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Results for “differentiable manifold”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1974

Perturbed nonlinear differential equations

For perturbed nonlinear systems, a norm, other than the supremum norm, is introduced on some spaces of continuous functions. This makes possible the study of new types of behavior. A study is presented on a perturbed nonlinear differential equation defined on a half line, and the existence of a family of solutions with special boundedness properties is established. The ideas developed are applied to the study of integral manifolds, and examples are given.

Proctor, T. G.↗

DOE OSTI2023

Mediating Photochemical Reaction Rates at Lewis Acidic Rare Earths by Selective Energy Loss to 4f-Electron States

Manifesting chemical differences in individual rare earth (RE) element complexes is challenging due to the similar sizes of the tripositive cations and the corelike 4f shell. In this work, we disclose a new strategy for differentiating between similarly sized Dy 3+ and Y 3+ ions through a tailored photochemical reaction of their isostructural complexes in which the f-electron states of Dy 3+ act as an energy sink. Complexes RE(hfac) 3 (NMMO) 2 (RE = Dy (2-Dy) and Y (2-Y), hfac = hexafluoroacetylacetonate, and NMMO = N-methylmorpholine-N-oxide) showed variable rates of oxygen atom transfer (OAT) to triphenylphosphine under ultraviolet (UV) irradiation, as monitored by 1 H and 19 F NMR spectroscopies. Ultrafast transient absorption spectroscopy (TAS) identified the excited state(s) responsible for the photochemical OAT reaction or lack thereof. Competing sensitization pathways leading to excited-state deactivation in 2-Dy through energy transfer to the 4f electron manifold ultimately slows the OAT reaction at this metal cation. The measured rate differences between the open-shell Dy 3+ and closed-shell Y 3+ complexes demonstrate that using established principles of 4f ion sensitization may deliver new, selective modalities for differentiating the RE elements that do not depend on cation size.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1992

Control and stabilization of nonholonomic dynamic systems

The paper establishes a theoretical framework for the control of nonholonomic dynamic systems, i.e., dynamic systems with nonintegrable constraints, with emphasis on control properties for nonholonomic systems that have no counterpart in holonomic systems. A model for nonholonomic dynamic systems is presented in terms of differential-algebraic equations defined on a phase space. A reduction procedure is carried out to obtain reduced-order state equations. Conditions for smooth asymptotic stabilization to an m-dimensional equilibrium manifold are presented. It is also demonstrated that a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback. An approach using geometric phases is developed as a basis for the control of Caplygin systems.

Bloch, Anthony M.↗

NASA NTRS2012

Current-State Constrained Filter Bank for Wald Testing of Spacecraft Conjunctions

We propose a filter bank consisting of an ordinary current-state extended Kalman filter, and two similar but constrained filters: one is constrained by a null hypothesis that the miss distance between two conjuncting spacecraft is inside their combined hard body radius at the predicted time of closest approach, and one is constrained by an alternative complementary hypothesis. The unconstrained filter is the basis of an initial screening for close approaches of interest. Once the initial screening detects a possibly risky conjunction, the unconstrained filter also governs measurement editing for all three filters, and predicts the time of closest approach. The constrained filters operate only when conjunctions of interest occur. The computed likelihoods of the innovations of the two constrained filters form a ratio for a Wald sequential probability ratio test. The Wald test guides risk mitigation maneuver decisions based on explicit false alarm and missed detection criteria. Since only current-state Kalman filtering is required to compute the innovations for the likelihood ratio, the present approach does not require the mapping of probability density forward to the time of closest approach. Instead, the hard-body constraint manifold is mapped to the filter update time by applying a sigma-point transformation to a projection function. Although many projectors are available, we choose one based on Lambert-style differential correction of the current-state velocity. We have tested our method using a scenario based on the Magnetospheric Multi-Scale mission, scheduled for launch in late 2014. This mission involves formation flight in highly elliptical orbits of four spinning spacecraft equipped with antennas extending 120 meters tip-to-tip. Eccentricities range from 0.82 to 0.91, and close approaches generally occur in the vicinity of perigee, where rapid changes in geometry may occur. Testing the method using two 12,000-case Monte Carlo simulations, we found the method achieved a missed detection rate of 0.1%, and a false alarm rate of 2%.

Carpenter, J. Russell↗

DOE OSTI2025

Generative learning of densities on manifolds

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps , a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Double diffusion maps↗

DOE OSTI2021

A Tailored Convolutional Neural Network for Nonlinear Manifold Learning of Computational Physics Data Using Unstructured Spatial Discretizations

In this work, we propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov n-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

An adaptive and stability-promoting layerwise training approach for sparse deep neural network architecture

This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training data set. In the first stage, a layerwise training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. We impose desirable structures on the DNN by employing manifold regularization, sparsity regularization, and physics-informed terms. We introduce a ε – δ – stability-promoting concept as a desirable property for a learning algorithm and show that employing manifold regularization yields a ε – δ stability-promoting algorithm. Further, we also derive the necessary conditions for the trainability of a newly added layer and investigate the training saturation problem. In the second stage of the algorithm (post-processing), a sequence of shallow networks is employed to extract information from the residual produced in the first stage, thereby improving the prediction accuracy. Numerical investigations on prototype regression and classification problems demonstrate that the proposed approach can outperform fully connected DNNs of the same size. Moreover, by equipping the physics-informed neural network (PINN) with the proposed adaptive architecture strategy to solve partial differential equations, we numerically show that adaptive PINNs not only are superior to standard PINNs but also produce interpretable hidden layers with provable stability. As a result, we also apply our architecture design strategy to solve inverse problems governed by elliptic partial differential equations.

42 ENGINEERING↗

DOE OSTI2021

Generalising G 2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E 7(7) × $\mathbb{R}$ + generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E 7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G 2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G 2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Geometry optimization speedup through a geodesic approach to internal coordinates

We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Overall, our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1993

Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems

In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.

Vaughan, William W.↗

NASA NTRS1949

Investigation of Aerodynamic and Icing Characteristics of Recessed Fuel-Vent Configurations

An investigation has been conducted in the NACA Cleveland icing research tunnel to determine the aerodynamic and icing characteristics of several recessed fuel-vent configurations. The vents were investigated aerodynamically to obtain vent-tube pressures and pressure distributions on the ramp surface as functions of tunnel-air velocity and angle of attack. Icing investigations were made to determine the vent-tube pressure losses for several icing conditions at tunnel-air velocities ranging from 220 to 440 feet per second. In general, under nonicing conditions, the configurations with diverging ramp walls maintained, vent-tube pressures greater than the required marginal value of 2 inches of water positive pressure differential between the fuel cell and the compartment containing the fuel cell for a range of angles of attack from 0 to 14deg at a tunnel-air velocity of approximately 240 feet per second. A configuration haying divergIng ramp sldewalls, a 7deg ramp angle; and vent tubes manifold,ed to a common plenum chamber opening through a slot In the ramp floor gave the greatest vent-tube pressures for all the configurations investigated. The use of the plenum chamber resulted in uniform pressures in all vent tubes. In a cloud-icing condition, roughness caused by ice formations on the airfoil surface ahead of the vent ramp, rather than icing of the vent configuration, caused a rapid loss in vent-tube pressures during the first few minutes of an icing period. Only the configuration having diverging ramp sidewalls, a 7 ramp angle, and a common plenum chamber maintained the required vent-tube pressures throughout a 60-minute icing period, although the ice formations on this configuration were more severe than those observed for the other configurations. No complete closure of vent-tube openings occurred for the configurations investigated. A simulated freezing-rain condition caused a greater and more rapid vent-tube pressure loss than was observed for a cloud-icing condition.

Ruggeri, Robert S.↗

DOE OSTI2021

A general framework for substructuring‐based domain decomposition methods for models having nonlocal interactions

Abstract A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods.

Capodaglio, Giacomo↗

DOE OSTI2020

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions [including apppendix B]

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A general framework for substructuring-based domain decomposition methods for models having nonlocal interactions [minus appendix B]

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2014

Airfoil-Shaped Fluid Flow Tool for Use in Making Differential Measurements

A fluid flow tool includes an airfoil structure and a support arm. The airfoil structure's high-pressure side and low-pressure side are positioned in a conduit by the support arm coupled to the conduit. The high-pressure and low-pressure sides substantially face opposing walls of the conduit. At least one measurement port is formed in the airfoil structure at each of its high-pressure side and low-pressure side. A first manifold, formed in the airfoil structure and in fluid communication with each measurement port so-formed at the high-pressure side, extends through the airfoil structure and support arm to terminate and be accessible at the exterior wall of the conduit. A second manifold, formed in the airfoil structure and in fluid communication with each measurement port so-formed at the low-pressure side, extends through the airfoil structure and support arm to terminate and be accessible at the exterior wall of the conduit.

England, John Dwight↗

NASA NTRS1984

An intercomparison of carbon monoxide, nitric oxide, and hydroxyl measurement techniques - Overview of results

Results from an intercomparison of methods to measure carbon monoxide (CO), nitric oxide (NO), and the hydroxyl radical (OH) are discussed. The intercomparison was conducted at Wallops Island, Virginia, in July 1983 and included a laser differential absorption and three grab sample/gas chromatograph methods for CO, a laser-induced fluorescence (LIF) and two chemiluminescence methods for NO, and two LIF methods and a radiocarbon tracer method for OH. The intercomparison was conducted as a field measurement program involving ambient measurements of CO (150-300 ppbv) and NO (10-180 pptv) from a common manifold with controlled injection of CO in incremental steps from 20 to 500 ppbv and NO in steps from 10 to 220 pptv. Only ambient measurements of OH were made. The agreement between the techniques was on the order of 14 percent for CO and 17 percent for NO. Hardware difficulties during the OH tests resulted in a data base with insufficient data and uncertanties too large to permit a meaningful intercomposition.

Hoell, J. M.↗

NASA NTRS2014

Star-Shaped Fluid Flow Tool for Use in Making Differential Measurements

A fluid flow tool's plate-like structure has a ring portion defining a flow hole, a support portion extending radially away from the ring portion and adapted to be coupled to conduit wall, and extensions extending radially away from the ring portion such that a periphery of the plate-like structure is defined by the extensions and trough regions between adjacent extensions. One or more ports formed in the ring portion are in fluid communication with the flow hole. A first manifold in the plate-like structure is in fluid communication with each port communicating with the flow hole. One or more ports are formed in the periphery of the plate-like structure. A second manifold in the plate-like structure is in fluid communication with each port formed in the periphery. The first and second manifolds extend through the plate-like structure to terminate and be accessible at the conduit wall.

England, John Dwight↗

DOE OSTI2022

Approximating the linear response of physical chaos

Abstract Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗