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Results for “Variational formulation”

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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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Modeling of Lithium Nucleation and Plating Kinetics Under Fast Charge Conditions

The onset of lithium nucleation and subsequent plating on the graphite particles is a challenging problem in the design of fast charging batteries. Here, a variational phase field formulation has been developed to initiate the nucleation of lithium particles based on the local plating overpotential conditions and allowing to predict the lithium plating under fast charge conditions. The model demonstrates that (1) at small charging current density, there is no nucleation and plating of metallic lithium, and no effect on the intercalation kinetics; (2) at intermediate charging current densities, initially a stagnated growth from isolated lithium nuclei is observed followed by a continuous growth of metallic lithium; (3) at large charging current densities, a continuous growth of metallic lithium resulting from coalescence of small nuclei into large nuclei decreases the excess interfacial energy, which in turn slows down the intercalation kinetics. The formulation sets the stage for predicting and evaluating the influence of various electrolyte compositions, graphite particle morphologies and electrode loadings on the nucleation and plating dynamics under fast charge conditions.

25 ENERGY STORAGE↗
NASA NTRS1979

A theoretical investigation of ground effects on USB configurations

A formulation predicts the variation of circulation forces and jet reaction forces in ground proximity as a function of ground height. The predicted results agree well with available experimental data. It is shown that the wing-alone theory is not capable of predicting the ground effect for USB configurations.

Lan, C. E.↗
NASA NTRS1986

Probabilistic structural analysis methods for critical SSME propulsion components

The development of a three-dimensional inelastic analysis methodology for the Space Shuttle main engine (SSME) structural components is described. The methodology is composed of: (1) composite load spectra, (2) probabilistic structural analysis methods, (3) the probabilistic finite element theory, and (4) probabilistic structural analysis. The progress in the development of generic probabilistic models for various individual loads which consist of a steady state load, a periodic load, a random load, and a spike, is discussed. The capabilities of the Numerical Evaluation of Stochastic Structures Under Stress finite element code designed for probabilistic structural analysis of the SSME are examined. Variation principles for formulation probabilistic finite elements and a structural analysis for evaluating the geometric and material properties tolerances on the structural response of turbopump blades are being designed.

Chamis, C. C.↗
NASA NTRS1990

Modal identification of gyroscopic distributed-parameter systems

A new modal identification method for gyroscopic distributed-parameter systems is presented. The method represents an extension of previous work for the class of self-adjoint distributed-parameter systems. The modal identification method is formulated as a variational problem in which stationary values of a functional quotient are sought. The computation of the functional quotient is carried out using a set of admissible functions defined over the spatial domain of the system. As an illustration, the modal identification of a whirling shaft undergoing bending vibration is carried out and the effectiveness of the method is verified.

Norris, Mark A.↗
NASA NTRS1990

A user guide for the EMTAC-MZ CFD code

The computer code (EMTAC-MZ) was applied to investigate the flow field over a variety of very complex three-dimensional (3-D) configurations across the Mach number range (subsonic, transonic, supersonic, and hypersonic flow). In the code, a finite volume, multizone implementation of high accuracy, total variation diminishing (TVD) formulation (based on Roe's scheme) is used to solve the unsteady Euler equations. In the supersonic regions of the flow, an infinitely large time step and a space-marching scheme is employed. A finite time step and a relaxation or 3-D approximate factorization method is used in subsonic flow regions. The multizone technique allows very complicated configurations to be modeled without geometry modifications, and can easily handle combined internal and external flow problems. An elliptic grid generation package is built into the EMTAC-MZ code. To generate the computational grid, only the surface geometry data are required. Results obtained for a variety of configurations, such as fighter-like configurations (F-14, AVSTOL), flow through inlet, multi-bodies (shuttle with external tank and SRBs), are reported and shown to be in good agreement with available experimental data.

Szema, Kuo-Yen↗
NASA NTRS1991

A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations

A space-time element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms and boundary conditions. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. Numerical results are presented to demonstrate the performance of the method.

Shakib, Farzin↗
NASA NTRS1993

Implicit multigrid Euler solutions with symmetric Total-Variation-Diminishing dissipation

A symmetric Total-Variation-Diminishing (TVD) formulation of the numerical dissipation terms has been incorporated into a diagonalized alternating direction implicit multigrid algorithm to solve the Euler equations of inviscid compressible flow. The new treatment of the dissipation makes is possible to capture both very strong and very weak shocks, virtually without oscillation for the steady flows of interest here. In addition, the TVD constraint fixes one of the two previously arbitrary constants in the formulation of the dissipation, and results in both converged solutions and convergence rates which are relatively insensitive to the choice of the remaining dissipation parameter.

Caughey, David A.↗
NASA NTRS2004

Simulations of Cavitating Cryogenic Inducers

Simulations of cavitating turbopump inducers at their design flow rate are presented. Results over a broad range of Nss, numbers extending from single-phase flow conditions through the critical head break down point are discussed. The flow characteristics and performance of a subscale geometry designed for water testing are compared with the fullscale configuration that employs LOX. In particular, thermal depression effects arising from cavitation in cryogenic fluids are identified and their impact on the suction performance of the inducer quantified. The simulations have been performed using the CRUNCH CFD[R] code that has a generalized multi-element unstructured framework suitable for turbomachinery applications. An advanced multi-phase formulation for cryogenic fluids that models temperature depression and real fluid property variations is employed. The formulation has been extensively validated for both liquid nitrogen and liquid hydrogen by simulating the experiments of Hord on hydrofoils; excellent estimates of the leading edge temperature and pressure depression were obtained while the comparisons in the cavity closure region were reasonable.

Dorney, Dan↗
NASA NTRS2021

AIRS Point Spread Function Reconstruction using AIRS and MODIS Data

The purpose of this work is to use data from the Atmospheric Infrared Sounder (AIRS) and the Moderate Resolution Imaging Spectroradiometer (MODIS) to refine our knowledge of post-launch AIRS point spread functions (PSFs), including suspected changes over the mission. We develop methodology, by deriving mathematical optimization formulation based on variational principles and Sobolev gradient descent, for reconstruction of AIRS spatial response functions. We use the data over the ocean, collected for the duration of a day, to reconstruct a single PSF. We examine the repeatability of our reconstructions by computing PSFs based on data collected during two consecutive days, and also investigating the change in the reconstructions by comparing the reconstructed PSF based on data collected in the beginning and the middle of the mission. We also quantify uncertainties in our reconstruction results.

Vese, Luminita↗
DOE OSTI2026

A 3D-Printed Millimeter-Wave Inline Waveguide-to-Coplanar-Waveguide Transition to Enable Dense Spectrometer Arrays for Intensity Mapping Surveys

We present a 3D-printed millimeter-wave, octave-bandwidth, in-line waveguide-to-coplanar-waveguide transition designed to enable focal planes with dense arrays of on-chip spectrometers. These arrays will enable compelling surveys of the large-scale structure of the universe through millimeter-wave intensity mapping. The transition consists of a four-step ridge-waveguide transformer that couples light from a rectangular waveguide onto a coplanar waveguide via an electrical connection made with indium bump bonds. We develop a tolerance-aware optimization approach to identify high-performance transition geometries that are robust to manufacturing variations; the same formulation can be applied to other tolerance-sensitive design problems. We also describe the implementation of a custom apparatus and procedure for bump-bonding a silicon chip to a metallized 3D-printed component. We detail the fabrication of the coplanar waveguide chip and three-dimensional waveguide structure, simulations and metrology of a test device, and room temperature reflectance measurements of this device. The room temperature metrology and reflection measurements are consistent with a model that predicts a coupling efficiency of $\mathord{\sim} 95\%$ at cryogenic temperatures in the 85-170 GHz frequency range.

Stover, Austin [Chicago U.; Chicago U., KICP] (ORC↗
NASA NTRS1993

Present-day secular variations in the zonal harmonics of earth's geopotential

The mathematical formulation required for predicting secular variation in the geopotential is developed for the case of a spherically symmetric, self-gravitating, viscoelastic earth model and an arbitrary surface load which can include a gravitational self-consistent ocean loading component. The theory is specifically applied to predict the present-day secular variation in the zonal harmonics of the geopotenial arising from the surface mass loading associated with the late Pleistocene glacial cycles. A procedure is outlined in which predictions of the present-day geopotential signal due to the late Pleistocene glacial cycles may be used to derive bounds on the net present-day mass flux from the Antarctic and Greenland ice sheets to the local oceans.

Mitrovica, J. X.↗
DOE OSTI2025

VitriEdge: Repairable & Durable Vitrimer Coatings for Wind Turbine Blade Leading Edges

The primary goal of this Level 1b incubator project was to explore the use of vitrimer coatings for repair of leading-edge erosion on end-of-life wind turbine blade surfaces, beyond coating strength of adhesion which has previously been demonstrated in the Level 1a project. Uniform vitrimer coatings (thickness: 400 µm) were applied to two end-of-life wind turbine blades for flexural, fatigue, and laminate tensile testing where the addition of the coating did not produce any statistical variation in tensile properties with minor drops in flexural strength for some laminate formulations. However, a <2% variation in storage modulus was measured for laminate structures (i.e., blade samples with vitrimer coatings) across 100,000 flexural cycles and upon laminate tensile failure, the vitrimer coatings displayed no visible signs of delamination. In addition, three methods to heal vitrimer coating damage was displayed: oven heating, addition of hot water, and a forced convection heat gun. All three heating and healing mechanisms demonstrated significant healing with scratch depths decreasing between 79-91% at healing times ranging between 1-min and 10-minutes. Finally, a water jet machine was used to simulate rain erosion for both the blade surfaces and vitrimer-coated blade surfaces where the diameter and depth of the damage was recorded as a function of exposure time, water pressure, height of exposure, and angle of exposure. Of interest, while the vitrimer coating did not significantly lessen the damage experienced during rain erosion, the addition of vitrimer composite coatings(5 wt.% mica addition) did result in a crack-resistant, durable coating capable of self-healing behavior and in all cases the angle of rain exposure was the most critical parameter explored. It is crucial to continue exploring this space where vitrimer coatings are of interest for both their self-healing properties and potential use as reversible adhesives.

17 WIND ENERGY↗
DOE OSTI2025

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗
DOE OSTI2024

A phase-field diffraction model for thermo-hydro-mechanical propagating fractures

This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. Furthermore, these new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

Diffraction systems↗
NASA NTRS1993

A nonreflecting upper boundary condition for anelastic nonhydrostatic mesoscale gravity-wave models

A sponge layer is formulated to prevent spurious reflection of vertically propagating quasi-stationary gravity waves at the upper boundary of a two-dimensional numerical anelastic nonhydrostatic model. The sponge layer includes damping of both Newtonian-cooling type and Rayleigh-friction type, whose coefficients are determined in such a way that the reflectivity of wave energy at the bottom of the layer is zero. Unlike the formulations in earlier studies, our formulation includes the effects of vertical discretization, vertical mean density variation, and nonhydrostaticity. This sponge formulation is found effective in suppressing false downward reflection of waves for various types of quasi-stationary forcing.

Kim, Young-Joon↗
DOE OSTI2021

A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows

A variation of moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is developed for solving the compressible Euler equations. The MDG-ICE method, originating from the work of Corrigan et al. [1], [2], [3], [4], is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a continuous space to determine discrete grid geometry. The resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation can then be solved in a least-squares sense, leading to an unconstrained nonlinear least-squares problem that is regularized and solved by Levenberg-Marquardt method. A number of numerical experiments for both 1D unsteady and 2D steady state compressible flow problems are conducted to assess the accuracy and robustness of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to implicitly detect and track all types of discontinuities via interface conservation enforcement and satisfy the conservation law on both elements and interfaces via grid movement and grid management, demonstrating that an exponential rate of convergence for Sod and Lax-Harden shock tube problems can be achieved and highly accurate solutions without overheating to both double-rarefaction wave and Noh problems can be obtained.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

hp -VPINNs: Variational physics-informed neural networks with domain decomposition

We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto the space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the entire computational domain, while the test space contains piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with a local learning algorithm that can efficiently localize the network parameter optimization. Here, we demonstrate the advantages of hp-VPINNs in both accuracy and training cost for several numerical examples of function approximation and in solving differential equations.

42 ENGINEERING↗
NASA NTRS1992

Improved assumed-stress hybrid shell element with drilling degrees of freedom for linear stress, buckling, and free vibration analyses

An improved four-node quadrilateral assumed-stress hybrid shell element with drilling degrees of freedom is presented. The formulation is based on Hellinger-Reissner variational principle and the shape functions are formulated directly for the four-node element. The element has 12 membrane degrees of freedom and 12 bending degrees of freedom. It has nine independent stress parameters to describe the membrane stress resultant field and 13 independent stress parameters to describe the moment and transverse shear stress resultant field. The formulation encompasses linear stress, linear buckling, and linear free vibration problems. The element is validated with standard tests cases and is shown to be robust. Numerical results are presented for linear stress, buckling, and free vibration analyses.

Rengarajan, Govind↗