Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Lagrangians”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

An Extended Lagrangian Method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method,' is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. The present method and the Arbitrary Lagrangian-Eulerian (ALE) method have a similarity in spirit-eliminating the cross-streamline numerical diffusion. For this purpose, we suggest a simple grid constraint condition and utilize an accurate discretization procedure. This grid constraint is only applied to the transverse cell face parallel to the local stream velocity, and hence our method for the steady state problems naturally reduces to the streamline-curvature method, without explicitly solving the steady stream-coordinate equations formulated a priori. Unlike the Lagrangian method proposed by Loh and Hui which is valid only for steady supersonic flows, the present method is general and capable of treating subsonic flows and supersonic flows as well as unsteady flows, simply by invoking in the same code an appropriate grid constraint suggested in this paper. The approach is found to be robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multi-dimensional discontinuities with a high level of accuracy, similar to that found in one-dimensional problems.

Liou, Meng-Sing↗
NASA NTRS2003

A Eulerian-Lagrangian Model to Simulate Two-Phase/Particulate Flows

Figure 1 shows a snapshot of liquid fuel spray coming out of an injector nozzle in a realistic gas-turbine combustor. Here the spray atomization was simulated using a stochastic secondary breakup model (Apte et al. 2003a) with point-particle approximation for the droplets. Very close to the injector, it is observed that the spray density is large and the droplets cannot be treated as point-particles. The volume displaced by the liquid in this region is significant and can alter the gas-phase ow and spray evolution. In order to address this issue, one can compute the dense spray regime by an Eulerian-Lagrangian technique using advanced interface tracking/level-set methods (Sussman et al. 1994; Tryggvason et al. 2001; Herrmann 2003). This, however, is computationally intensive and may not be viable in realistic complex configurations. We therefore plan to develop a methodology based on Eulerian-Lagrangian technique which will allow us to capture the essential features of primary atomization using models to capture interactions between the fluid and droplets and which can be directly applied to the standard atomization models used in practice. The numerical scheme for unstructured grids developed by Mahesh et al. (2003) for incompressible flows is modified to take into account the droplet volume fraction. The numerical framework is directly applicable to realistic combustor geometries. Our main objectives in this work are: Develop a numerical formulation based on Eulerian-Lagrangian techniques with models for interaction terms between the fluid and particles to capture the Kelvin- Helmholtz type instabilities observed during primary atomization. Validate this technique for various two-phase and particulate flows. Assess its applicability to capture primary atomization of liquid jets in conjunction with secondary atomization models.

Apte, S. V.↗
DOE OSTI2025

A Smoothed Augmented Lagrangian Framework for Convex Optimization with Nonsmooth Constraints

Augmented Lagrangian (AL) methods have proven remarkably useful in solving optimization problems with complicated constraints. The last decade has seen the development of overall complexity guarantees for inexact AL variants. Yet, a crucial gap persists in addressing nonsmooth convex constraints. To this end, we present a smoothed augmented Lagrangian (AL) framework where nonsmooth terms are progressively smoothed with a smoothing parameter $\eta _k$ . The resulting AL subproblems are $\eta _k$ -smooth, allowing for leveraging accelerated schemes. By a careful selection of the inexactness level $\epsilon _k$ (for inexact subproblem resolution), the penalty parameter $\rho _k$ , and smoothing parameter $\eta _k$ at epoch k, we derive rate and complexity guarantees of $\tilde{\mathcal {O}}(1/{\varepsilon }^{3/2})$ and $\tilde{\mathcal {O}}(1/{\varepsilon })$ in convex and strongly convex regimes for computing an ${\varepsilon }$ -optimal solution, when $\rho _k$ increases at a geometric rate, a significant improvement over the best available guarantees for AL schemes for convex programs with nonsmooth constraints. Analogous guarantees are developed for settings with $\rho _k = \rho$ as well as $\eta _k = \eta$ . Preliminary numerics on a fused Lasso problem display promise.

augmented Lagrangian↗
DOE OSTI2026

Seasonal variation of the surface cross-shelf exchange in the northern South China Sea: a Lagrangian perspective

Previous studies on cross-shelf exchange, predominantly adopted an Eulerian perspective, struggled to identify water sources and pathways. Using a high-resolution regional ocean modeling system (ROMS) and Lagrangian particle tracking, this study systematically investigates the seasonal variation and dynamics of surface cross-shelf exchange in the northern South China Sea (NSCS) from a Lagrangian perspective. Based on daily released 30-day drifter trajectories we identify the key pathways, source regions for surface cross-shelf exchange, revealing pronounced seasonal variability. Results show the surface cross-shelf exchange generally following monsoon-driven Ekman transport. However, local dynamics, especially topographic modulation, can outweigh the expected Ekman-driven transport, producing surface exchange patterns opposite to that predicted from the prevailing winds. Topographic effects vary across different scales. In the coastal waters of western Guangdong during winter, despite downwelling-favorable winds, the modulation of alongshore currents by island topography induces an eastward pressure gradient. This gradient, through geostrophic balance, drives offshore flows opposite to wind-driven Ekman onshore transport. Furthermore, the eastern widened shelf exhibits a distinct seasonal variation of cross-shelf exchange, with strong offshore transport (opposite to the direction of Ekman transport) in winter and exceedingly weak exchange in summer. Analysis of the underlying mechanisms reveals that this winter offshore transport is primarily attributed to geostrophic flows driven by surface pressure gradient that is jointly modulated by the Kuroshio intrusion and local widened shelf topography, and enhanced by cumulative submesoscale processes. In summer, a persistent and strong along-isobath jet acts as a dynamic barrier, effectively suppressing the exchange. These findings highlight the important role of topography in regulating surface material transport, and have important implications for predicting the advection and dispersion of plankton or oil spills over the continental shelf influenced by monsoon.

Hao, Dongliang↗
DOE OSTI2025

Differentiable lagrangian shock hydrodynamics with application to stable shock acceleration of density interfaces

We develop a gradient based optimization approach for the equations of compressible, Lagrangian hydrodynamics and demonstrate how it can be employed to automatically uncover strategies to control hydrodynamic instabilities arising from shock acceleration of density interfaces. Strategies for controlling the Richtmyer-Meshkov instability (RMI) are of great benefit for inertial confinement fusion (ICF) where shock interactions with many small imperfections in the density interface lead to instabilities which rapidly grow over time. These instabilities lead to mixing which, in the case of laser driven ICF, quenches the runaway fusion process ruining the potential for positive energy return. Here, we demonstrate that control of these instabilities can be achieved by optimization of initial conditions with ( > 100) parameters. Optimizing over a large parameter space like this is not possible with gradient-free optimization strategies. This requires computation of the gradient of the outputs of a numerical solution to the equations of Lagrangian hydrodynamics with respect to the inputs. We show that the efficient computation of these gradients is made possible via a judicious application of (i) adjoint methods, the exact formal representation of sensitivities involving partial differential equations, and (ii) automatic differentiation (AD), the algorithmic calculation of derivatives of functions. Careful regularization of multiple operators including artificial viscosity and timestep control is required. We perform design optimization of > 100 parameter energy field driving the Richtmyer Meshkov instability showing significant suppression while simultaneously enhancing the acceleration of the interface relative to a nominal baseline case.

Hydrophysics↗
DOE OSTI2024

CFD unified approach under Eulerian–Lagrangian framework for methanol and gasoline direct injection sprays in evaporative and flash boiling conditions

Innovative synthetic fuels for advanced propulsion systems, such as methanol and ammonia, and synthetic blended fuels (E00, E10, and E30), known for their high volatility, are often injected directly into combustion chambers. It follows that Eulerian–Lagrangian spray models need to accurately capture the spray collapse as a consequence of flash boiling onset and be capable of proficiently handling the preferential evaporation of multi-component fuels in evaporative scenarios. So, we performed the assessment of an Eulerian–Lagrangian CFD code for simulating methanol and E00 gasoline blend sprays in both early and late injection conditions involving flash boiling conditions and preferential evaporation. The adoption of an effervescent breakup model and of a non-equilibrium phase transition model for the discrete phase allows the adoption of a setup that is almost completely free from specific constant tuning, especially for what concerns the breakup model. We validated the simulations using experimental PLV maps of methanol and E00 sprays issued from the ECN Spray M injector. The results highlight a significantly different morphology of the methanol spray compared to the E00 one under late injection conditions. Under stratified combustion, low-volatile fuels are likely to be ignited first, and the flame propagates toward the high-volatile fuels. In conclusion, the spray collapse was also correctly reproduced, inducing the presence of a low-pressure zone and modifying the spray morphology.

E00↗
DOE OSTI2024

Lagrangian formulation of nuclear–electronic orbital Ehrenfest dynamics with real-time TDDFT for extended periodic systems

Here, we present a Lagrangian-based implementation of Ehrenfest dynamics with nuclear–electronic orbital (NEO) theory and real-time time-dependent density functional theory for extended periodic systems. In addition to a quantum dynamical treatment of electrons and selected protons, this approach allows for the classical movement of all other nuclei to be taken into account in simulations of condensed matter systems. Furthermore, we introduce a Lagrangian formulation for the traveling proton basis approach and propose new schemes to enhance its application for extended periodic systems. Validation and proof-of-principle applications are performed on electronically excited proton transfer in the o-hydroxybenzaldehyde molecule with explicit solvating water molecules. These simulations demonstrate the importance of solvation dynamics and a quantum treatment of transferring protons. This work broadens the applicability of the NEO Ehrenfest dynamics approach for studying complex heterogeneous systems in the condensed phase.

Calculus of variations↗
DOE OSTI2026

Code for the manuscript "Lagrangian Attention Tensor Networks for Velocity Gradient Statistical Mode

We disclose a python/pytorch implementation of the physics-informed machine learning algorithm described in "Lagrangian Attention Tensor Networks for Velocity Gradient Statistical Modeling", LA-UR-24-30678. Direct numerical simulation (DNS) of ubiquitous turbulence phenomena is computationally infeasible for realistic flows. As a result, reduced modeling for turbulent flows aim to reduce the number of resolved scales while retaining accurate representations of the small-scale physics. The dynamics of the velocity gradient tensor (VGT) is a key ingredient in reduced or subgrid turbulence models. The evolution equation for the VGT involves nonlocal terms, requiring closure modeling. This implementation of the novel methodology of Lagrangian Attention Tensor Networks (LATN), utilizes a structured representation of the history of the VGT to inform a physics-informed machine learning algorithm. This addition of structured memory terms is shown to outperform previous models when trained and evaluated on DNS data.

Livescu, Daniel [LANL]↗
NASA NTRS1978

Lagrangian computation of inviscid compressible flows

A Lagrangian method is developed to solve the Euler equations of gas dynamics. The solution of the equations is obtained by a numerical computation with the well-known Flux-Corrected-Transport (FCT) numerical method. This procedure is modified so that the boundary treatment is accurate and relatively simple. Shock waves and other flow discontinuities are captured monotonically without any type of fitting procedures. The Lagrangian method is employed so that the problem of mesh generation is completely avoided. The method is applicable to all Mach numbers except the low subsonic range where compressibility effects are small. The method is applied to a one-dimensional Riemann problem (shock tube) and to a two-dimensional supersonic channel flow with reflecting shock waves.

Klopfer, G. H.↗
NASA NTRS1978

CELFE: Coupled Eulerian-Lagrangian Finite Element program for high velocity impact. Part 1: Theory and formulation

A 3-D finite element program capable of simulating the dynamic behavior in the vicinity of the impact point, together with predicting the dynamic response in the remaining part of the structural component subjected to high velocity impact is discussed. The finite algorithm is formulated in a general moving coordinate system. In the vicinity of the impact point contained by a moving failure front, the relative velocity of the coordinate system will approach the material particle velocity. The dynamic behavior inside the region is described by Eulerian formulation based on a hydroelasto-viscoplastic model. The failure front which can be regarded as the boundary of the impact zone is described by a transition layer. The layer changes the representation from the Eulerian mode to the Lagrangian mode outside the failure front by varying the relative velocity of the coordinate system to zero. The dynamic response in the remaining part of the structure described by the Lagrangian formulation is treated using advanced structural analysis. An interfacing algorithm for coupling CELFE with NASTRAN is constructed to provide computational capabilities for large structures.

Lee, C. H.↗
NASA NTRS1980

A note on a Lagrangian formulation for motion about the collinear points

A Lagrangian formulation for the three-dimensional motion of a satellite in the vicinity of the collinear points of the circular-restricted problem is reconsidered. It is shown that the influence of the primaries can be expressed in the form of two third-body disturbing functions. By use of this approach, the equations for the Lagrangian and for the motion itself are readily developed into highly compact expressions. All orders of the non-linear developments are shown to be easily obtainable using well-known recursive relationships. The resulting forms for these equations are well suited for use in the initial phase of canonical or non-canonical investigations.

Richardson, D. L.↗
NASA NTRS1980

Lagrangian methods in nonlinear plasma wave interaction

Analysis of nonlinear plasma wave interactions is usually very complicated, and simplifying mathematical approaches are highly desirable. The application of averaged-Lagrangian methods offers a considerable reduction in effort, with improved insight into synchronism and conservation (Manley-Rowe) relations. This chapter indicates how suitable Lagrangian densities have been defined, expanded, and manipulated to describe nonlinear wave-wave and wave-particle interactions in the microscopic, macroscopic and cold plasma models. Recently, further simplifications have been introduced by the use of techniques derived from Lie algebra. These and likely future developments are reviewed briefly.

Crawford, F. W.↗
NASA NTRS1981

The Eulerian- and Lagrangian-mean flows induced by stationary, dissipating planetary waves

The Eulerian- and the Lagrangian-mean flows induced by stationary, dissipating planetary waves are discussed by employing a simple channel model on a beta-plane. It is assumed that the wave is excited by the bottom undulation and dissipated by Newtonian cooling with relaxation time alpha and by Rayleigh friction with (lambda)(alpha), lambda being constant. Three cases where lambda is equal to one are discussed: (1) the basic zonal wind U sub 0 and the dissipation rate alpha are both constant; (2) U sub 0 varies with height while alpha is constant; and (3) U sub 0 and alpha both vary with height. In case (1), the Eulerian- and the Lagrangian-mean fields are shown to depend on the difference between the dissipation scale-height and the density scale-height. In case (2) and case (3), it is shown that the results for case (1) are modified under slightly more realistic situations.

Takahashi, M.↗
NASA NTRS1983

Numerical simulations of fuel droplet flows using a Lagrangian triangular mesh

The incompressible, Lagrangian, triangular grid code, SPLISH, was converted for the study of flows in and around fuel droplets. This involved developing, testing and incorporating algorithms for surface tension and viscosity. The major features of the Lagrangian method and the algorithms are described. Benchmarks of the algorithms are given. Several calculations are presented for kerosene droplets in air. Finally, extensions which make the code compressible and three dimensional are discussed.

Fritts, M. J.↗
NASA NTRS1983

Macroscopic Lagrangian description of warm plasmas. II Nonlinear wave interactions

A macroscopic Lagrangian is simplified to the adiabatic limit and expanded about equilibrium, to third order in perturbation, for three illustrative cases: one-dimensional compression parallel to the static magnetic field, two-dimensional compression perpendicular to the static magnetic field, and three-dimensional compression. As examples of the averaged-Lagrangian method applied to nonlinear wave interactions, coupling coefficients are derived for interactions between two electron plasma waves and an ion acoustic wave, and between an ordinary wave, an electron plasma wave, and an ion acoustic wave.

Kim, H.↗
NASA NTRS1984

A Lagrangian theory of the classical spinning electron

A Lagrangian is defined that governs the dynamics of a classical electron with spin, moving under the influence of electromagnetic forces. The Euler-Lagrange equations associated with this Lagrangian for space-time position x exp-alpha provide a generalization of the Lorentz force law. The remaining Euler-Lagrange equations lead directly to the (generalized) Frenkel (1926)-Thomas (1927)-BMT (1959) equations.

Nash, P. L.↗
NASA NTRS1987

Phase-space-Lagrangian action principle and the generalized K-X theorem

The covariant coupled equations for plasma dynamics and the Maxwell field are expressed as a phase-space-Lagrangian action principle. The linear interaction is transformed to the bilinear beat Hamiltonian by a gauge-invariant Lagrangian Lie transform. The result yields the generalized linear susceptibility directly.

Kaufman, Allan N.↗
NASA NTRS1987

Lagrangian mean circulations in the stratosphere

The mean Lagrangian motion in the winter stratosphere is computed on the basis of three-dimensional winds derived from the 1978/1979 winter LIMS satellite observations, initializing material tubes of air parcels every 4 days and following them for periods of 10 days each. When the wave amplitudes are large, the tubes break and the parcels disperse; during the major sudden warming, every material tube in the stratosphere was broken, leading to extensive mixing between low and high amplitude parcels. The Lagrangian mean vertical motion tended to be smaller than the motion in the transformed Eulerian coordinate system.

Lyjak, Lawrence V.↗