to the suppression of unwanted large-amplitude oscillations around the equilibria of the stochastically forced Verhulst model with noisy observations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

The method of complex characteristics for transonic airfoil design, with an application to compressors

The use of mathematical models to study physical problems of current interest to aeronautical engineers has been made possible by the development of numerical techniques to compute solutions of the differential equations of transonic aerodynamics. These advances have encouraged the improvement of supercritical wing technology. A method to determined steady, shockless flow of an inviscid, compressible fluid past a cascade of airfoils in the (x,y)-plane is considered, taking into account also the case of an isolated airfoil. The method of complex characteristics solves the equations in the hodograph plane by extending all variables into the complex domain, where the notion of type is no longer significant. Attention is given to the mathematical background, the method of complex characteristics, and numerical calculations.

Bledsoe, M.↗

Mathematical analysis of the dimensional scaling technique for the Schroedinger equation with power-law potentials

The dimensional scaling (D-scaling) technique is an innovative asymptotic expansion approach to study the multiparticle systems in molecular quantum mechanics. It enables the calculation of ground and excited state energies of quantum systems without having to solve the Schroedinger equation. In this paper, we present a mathematical analysis of the D-scaling technique for the Schroedinger equation with power-law potentials. By casting the D-scaling technique in an appropriate variational setting and studying the corresponding minimization problem, the D-scaling technique is justified rigorously. A new asymptotic dimensional expansion scheme is introduced to compute asymptotic expansions for ground state energies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Correspondence between open bosonic systems and stochastic differential equations

Bosonic mean-field theories can approximate the dynamics of systems of $n$ bosons provided that $n \gg 1$. Here, we show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment and the mean-field theory is replaced by a stochastic differential equation. When the $n \to \infty$ limit is taken, the stochastic terms in this differential equation vanish, and a mean-field theory is recovered. Besides providing insight into the differences between the behavior of finite quantum systems and their classical limits given by $n \to \infty$, the developed mathematics can provide a basis for quantum algorithms that solve some stochastic nonlinear differential equations. We discuss conditions on the efficiency of these quantum algorithms, with a focus on the possibility for the complexity to be polynomial in the log of the stochastic system size. A particular system with the form of a stochastic discrete nonlinear Schrödinger equation is analyzed in more detail.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

The origin of holomorphic states in Landau levels from non-commutative geometry and a new formula for their overlaps on the torus

Holomorphic functions that characterize states in a two-dimensional Landau level have been central to key developments such as the Laughlin state. Their origin has historically been attributed to a special property of “Schrödinger wavefunctions” of states in the “lowest Landau level.” In this work, it is shown that they instead arise in any Landau level as a generic mathematical property of the Heisenberg description of the non-commutative geometry of guiding centers. When quasiperiodic boundary conditions are applied to compactify the system on a torus, a new formula for the overlap between holomorphic states, in the form of a discrete sum rather than an integral, is obtained. The new formula is unexpected from the previous “lowest-Landau level Schrödinger wavefunction” interpretation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

The combined approach to obtain approximate analytical solution of instability phenomenon arising in secondary oil recovery process

Instability phenomenon mainly occurs in secondary oil recovery process due to viscosity difference of water and oil. Mathematical modeling of this phenomenon yields the non-linear partial differential equation. The homotopy perturbation new integral transform method has been proposed to solve this non-linear partial differential equation subject to appropriate initial and boundary conditions. It is concluded that the saturation of injected water S{sub w}(X,T) is increasing during instability phenomenon, when length of fingers(perturbance) and time increases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Assessments of Physiology and Cognition in Hybrid-Reality Environments (APACHE)

NASA is planning to return to the Moon in the mid-2020s as a stepping stone to Mars missions in the 2030s. Spacewalks, or extravehicular activities (EVAs), performed on the Moon and Mars will differ in a variety of ways from those that have been performed in decades past. NASA has identified multiple risks to human health and performance associated with a crewed mission to Mars, especially those associated with exploration EVAs which are expected to be a primary mission activity. Crew may be expected to conduct up to 24 hours of EVA per person per week, where the likelihood of injury and/or mental mistakes are increased compared to ground-based training or current microgravity EVAs and the consequences of which can be catastrophic. Current test environments for exploration EVA research and technology development are large, costly facilities that are limited in their availability or capabilities. Spacesuit testing in a reduced gravity environment such as NASA’s Neutral Buoyancy Laboratory, while a good representation of the crew’s physical workload during exploration EVAs, typically has small datasets and is difficult to integrate physiological sensors or other types of crew performance measures. Meanwhile, scientific field-based testing such as NASA’s Desert Research and Technology Studies offers an operationally relevant environment for exploration EVAs, particularly for cognitive workload, but is also limited by small datasets, lack of a pressurized spacesuit, and obtrusive measures. The limitations of current analogs for exploration EVAs identify a need for a new test environment that can approximate both the physical and cognitive demands associated with exploration EVAs to enable rapid, controlled, and repeatable evaluations of human health and performance risks of exploration missions. In response, the Human Physiology, Performance, Protection, and Operations Laboratory (H-3PO) at NASA Johnson Space Center has developed a hybrid reality exploration EVA analog named the Assessments of Physiology And Cognition in Hybrid-reality Environments (APACHE) to address these limitations using a combination of virtual, physical, and hybrid reality techniques. The APACHE facility resides at NASA Johnson Space Center and serves as a large “sandbox” for EVA research and simulation. At its center is a roughly 15x20ft space surrounded by a 14” tall sandbox partially filled with lunar regolith simulant to emulate the physical feeling of walking on a planetary surface and to allow for simulated geology operations. Nearby, a curved passive treadmill (Skillmill Connect, Technogym, Fairfield, NJ) and an omnidirectional treadmill (Infinadeck, Infinadeck, Rocklin, CA) are included to enable exploration of these large virtual environments while also imposing the physical demands, representative timelines, and cognitive burdens required to navigate and traverse these distances during exploration EVA. A 6DOF motion platform is used to simulate rover operations and supports various human performance evaluations and associated risks. Lastly, APACHE can support two extravehicular (EV) crewmembers working in tandem. A computer workstation is located nearby and also supports an intravehicular (IV) crewmember as part of a full mission simulation. The IV crewmember has direct video and audio communication with the EV crew in VR to provide operational and procedural support. The software used in APACHE was created by the JSC Engineering Directorate, in partnership with Buendea, powered by a custom Unreal Engine 5 (UE5.3, Epic Games) project. APACHE currently utilizes the HTC Vive Pro Eye in a wireless configuration for VR simulations. There are two virtual environments that subjects can explore within APACHE, a Lunar and Martian surface. The virtual Lunar surface was created from LIDAR data of the Lunar South Pole to create roughly 16 sq km of explorable terrain. The virtual Martian surface contains roughly 400 sq km of explorable terrain derived from Mars Reconnaissance Orbiter LIDAR data of the Jezero Crater. The immersion and related cognitive burdens of conducting a planetary EVA is simulated through a series of EVA-relevant tasks performed in the VR environment, using these high-fidelity visual representations. Additionally, APACHE includes biosensor driven informatics, such as real-time heart rate monitoring and/or derived values from model simulations, for active monitoring by the EV crew and added cognitive demand. A “Wizard of Oz” control panel enables test operators to activate contingency events such as simulated spacesuit malfunctions, loss of communications, and/or limited visibility. Embedded performance measures such as accuracy, completeness, and execution time have been developed for various exploration tasks to objectively quantify crew performance during an EVA and compare impacts to performance when different environmental stressors, both physical and cognitive, are added to or removed from the simulation. Additionally, validated cognitive and operational performance measures such as the Digit Symbol Substitution Task have been recreated and embedded in VR for direct and relatively unobtrusive measurement of motor perception. The APACHE environment currently supports multiple research studies at NASA. Examples include the CHAPEA project, a series of simulated year-long missions on Mars by a 4-person crew; and the CO2 Contingency Walk Back Study, an investigation of elevated CO2 exposure on crew performance during a contingency EVA scenario. APACHE also provides a test environment to support the development of the Crew State and Risk Model, which is a collection of individualized, mathematical models of crew physical and cognitive state; and the Personalized EVA Informatics and Decision Support system, an operational tool for flight controllers, and eventually a self-reliant Martian crew, to make biomedically-informed decisions in real-time to optimize the EVA planning and execution with respect to crew health and performance. Some technical challenges associated with developing the APACHE environment, as well as current limitations, include VR limitless natural walking with a hybrid spacesuit simulator, optimizing performance for wireless PC VR streaming while maintaining a high degree of visual fidelity, and the integration of various physiological (metabolic masks) and psychometric (eye tracking) sensors with the VR headset.

Human Performance↗

JCZ3: An Equation of State Based on the Exponential-Six Intermolecular Potential – Update: Formulation

This report presents a comprehensive analysis of the JCZ3 equation of state (EOS) as implemented in the TIGER code, focusing on its thermodynamic framework, mathematical formulations, and implications for modeling gas phase behavior under varying conditions. Overall, the report affirms the TIGER code's foundational robustness and its potential as a valuable tool for simulating high-pressure, high-temperature gas mixtures, while also highlighting areas for further refinement and validation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Macromolecular Crystallization in Microgravity

The key concepts that attracted crystal growers, macromolecular or solid state, to microgravity research is that density difference fluid flows and sedimentation of the growing crystals are greatly reduced. Thus, defects and flaws in the crystals can be reduced, even eliminated, and crystal volume can be increased. Macromolecular crystallography differs from the field of crystalline semiconductors. For the latter, crystals are harnessed for their electrical behaviors. A crystal of a biological macromolecule is used instead for diffraction experiments (X-ray or neutron) to determine the three-dimensional structure of the macromolecule. The better the internal order of the crystal of a biological macromolecule then the more molecular structure detail that can be extracted. This structural information that enables an understanding of how the molecule functions. This knowledge is changing the biological and chemical sciences with major potential in understanding disease pathologies. Macromolecular structural crystallography in general is a remarkable field where physics, biology, chemistry, and mathematics meet to enable insight to the basic fundamentals of life. In this review, we examine the use of microgravity as an environment to grow macromolecular crystals. We describe the crystallization procedures used on the ground, how the resulting crystals are studied and the knowledge obtained from those crystals. We address the features desired in an ordered crystal and the techniques used to evaluate those features in detail. We then introduce the microgravity environment, the techniques to access that environment, and the theory and evidence behind the use of microgravity for crystallization experiments. We describe how ground-based laboratory techniques have been adapted to microgravity flights and look at some of the methods used to analyze the resulting data. Several case studies illustrate the physical crystal quality improvements and the macromolecular structural advances. Finally, limitations and alternatives to microgravity and future directions for this research are covered.

Snell, Edward H.↗

Discovering Strong Gravitational Lenses in the Dark Energy Survey with Interactive Machine Learning and Crowd-sourced Inspection with Space Warps

We conduct a search for strong gravitational lenses in the Dark Energy Survey (DES) Year 6 imaging data. We implement a pre-trained Vision Transformer (ViT) for our machine learning (ML) architecture and adopt interactive machine learning to construct a training sample with multiple classes to address common types of false positives. Our ML model reduces ∼236 million DES cutout images to 22,564 targets of interest, including ∼85% of previously reported galaxy–galaxy lens candidates discovered in DES. These targets were visually inspected by citizen scientists, who ruled out ∼90% as false positives. Of the remaining 2618 candidates, 149 were expert-classified as “definite” lenses and 516 as “probable” lenses, for a total of 665 systems, with 147 of these candidates being newly identified. Additionally, we trained a second ViT to find double-source plane lens systems, finding at least one double-source system. Our main ViT excels at identifying galaxy–galaxy lenses, consistently assigning high scores to candidates with high expert assessments. The top 800 ViT-scored images include ∼100 of our “definite” lens candidates. This selection is an order of magnitude higher in purity than previous convolutional neural-network-based lens searches and demonstrates the feasibility of applying our methodology for discovering large samples of lenses in future surveys.

79 ASTRONOMY AND ASTROPHYSICS↗

Final Technical Report for "Applied Mathematics Research: Simulation Based Optimization and Application to Electromagnetic Inverse Problems"

The focus of research was: Developing adaptive mesh for the solution of Maxwell's equations; Developing a parallel framework for time dependent inverse Maxwell's equations; Developing multilevel methods for optimization problems with inequality constraints; A new inversion code for inverse Maxwell's equations in the 0th frequency (DC resistivity); A new inversion code for inverse Maxwell's equations in low frequency regime. Although the research concentrated on electromagnetic forward and in- verse problems the results of the research was applied to the problem of image registration.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗